ADVANCES IN HEAT TRANSFER Volume 37
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Advances in
HEAT TRANSFER Serial Editors James P. Hartnett
Thomas F. Irvine, Jr.
Energy Resources Center University of Illinois at Chicago Chicago, Illinois
Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, New York
Young I. Cho
George A. Greene
Department of Mechanical Engineering Drexel University Philadelphia, Pennsylvania
Energy Sciences and Technology Department Brookhaven National Laboratory Upton, New York
Volume 37
Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
Academic Press An imprint of Elsevier Elsevier Inc., 525 B Street, Suite 1900, San Diego, California 92101-4495, USA Elsevier Ltd., The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK ß 2003 Elsevier Inc. All rights reserved. This work is protected under copyright by Elsevier, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@ elsevier.com. You may also complete your request on-line via the Elsevier homepage (http:// www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2003 ISBN: 0-12-020037-6 ISSN: 0065-2717 The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in Great Britain.
CONTENTS Contributors . . . . . . . . . . . . . . . . . . . . . . .
ix
Preface
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xi
I. Introduction . . . . . . . . . . . . . . . . . . . . II. Development of Transparent Heated Tube . . . . . . .
1 4
A. Structure of Transparent Heated Tube . . . . . . . . . B. Electric Resistance of Thin Gold Film . . . . . . . . . C. Performance of Thin Gold Film as a Temperature Thermometer
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4 4 7
III. Experimental Apparatus and Procedure . . . . . . . . .
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Microgravity Heat Transfer in Flow Boiling HARUHIKO OHTA
A. B. C. D.
Experimental Apparatus . . . . . . . . Outline of Aircraft Experiments . . . . . Experimental Conditions and Procedure . . Preliminary Experiments on Ground . . . .
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10 12 13 14
IV. Effect of Gravity on Flow Boiling Heat Transfer in Circular Tubes . . . . . . . . . . . . . . . . . . .
15
A. Flow Pattern Change . . . . . . . . . . . . . . . . . B. Heat Transfer . . . . . . . . . . . . . . . . . . . C. Summary of Gravity Effect on Liquid–Vapor Behavior and Heat Transfer
V. Mechanisms of Gravity-dependent Heat Transfer due to Two-phase Forced Convection in Annular Flow Regime
.
A. Analytical Model . . . . . . . . . . . . . . . . . . B. Gravity Effect on Interfacial Friction Factor . . . . . . . . . C. Mechanisms of Gravity Affecting on Heat Transfer due to Two-phase Forced Convection . . . . . . . . . . . . . . . . . . D. Effect of Liquid Flow Rate due to Disturbance Wave on Heat Transfer E. Effect of Thermal Entrance Region on Heat Transfer . . . . . . F. Prediction of Gravity Effect on Heat Transfer due to Two-phase Forced Convection for Water from Pressure Drop Data . . . . . . . . G. Summary of Analytical Model . . . . . . . . . . . . . .
15 16 21
23 23 26 28 32 33 33 38
VI. Experiments on Dryout Phenomena under Microgravity Conditions . . . . . . . . . . . . . . . . . . . . .
39
A. Methods for CHF Experiments under Microgravity Conditions B. Results of CHF Measurement . . . . . . . . . . . .
39 41
v
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C. Temperature Oscillation and Liquid–Vapor Behavior just before CHF D. Summary and Direction of Further Investigation . . . . . . .
48 58
VII. Experiments on Flow Boiling Heat Transfer in Narrow Channels . . . . . . . . . . . . . . . . . . . . . .
60
A. B. C. D.
Background for Boiling Experiments in Narrow Experimental Apparatus and Procedure . . . Experimental Results and Discussion . . . Summary of Experimental Results . . . .
Gaps . . . . . .
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60 61 63 70
VIII. Future Investigations for Microgravity Flow Boiling . . . Nomenclature . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
71 73 74
Fluid Mechanics and Heat Transfer with Non-Newtonian Liquids in Mechanically Agitated Vessels R. P. CHHABRA I. Introduction . . . . . . . . . . . . . . . . . . . . II. Scope . . . . . . . . . . . . . . . . . . . . . . . III. Rheological and Thermo-physical Properties . . . . . . A. Rheological Properties . . . . . . . . B. Thermo-physical Properties . . . . . . .
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81 84
IV. Non-Newtonian Effects in Agitated Vessels . . . . . . . V. Mechanisms of Mixing . . . . . . . . . . . . . . . .
86 87
A. Laminar Mixing B. Turbulent Mixing
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87 90
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91 108 124 133 139
VII. Heat Transfer . . . . . . . . . . . . . . . . . . . .
141
VI. Fluid Mechanics A. B. C. D. E.
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77 81 81
Scale Up . . . . . . . . Power Input . . . . . . . Flow Patterns and Flow Fields Mixing and Circulation Times . Numerical and CFD Modelling
A. Class I Impellers . . . B. Class II Impellers . . . C. Class III Impellers . . .
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VIII. Mixing Equipment and its Selection A. Tank or Vessel . . . . B. Baffles . . . . . . . C. Impellers . . . . . .
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143 144 147
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150
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150 151 151
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IX. Concluding Summary . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
156 160 160
Optical and Thermal Radiative Properties of Semiconductors Related to Micro/Nanotechnology Z. M. ZHANG, C. J. FU
AND
Q. Z. ZHU
I. Introduction . . . . . . . . . . . . . . . . . . . . II. Fundamentals of Optical Properties of Semiconductors . . A. B. C. D. E.
Electronic Band Structures . . . Phonons . . . . . . . . . Scattering of Electrons and Phonons Absorption and Emission Processes Dielectric Functions . . . . .
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III. Radiative Properties of Layered Structures A. Reflection and Refraction at an Interface . . B. Radiative Properties of a Single layer . . . C. Radiative Properties of Multilayer Structures .
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179 182
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186 197 202 206 219
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226
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227 231 240
IV. Radiative Properties of Rough and Microstructured Surfaces
246
A. Surface Roughness Characterization . . . . . . . . . . B. Bidirectional Scattering Distribution Functions . . . . . . C. Radiative Properties of Microstructured Surfaces . . . . .
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247 258 267
V. Quantum Confinement and Photonic Crystals . . . . . .
269
A. Quantum Confinement . B. Photonic Crystals . . .
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269 274
VI. Concluding Remarks . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
275 277 278
Microchannel Heat Exchanger Design for Evaporator and Condenser Applications MAN-HOE KIM, SANG YONG LEE, SUNIL S. MEHENDALE AND RALPH L. WEBB I. Introduction . . . . . . . . . . . . . . . . . . . . II. Single- and Two-phase Flows in Microchannels . . . . . A. Introduction . . . . .
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297 298 298
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B. Single-phase Flows . . . C. Two-phase Flows . . . D. Concluding Remarks . .
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300 306 337
III. Two-phase Flow Mal-distribution in Microchannel Headers and Heat Exchangers . . . . . . . . . . . . .
338
A. Introduction . . . . . . . . . . . . B. Review of Relevant Literature . . . . . . C. Concluding Remarks . . . . . . . . .
IV. Air-side Performance A. B. C. D. E.
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Louver Fin . . . . . . . . . . . . . . . .
Array . . . . . . . . . . . .
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338 339 348
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350 366 386 396 400
V. Heat Exchanger Applications . . . . . . . . . . . . .
401
A. B. C. D. E. F. G. H.
Flow Structure in the Dry Conditions . . Wet Conditions . . Frosting Conditions Concluding Remarks
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Brazed Aluminum Condensers . . . . . . Tube-side Design of the Automotive Condenser Brazed Copper Air-cooled Heat-exchangers . Electronic Equipment Cooling . . . . . . Working Fluids . . . . . . . . . . . Flow Distribution Concerns . . . . . . Model for Microchannel Heat Exchangers . . Concluding Remarks . . . . . . . . .
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402 404 405 405 406 407 408 410
VI. Conclusion . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
411 412 414
Author Index . . . . . . . . . . . . . . . . . . . . . . .
431
Subject Index . . . . . . . . . . . . . . . . . . . . . . .
457
CONTRIBUTORS
Numbers in parentheses indicate the pages on which the author’s contributions begin.
RAJ P. CHHABRA (77), Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India C. J. FU (179), George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA MAN-HOE KIM (297), R&D Center, Digital Appliance Network Business, Samsung Electronics Co., Ltd., 416 Maetan-3Dong, Suwon 442-742, South Korea SANG YONG LEE (297), Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Science Town, Daejeon 305-701, South Korea SUNIL S. MEHENDALE (297), Delphi Harrison Thermal Systems, 200 Upper Mountain Road, Lockport, NY 14094, USA HARUHIKO OHTA (1), Department of Aeronautics and Astronautics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan RALPH L. WEBB (297), Department of Mechanical Engineering, Pennsylvania State University, 206 Reber Building, University Park, PA 16802, USA Z. M. ZHANG (179), George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Q. Z. ZHU (179), George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA
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PREFACE
For more than a third of a century, the serial publication Advances in Heat Transfer has filled the information gap between regularly published journals and university-level textbooks. The series presents review articles on topics of current interest. Each contribution starts from widely understood principles and brings the reader up to the forefront of the topic being addressed. The favorable response by the international scientific and engineering community to the 37 volumes published to date is an indication of the success of our authors in fulfilling this purpose. In recent years, the editors have published topical volumes dedicated to specific fields of endeavor. Examples of such topical volumes are Volume 22 (Bioengineering Heat Transfer), Volume 28 (Transport Phenomena in Materials Processing) and Volume 29 (Heat Transfer in Nuclear Reactor Safety). As a result of the enthusiastic response of the readers, the editors intend to continue the practice of publishing topical volumes as well as the traditional general volumes in the future. Volume 32, a cumulative author and subject index for the first 32 volumes, has become a valuable guide for our readers to search the series for contributions relevant to their current research interests. The editorial board expresses its appreciation to the contributing authors of Volume 37 who have maintained the high standards associated with Advances in Heat Transfer. Lastly, the editors would like to acknowledge the efforts of the staff at Elsevier who have maintained the attractive presentation of the volumes over the years.
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ADVANCES IN HEAT TRANSFER VOL. 37
Microgravity Heat Transfer in Flow Boiling
HARUHIKO OHTA Department of Aeronautics and Astronautics, Kyushu University, Fukuoka, Japan
Abstract To investigate flow boiling in microgravity, test sections of transparent heated tube and transparent heating surface were developed, and heat transfer characteristics were directly related to the liquid–vapor behaviors observed. The experiments were performed on board aircraft where the boiling system was exposed in series to normal, hyper and reduced gravity fields along a parabolic trajectory. In the experiments using a round tube and an analytical model, an important gravity effect on two-phase forced convective heat transfer where heat transfer is deteriorated in microgravity at low mass velocity was clarified. As regards the dryout phenomenon, measurement of critical heat flux was attempted in a short microgravity duration and the process of dryout was investigated for a moderate quality region based on the measured temperature fluctuation and corresponding liquid–vapor behaviors. Flow boiling in narrow channels was also investigated as one of the systems to be applied to space heat exchangers, and a few important characteristics were clarified concerning the gravity effect. Because of the limited opportunity for experiments and the short microgravity duration created by aircraft, the results obtained here could not cover all aspects of the phenomenon for the gravity effects on flow boiling for different systems and parameters, but the results are intended to become a powerful aid for further investigation in the present discipline utilizing longer microgravity periods in a new space platform to be realized soon. I. Introduction Recent increases in spacecraft size and power requirements for advanced satellites and other orbiting platforms have increased the demands for more effective thermal management and thermal control systems. Thermal systems utilizing boiling and two-phase flow are effective means for the development Advances in Heat Transfer Volume 37 ISSN 0065-2717
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Copyright ß 2003 Elsevier Inc., All rights reserved
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HARUHIKO OHTA
of high-performance, reliable and safe heat transport systems for future space missions. Boiling heat transfer offers high heat transfer rates associated with the transport of latent heat of vaporization and has the potential to significantly reduce the required size and weight of heat exchangers. The latent heat transport in two-phase flow reduces the flow rate of liquid circulated in the loop for the same amount of heat transport and, in turn, reduces the pump power requirement. Furthermore, two-phase fluids allow for precise adjustment of the fluid’s temperature responding to the thermal load by simply pressurizing the system using an accumulator. Despite their acknowledged importance, boiling and two-phase flow systems have not yet been fully implemented in new spacecraft except for small-scale heat pipes and a thermal transport loop planned in the Russian module of the International Space Station. This is partially attributed to the lack of a reliable database for the operation of such systems in microgravity. In addition, the uncertainty in the critical heat flux (CHF) conditions discourages space system designers from introducing such systems. Singlephase liquid cooling systems are favored despite the large mass penalty. But even with single-phase systems, boiling and two-phase flow would inevitably occur as a result of, for example, accidental increase in the heat generation rate, or a sudden system depressurization caused by valve operation. It is safe to say that, to date, there is no cohesive database for microgravity boiling and two-phase flow (reduced gravity is referred to as microgravity or g here). There is also a prevailing misconception that few differences actually exist between normal and microgravity heat transfer coefficients in flow boiling in the existence of bulk flow. But this is not true, as is shown in the following section when bulk flow is not so large. In addition to the clarification of phenomena in microgravity, the establishment of a coherent database for microgravity flow boiling and two-phase flow provides fundamental information for the development of large-scale two-phase thermal management systems for possible implementation in future spacecraft and earth orbiting satellites. Research on microgravity boiling has a history of more than 40 years with a short pause in the 1970s and has been advanced with the development of various microgravity facilities and with increased experimental opportunities, especially in the last 15 years. Most boiling experiments in microgravity, however, have been conducted for pool boiling, while the data on flow boiling experiments are very limited except those for isothermal twophase flow concerning the gravity-dependent flow pattern change and pressure drop. This is partially due to the practical difficulties in adapting the flow boiling apparatus with its various components to the microgravity facilities such as drop towers, aircraft, ballistic rockets and space shuttles with limited capacities in both integration volume and power supply.
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
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Misawa and Anghaie [1] introduced two different test sections for boiling experiments, i.e. a transparent square channel of pyrex glass with a coating of transparent heating films for flow pattern observation and a copper tube with a nichrome coil on the outer surface for the pressure drop measurements. Drop experiments were conducted for Fron113 flowing in vertical test sections. It was clarified that the slip ratio under microgravity is less than unity and the pressure drop is larger than the values predicted by the homogeneous model because of the increased contribution of acceleration resulting from the increase of void fraction. Kawaji et al. [2] investigated on board KC-135 aircraft the behavior of two-phase flow and heat transfer during the quenching of a preheated quartz tube. The tube, heated externally by a spiral nichrome tape, was initially empty and Fron113 was pumped into it. In microgravity, a thicker vapor film is formed on the tube wall making the rewetting of the wall more difficult and resulting in the reduction of the heat transfer rate. They observed flow patterns for flow boiling of subcooled Fron113 and saturated LN2 both on the ground and in microgravity, and reported marked differences in the shapes of liquid droplets in the dispersed flow region [3]. Saito et al. [4], using Caravelle aircraft, performed flow boiling experiments for water under subcooled and saturated conditions in a horizontal transparent duct with a concentric heater rod. In microgravity, generated bubbles move along the heating rod without detachment and grow and coalesce to become large bubbles, while the local heat transfer coefficients along the periphery of the heater rod, however, are quite insensitive to gravity levels. Lui et al. [5] presented experimental results on subcooled flow boiling in a horizontal tube, where the heat transfer coefficients due to nucleate boiling in microgravity increase up to 20% from those in normal gravity if subcooling is low. Rite and Rezkallah [6,7] investigated heat transfer in bubbly to annular flow regimes of air–water two-phase flow. The method is useful for the investigation of heat transfer mechanisms for two-phase forced convection under various flow rate combinations of both phases, involving those not easily realized by the single-component system, if the differences between the single-component and binary systems in the interaction of liquid and vapor phases are taken into consideration. To improve the approach for the clarification of phenomena in microgravity, the present author developed the observation technique, i.e. transparent heated tubes and transparent heating surfaces employed in the flow boiling using round tubes and narrow channels, respectively. In the experiments for flow boiling in a tube, the effect of gravity on the heat transfer was clarified by making reference to the observed liquid–vapor behaviors in a wide quality range covering the bubble to the annular flow regime. Gravity effects on heat transfer due to two-phase forced convection in the annular flow regime were analytically investigated to clarify the mechanisms
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relating to the gravity-dependent behaviors of annular liquid film. Acquisition of CHF data was attempted in microgravity and one of the major dryout mechanisms was investigated based on the temperature fluctuations obtained at heat fluxes just lower and higher than the critical value. For flow boiling in a narrow gap, a transparent flat heating surface was developed and integrated in a narrow channel, and some heat transfer characteristics inherent in microgravity conditions were clarified. II. Development of Transparent Heated Tube A. STRUCTURE OF TRANSPARENT HEATED TUBE The heated tube is made from a pyrex tube of I.D. 8 mm with a wall thickness of 1 mm to minimize the heat capacity for the effective use of short microgravity duration. The heated length is varied from 17 mm to 260 mm depending on the purposes of individual experiments. The heater is made of a thin gold film and the heating is conducted by the application of DC electric current directly thorough it. The film has a thickness of the order of 0.01 mm and it is transparent to allow the observation of liquid–vapor behavior through the glass tube wall. At the same time the film is utilized as a resistance thermometer to evaluate directly the inner wall temperature averaged over the entire heated length. The gold film is coated uniformly along the heated length by the pulse magnetron spattering technique and therefore has sufficient mechanical toughness against the thermal stress caused by the difference in the linear expansion between the film and substrate glass. At both ends of the heated tube, silver films of quite large thickness are coated to be used as electrodes and are contacted to copper flanges as shown in Fig. 1. Several ring sheets made of aluminum foil are inserted between the tube and the copper flanges to remove additional electrical resistances and to solve the problem of thermal expansion. A test section consists of a heated tube and two unheated tubes of the same inner diameter connected at upstream and downstream locations as shown in the figure. The unheated tube in the upstream is used as an entrance section and its length is so decided that it takes the maximum under the restriction of apparatus height inherent in the microgravity facilities employed. The copper flanges are used for power supply and for the sealing of tubes by the aid of O-rings involved in them. B. ELECTRIC RESISTANCE OF THIN GOLD FILM For the evaluation of inner wall temperature, high accuracy is required in the measurement of electric resistance of the thin gold film coated there. The
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
5
FIG. 1. Test section of transparent heated tube.
FIG. 2. Electric circuit for the measurement of heated tube resistance.
electric resistance R is directly calculated from R ¼ (V/Vo)Ro by using a simple circuit as shown in Fig. 2, where Ro is the value of standard resistance connected in series, and V and Vo are the voltages across the resistances R and Ro, respectively. The specific electric resistance of the thin metal film is, in general, higher than that of bulk metal and is less sensitive to the temperature change. Furthermore, in the present case, the resistance value is quite unstable and changes with repeated heating and cooling. To obtain the relation between the resistance R and temperature t, a very low
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electric current is applied so as not to increase the wall temperature, where the temperature of the film is assumed to be the same as that of the test liquid flowing in the tube. Annealing of the film at high temperature improves the stability of the resistance–temperature relation, but the level of resistance gradually decreases during a series of experiments as shown in Fig. 3. The figure indicates that the gradient of the resistance–temperature curve remains almost constant. The temperature coefficient of the film obtained from the figure is 5.7 104/K which is about one-seventh of the value for bulk gold, 3.9 103/K. Since the decrease in electric resistance depends not only on the time elapsed from the manufacturing and the history of the heating and cooling but is strongly dependent on the conditions of the coating process, the prediction of the transient nature of the resistance is impossible. Another behavior is also recognized for the thin film. The value of electric resistance falls seriously just after the heating despite the wall temperature being still higher than that of the liquid. The resistance value gradually recovers to become a value corresponding to the liquid temperature. Figure 4 shows the unrealistic wall temperature calculated by the substitution of indicated transient resistance values into the relation between the resistance and temperature calibrated after 2 h has elapsed from the heating. The difference between the present superficial wall temperature and measured liquid temperature gradually reduces. In the aircraft experiments, however, the test runs at various heat flux levels are performed successively before the complete recovery of the film resistance. To confirm the validity of the measured wall temperature after the heating, heat flux was supplied in advance at qo ¼ 4 104 W/m2 for 30 min followed by 7 min pause, then heat flux at the prescribed level is supplied again. The temperature differences between the wall and the liquid Tb before and after the aging are
FIG. 3. An example of the change in the relation between resistance and temperature for thin gold film coated on the inner tube wall.
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
7
FIG. 4. Drop and recovery of thin gold film resistance after aging.
FIG. 5. Temperature difference between wall and bulk liquid at heat flux q before and after aging at q0.
compared in Fig. 5, where the transient data for heating after the aging is plotted for single-phase forced convection at q ¼ 5 103 W/m2 and for nucleate boiling at q ¼ 4 104 W/m2. Fron113 in the saturation state was used and the test was conducted at mass velocity G ¼ 150 kg/m2 under atmospheric pressure P ¼ 0.01 MPa. It is clear that no difference between the wall temperature data before and after the aging is observed for both heat flux levels. Hence, the transient nature of the value of the electric resistance can be eliminated even if heat flux is supplied successively in a series of experimental runs. C. PERFORMANCE OF THIN GOLD FILM AS A TEMPERATURE THERMOMETER In the aircraft experiments, the gravity level changes stepwise along a parabolic trajectory. The acquisition of steady state data at different gravity levels and of the data for rapid phenomena requires high response of the wall temperature. Figure 6 shows the transition of wall temperature to after
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FIG. 6. Comparisons between measurement and calculation for wall temperature transition after step power input.
FIG. 7. Calculation model to examine temperature response of transparent heated tube.
stepwise power input. Symbols in the figure represent the experimental results for q ¼ 5 103 W/m2 and q ¼ 4 104 W/m2, where heat flux is increased suddenly from the value zero. The time required for constant wall temperatures is shorter for higher heat flux. The solid line in the figure is calculated by one-dimensional transient heat conduction thorough the pyrex glass substrate, where the existence of thin gold film is neglected because of its extremely low heat capacity. As shown in Fig. 7, the outer surface exposed to air is assumed to be adiabatic and an inner heat transfer coefficient is evaluated from auxiliary experiments as ¼ 2.75 102 W/m2 K for singlephase forced convection and ¼ 0.52q0.75 ( in W/m2 K and q in W/m2) for nucleate boiling under G ¼ 150 kg/m2 and P ¼ 0.01 MPa. The criterion of the initiation of boiling is given by to ¼ 77 C in the present calculation. As is known from Fig. 6, the calculation agrees well with the experimental data for both heat fluxes. Then the predicting method is extended to examine the
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
9
response of wall temperature in the actual case where the heat transfer coefficient changes stepwise or oscillates. Figure 8 shows the time o required for the difference in wall temperatures before and after the change to become within 5% of the asymptotic value when the heat transfer coefficient abruptly changes to the value . The results are practically independent of the initial heat transfer coefficient before the change and of the level of heat flux supplied. The figure implies that almost steady-state heat transfer data can be obtained at the end of microgravity duration of 20 s if 3 102 W/m2 K, and the inequality holds true for the phenomena to be investigated here. The frequency response is also examined assuming cosine variation of heat transfer coefficient at frequency f. As shown in Fig. 9, the time lag is 10 ms for f ¼ 10 Hz, the order of passing frequency of disturbance waves at low quality as mentioned in a later section. Such high response of the wall temperature makes possible the exact comparison of measured temperature fluctuation with observed liquid–vapor behaviors.
FIG. 8. Step response of transparent heated tube.
FIG. 9. Delay time of wall temperature when heated transfer coefficient oscillates.
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III. Experimental Apparatus and Procedure A. EXPERIMENTAL APPARATUS A test section is connected to the test loop so that the upward flow is realized in the gravity fields. As shown in Fig. 10, the components of the test loop are the circulating pump, bypass loop, flow meter, inlet mixing chamber, preheater, test section, outlet mixing chamber, condenser, liquid– vapor separator and cooler. The test loop is integrated together with power units and data acquisition systems in one or two racks of 700 mm width, 900 mm height and 450 mm depth prepared for the flight experiment by MU-300 jet aircraft. The payload for one rack is restricted to 100 kg, and the maximum power supply is 3 kW in total. A magnetic gear pump is introduced to maintain a constant flow rate against the change of gravity level. Pulse signals from the oval gear flow meter are input to a PID controller, which gives feedback to the inverter regulating the revolution of the pump motor for an optimum gain of response/stability setting. The control system minimizes the time required to stabilize the flow in the loop against the sudden change of gravity level along a trajectory. Vapor quality at the inlet of the test section is adjusted by the power input to the preheater. Since the use of a long heated tube is impossible in the aircraft cabin, bending of pipes is unavoidable in the preheater. To prevent the
FIG. 10. Test loop for parabolic flight experiments.
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
11
thermal decomposition of Fron experienced at high temperature, which is easily encountered during the burnout, the heat generation from sheath heaters is lowered at the bends. Effective heat removal is required to keep the temperature of the system almost constant and to prevent excessive pressure increase. In the cabin, however, only air is available as a cooling medium. Peltier elements are utilized to transport the waste heat from the condenser to the heat sinks of the fin structure, where the elevated surface temperature enhances the heat removal by the airflow. The same structure is applied to the cooler, which is introduced to prevent cavitations in the pump since pressurization by the hydraulic head is impossible in the absence of gravity. To prevent the flow oscillation due to the vapor inflow to the circulating pump, a liquid–vapor separator was developed. The operation is illustrated in Fig. 11, where the cylindrical wall is wetted by liquid even in the microgravity period, if the wall material, cylinder dimensions and volumetric ratio of both phases are selected appropriately. Microgravity environment breaks vapor, accumulated in the upper part of the cylinder, into small bubbles. Liquid is penetrated through the clearance between the cylinder wall and the peripheries of circular baffle plates inserted. When the microgravity period is ended, a few bubbles moved finally beyond the baffle plates are returned to the upper part of the cylinder by inclination of the plates. The temperatures at inlet and exit mixing chambers are measured by C-A thermocouples. Both normal CCD cameras and a high-speed video camera are employed to record the liquid–vapor behaviors. High-frequency voltage is supplied to fluorescence lights to avoid the non-uniform exposure among frames due to interference between the blinking frequency and high shutter speeds. Voltage recorders with 8 or 12 isolated channels are used for the measurement of voltages with a sampling frequency of 9 or 135 Hz. All the equipment for the measurement and power supply is controlled by a personal computer via a GPIB interface. The acquired data and the pictures of liquid–vapor behavior are synchronized by the aid of a microgravity starting signal generated from the support system installed in the aircraft.
FIG. 11. Liquid–vapor separator for parabolic flight experiments.
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B. OUTLINE OF AIRCRAFT EXPERIMENTS The flow boiling experiment using drop tower facilities is quite difficult because it requires an additional period to stabilize the system of two-phase flow, especially where a test loop is employed to supply two-phase mixtures to a test section. Also a ballistic rocket is not suitable because of significant limitation in the apparatus size and power supply. The aircraft is one of the useful means for flow boiling experiments because it provides a 20 s microgravity period and a larger power supply. Furthermore, the gravity effects can be investigated systematically for nearly the same experimental conditions realized successively along a parabolic trajectory as illustrated in Fig. 12. The gravity level created is around 0.01–0.03g, i.e. an order of 102g, accompanied by the fluctuation well known as g-jitter mainly due to turbulence. An example of measured gravity levels in three directions is shown in Fig. 13. In flow boiling the shear force acting on the surface of bubbles or of the annular liquid films becomes quite larger than the buoyancy at the gravity level less than 101g except in the case where extremely low values of mass velocity and quality are simultaneously concerned. Hence, the difference in
FIG. 12. Parabolic flight trajectory by MU-300.
FIG. 13. Typical g-jitter in microgravity for MU-300.
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
13
the gravity level between 102g and 104g, for an example, seems to have no serious influence on the phenomenon in most cases. Furthermore, the influence of g-jitter is hidden by large disturbances observed in the annular flow regime. The situation is quite different from that in pool boiling where the bubble detachment from the heating surface is substantially promoted by the residual gravity or g-jitter, and this in turn influences significantly the heat transfer characteristics and CHF values. In the present study, the test section is vertically oriented and upward flow is realized. If the test section is installed horizontally, the effect of gravity is outstanding especially at the low quality region because the distribution of both phases changes drastically, for example, from the stratified flow in normal gravity to the annular flow in microgravity. But at the same time, the change in the elementary processes affecting the heat transfer such as bubble behaviors or turbulence in the liquid film becomes unclear owing to the large change in bulk phases. For fundamental studies, a simplified boiling system of vertical flow is desirable, and the technique for detailed measurement and observation is applied to the boiling system, which is regarded approximately as axisymmetric regardless of gravity level. The experiments were conducted on board MU-300 aircraft of DAS (Diamond Air Service) under the research programs of NASDA (National Space Development Agency of Japan) and ISAS (Institute of Space and Astronautical Science). During one flight for 1 h, test runs are repeated six to ten times along parabolic trajectories, where the intervals between test runs are utilized for the adjustment of experimental conditions. C. EXPERIMENTAL CONDITIONS AND PROCEDURE The heat flux is calculated from the power input to the heated tube. The heat transfer coefficient is defined by the use of averaged inner wall temperature along the heated length, or more precisely, the averaged wall temperature weighted by the lengths along the flow direction assuming uniform temperature along the circumferential direction. The liquid temperature is estimated from the heat balance equation based on the measured temperatures in the test loop. Flight experiments were conducted using Fron113 at mass velocity G ¼ 150–600 kg/m2 s and heat flux from q ¼ 2.5 103 to 1.2 105 W/m2. The system pressure is kept constant at P ¼ 0.1 MPa except in the case of quite large power input to both the preheater and the test section, where the system pressure increases up to 0.2 MPa due to the limitation in power supply to the Peltier elements attached to the condenser. A test run consists of around 7 min preparation and 3 min data acquisition. The
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temperature–resistance characteristics of the thin gold film are calibrated at different temperatures before the flight and around 6 h after the flight. Since both calibration curves deviate significantly, and in the extreme case the obtained heat transfer coefficients differ as much as 50% depending on the choice of these calibrations, the level of the resistance–temperature curve is adjusted during the experiments: between the two successive test runs, forced convection of single-phase liquid is realized at constant mass velocity and the calibration curve is shifted so that the obtained heat transfer coefficient takes a constant value determined experimentally in advance. D. PRELIMINARY EXPERIMENTS ON THE GROUND Terrestrial experiments were conducted to confirm the performance of the transparent heated tube before the flight. Figure 14 shows the heat transfer coefficients due to single-phase forced convection and saturated nucleate boiling versus heat flux q for mass velocities G ¼ 150, 300 and 600 kg/m2 s. In the experiments, a tube with 68 mm heated length was employed. No oxidization or deposition of scale on the inner tube wall was observed. The reproducibility of the data is quite good. The level of heat transfer coefficient in the nucleate boiling region is around half of that obtained from an upward-facing copper flat surface with roughness Rp ¼ 0.125 mm in pool boiling [8]. This is because the surface of the substrate glass tube has lower nucleation ability and even a coating of the gold film does not change it. By using such a smooth surface, heat transfer
FIG. 14. Heat transfer characteristics of transparent heated tube in nucleate boiling and single-phase forced convection on the ground.
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
15
FIG. 15. Heat transfer coefficients versus mass velocity for single-phase forced convection.
characteristics in microgravity are well discussed without any change in the essentials of the phenomenon. The effect of mass velocity on the level of convective heat transfer is examined in Fig. 15, which shows reasonable trends despite data scattering.
IV. Effect of Gravity on Flow Boiling Heat Transfer in Circular Tubes A. FLOW PATTERN CHANGE For three different inlet conditions, Fig. 16 shows the effect of gravity on the flow pattern change by using a long transparent tube of 260 mm heated length at mass velocity G ¼ 150 kg/m2 s: (i) heat flux q ¼ 2 104 W/m2, inlet subcooling Tsub,in ¼ 7.2 K and exit quality xex ¼ 0.07; (ii) q ¼ 1 104 W/m2, inlet quality xin ¼ 0.21, xex ¼ 0.27; (iii) q ¼ 1 104 W/m2, xin ¼ 0.47, xex ¼ 0.53. For the subcooled inlet condition under normal gravity (denoted as 1g), bubbly flow is observed in the upstream location, while alternation of froth flow and the annular flow is recognized in the downstream location. At hyper gravity (2g) the diameter of the bubbles is decreased. The bubble velocity relative to the liquid velocity increases, and void fraction compared at the same location, i.e. at the same thermodynamic equilibrium quality, decreases. In microgravity (g), on the other hand, the decrease in bubble velocity increases the void fraction, resulting in the transition to froth and to annular flow at lower quality. The trend coincides, as a result, with that in isothermal two-phase flow in microgravity [9]. But distinct slug flow is scarcely realized in the heated system and the mechanisms of transition via slug flow regime require some modification. For xin ¼ 0.47, where the annular flow was observed along the entire tube length, the flow pattern is almost independent of gravity.
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FIG. 16. Change in flow patterns at different gravity levels. (P ¼ 0.1 MPa, G ¼ 150 kg/m2 s)
B. HEAT TRANSFER 1. Bubble Flow Regime A test section with a heated length of 68 mm was employed in a series of experiments. The liquid–vapor separator operated correctly during the g period. Flow rate did not fluctuate during the transition from 1g to 2g even at low mass velocity G ¼ 150 kg/m2 s, while it fluctuated up to 10% in the first half of the g period due to the change of gravity level from 2g to g in a few seconds. To decrease the liquid subcooling without generating bubbles before the entrance of the test section, the power input to the preheater required for the initiation of bubble nucleation was gradually reduced. Figure 17 shows the bubble behaviors and heat transfer coefficients defined by using averaged liquid temperature under the conditions of mass
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
17
FIG. 17. Heat transfer coefficients and liquid–vapor behaviors at different gravity levels (subcooled region, bubble flow regime, low mass velocity).
velocity G ¼ 150 kg/m2 s, inlet and exit liquid subcooling Tsub,in ¼ 13.0 K and Tsub,ex ¼ 8.5 K, respectively, and heat flux q ¼ 2 104 W/m2. Bubbles are isolated for all gravity levels and distributed uniformly on the inner tube wall. The number of active sites seems to be insensitive to the gravity level. At 1g small bubbles, generated from the thin gold film, move along the tube wall in the flow direction and grow, absorbing the latent heat from the wall. After the bubbles move a certain distance downstream they detach due to the buoyancy and shear force acting on the bubble surface. At 2g bubbles move along the tube wall with higher velocity because of the enhanced buoyancy. The frequency of detachment is increased to some extent. The increase in moving velocity reduces the evaporation period before the detachment, which, in turn, decreases the bubble volume. All such situations are reversed under microgravity conditions. A reduction of bubble moving velocity and an increase in individual bubble volume are observed. The increase in bubble size in g is not attributed to the decrease of static pressure, because the liquid head acting on the test section is minimized by the reduction of distance between the exit of the test section and the top part of the test loop. The heat transfer coefficient is, however, rather insensitive to gravity despite the distinct change of bubble behavior. The same trend is obtained if the heat transfer coefficient is defined by the degree of wall superheat. The trend is consistent with the results often encountered in pool boiling [10]. But research by the author clarified that either enhancement or deterioration of the heat transfer is possible in nucleate pool boiling at low liquid subcooling [11]. The present results seem to be inconsistent with the observed gravity-dependent bubble attached area, attached period and detachment frequency. It could be said that, in addition to liquid inertia,
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FIG. 18. Heat transfer coefficients and liquid–vapor behaviors at different gravity levels (subcooled region, bubble flow regime, high mass velocity).
liquid subcooling decreases the gravity effect, which is deduced from the fact that the behavior of bubbles attached on the heating surface does not vary with gravity under high liquid subcooling conditions [12]. Because of the high sensitivity of measured wall temperature as examined in Section II.C, a few data points of heat transfer coefficient fall during the g period, corresponding to the instantaneous emergence of a dry patch due to flow oscillation in a short time. Figure 18 shows the results at higher mass velocity G ¼ 600 kg/m2 s, liquid subcooling Tsub,in ¼ 6.7 K, Tsub,ex ¼ 4.3 K, and heat flux q ¼ 4 104 W/m2. No distinct effect of gravity on either the bubble behavior or heat transfer is observed. This is a result of the contribution of inertia by the bulk liquid flow becoming dominant compared to other forces. The bubble behaviors, including the size and the detachment frequency, are dominated by the shear force exerted by the bulk flow, and the role of buoyancy is weakened. 2. Annular Flow Regime Figure 19 shows the result for mass velocity G ¼ 150 kg/m2 s, inlet and exit vapor qualities xin ¼ 0.28 and xex ¼ 0.29, respectively and heat flux q ¼ 1 104 W/m2. The annular flow is realized. Nucleate boiling is completely suppressed at low heat flux, and the heat transfer is dominated by the two-phase forced convection. At 1g the surface of the annular film is quite wavy, and disturbance waves are periodically passing on the film surface, which is clearly recognized by the reduction in transparency due to the enhanced turbulence at this moment. At 2g the passage frequency of the disturbance wave increases and its average length becomes longer. The turbulence in the surface of the base film between the disturbance waves is
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
19
FIG. 19. Heat transfer coefficients and liquid–vapor behaviors at different gravity levels (low quality region, annular flow regime, low heat flux).
FIG. 20. Velocity, length and frequency of passing disturbance waves (low quality region, annular flow regime, low heat flux).
also increased. At g, on the contrary, the frequency of the disturbance wave reduces and the surface of the annular liquid film becomes smooth. As a consequence, the annular film becomes transparent because the random refraction of the backlight on the wavy liquid surface is significantly weakened. Heat transfer coefficient obviously increases by 25% at 2g and decreases by 7% at g in comparison with its value at 1g. At 2g, two levels of heat transfer coefficient are clearly recognized, and are supposed to be due to the two different instances with and without disturbance waves on the annular liquid film. To investigate quantitatively the effect of gravity on the behavior of the disturbance wave, as one of the reasons for the turbulence in the annular liquid film, passing frequency fd, velocity cd and length in the flow direction ld are measured from video pictures for the tube of 260 mm heated length as shown in Fig. 20. With the reduction of gravity level, fd decreases and cd
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increases. No simple trend is obtained for ld, partially due to the accuracy of identifying the boundaries of waves. The difference between 1g and g is not so large, while the length obviously increases at 2g. Figure 21 shows the result for G ¼ 150 kg/m2 s, xin ¼ 0.26, xex ¼ 0.33, and q ¼ 4 104 W/m2, where a larger heat flux is supplied than that in Fig. 19, keeping other conditions almost unchanged. The disturbance waves are passing through the tube for all gravity levels. Since boiling nucleation from the wall is not completely suppressed because of high heat flux, many bubbles are observed in the annular liquid film at 1g. The turbulence of the annular film seems to be larger owing to the bubble generation. The transparency is high at bubble base areas where the microlayer extends on the tube wall and at the areas of liquid film surrounding bubbles and supplying liquid for their growth. Similar bubble generation in the annular liquid film is also observed for 2g and g. In g, however, a larger void fraction in the annular liquid film results in the frequent emergence of dry patches and their extension in the base film around the bubbles. At the beginning of the g period, a large dry patch extends instantaneously due to the flow instability, resulting in significant deterioration of heat transfer as is observed in the figure, while the flow became stable in the latter half of the g period. The gradual increase in heat transfer coefficient is caused by the increase in saturation temperature and is interpreted as the effect of pressure on the heat transfer due to nucleate boiling. The situation was inevitable for the shortage of cooling capacity under the limitation of power supply to Peltier cooling elements. If such an effect is taken into consideration, it is concluded that the heat transfer in the annular flow regime is not influenced by gravity when it is dominated by the nucleate boiling under high heat flux
FIG. 21. Heat transfer coefficients and liquid–vapor behaviors at different gravity levels (low quality region, annular flow regime, high heat flux).
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
21
FIG. 22. Heat transfer coefficients and liquid–vapor behaviors at different gravity levels (moderate quality region, annular flow regime, low heat flux).
conditions. The discrete values of heat transfer coefficients observed in the figure are caused by the coarse resolution in the measured electric resistance of gold film and the high sensitivity of wall temperature to this value. Figure 22 shows the result for G ¼ 150 kg/m2 s, xin ¼ 0.47, xex ¼ 0.48, and q ¼ 1 104 W/m2, where higher inlet quality is realized than in Fig. 19, keeping other conditions almost constant. In the present case, nucleate boiling is completely suppressed for all gravity levels. Though trends in the behavior of annular liquid film are similar to those for lower quality, the effect of gravity on the heat transfer is reduced. Increase in quality elevates the velocity of vapor core flow and thus enhances the shear level in the annular liquid film and its inertia, resulting in the heat transfer being almost unaffected by gravity. C. SUMMARY OF GRAVITY EFFECT ON LIQUID–VAPOR BEHAVIOR AND HEAT TRANSFER The effects of gravity on the liquid–vapor behavior and heat transfer for flow boiling in a tube are clarified by using transparent heated tubes as summarized in Table I, where mass velocity, quality and heat flux are selected parameters. From the present experiments, the boundary of low and high mass velocity in the table is around G ¼ 300 kg/m2 s. The effects of gravity in extremely low mass velocity G 100 kg/m2 s have not yet been clarified. The following is obtained from the experiments: (i) At high mass velocity, neither the liquid–vapor behavior nor heat transfer are influenced by gravity for any combinations of quality and heat flux. (ii) At high quality where annular flow is realized, the behavior of annular liquid film is dominated by
22
TABLE I INFLUENCE OF GRAVITY REDUCTION ON LIQUID–VAPOR BEHAVIOR AND HEAT TRANSFER CLASSIFIED BY MASS VELOCITY, QUALITY AND HEAT FLUX Bubbly flow regime Low subcooling or Low quality
Low heat flux
Increase of detached bubble size in microgravity
High heat flux High mass velocity
No gravity effect
Heat transfer (dominating mode) Low mass velocity Low heat flux
High heat flux
High mass velocity
[Nucleate boiling] No large gravity effect [Nucleate boiling] No large gravity effect No gravity effect
Moderate quality
Decrease of turbulence in annular liquid film No large gravity effect No gravity effect
[Two-phase forced convection] Heat transfer deterioration in microgravity [Nucleate boiling in annular liquid film] No large gravity effect No gravity effect
High quality
No gravity effect No gravity effect No gravity effect
[Two-phase forced convection] No gravity effect [Nucleate boiling in annular liquid film] No gravity effect No gravity effect
HARUHIKO OHTA
Liquid–vapor behavior Low mass velocity
Annular flow regime
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
23
the interfacial shear stress exerted by the vapor core flow at high velocity, and no gravity effect is observed in either flow behaviors or heat transfer. (iii) At low quality, bubble flow is realized, and bubble size increases in microgravity if mass velocity is low. Heat transfer is dominated by nucleate boiling and is almost not influenced by the gravity level. (iv) At moderate quality, where annular flow is observed, nucleate boiling is completely suppressed when heat flux is not high enough to initiate nucleate boiling in the liquid film. Heat transfer is dominated by two-phase forced convection and is decreased by the reduction of gravity and vice versa. The influence of gravity on heat transfer is related to the disturbance in the annular liquid film and is analyzed in the next section. (iv) At moderate quality and high heat flux, nucleate boiling occurs in the annular liquid film and the behaviors of annular liquid film and bubbles are not markedly influenced by gravity. Heat transfer dominated by nucleate boiling is insensitive to gravity. (v) In microgravity, the transition to an annular flow regime occurs at lower quality as was confirmed in the existing researches on the isothermal twophase flow.
V. Mechanisms of Gravity-dependent Heat Transfer due to Two-phase Forced Convection in Annular Flow Regime A. ANALYTICAL MODEL To clarify the mechanisms for the influence of gravity on the two-phase forced convective heat transfer observed in Fig. 19, a simple model was analyzed for the behavior of annular liquid film. The validity of experimentally observed trends of heat transfer by using a short heated tube is confirmed in a later section. The outline of the model is illustrated in Fig. 23. The following assumptions are made: (i) Two-phase flow is composed of an annular liquid film, disturbance waves and vapor core flow, and the annular substrate (base) film has an uniform thickness in the circumferential direction and the thickness does not change with time even during the passage of disturbance waves. (ii) Velocity and temperature profiles are symmetric with respect to the tube axis, and not influenced by the passage of disturbance waves. (iii) Influence of disturbance is reflected only by a change of liquid flow rate due to annular liquid film under a constant liquid flow rate. To determine the velocity and temperature profiles in the annular liquid film, a method analogous to those concerning single-phase flow near a heated wall [13–15] is attempted. The volumetric flow rate of vapor Qv and
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FIG. 23. A simplified system for heat transfer simulation in annular flow regime.
the volumetric flow rate of liquid Ql are represented by mass velocity G and quality x: Gxr2o v
ð1Þ
Gð1 xÞr2o l
ð2Þ
Qv ¼
Ql ¼
where ro ¼ inner tube radius, l ¼ liquid density, and v ¼ vapor density. Ql consists of the flow rate due to annular substrate film Qlf and due to disturbance waves Qld. In the present analysis, the liquid flow rate due to entrained droplets Qle in the vapor core is included in Qld: Ql ¼ Qlf þ Qld
ð3Þ
Momentum and energy equations are described for the fully developed annular liquid film: 1 @ dP þ l g ¼ 0 ðrr Þ þ r @r dz
ð4Þ
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
25
1 @ ðrqr Þ ¼ 0 r @r
ð5Þ
where r ¼ radius, r ¼ shear stress, dP/dz ¼ pressure gradient, and qr ¼ heat flux. One-dimensional axisymmetric velocity and temperature profiles in steady state are assumed. The integration of Eq. (4) across the annular liquid film gives shear stress o at the tube wall: r2o r2i dP ri þ l g o ¼ ð6Þ i dz ro 2ro where ri ¼ radius of interface (surface of annular liquid film) and i ¼ interfacial shear stress. The pressure gradient dP/dz is related to the interfacial shear stress by the integration of Eq. (4) using v instead of l for the vapor core flow: dP 2i ¼ v g ri dz
ð7Þ
Shear stress for turbulent flow of liquid film is represented as r ¼ l ð þ "mr Þ
@u @r
ð8Þ
where is kinematic viscosity, and eddy viscosity "mr is evaluated from the relation by Sleicher [16] in the range yþ<30–50 for single-phase flow: ð9Þ "mr = ¼ b2 ðyþ Þ2 ; b ¼ 0:091 pffiffiffiffiffiffiffiffiffiffiffi yþ is defined by friction velocity o =l and distance from the inner tube wall y ¼ ror as: rffiffiffiffiffi y o yþ ¼ ð10Þ l l The velocity profile in the annular liquid film is determined by the integration of Eq. (4) under the boundary conditions: r ¼ ro ;
u ¼ 0;
r ¼ ri ;
¼ i
ð11Þ
Volumetric flow rate of annular liquid film Qlf is obtained from the integration of velocity profile: Z ro Qlf ¼ 2 ur dr ð12Þ ri
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In the analysis, there exist six unknown parameters ri, o, i, dP/dz, Qld, Qlf while four equations, (3), (6), (7) and (12), are given. The solution is obtained if two unknowns, for example Qld and i, are given. The temperature profile is obtained by the substitution of the following equation into Eq. (5): qr ¼ l cpl ðal þ "hr Þ
@T @r
ð13Þ
where cpl ¼ isobaric heat capacity and al ¼ thermal diffusivity. Eddy thermal diffusivity "hr is equated to "mr, and Eq. (5) is integrated under the following boundary conditions: r ¼ ro ;
qr ¼ qo ;
r ¼ ri ;
t ¼ tsat
ð14Þ
where qo ¼ heat flux at the tube wall and tsat ( ¼ ti) ¼ saturation temperature (interfacial temperature). The heat transfer coefficient is defined as: ¼
qo to tsat
ð15Þ
where to represents wall temperature. B. GRAVITY EFFECT ON INTERFACIAL FRICTION FACTOR For the vapor core flow, the wavy surface of the annular liquid film is regarded as wall roughness to cause pressure drops, and thus interfacial friction factor fi and corresponding shear stress i are evaluated. From the experimental results: (i) the surface of the annular liquid film becomes smooth as gravity level decreases and vice versa; (ii) the influence of gravity on the roughness of annular liquid film reduces when mass velocity or quality is increased. With reference also to the trends of obtained heat transfer data, fi is finally given by a function of gravitational acceleration g, quality x and mass velocity G as follows: fi 1 x 0:9 1 Fr : 0 < g=ge 2 for Fron113 at 0:1 MPa ¼ 1 þ 0:08 fi0 x ð16Þ where fi0 indicates the interfacial friction factor at assumed zero gravity (g ¼ 0 m/s2). Froude number Fr is defined as Fr ¼
G2 2l gD
ð17Þ
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
27
In Eq. (16), fi1g ¼ fi2g ¼ fi0 holds for x ¼ 1, which means that the effect of gravity on fi disappears for vapor single-phase flow. The influence of gravity on fi in the above relation under the assumed zero entrainment condition Qld ¼ 0 is represented against the quality x and mass velocity G in Fig. 24(a) and (b), respectively. Here, the friction factor under normal gravity fi1g was evaluated so that the pressure gradient coincides with the values predicted using the Chisholm and Laird correlation [17,18]. The correlation gives the reference values to evaluate the deviation of friction factors from those at 1g under similar flow conditions. As observed in the figure, the friction factor from Eq. (16) becomes smaller with the reduction of gravity, and the influence of gravity on the factor becomes smaller with increase in quality. The friction factor for 2g significantly decreases with the increase in mass velocity, while the friction factor for g is quite insensitive to it. These results are consistent with the observed disturbance on the annular liquid film for different combinations of mass velocity and vapor quality. In Fig. 25(a) and (b), calculated values of heat transfer coefficient and thickness of annular liquid film ( ¼ rori) for three different gravity levels of 1g, 2g and 0.01g are shown against quality x for given mass velocities G ¼ 150 and 300 kg/m2 s, respectively. In the preset calculation, the liquid film becomes thicker in microgravity. The observed trends that the effect of gravity on heat transfer becomes small with increase in mass velocity or
FIG. 24. Ratios of interfacial friction factor to that for assumed zero gravity calculated from Eq. (16) for the change of quality and mass velocity.
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FIG. 25. Effect of gravity on heat transfer coefficient due to two-phase forced convection and on annular liquid film thickness calculated from Eq. (16) for a tube with I.D. 8 mm.
quality are well reproduced, while the level of calculated heat transfer coefficients is higher by 50% than the measured values. Also the gravity effect on the annular film thickness is decreased with increase in mass velocity or quality.
C. MECHANISMS OF GRAVITY AFFECTING HEAT TRANSFER DUE TO TWO-PHASE FORCED CONVECTION 1. Effect of Gravity Reduction on Annular Liquid Film Thickness Figure 26 represents schematically how the gravity reduction influences key parameters dominating the mechanism of two-phase forced convective heat transfer. Decrease in gravity reduces the body force acting opposite to the flow direction and increases the velocity ul in the annular liquid film. Under the constant volumetric flow rate Qlf in the annular liquid film, the film thickness becomes small h1i. From observation, on the other hand, microgravity makes the film surface smooth and reduces the value of the friction
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
FIG. 26. An example of trace for the effect of gravity reduction on the heat transfer due to two-phase forced convection.
29
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factor fi h2i and then of interfacial shear stress i h3i, which in turn results in the reduction of velocity ul and increase in film thickness h4i. Under the experimental conditions in Fig. 19, the latter effect seems to be dominant and increases. The increase in results in the decrease in mean diameter of vapor core flow, and the increase in uv enhances i h5i. But this effect is weak because the change of is small relative to the tube radius ro. 2. Effect of Gravity Reduction on Wall Shear Stress o By the elimination of dP/dz from Eqs. (6) and (7), the contribution of interfacial shear stress i, gravity g, and annular film thickness ( ¼ rori) on the wall shear stress o is obtained: o ¼
ro r2 r2i ðl v Þg i o 2ro ri
ð18Þ
In Fig. 27, the relation between these parameters is shown. Three trends are known and are reflected in Fig. 26: (i) increase in wall shear stress o h6i due to gravity reduction; (ii) decrease in o caused directly by the reduction of fi, then of i h7i; (iii) decrease in o due to increase in r2o r2i , i.e. h8i. The increase in wall shear by the body force reduction should be completely cancelled by the other two mechanisms because resulting heat transfer enhancement is not true (see the next section). Since the change of film thickness has small effect on the wall shear, as observed in Fig. 27, the decrease in shear level in the film affects directly the reduction of wall shear.
FIG. 27. Relationships between wall shear stress, interfacial shear stress, annular liquid film thickness for different gravity.
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
31
3. Effect of Gravity Reduction on Heat Transfer Coefficient Since the relation <
1 dr=l cp ðal þ "hr Þ
ð19Þ
After the substitution of eddy thermal diffusivity "hr ( ¼ "mr), the integration is performed between y ¼ 0 and ( ¼ rori) to obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l cp al b2 o =l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ arctan ðb2 o Þ=ðal l Þ
ð20Þ
In this equation, the heat transfer coefficient is described only by o and . As shown in Fig. 28, the value increases with increase in o, while it decreases with increase in . The influence of is large only if film thickness is less than 0.1 mm, but such a thin film is not observed. For the experimental conditions in Fig. 19, G ¼ 150 kg/m2 s and x ¼ 0.29, values of evaluated from Fig. 25(a) are 0.25, 0.2 and 0.3 mm for 1g, 2g and g ( ¼ 0.01g), respectively. Because these values are quite large, the observed heat transfer deterioration in microgravity is caused not by the increase in annular liquid film thickness h9i but by the decrease in wall shear stress o h10i.
FIG. 28. Relationships between heat transfer coefficient due to two-phase forced convection, wall shear stress and annular liquid film thickness.
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D. EFFECT OF LIQUID FLOW RATE DUE TO DISTURBANCE WAVE ON HEAT TRANSFER In the above analysis, volumetric flow rate due to the disturbance wave Qld is assumed to be zero. But Qld sometimes occupies a large part in the total liquid flow rate and furthermore it varies with gravity. Here, the influence on the heat transfer is examined when liquid flow rate in the annular liquid film Qlf is reduced as much as Qld which was assumed to be zero in the above analysis. If triangular cross section of disturbance waves in the r–z plane is assumed as illustrated in Fig. 29, the values of Qld are evaluated from the measured length ld, passing frequency fd and height, i.e. maximum liquid film thickness max at the location of a disturbance wave. An example of the gravity effects on the values of ld and fd has already been shown in Fig. 20. The value max is evaluated from the Reynolds number of vapor core flow Rev ¼ 3.2 104 and film Reynolds number Ref ¼ 1.7 103 for the conditions in Fig. 19, and is given by the relation to a tube radius ro as max/ro ffi 0.15 for both 1g and g (Fujii et al. [19]). In Fig. 29, the heat transfer coefficients, calculated for common values of max/ro independent of gravity, are shown by lines. Along these lines the value of Qld changes from zero for max ¼ to the values corresponding to the difference between max and . Even in the case of max/ro ¼ 0.15, the deviation of from those for Qld ¼ 0 (max ¼ ), is less than 2% at 1g and g, and less than 8% at 2g. Hence, the liquid flow rate transported by the disturbance waves has a weak effect on the heat transfer and is neglected without resulting serious error in the present analysis. Dynamic effects by disturbance waves on the heat transfer remains an important problem for further investigation.
FIG. 29. Effect of liquid flow rate due to disturbance wave on heat transfer.
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
33
FIG. 30. Transition of wall temperature and heat transfer coefficient along a heated length.
E. EFFECT OF THERMAL ENTRANCE REGION ON HEAT TRANSFER The heated length for the present experiments is only 68 mm. To examine the deviation of heat transfer coefficients from those obtained by using a tube with a longer heated length, the transition of temperature profile across the annular liquid film with a fully developed velocity profile is calculated along the heated length. Figure 30 shows the calculated transition of wall temperature to and heat transfer coefficient along the flow direction for the conditions in Fig. 19. In the calculation, Eq. (16) was used for the evaluation of the interfacial shear stress. Zero entrainment flow rate Qld ¼ 0 is assumed as the most strict case because the length of the thermal entrance region becomes long with increase in the flow rate Qlf in the annular liquid film and then of the film thickness. The difference between the positions where the temperature profile across the film becomes asymptotic and wall temperature saturates is negligible. It is confirmed that the entrance region where the heat transfer is seriously enhanced from that in the fully developed region is short in the entire heated length. The averaged heat transfer coefficients along the heated length of 68 mm are only 3% higher than those for the fully developed region for all gravity levels. F. PREDICTION OF GRAVITY EFFECT ON HEAT TRANSFER DUE TO TWOFORCED CONVECTION FOR WATER FROM PRESSURE DROP DATA
PHASE
1. Microgravity Pressure Drop Data Existing studies on isothermal two-phase flow in microgravity are focused excessively on the flow pattern classification, while it is still hard to find systematic data on pressure drop. Chen et al. [20] measured the data for
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horizontal flow of saturated single-component R114 on the ground and on board KC-135. Values of quality were varied, from stratified to annular flow regime on the ground, and from slug to annular flow regime in microgravity. A tube with I.D. 15.8 mm and 1830 mm in length was employed. Differences in flow patterns between terrestrial and microgravity conditions were described for various quality levels. A larger pressure drop than terrestrial values was reported under microgravity conditions. At moderate quality, stratified flow is changed to annular flow by the reduction of gravity. The difference in pressure drop, however, is still large even at quality 0.86 where annular flow is observed for both gravity levels. The use of singlecomponent two-phase flow, although it accompanies in general the difficulty of evaluating accurate quality values, seems to be very important for reliable information for the practical application. Zhao and Rezkallah [21] proposed the pressure drop measurement for a vertical tube realizing similar flow structure independent of gravity levels. At that time, the method using a vertical tube for different gravity levels was not adopted in most existing studies, e.g. [20,22,23]. In the experiments by Zhao and Rezkallah, an air–water mixture was tested by using a tube with I.D. 9.5 mm and 690 mm in length for wide ranges of flow rates in both phases, from bubble to annular flow regime. They examined the effect of gravity on the frictional contribution of pressure drop evaluated from the estimation of void fraction, and concluded that the frictional pressure drop in microgravity is very comparable with that at normal gravity under the same flow conditions. In the present experiments, to measure the pressure drop, a test section made of an acrylic tube with I.D. 8 mm and 723 mm length was introduced and was installed in the aircraft cabin so that vertical upward flow was realized under the gravity fields. Pressure drop for the air–water mixture is measured by two transducers with appropriate diaphragms interchangeable. Gas flow from a controller is mixed with liquid flow from the circulation pump through the porous metal wall of a cylindrical mixer located below the test section. Experimental conditions are: pressure P ¼ 0.1 MPa, liquid and gas superficial velocity JL ¼ 0.0497–0.199 m/s and JG ¼ 1.99–15.9 m/s, respectively. Experiments are conducted on board a DAS MU-300. Under most experimental conditions annular flow is realized independent of gravity levels, which is confirmed by the flow regime map shown in Fig. 31, proposed by Dukler et al. [24] for microgravity conditions. Figure 32 shows the liquid–gas behaviors. For JL ¼ 0.0663 m/s and JG ¼ 1.99 m/s (Fig. 32(a)), an annular liquid film contains small bubbles at 1g. At 2g both the disturbance on the interface and the number of bubbles entrapped by the annular film are increased. At g the film becomes smooth without bubbles as if it is slug flow. This could be predicted from the
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FIG. 31. Experimental combinations of liquid and gas superficial velocities on flow regime map for microgravity by Dukler et al. [24].
FIG. 32. Behaviors of liquid–gas interface. (a) JG ¼ 1.99 m/s, JL ¼ 0.0663 m/s; (b) JG ¼ 15.9 m/s, JL ¼ 0.0663 m/s.
location near the boundary between the slug and annular flow regimes on the flow regime map. But the trend contradicts the fact that the transition to annular flow in microgravity occurs at lower gas velocity. When gas velocity is increased up to JG ¼ 15.9 m/s, keeping the same value JL ¼ 0.0663 m/s (Fig. 32(b)), the disturbance on the annular liquid film
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becomes larger in the order of 2g, 1g and g, but the difference in the film behaviors is smaller than that observed at low JG. The passing frequency of the disturbance wave is markedly increased at 2g. The thickness of the annular liquid film is decreased for larger JG. Even though the liquid superficial velocity is increased up to JL ¼ 0.199 m/s, the liquid–gas behaviors are similar to those for JL ¼ 0.0663 m/s in the present experimental range of JG. Values for time-averaged frictional pressure drop are plotted in Fig. 33. In an annular flow regime, pressure drop increases in 2g and decreases in g. The gravity effect becomes small at higher JL or JG. At g pressure drop increases markedly with increase in JL or JG. At low JL, values of pressure drop for 1g and 2g are not sensitive to JG. The effect of gravity on the frictional pressure drop is reflected to the correlation under the assumption that roughness on the surface of annular liquid film changes with gravity for the gas core flow. Using interfacial friction factor fi, the force balance becomes ðuG ui Þ2 u2 ðJG = Þ2 i ¼ fi G ffi fi G G ¼ fi G ð21Þ 2 2 2
dP 4i 4i 0:5 ¼ ¼ dz D 2 D
FIG. 33. Effects of gravity on pressure gradient.
ð22Þ
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
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where i ¼ interfacial shear stress, ¼ thickness of annular liquid film, ¼ void fraction, uG and ui ¼ mean gas core velocity and velocity at the surface of annular liquid film, respectively. It is noteworthy that the error in neglecting interfacial velocity in the definition of i in Eq. (21) is reflected in the values of friction factors. The validity at low gas velocity under microgravity conditions should be examined in detail for further investigation. Substitution of Eq. (21) into Eq. (22) gives a relation between pressure gradient dp/dz and fi for given JG and , where the void fraction is assumed to be unchanged with gravity and evaluated from no slip conditions between both phases. Data are well reproduced by a method similar to Eq. (16), which gives a ratio of interfacial friction factors for an arbitrary gravity level fi to that for assumed zero gravity fi0: fi 1 x 0:65 1 Fr ¼ 1 þ 0:035 x fi0 Fr:
ðG JG þ L JL Þ2 ; 2L gD
for water at 0:1 MPa
ð23Þ
0 < g=ge 2
ð24Þ
where x is quality, defined as x:
GG G JG ¼ GG þ GL G JG þ L JL
ð25Þ
Values of pressure gradient for 2g or g are calculated as the ratios to the value for 1g by using Eq. (23) twice. As shown in Fig. 34, the gravity effects on the ratios of interfacial friction factors from the experiments, in spite of their scattering, are well reproduced by the above correlation. The effect of gravity on the interfacial friction factor decreases with increase in JL or JG. 2. Predicted Gravity Effects on Heat Transfer for Water In Fig. 35(a) and (b), predicted values of the heat transfer coefficient due to two-phase forced convection and thickness of annular liquid film are shown against quality x for G ¼ 150 kg/m2 s and 300 kg/m2 s, respectively. The trends are qualitatively consistent with those observed for Fron113 in Fig. 25, but the effect of gravity become quite smaller for water under the same conditions of pressure, mass velocity and quality. In the figure, the heat transfer coefficients predicted from the correlation by Dengler and Addoms [25] are also indicated by lines which gives the same level of values as the prediction except at high quality larger than 0.7.
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FIG. 34. Measured and calculated ratios of interfacial friction factors. (a) fi,g/fi,1g; (b) fi,2g/fi,1g.
G. SUMMARY OF ANALYTICAL MODEL To simulate the effect of gravity observed in the heat transfer due to twophase forced convection at moderate quality in a vertical tube, a simple approach was tested to obtain the velocity and temperature profiles in the annular liquid film. (i) By the introduction of an interfacial friction factor reflecting the change of disturbance on the liquid film surface, the compound effects of gravity, quality and mass velocity on the heat transfer are reproduced. (ii) Heat transfer is deteriorated in microgravity as a result of reduced turbulence in the annular liquid film. (iii) The effect of gravity on the heat transfer due to two-phase forced convection is smaller for water than those for Fron113 as far as the same mass velocity and quality are concerned under atmospheric pressure. The actual heat transfer process is never steady but transient or quasisteady and characterized by the periodical passage of disturbance waves. The detailed analysis of velocity and temperature profiles across the annular liquid film influenced by the passage of disturbance waves is required for further investigation. The accuracy of the predictive method is to be improved to reproduce the experimental values of heat transfer coefficients.
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FIG. 35. Effect of gravity on heat transfer coefficient due to two-phase forced convection and on annular film thickness predicted from measured pressure drop data.
As for the effect of gravity on the annular liquid film, three major influences are to be considered: a change of liquid velocity in an annular film under a constant film flow rate, a change of shear level in the film as a result of the change of interfacial disturbance, and a change of film flow rate by disturbance waves and entrainment in the vapor core. The existing experimental results for film thickness are still conflicting and new sets of data are required.
VI. Experiments on Dryout Phenomena under Microgravity Conditions A. METHODS FOR CHF EXPERIMENTS UNDER MICROGRAVITY CONDITIONS Systematic experiments on CHF under microgravity conditions do not exist for the fundamental flow boiling systems. But knowledge of CHF might be more important than that for heat transfer rate because it is
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concerned directly with safety in the orbital heat management systems, even for those handling single-phase liquid flow in the case of an accidental increase in heat generation density. Transparent heated tubes are also used in this experiment. Three transparent heated tubes with different heated lengths of 17 mm, 68 mm and 260 mm are employed for different purposes of individual measurement. Because of the short microgravity duration created during one parabola, the method of CHF measurement by increasing heat flux q in steps by a small increment (Fig. 36(a)), very popular on the ground, cannot be applied. Two different methods are introduced. One is to detect the CHF condition during the increase in heat flux at a constant rate (Fig. 36(b)), and the other is to find the occurrence of CHF conditions when heat flux is increased by one step from zero to a prescribed value and thereafter is kept constant (Fig. 36(c)). In the latter case, multiple trials of stepwise increases at different heat flux levels are required to determine a heat flux range involving an exact CHF value. CHF conditions are determined from the detection of deteriorated heat transfer as a result of rapid increase in wall temperature, often referred to as temperature excursion. To avoid the measurements during the flow oscillation inevitably encountered at the beginning of g
FIG. 36. Methods for CHF measurement in terrestrial and parabolic flight experiments.
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41
period due to the abrupt gravity change, the heating is delayed for 5–10 s until the flow is stabilized in the latter half of the period. Power input to the heated tube is interrupted in case the wall superheat exceeds 80 K in order to protect the gold film from circumferential cracks or from serious change in its electric resistance. Because of the limitation in the power supply from the aircraft, CHF values corresponding to total power consumption above 3 kW cannot be realized. The heat transfer coefficient is defined by using the difference between mean wall temperature along the entire heated length and the fluid saturation temperature. Heat flux is calculated from the power input to the gold film and wall temperature to is estimated by a calibration curve obtained in advance. Since the relation between electric resistance of thin gold film and its temperature varies with time elapsed and with the iteration of heating and cooling, a large change in the calibration curve is confirmed when the wall temperature is increased markedly under CHF conditions. The calibration curve is modified during every test run in the manner described in Section III.C. The data are acquired at a frequency of 9 Hz or 135 Hz. The liquid–vapor behavior in the heated tube is recorded by either CCD camera or high-speed video camera with 372 fps. The experiments are conducted using R113 at mass velocity G ¼ 150 kg/m2 s under pressure P ¼ 0.1–0.2 MPa. The variation in system pressure is due to the shortage of cooling capacity of the condenser, where limitation in the power supply deteriorates the performance of Peltier modules.
B. RESULTS OF CHF MEASUREMENT 1. Critical Heat Flux in High Quality Region A tube with 260 mm heated length is employed so that CHF conditions are easily established at a certain location of the long heated section. Figure 37(a) shows the transition of heat transfer coefficient measured in microgravity by increasing heat flux at a constant rate under pressure P ¼ 0.19 MPa, mass velocity G ¼ 150 kg/m2 s and inlet quality xin ¼ 0.71. The time ¼ 0 implies the start of the measurement. Annular flow is observed, and heat transfer is dominated by two-phase forced convection with weak dependence on gravity. Heating is delayed 6 s from the beginning of the microgravity period and heat flux is increased at a constant rate dq/d ¼ 5.7 103 W/m2/s. Increase in is observed as the heat flux is increased in the early stage. The value of , however, soon takes an almost constant value despite the successive increase of q, and then it decreases
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FIG. 37. Transition of heat transfer coefficient and wall temperature for linear heat flux increase and liquid–vapor behaviors at high quality.
gradually due to the extension of the dry area on the inner wall of the tube. The observed trend of constant is interpreted as follows: (i) The increase in nucleate boiling heat transfer coefficient is canceled by the heat transfer deterioration due to the dry patch extension toward the upstream. (ii) Nucleate boiling in the annular liquid film is suppressed by the increased velocity of liquid film flow due to the increased vapor flow rate as a result of enhanced vaporization, and the heat transfer due to two-phase forced convection is not influenced by heat flux. Since the decrease in with increase in q is interpreted as a result of CHF conditions, the CHF value is evaluated as q ¼ 3 104–4 104 W/m2. The stabilized heat transfer after ¼ 38 s is not taken into consideration because it is obtained under the hyper-gravity condition at 1.5g after the recovery of gravity level. An example of liquid–vapor behavior at CHF, obtained in the same test run, is shown in the figure. The dry area is extended on the inner wall of the tube and many liquid droplets as a trace of annular base film are
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
43
attached to it. These droplets are carried downstream along the tube wall by the shear force exerted by the vapor core flow. The dried areas are not distributed uniformly on the inner wall, but rivulets of liquid film are still recognized. Small disturbance waves sweep the droplet and rivulets periodically to reproduce instantaneously the annular liquid film on the wall. Despite CHF conditions being triggered by the disappearance of annular liquid film, the accurate CHF point cannot be specified because disturbance waves periodically re-wet all surfaces even once dried, at least in the early stage of temperature excursion. The terrestrial result is shown in Fig. 37(b) for the same flow conditions. Similar trends are observed in the transition of versus q. The value of CHF is estimated as q ¼ 3 104–4 104 W/m2. No marked difference in CHF values under microgravity and normal gravity conditions is recognized within the accuracy of the present experimental method. Such a trend is confirmed by the comparison of liquid–vapor distributions between both gravity levels. The method of stepwise heat flux increase was also employed in order to confirm the results of the above CHF measurement by linear heat flux increase. Figure 38 summarizes the results when different heat fluxes, (a) q ¼ 3 104 W/m2, (b) q ¼ 4 104 W/m2 and (c) q ¼ 5 104 W/m2, are supplied by one step at high quality under microgravity conditions. At q ¼ 3 104 W/m2 dry patches are not observed on the inner tube wall, and the value of assumes a constant value which is nearly equal to that of nucleate boiling obtained in the bubbly flow regime. On the other hand, at q ¼ 4 104 and 5 104 W/m2, the dry patches extend on the inner tube wall and deteriorates under the constant values of supplied heat flux. At q ¼ 4 104 W/m2, takes a nearly constant value after it deteriorates due to CHF. The situation seems to be caused by the periodical passage of disturbance waves resulting in quasi-static conditions by quick alternation of wet and dry periods on most parts of the tube wall. In the meantime, a slight increase in wall temperature to is recognized as shown in the figure. But the phenomenon could be traced no more than several seconds before the end of microgravity duration. At q ¼ 5 104 W/m2, the wall temperature to increases rapidly due to extending dry patches. Heat input was suspended immediately because the wall superheat exceeded 80 K. Figure 39 summarizes the CHF data obtained by the method of stepwise heat flux increase under both microgravity and normal gravity conditions. It is difficult, also in this manner of heating, to find the CHF points defined by the location where a marked temperature jump is recognized. Therefore, dryout condition is represented by the heat flux versus range of quality between the inlet and outlet of the heated tube. Bold and thin lines in the figure represent the conditions where dryout occurs and does not occur,
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FIG. 38. Transition of heat transfer coefficient and wall temperature by stepwise heat flux increase at high quality under microgravity conditions. (a) q ¼ 3 104 W/m2; (b) q ¼ 4 104 W/m2 (CHF conditions); (c) q ¼ 5 104 W/m2 (CHF conditions).
respectively. In the figure, terrestrial CHF values calculated from the correlations by Katto and Ohno [26] are shown by a curved line for reference. The CHF value lies between 3 104 W/m2 and 4 104 W/m2 under both gravitational conditions. Thus, again no distinct effect of gravity
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
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FIG. 39. Comparison of CHF data by stepwise heat flux increase in high quality region at 1g and g.
on CHF value is recognized in the high quality region within the accuracy of the present method. These results are consistent with the nearly gravityindependent behavior of annular liquid film due to the enhanced interfacial shear stress exerted by the vapor core flow with high velocity. 2. Critical Heat Flux in Moderate Quality Region In the moderate quality region, x ¼ 0.3–0.5, the CHF values are quite larger than those at high quality. Therefore, the shortest tube with heated length 17 mm was employed to realize large heat flux under the limitation of power supply from the aircraft. Figure 40 shows the transition of heat transfer coefficient under microgravity conditions when the heat flux is increased up to the maximum value q ¼ 1.6 105 W/m2 at a constant rate dq/d ¼ 2.4 104 W/m2/s at pressure P ¼ 0.20 MPa, mass velocity G ¼ 150 kg/m2 s and inlet quality xin ¼ 0.19. The heat flux is kept at a small value q ¼ 2 104 W/m2 for 8 s from the beginning of the microgravity period until the flow oscillation disappears. A large increasing rate of heat flux is unavoidable in short microgravity duration because of the large CHF value predicted. In this quality annular flow is realized and nucleate boiling is completely suppressed at q ¼ 2 104 W/m2, while bubbles are observed in the liquid film after increase in heat flux. As shown in the figure, the value of increases with the increase in q, because the heat transfer dominated by nucleate boiling is seriously influenced by the increase in heat flux along a relation / q0.8. The value takes a maximum value at q ¼ 1.5 105 W/m2 because of the CHF condition and decreases rapidly at higher heat flux due
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FIG. 40. Transition of heat transfer coefficient and wall temperature by linear heat flux increase in moderate quality region at g (CHF conditions).
to dryout. Liquid–vapor behavior near CHF conditions is similar to that observed in the high quality region except that many bubbles due to nucleate boiling are recognized in a thicker annular liquid film and disturbance waves transport a larger amount of liquid and periodically quench the dried areas. Small dry patches distributed on the heated wall seem to originate from an annular base film surrounding small bubbles. The fluctuation of around the CHF point is larger than that in the high quality region. The mechanisms of CHF accompanied by such temperature fluctuation are discussed in detail in Section IV.C. To determine the influence of the increasing rate of heat flux on CHF values, CHF conditions are established by different rates under terrestrial conditions. The experiments were conducted in the moderate quality region. As shown in Fig. 41, a larger CHF value resulted from higher heating rate because of the heat capacity of the tube employed. The value of qCHF for a limit dq/d ¼ 0 W/m2/s should coincide with the value from the conventional method of CHF experiments on the ground depicted in Fig. 36(a), where enough time is given before a small increment in heat flux. The influence of high heating rate on CHF is negligible if dq/d<3 104 W/m2/s. The criterion was satisfied in the previous experiments. Figure 42 summarizes CHF data in the moderate quality region under both microgravity and normal gravity conditions. There is little effect of gravity on the CHF values. The result seems to contradict the observation at heat flux quite lower than CHF, where interfacial disturbance on the annular liquid film varies seriously with gravity when nucleate boiling is
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
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FIG. 41. Effect of increasing rate of heat flux on CHF values.
FIG. 42. Comparisons of CHF data by linear heat flux increase in moderate quality region at 1g and g.
completely suppressed. But the result is consistent with almost no gravitydependent behaviors of phase distribution and heat transfer coefficient once nucleate boiling occurs in the annular liquid film, as already shown in Fig. 21. Nucleate boiling occurs in the annular liquid film, and collapse and coalescence are influenced not by gravity but mainly by the surface tension force because of the small bubble size in the annular liquid film. The present CHF data are much higher than those obtained at high quality shown in Fig. 39 but much lower than the prediction by Katto and Ohno [26]. Further discussion is required on the differences in nucleation characteristics, surface wettability, heat capacity between the smooth glass surface and the ordinary metal surfaces, and on the effect of extremely short heated length inevitably adopted in the present experiment. The present CHF values under normal gravity conditions could, however, be possible when the reduction from the pool boiling CHF value, i.e. 2.5 105 W/m2 at 0.1 MPa for Fron113, due to the boiling in the thin annular liquid film, overcomes the increase due to the existence of bulk flow in the film.
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FIG. 43. Inverted annular flow at 1g and g (R113, P ¼ 0.10 MPa, G ¼ 150 kg/m2 s, xin ffi 0, q ¼ 3 105 W/m2).
3. Burnout in Low Quality Region In the low quality region, CHF conditions are established by DNB also in microgravity. Since it accompanies high wall temperature, a gold film heater coated on the inner wall of tube is damaged for one heating. Because the temperature–resistance relation changes seriously during a possible short heating period, heat transfer data could not be obtained in the present CHF condition. Experiments were performed by using heated tubes with 17 mm heated length. In Fig. 43, pictures of inverted annular flow are compared between 1g and g for stepwise power input at q ¼ 3 105 W/m2 under P ¼ 0.10 MPa, G ¼ 150 kg/m2 s and xin ffi 0. At 1g, a wavy annular vapor film with a wavelength of 10–20 mm was observed and many disturbances due to the generation of fine bubbles at the inlet periphery of the heated tube are recognized on the surface of the liquid core flow, while large bubbles from the inlet are entrapped in the liquid core flow having smoother surface and thinner diameter, i.e. thicker vapor film, at g.
C. TEMPERATURE OSCILLATION AND LIQUID–VAPOR BEHAVIOR JUST BEFORE CHF 1. Experimental Methods To find appropriate experimental conditions for the future experiments using longer microgravity duration, the clarification of CHF mechanisms becomes inevitable. On the ground, Ueda and Kim [27], performed experiments using a vertical annulus with a heated core rod in a bubbly flow regime, and concluded that CHF conditions are established when the increase in wall temperature due to the partial disruption of the liquid film becomes greater than the temperature decrease due to the quenching during the following period of liquid passage.
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In the present section, the process of dryout in the moderate quality region is investigated in detail. The identification or regulation of the dryout point on the heated tube is required to relate the CHF value to the dryout quality, and the distribution of wall temperature along the flow direction could be evaluated by dividing the resistance film into segments by the attachment of additional thin electrodes [28]. According to the previous experiments, the thin gold film is cracked along the circumferential direction if the tube is exposed once to extremely high temperature due to dryout. If a long heated tube is introduced, it becomes quite difficult to control the dryout point during a short microgravity duration without resulting serious damage of the tube. To remove such difficulties, in the present experiments a short tube with 17 mm heated length is employed. By the introduction of small heated length, the fluctuation of local wall temperature and periodical liquid–vapor behaviors due to the passage of disturbance waves are directly compared. An unheated glass tube is connected in the upstream location of the heated tube in order to eliminate the entrance effect on the flow. The experiments are conducted using R113 as the test fluid under the conditions: system pressure P ¼ 0.093 MPa, mass velocity G ¼ 150 kg/m2 s, inlet subcooling xin ¼ 0.2, 0.5, heat flux qo ¼ 4.0 104–3.0 105 W/m2. To avoid flow oscillation by the transition from hyper-gravity to microgravity, heating of the tube is started after the flow becomes stable for several seconds. Voltage drops across the gold film and across the standard resistance for evaluating the electrical current passing through the film, and e.m.f. of thermocouples, are measured at a frequency of 135/s by the use of a digital recorder. A high-speed video camera is used at 240 fps for the observation. Because of the size of the digital memory installed, the maximum recording period is restricted to 4.5 s, and recording of pictures is conducted only in microgravity periods. Heat flux to the fluid is evaluated from the supplied heat flux qo and the transition of the inner wall temperature Tw() by the numerical solution of transient heat conduction across the glass tube substrate (Fig. 44): @t=@ ¼ aG r2 t Boundary conditions:
Initial condition:
>0 y ¼ 0; t ¼ Tw ðÞ y ¼ G ; @t=@y ¼ 0 G y 0 ¼ 0; t ¼ Tw ð0Þ
ð26Þ
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FIG. 44. Calculation of heat flux to fluid based on measured inner wall temperature.
where T ¼ temperature, r ¼ radius, ¼ time, G ¼ thickness of glass wall and aG ¼ thermal diffusivity of the glass tube. The initial condition is given at the start of heating, but it has no influence on the results after a short time has elapsed. Heat flux to the fluid qw is obtained from: qw ¼ qo G ð@t=@yÞy¼0
ð27Þ
where G is the thermal conductivity of glass tube. 2. Observation of Liquid–Vapor Behavior Behaviors of annular liquid film are shown in Fig. 45. All of the pictures A–E are obtained in microgravity at heat fluxes qo ¼ 1.6 105, 2.2 105 W/ m2, lower and higher than the CHF value. Inlet quality is xin ¼ 0.2, while exit qualities are 0.26 and 0.29 for these heat fluxes, respectively. Annular flow is observed under these conditions. At qo ¼ 1.6 105 W/m2, the passage of disturbance waves has a role to supply liquid to the tube wall (Picture A). Dry patches are extended locally and this is repeated between an interval of two consecutive disturbance waves (Picture B). The extension of dry patches is more marked for a longer interval (Picture C). At heat flux qo ¼ 2.2 105 W/m2 above the CHF value, a large dried area is observed just before the passage of a succeeding disturbance wave (Picture D). A much larger dry area is observed at higher wall temperature as time goes on (Picture E). Under these conditions, nucleate boiling is not suppressed and small bubbles are recognized in the annular liquid film. The bubble size is around 1 mm, the same size observed in saturated pool boiling. No qualitative difference between liquid–vapor behaviors at these heat fluxes is observed.
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FIG. 45. Behaviors of annular liquid film in microgravity.
Pictures F–I are obtained for higher inlet quality xin ¼ 0.5 at heat fluxes qo ¼ 2.2 105, 3.0 105 W/m2, lower and higher than a CHF value, respectively, where exit qualities are xex ¼ 0.59 and 0.62. In comparison with the meniscus behaviors for xin ¼ 0.2, even at heat flux qo ¼ 2.2 105 W/m2, lower than CHF, the annular liquid film is easier to evaporate and is disrupted to make rivulets. And just after the passage of a disturbance wave, dry patches are immediately formed (Picture F). The interval of disturbance waves in this quality becomes shorter than that at xin ¼ 0.2, and no large change is observed in the distribution of liquid–vapor meniscus even just before a disturbance wave followed (Picture G). The deposition of liquid droplets entrained by the vapor core flow is not recognized from the video pictures. Since the annular liquid film is thinner for xin ¼ 0.5, the bubbles due to nucleate boiling collapse at smaller diameter. Similar bubble behavior is observed also for qo ¼ 3.0 105 W/m2, higher than CHF (Picture H), but a clearly larger extent of dry patches is recognized just before the passage of the disturbance wave (Picture I). Also under the present quality conditions, no marked qualitative difference is noticeable between both heat flux levels. Even at a heat flux larger than CHF, however, small pieces of rivulets and droplets still remain on the dried area. 3. Heat Transfer Data Figure 46 shows the transition of inner wall temperature Tw, heat flux to fluid qw and heat transfer coefficient versus time under microgravity
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FIG. 46. Fluctuation of inner wall temperature, heat flux to fluid and heat transfer coefficient at low quality in microgravity.
conditions for inlet subcooling xin ¼ 0.2 at supplied heat flux qo ¼ 1.6 105, 2.2 105 W/m2, lower and higher than CHF, respectively. Abscissa is time elapsed from the start of the recording of video images. Vertical gray bands indicate the passing periods of disturbance waves obtained from the analysis of high-speed pictures. The instances corresponding to the pictures in Fig. 45 are indicated in the figures. At qo ¼ 1.6 105 W/m2 (Fig. 46(a)), values of and qw vary exactly with the passage of disturbance waves. As is observed in the case of a long interval between consecutive disterbance waves, for example, ¼ 3.2–3.3 s, the transition of takes minimum and maximum values. (i) The first decrease in is caused by the increase in annular liquid film just after the
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passage of the disturbance wave and also by the suppression of nucleate boiling. (ii) The increase in followed by the first decrease results from the decrease in the thickness of annular liquid film due to evaporation and also from the reactivation of nucleation sites. (iii) The second decrease in is due to the increased total area of dry patches. Such variation in heat transfer coefficient is caused primarily by the change of wall heat flux and not of wall temperature. The values of obtained here were confirmed to be similar to those in pool boiling from the horizontal copper plate [29]. At heat flux qo ¼ 2.2 105 W/m2 (Fig. 46(b)), only the trends (ii) and (iii) are observed because the thickness of annular liquid film rapidly decreases just after the passage of the disturbance wave under the higher heat flux condition. The transition of heat transfer coefficient takes a maximum value only between a large interval of disturbance waves. When the interval is large, the wall temperature is markedly increased because of the large extension of dry patches as shown in Pictures D and E. It is known that the temperature excursion is composed of the accumulation of temperature increments during large intervals of disturbance waves. The value qw is smaller than qo by the amount consumed for the heating of the tube substrate. The wall temperature Tw is increased up to 200 C. Figure 47 shows the results in microgravity for xin ¼ 0.5 at heat fluxes qo ¼ 2.2 105 and 3.0 105 W/m2, lower and higher than the CHF value. At this quality, the passing frequency of disturbance waves is quite high, and the variation of qw and is small at qo ¼ 2.2 105 W/m2 (Fig. 47(a)). At qo ¼ 3.0 105 W/m2 (Fig. 47(b)), a distinct decrease in and an increase in Tw are observed when the interval between disturbance waves becomes large. As is clear from Picture I, small liquid films or rivulets still remain during the extension of the dried area even just before the passage of a succeeding disturbance wave, because the ‘‘large’’ intervals of disturbance waves resulting in substantial wall temperature increment are smaller than those for xin ¼ 0.2. As a consequence, the heat flux to fluid qw is quite large and the deviation from the supplied heat flux qo is small, and the rate of increasing Tw is smaller than that for xin ¼ 0.2 and qo ¼ 2.2 105 W/m2. When the wall temperature becomes much higher under CHF conditions, its fluctuation becomes smaller and finally disappears resulting in a rapid increase in the wall temperature. Such a situation of wall temperature increase is also true for xin ¼ 0.2. 4. Detailed Analysis of Liquid–Vapor Behavior Liquid–vapor behavior at a local point on the heated tube wall is classified into four patterns based on the observation between the passage of consecutive disturbance waves. Case (L): the wall is covered by a thick liquid film after the passage of a disturbance wave and nucleate boiling is
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FIG. 47. Fluctuation of inner wall temperature, heat flux to fluid and heat transfer coefficient at moderate quality in microgravity.
suppressed. Case (B): nucleate boiling occurs from the tube wall and many bubbles are recognized in the annular base film. Case (F): the wall is covered by thin films or rivulets involving no bubbles. Case (D): the wall is completely dried. To examine the ratio occupied by these liquid–vapor behaviors, the wall area with 8 mm in the circumferential length and 15 mm in the axial length is divided into 7 15 segments and the phenomena on each segment wall are identified and classified into the above four patterns. Figure 48(a) and (b) show the results for xin ¼ 0.2, qo ¼ 1.6 105 W/m2 and qo ¼ 2.2 105 W/m2, respectively. The ordinate is time elapsed between the consecutive passage of two disturbance waves. The abscissa is the axial
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FIG. 48. Transition of liquid–vapor behaviors on the segment areas of tube in microgravity.
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distance of segment areas from near the heating edge in the upstream, and seven different segments along the wall periphery are indicated together in each band width, for convenience, as if they are distributed along the flow direction. In Fig. 48(a), for the segments located at the left and right edges of the individual band in the upstream region, Case (L) is frequent at the early stage. This is attributed to the compression of pictures in the sides of a round tube. In the figure, the trace of the disturbance wave movement downstream can be recognized. The top part of the figure represents the classification of liquid–vapor behaviors just before the passage of the next disturbance wave. There still remains liquid around bubbles. As shown in Fig. 48(b), at heat flux higher than CHF, the disappearance of areas covered by a thick film is very quick and most of the downstream surface is dried during an interval between consecutive disturbance waves. But in the upstream, there still remains liquid being sustained around the contact lines of small bubbles. Therefore, it can be said that small bubbles in the annular liquid film delay the occurence of CHF conditions. In Fig. 49(a) and (b), corresponding to Fig. 48(a) and (b), respectively, the ratios of area occupied by each pattern to the total are plotted against time, where transitions of heat transfer coefficient and wall heat flux qw are shown together. Partially due to the ambiguity in the identification of the liquid–vapor patterns, large fluctuation is observed for all lines. (i) For supplied heat flux levels both higher and lower than CHF, the total area of nucleate boiling is kept almost constant during the interval, despite the areas of pattern B being concentrated in the upstream at qo ¼ 2.2 105 W/m2. (ii) Just after the passing of the disturbance wave, a thin liquid film is formed, but the duration of this thin liquid film is extremely short at heat flux higher than CHF. (iii) Values of and qw take maxima when heat transfer enhancement due to the increase in the areas of pattern F, where the evaporation of thin films occurs, is replaced by the increase in dried areas of pattern D owing to their complete evaporation. 5. Effect of Gravity on Heat Transfer Data In Fig. 50(a) and (b), the transition of Tw, qw and at hyper-gravity 2g for xin ¼ 0.2, qo ¼ 1.6 105 and 2.2 105 W/m2, the same conditions as those in Fig. 46(a) and (b), respectively, is shown. Also in hyper-gravity, heat flux qo ¼ 1.6 105 W/m2 is lower than CHF, while CHF conditions are established at qo ¼ 2.2 105 W/m2 as observed in the excursion of Tw. No picture is available in this gravity condition because of the restrictions mentioned before, and the figures do not indicate the periods of disturbance waves. In hyper-gravity, detailed classification of the phenomena is inherently not possible due to the reduction of brightness of images caused
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FIG. 49. Area ratio of patterns for different liquid–vapor behaviors in microgravity. L: Thick liquid film or disturbance wave; B: Nucleate boiling; F: Thin liquid film; D: Dried area.
FIG. 50. Transition of fluctuation of inner wall temperature, heat flux to fluid and heat transfer coefficient at low quality in hyper-gravity (2g, xin ¼ 0.2).
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by the high turbulence in the annular liquid film as deduced from Fig. 19. At qo ¼ 1.6 105 W/m2, no appreciable effect of gravity is observed in the level of and qw, while the fluctuation of these values is much emphasized in hyper-gravity. Since the passing frequency of disturbance waves in the low and moderate quality regions increases markedly at hyper-gravity, the reason for such a low frequency of the fluctuation is not clear. The observed large fluctuation of heat transfer coefficients is exactly corresponding to the actual behaviors of liquid and vapor in a tube, and it is expected that the interaction between the phenomena near the wall and bulk flow result in a larger fluctuation with lower frequency. At qo ¼ 2.2 105 W/m2, after dry patches cover most parts of the surface at ¼ 2.7 s, wall temperature increases rapidly. The fluctuation of and qw during the excursion of wall temperature gradually become smaller with the decrease in wetted area at higher temperature. The mechanism to increase the wall temperature in hyper-gravity is supposed to be similar to that in microgravity. At higher inlet quality xin ¼ 0.5, no appreciable gravity effect was observed in the range of CHF values and in the nature of temperature fluctuation. 6. Mechanisms of CHF Figure 51(a)–(d) shows histograms for the interval between two consecutive disturbance waves f for the experimental conditions in Figs. 46 and 47. The number of data read from video images is not unified among them. The values f for xin ¼ 0.5 are, on average, smaller than those for xin ¼ 0.2. The effect of heat flux on f is not large, but detailed examination indicates the lack of short intervals at heat flux higher than CHF for both xin ¼ 0.2 and 0.5, which is also confirmed by Figs. 46 and 47. CHF conditions are established by the temperature increase in a long interval between succeeding disturbance waves becoming excessive beyond the ability of cooling by a disturbance wave following. It is difficult to specify a trigger of temperature excursion, but one longer interval, statistically encountered, results in higher wall temperature, and is not sufficiently quenched by a succeeding disturbance wave, resulting in quicker evaporation of thin liquid film to produce an excessive temperature increment again. D. SUMMARY AND DIRECTION OF FURTHER INVESTIGATION Acquisition of CHF data in a short microgravity duration created along a parabolic trajectory of an aircraft was attempted by using transparent tubes, where liquid–vapor behaviors such as the motion of disturbance waves and the distribution of liquid meniscus on the heated wall around the CHF point
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FIG. 51. Statistical representation for passage intervals of disturbance waves in microgravity.
were directly compared to the observed liquid–vapor behaviors. The results from the experiments using Fron113 at around 0.1 MPa are summarized as follows: (i) In the high quality region, high interfacial shear exerted by vapor core flow and high evaporation rate dominates the transient shape and distribution of annular liquid film and dry patches, and gravity has only a secondary effect on the CHF phenomena. As a consequence, no difference in CHF values between microgravity and normal gravity is observed. (ii) The gravity-independent CHF values also hold true for the moderate quality region, where nucleate boiling occurs in the annular liquid film and collapse and coalescence are influenced not by gravity but mainly by the surface tension force because of small bubble size in the annular liquid film. Small dry patches distributed on the heated wall seem to originate from a liquid film surrounding small bubbles. (iii) Even at CHF, the inner tube wall never completely dries, and the disturbance wave supplies liquid to wet the dry patches extended during an interval of consecutive disturbance waves. (iv) At heat flux around CHF in low and moderate quality regions, in general, four different heat transfer processes are possible during an interval between disturbance waves: forced convection due to the flow in a thick annular film during or just after the passage of the disturbance wave; nucleate boiling from the sites reactivated in the annular liquid film; the evaporation from thin liquid films or rivulets contacting to the heated wall; and local dryout by
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extending dry patches. (v) CHF occurs in low and moderate quality regions, when ‘‘heating’’ of the wall by the extension of dry patches in one interval between disturbance waves exceeds ‘‘cooling’’ by the passage of disturbance following. (iv) A longer interval between disturbance waves produces an excessive temperature increment and could become a trigger for temperature excursion if the heat flux is high enough. There are still inconsistencies among the values of CHF obtained in different series of experiments, partially due to the inferior accuracy in the setting of experimental conditions on board the aircraft. The acquisition of data with high reproducibility is expected in long-term experiments in the future. The difference in CHF conditions between the present results obtained from a very short heated tube and those which could be expected by using long heated tubes becomes larger at higher quality, where the difference in the upstream conditions directly influences the cross-sectional distribution of both phases and, in turn, on the behavior of annular liquid film including, for an example, the deposition of entrained droplets. In the present experiment, the burnout data under low quality or subcooled liquid conditions was not obtained. Liquid subcooling, as well as the velocity of bulk flow, is one of the useful means of regulating bubble size also in microgravity as confirmed by the present author [12] in pool boiling experiments, and significant effects on the CHF phenomena. In addition to the parametric study for CHF values, analysis concerning the gravity effect on the behavior of vapor film in inverted annular flow is required.
VII. Experiments on Flow Boiling Heat Transfer in Narrow Channels A. BACKGROUND FOR BOILING EXPERIMENTS IN NARROW GAPS For the effective utilization of space platforms and for decreases in the launch weight to reduce the cost, the small size of heat exchangers is one of the goals and the system of a narrow channel with a flat heating surface becomes the simplest example investigated here. The effect of surface orientation on the CHF for flow boiling in a duct was investigated on the ground by Brusstar and Merte [30]. The present author [31] reported the results of preliminary experiments for microgravity flow boiling in narrow channels where the deteriorated heat transfer in microgravity was shown. In this research, the effect of gravity on flow boiling in narrow channels is investigated systematically and characteristics of bubble behavior and of heat transfer peculiar to microgravity conditions are clarified. From microgravity experiments of nucleate pool boiling conducted so far by the present author, the behavior of the microlayer underneath bubbles
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attached to the heating surface plays a dominant role on the heat transfer [12]. Either of opposite trends of enhanced or deteriorated heat transfer was observed depending on the behavior of dry patches extending in microlayers [32]. Even under the microgravity conditions, however, the liquid–vapor actions underneath an attached bubble cannot be clearly observed because of small bubble size, especially under subcooled liquid conditions. In narrow channels between parallel flat plates, generated bubbles are enlarged by the deformation and increase the contact area on the heating surface, which emphasizes the nature of nucleate boiling and becomes advantageous for knowing fundamentals of heat transfer mechanisms in microgravity nucleate boiling. The knowledge from the present boiling system, on the other hand, has various practical applications, e.g. the cooling of electronic devices of high heat generation density. In existing research on the ground, the heat transfer enhancement is observed if the gap size is smaller to some extent than the bubble departure diameter in pool boiling, but further decrease in the gap size results in heat transfer deterioration as a result of vapor accumulation on the heating surface [33]. In these experiments, the test section assembly, composed of a flat heating and an unheated surface in parallel to confine the flow channel remaining outer peripheries open, is immersed in the liquid pool. Then the velocity and direction of liquid flow induced in the narrow channel is a function of the void fraction in the channel and, in turn, of the heat flux supplied. In the present experiments, the test section of the narrow channel is connected to the test loop to regulate the inlet temperature and velocity as shown in Ref. [34]. B. EXPERIMENTAL APPARATUS AND PROCEDURE The test section of the narrow channel is composed of a glass heating surface, an unheated glass plate and a stainless flange, assembled by the aid of O-rings to construct a rectangular duct as shown in Fig. 52. The test section is oriented vertically and upward flow is realized in the gravitational fields. The size of the channel in the test section is 48 mm in width and 81 mm in length. Gap sizes are adjusted as s ¼ 0.7 mm, 2 mm and 10 mm by the selection of spacers between both glass plates. Since the width of the inlet and exit ports of the test section is smaller than that of the inside channel due to a problem of sealing, curved vanes are inserted in the inlet port to realize the uniform liquid velocity flowing in the inside channel. A heating surface is made of sapphire glass plate of 3 mm thickness which is cut from a circular plate of 101 mm diameter as shown in Fig. 53. The surface is heated by the direct application of electric current through the ITO film coated on the surface opposite to the heat transfer side. The
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FIG. 52. Test section of narrow channel.
FIG. 53. Glass heating surface and sensor locations.
structure ensures the perfect electrical insulation between the heater and sensors on the heat transfer side, and makes possible the accurate evaluation of surface heat flux by the solution of conduction across the grass substrate. The heated area is 40 mm in width and 70 mm in length along the flow direction. The transparent material of the heating surface allows the direct observation of liquid–vapor behavior from underneath. The distribution of surface temperatures is determined by seventeen temperature sensors, i.e. three or five sensors along the flow direction and five sensors in the transverse direction as shown in the figure, where six sensors at the right and left sides are introduced to evaluate the temperature distributions with more accuracy, and thus to minimize the error in the evaluation of local heat flux distribution. The temperature sensor, made of thin platinum film directly coated on the glass substrate, operates as electric resistance thermometer, and it has resistance values of around 1500 . The local values of surface heat flux on the heat transfer side are calculated by three-dimensional transient heat conduction across the substrate, where the measured temperature distribution and a uniform temperature gradient corresponding
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to the heat flux supplied are used as boundary conditions. Local values of heat transfer coefficients are evaluated by using bulk fluid temperatures obtained from the heat balance equation. The heating surface has three additional sensors for the measurement of liquid film thickness along the centerline. The microlayer thickness could be evaluated from the measured electric resistance of liquid across the electrodes of specified arrangement [35]. Accurate measurement of the film thickness, however, was impossible in the present experiments owing to the superposition of noise on the signal of electric current. Two test sections are connected in parallel to the test loop to change the gap size easily during the flight experiments. The same loop as shown in Fig. 10 is employed and the test sections are installed perpendicular to the cabin floor of the aircraft to realize upward flow under gravity fields. Liquid–vapor behaviors are observed from the heater side opposite to the heat transfer side. Experiments were performed by using distilled water as the test fluid at pressure P ¼ 0.093 MPa, inlet liquid subcooling Tsub,in ¼ 0–10 K, average heat flux calculated from power input qo 3.2 105 W/m2. The inlet liquid velocity is kept constant at uin ¼ 0.06 m/s for all gap sizes tested. Nucleate boiling was initiated during the horizontal flight under a 1g gravity field. After confirmation that the supplied heat flux was to be lower than that at CHF conditions, the boiling system was exposed to different gravity levels along a parabolic trajectory. Also in this case, the duration of reduced gravity at 0.01–0.03g was succeeded by the hyper-gravity 2g period of 15–20 s. The digital data was acquired at 9 Hz. C. EXPERIMENTAL RESULTS AND DISCUSSION 1. Large Gap Size (s ¼ 10 mm) The effect of a narrow channel on the phenomenon is small in the present case because the gap size is larger or not much smaller than the bubble detachment diameter without such a restriction for each gravity level. The interference between the surfaces of the channel and bubbles is quite weak, especially under 1g and 2g conditions. At g, the rising velocity of attached bubbles is decreased due to the reduction of buoyancy and takes a value dominated by the shear force exerted by the bulk flow and by the surface tension force. As a consequence, the period for the bubble contacting and sliding along the heating surface becomes longer resulting in an increase in its volume. At the same time, bubble coalescence is promoted in the downstream region of the channel. To remove the variation of static pressure due to the change of gravity level or change in exit void fraction at different heat flux levels, the test section was installed at the top part of the test loop. Then bubble size is not influenced by the change of static pressure
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when gravity level is decreased. At 2g, the increased rising velocity due to the elevated buoyancy results in a decrease in bubble diameter, which is clearly observed at the end of the heated section. At g, a flattened bubble is not yet observed for the highest value of heat flux qo ¼ 3.2 105 W/m2 in the present experiment and near saturated liquid condition Tsub,in ¼ 2–10 K at uin ¼ 0.06 m/s. Even in the downstream, the effect of gravity on the heat transfer is not clearly recognized except that the fluctuation of local heat flux and of local heat transfer coefficient is larger to some extent because vigorous coalescence of bubbles occurs there. Heat transfer insensitive to the change of bubble size or to the occurrence of bubble coalescence is consistent with some previous experimental results, where heat transfer dominated by nucleate boiling is quite insensitive to gravity as already indicated in Section IV.B.1. Such insensitivity in heat transfer can be possible only if the distribution of the microlayer thickness around the bubble contacting lines is almost unchanged regardless of gravity under the experimental range tested. Further decrease in the inlet velocity would promote the bubble coalescence and, in general, two opposite trends of heat transfer enhancement and deterioration coexist, one of which becomes dominant depending on the liquid meniscus behaviors underneath the coalesced bubbles, as was already observed in pool boiling in microgravity [31]. 2. Small Gap Size (s ¼ 2 mm)
Experimental results for qo ¼ 1.4 105 W/m2, Tsub,in ¼ 0–2 K and uin ¼ 0.06 m/s are shown in Fig. 54: the typical bubble behaviors for three different gravity levels, and the transition of surface temperature Tw, surface heat flux qlocal, heat transfer coefficient local and gravity level g/ge. The abscissa is time elapsed from the start of measurement. It includes the heat transfer data at three different positions on the centerline of the heating surface along the flow direction, i.e. 19.5 mm (referred to as upstream location), 39.5 mm (middle location), and 59.5 mm (downstream location) from the bottom heater edge. For all gravity levels flattened bubbles are observed, which grow in various directions on the surface. As a result, the periphery of the bubble becomes longer and the shape is complicated in the presence of the bulk flow of liquid at the constant rate. The bubble can no longer stay on the heating surface and detaches to move downstream when the shear forces acting on the periphery overcome the surface tension force. The staying period of flattened bubbles is shorter in the upstream region than that in the downstream, and becomes longer with the reduction of gravity level. Fluctuation in the values of local heat flux and of heat transfer coefficient is larger in the upstream, where the distinct bubble detachment is observed and a large change in the heat transfer rate is concerned whether the heating
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FIG. 54. Liquid–vapor behavior and local heat transfer data for different gravity levels. (Large gap size s ¼ 2 mm, qo ¼ 1.4 105 W/m2, Tsub,in ¼ 0–2 K, uin ¼ 0.06 m/s).
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surface is covered by a flattened bubble or is exposed to the bulk liquid. The fluctuation of qlocal and local is smaller in the middle location, where exchange of bubbles and bulk liquid on the same surface position occurs so frequently that heat transfer rate is averaged. In the middle and downstream locations, the symptom of heat transfer deterioration is already observed at 1g when a large dry patch extends underneath a large coalesced flattened bubble. At g, the deteriorated heat transfer is much more emphasized when large dry patches extend frequently due to the longer staying period of the flattened bubble and the decrease in the interval between the bubble detachments. It is noteworthy that even in such periods of deteriorated heat transfer, instantaneous values of qlocal and local in g are quite frequently increased from the values for 1g and 2g. The increased surface temperature due to heat transfer deterioration, in turn, increases the heat flux at the instances of re-wetting of large dry patches by the penetration of bulk liquid. Thus, the larger fluctuation in the values of qlocal and local in g is caused by both heat transfer deterioration and enhancement. Such instantaneous enhancement of heat transfer is not noticed at 2g where bubble detachment occurs at high frequency. The coexistence of opposite trends of enhanced and deteriorated heat transfer is confirmed also in the system of the narrow channel. Such a characteristic is one of the important features in microgravity boiling, but at the same time could be said to be one of the general heat transfer characteristics in nucleate boiling which is exaggerated by the aid of gravity reduction. Figure 55 shows the results for increased inlet subcooling up to Tsub,in ¼ 10 K keeping other parameters unchanged. A large flattened bubble is scarcely observed in the upstream location at 1g and 2g, because the liquid temperature is below the saturation and generated bubbles which overcome the severe nucleation criteria in this location cannot grow to become a large bubble and sometimes disappear immediately due to condensation. No large flattened bubble is formed even in the downstream location at 2g. On the other hand, at g, a large flattened bubble with a round shape stays for a long time on the surface and covers almost the entire surface area, where liquid flows at the constant rate along the sidewalls hidden by the front flange. The large flattened bubble detaches at very low frequency and the interval increases up to several seconds. During the time of its stay on the surface, the rate of evaporation from microlayers along the bubble contact line in the middle and downstream balances with the rate of condensation at the front of the bubble in the upstream location. The emergence of a large bubble results in the increase of wall temperature in the middle location in microgravity just at the center of the large flattened bubble. The trend of heat transfer deterioration observed under the saturated inlet conditions is still observed despite the increase in inlet subcooling, and is
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FIG. 55. Liquid–vapor behavior and local heat transfer data for different gravity levels. (Small gap size s ¼ 2 mm, qo ¼ 1.4 105 W/m2, Tsub,in ¼ 10 K, uin ¼ 0.06 m/s).
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emphasized at g, especially in the middle location which implies the emergence of a large dry patch underneath the large flattened bubble. Under some conditions at g, the heat transfer deterioration could be promoted with the increase of liquid inlet subcooling. Such a curious trend is caused by the size of bubbles elongating from the upstream to the downstream by the reduction of gravity. When heat flux is reduced to qo ¼ 1.1 105 W/m2 under the same subcooled condition and other parameters, at g, instantaneous enhancement of heat transfer coefficients due to the wetting of a large dry patch by the bulk liquid is frequently observed in addition to the deterioration accompanied by the emergence of a quite stable large bubble observed at the higher heat flux. 3. Extremely Small Gap Size (s ¼ 0.7 mm)
The gap size is decreased further to s ¼ 0.7 mm for qo ¼ 1.1 105 W/m2, Tsub,in ¼ 0–2 K, uin ¼ 0.06 m/s. Flattened bubbles extend at high growth rate for all gravity levels as shown in Fig. 56. The reduction of gap size increases the growth rate. On the other hand, inlet liquid velocity is kept constant regardless of gap size, which causes the rapid penetration of liquid into the flattened bubbles resulting in the shape of bubble peripheries being quite complicated. When observed at the fixed point on the heating surface, a series of instantaneous processes, i.e. extension of microlayer in the bubble growth period, its dryout, and rewetting of dry patches by the penetration of bulk liquid, is repeated at high frequency resulting in serious fluctuations of Tw, qlocal and local. But most data points are involved in a certain range as if steady state is established, and stable heat removal seems to be possible without resulting temperature excursion provided that the operating parameters are strictly controlled. No distinct gravity effect is observed, and terrestrial results could be applied to predict the heat transfer characteristics in space without uncertainty as a result of inertia and surface tension forces exceeding the buoyancy for this gap size and heat flux level. But some additional improvement, e.g. liquid bypass, is required to ensure the safe operation of a longer channel having such an extremely small gap size. Also a round periphery profile of bubbles can be obtained even at this gap size. Bubble behavior for qo ¼ 8.0 104 W/m2, Tsub,in ¼ 10 K, uin ¼ 0.06 m/s is shown in Fig. 57. Higher liquid subcooling and lower heat flux reduce growth rate of bubble resulting in round bubble shapes regardless of gravity level. The situation that rate of evaporation balances with that of condensation at the front of the bubble in the upstream location is again observed here. The periods of bubble generation and movement toward the downstream is short and not repeated at high frequency in this case. The frequency becomes lower as gravity level decreases. There is also no distinct gravity effect in this case.
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FIG. 56. Liquid–vapor behavior and local heat transfer data for different gravity levels. (Extremely small gap size s ¼ 0.7 mm, qo ¼ 1.1 105 W/ m , Tsub,in ¼ 0–2 K, uin ¼ 0.06 m/s). 2
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FIG. 57. Liquid–vapor behavior for different gravity levels (Extremely small gap size s ¼ 0.7 mm, qo ¼ 8.0 104 W/m2, Tsub,in ¼ 10 K, uin ¼ 0.06 m/s).
D. SUMMARY OF EXPERIMENTAL RESULTS Flow boiling in a rectangular narrow channel was investigated for different gravity levels along the parabolic trajectory of an aircraft. A glassheating surface with sensors for the measurement of surface temperature distribution was introduced to relate the measured heat transfer data directly to the observed bubble behaviors. (i) At a large gap size, the deformation of bubbles to generate flattened bubbles is not seen for all gravity levels and the increase in bubble size and the bubble coalescence in the downstream are promoted to some extent in microgravity resulting, however, in no appreciable change in the heat transfer dominated by nucleate boiling. (ii) At a small gap size, marked heat transfer deterioration is
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observed in microgravity due to the extension of dry patches underneath large flattened bubbles. But, at the same time, instantaneous heat transfer enhancement occurs frequently when the dried surfaces with higher temperature are re-wetted by the penetration of bulk liquid. (iii) At a small gap size, in microgravity, increase in inlet liquid subcooling sometimes deteriorates the heat transfer at low mass velocity because it makes a large flattened bubble covering almost the entire heating surface for a long staying period, resulting in the emergence of a large dry patch underneath the bubble. (iv) At an extremely small gap size, enhanced growth rate of flattened bubbles promotes the penetration of bulk liquid into these bubbles and instantaneous displacement of flattened bubbles by the bulk liquid is repeated at high frequency. Local values of heat flux and the heat transfer coefficient oscillate within a certain limited range as if steady state is established, and no distinct gravity effect is observed.
VIII. Future Investigations for Microgravity Flow Boiling Recent interest in the feasibility of SSPS (Space Solar Power System), which transports electric power generated in the solar panels on a satellite directly to the earth’s surface, enhances the requirement for knowledge of microgravity boiling heat transfer as a powerful means for cooling of electric devices involved in the system. Two-phase flow loops applied to space are classified into two types of loops operated by mechanical pump and by capillary pump. One more variation of the loop is called a hybrid loop, and this operates under active or passive pumping conditions, depending on the thermal load. The load varies with the generation of electricity along an orbit and with the operation of machines installed on the space platform. The subjects of microgravity boiling researches have extensive variations for different systems and different operational parameters. Under limited opportunities of microgravity experiments with short microgravity duration, the objective of research should be to find the phenomena characterized by microgravity environment and not to establish the complete database required for the design of boiling heat exchangers applied to space. But the situation of the investigation in the present discipline would be greatly improved if the experiments on the ISS (International Space Station) become possible in the near future, where a series of experimental periods as much as 100 h or more is available under a microgravity environment of better quality. Flow observation and flow parameter measurements are planned to be made for many subjects including phase distribution, classification of flow patterns, behavior of bubbles and liquid films, mechanisms of heat transfer, CHF conditions, and
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flow characteristics such as pressure drop and velocity profiles. Dependent parameters include, among others, the gravity field, phase velocity, channel geometry and dimensions, bubble diameter and the degree of liquid subcooling. A dimensional analysis approach should be included to determine the most pertinent dimensionless groups representing such parameters, i.e. Froude number, Bond number and Weber number, with appropriate definitions of the parameters involved. In addition to the macroscopic approach, detailed studies are to be done on bubble growth and detachment in the presence of external flow, the behavior of the microlayer underneath the bubbles, and flattened bubbles in narrow channels. Accurate measurements could easily be made for simpler phenomena in the absence of gravity. Even for the experimental system of flow boiling, reduction of inflow velocity to the test section would realize the conditions near those of pool boiling, and also becomes a means of realizing subcooled ‘‘pool boiling’’ without changing liquid conditions. Accumulated knowledge of pool boiling research [10,36–38] should be reflected in flow boiling studies, especially in the low quality or subcooled region, in addition to the results from microgravity isothermal two-phase flow. Experimental results in pool boiling on, for example, bubble growth [39], thermocapillary flow [40], heat transfer in subcooled liquid [41], CHF and micro-bubble emission boiling [42] will give new subjects for flow boiling investigation. The use of the binary system [43], in particular, will derive new phenomena characterized by Marangoni force due to concentration and temperature gradients which become effective in microgravity environment at low mass velocity. For further microgravity investigations on boiling and two-phase flow with hidden interesting phenomena, which deviate from those on the ground, the opportunity of orbital experiments with long-term microgravity duration should be open to many investigators having interest in the present discipline. Acknowledgements The present chapter summarize the results of microgravity flow boiling experiments conducted in 1993–1999 under the support of ‘‘Space Utilization Joint Research Projects’’ promoted by NASDA (National Space Development Agency of Japan)/JSUP (Japan Space Environment Utilization Center), ‘‘Fund for Basic Experiments Oriented to Space Station Utilization’’ by ISAS (Institute of Space and Astronautical Science) and ‘‘Ground Research Announcement for Space Utilization’’ by NASDA/ JSF (Japan Space Forum). The research began after the opportunities of my stay in the laboratory of Prof. Johannes Straub, T.U. Muenchen, where I saw a
MICROGRAVITY HEAT TRANSFER IN FLOW BOILING
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transparent heating surface developed by Prof. Herman Merte, Jr. in the University of Michigan. The author expresses appreciation to these leaders in the present research discipline, Prof. Hiroshi Fujiyama in Nagasaki University for consistent support of the present experiments, Dr. Koichi Inoue for the collaboration when he was my student, and Mr. Yasuhisa Shinmoto in Kyushu University for preparing the manuscript. The author appreciates very much the financial and technological support of the above organizations and the great assistance given by DAS (Diamond Air Service) in the aircraft experiments. Nomenclature a b C cp cd d dP/dz Fr fd fi fi0 G g g/ge JL JG ld n N P Q q qCHF qw q0 R r ro s T t tb to
thermal diffusivity, m2/s constant constant specific heat, J/kg K velocity of disturbance wave, m/s inner tube diameter, m pressure gradient, Pa/m Froude number :ðL JL þ G JG Þ2 =2L gD passing frequency of disturbance wave, 1/s interfacial friction factor interfacial friction factor at assumed zero gravity mass velocity, kg/m2 s gravity, m/s2 ratio of gravity to the earth value liquid superficial velocity, m/s gas superficial velocity, m/s length of disturbance wave, m number total number pressure, MPa volumetric flow rate, m3/s heat flux, W/m2 critical heat flux, W/m2 heat flux to fluid, W/m2 heat flux supplied, W/m2 electric resistance,
radius, m inner radius, m gap size, m temperature, C temperature, C bulk temperature, C wall temperature, C
tsat u V x y z
saturation temperature, C velocity, m/s voltage, V quality distance from tube wall ( ¼ ror), m length along flow direction, m
GREEK SYMBOLS T Tsub f "m "h
heat flux coefficient, W/m2 K void fraction temperature difference degree of liquid subcooling time interval between two consecutive disturbance waves, s thickness of annular liquid film, m eddy viscosity, m2/s eddy thermal diffusivity, m2/s kinematic viscosity, Pas density, kg/m3 shear stress, N/m2, or time, s thermal conductivity, W/mK
SUBSCRIPTS b D ex G i in L l lf
liquid bulk downstream location, or disturbance wave exit gas or glass interface inlet liquid (binary system) liquid (single-component system) annular liquid film
74 ld le M o
HARUHIKO OHTA
liquid transported by disturbance wave liquid entrainment middle location tube wall
r U v w
radial direction upstream location vapor wall
References 1. Misawa, M. and Anghaie, S. (1991) Zero-gravity Void Fraction and Pressure Drop in a Boiling Channel. AIChE Symp. Ser. 87(283), 226–235. 2. Kawaji, M., Westbye, C. J., and Antar, B. N. (1991) Microgravity Experiments on Twophase Flow and Heat Transfer during Quenching of a Tube and Filling of a Vessel. AIChE Symp. Ser. 87(283), 236–243. 3. Antar, B. N., Collins, F. G., and Kawaji, M. (1992). ‘‘Flow Boiling in Low Gravity Environment’’. Proc. 9th Symp. on Space Nuclear Power Systems, New Mexico, pp. 1210– 1215. 4. Saito, M., Miyazaki, K., Kinoshita, M., and Abe, Y. (1992). ‘‘Experiments of Boiling TwoPhase Flow under Microgravity’’. Proc. 5th Int. Topical Meeting on Reactor Thermal Hydraulics, Salt Lake City, pp. 1738–1748. 5. Lui, R. K., Kawaji, M., and Ogushi, T. (1994). ‘‘An Experimental Investigation of Subcooled Flow Boiling Heat Transfer under Microgravity Conditions’’. Proc. 10th Int. Heat Transfer Conf., Vol. 7, 18-FB-16, pp. 497–502. 6. Rite, R. W. and Rezkallah, K. (1993). ‘‘Heat Transfer Coefficients during Two-phase, Gasliquid Flow in a Circular Tube under Microgravity Conditions’’. AIAA 28th Thermophysics Conf., AIAA 93-2851. 7. Rite, R. W. and Rezkallah, K. (1995). ‘‘New Heat Transfer data for Two-phase, Gas-liquid Flows under Microgravity Conditions’’. Proc. 2nd Int. Conf. Multiphase Flow, pp. P6.7–6.12. 8. Nishikawa, K., Fujita, Y., Ohta, H., and Hidaka S. (1982). ‘‘Effect of the Surface Roughness on the Nucleate Boiling Heat Transfer over the Wide Range of Pressure’’. Proc. 7th Int. Heat Transfer Conf., Vol. 4, pp. 61–65. 9. Dukler, A. E., Fabre, J. A., McQuillen, J. B., and Vernon, R. (1988) Gas-liquid Flow at Microgravity Conditions: Flow Patterns and Their Transitions. Int. J. Multiphase Flow 14(4), 389–400. 10. Straub, J., Zell, M., and Vogel, B. (1990). ‘‘Pool Boiling in a Reduced Gravity Field’’. Heat Transfer 1990, Proc. 9th Int. Heat Transfer Conf., Vol. 1, pp. 91–112. 11. Ohta, H., Kawaji, M., Azuma, H., Kawasaki, K., Tamaoki, H., Ohta, K., Takada, T, Okada, S., Yoda, S., and Nakamura, T. (1997). ‘‘TR-1A Rocket Experiment on Nucleate Pool Boiling Heat Transfer Under Microgravity’’. Heat Transfer in Microgravity Systems, ASME HDT-Vol. 354, pp. 249–256. 12. Ohta, H., Kawaji, M., Azuma, H., Inoue, K., Kawasaki, K., Okada, S., Yoda, S., and Nakamura, T. (1998). ‘‘Heat transfer in Nucleate Pool Boiling under Microgravity Condition’’. Heat Transfer 1998, Proc. 11th Int. Heat Transfer Conf., Vol. 2, pp. 401–406. 13. Rohsenow, W. M., Weber, J. H., and Ling, A. T. (1956) Effect of Vapor Velocity on Laminar and Turbulent-film Condensation. Trans. ASME 78, 1637–1643. 14. Dukler, A. E. (1960) Fluid Mechanics and Heat Transfer in Vertical Falling Film Systems. Chem. Engng. Prog. Symp. Ser. 56(30), 1–10. 15. Brumfield, L. K. and Theofanous, T. G. (1976) On the Prediction of Heat Transfer Across Turbulent Films. J. Heat Transfer 98, 496–502. 16. Sleicher, C. A., Jr. (1958) Experimental Velocity and Temperature Profiles for Air in Turbulent Pipe Flow. Trans. ASME 80, 693–704.
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17. Lockhart, R. W. and Martinelli, R. C. (1949) Proposed Correlations of Data for Isothermal Two-phase Two-component Flow in Pipes. Chem. Engng. Prog. 45, 39–48. 18. Chisholm, D. and Laird, A. D. (1958) Two-Phase Flow in Rough Tubes. Trans. ASME 80, 276–286. 19. Fujii, T., Asano, H., Yamaoka, M., Nakazawa, T., Yamada, H., and Yoshiyama, T. (1996). ‘‘Flow Characteristics of Gas-Liquid Two-phase Annular Flow under Microgravity’’. Space Utilization Research, Proc. 13th Space Utilization Symposium, Tokyo, Vol. 13, pp. 183–186 (in Japanese). 20. Chen, I., Dowing, R., Keshock, E. G., and AI-Sharif, M. (1991) Measurements and Correlation of Two-Phase Pressure Drop Under Microgravity Conditions. J. Thermophysics 5(4), 514–523. 21. Zhao, L. and Rezkallah, K. S. (1995) Pressure Drop in Gas-liquid Flow at Microgravity Conditions. Int. J. Multiphase Flow 21(5), 837–849. 22. Heppner, D. B., King, C. D., and Littles, J. W. (1975). ‘‘Zero-G Experiments in Two-phase Fluids Flow Patterns’’. ASME Paper, No. TS-ENAs-24. 23. Colin, C., Fabre, J. A., and Dukler, A. E. (1991) Gas-liquid Flow at Microgravity Conditions—I. Dispersed Bubble and Slug Flow. Int. J. Multiphase Flow 17, 533–544. 24. Dukler, A. E., Fabre, J. A., McQuillen, J. B., and Vernon, R. (1988) Gas-liquid Flow at Microgravity Conditions: Flow Patterns and Their Transitions. Int. J. Multiphase Flow 14(4), 389–400. 25. Dengler, C. E. and Addoms, J. N. (1956) Heat Transfer Mechanism for Vaporization of Water in a Vertical Tube. Chem. Eng. Prog. Symp. Ser. 52(18), 95–103. 26. Katto, Y. and Ohno, H. (1984) An improvement version of the generalized correlation of critical heat flux for the forced convective boiling in uniformly heated tubes. Int. J. Heat Mass Transfer 27, 1641–1648. 27. Ueda, T. and Kim, K. (1986). ‘‘Heat Transfer Characteristics During the Critical Heat Flux Condition in a Subcooled Flow Boiling System’’, Heat Transfer 1986, Proc. 8th Int. Heat Transfer Conf., Vol. 5, pp. 2203–2208. 28. Ohta, H. and Fujiyama, H. (1999). ‘‘Local Heat Transfer Measurement and Visualization of Liquid–vapor Behavior for Microgravity Flow Boiling Experiments’’. Two-Phase Flow Modeling and Experimentation 1999, Vol. 1, pp. 255–262. 29. Nishikawa, K., Fujita, Y., Ohta, H., and Hidaka, S. (1982). ‘‘Effect of the Surface Roughness on the Nucleate Boiling Heat Transfer over the Wide Range of Pressure’’. Heat Transfer 1982, Proc. 7th Int. Heat Transfer Conf., Vol. 4, pp. 61–66. 30. Brusstar, M. J., Merte, H., Jr., and Keller, R. B. (1995). ‘‘Relative Effect of Flow and Orientation on the Critical Heat Flux in Subcooled Forced Convection Boiling’’. NASA Rep. UM-MEAM-95-15. 31. Ohta, H., Watanabe, H., Sabato, T., Okada, S., Takasu, S., and Kawasaki, H. (1999). ‘‘Gravity Effect on Flow Boiling Heat Transfer in Narrow Gaps’’. Proc. 5th ASME-JSME Thermal Engineering Joint Conference, CD Rom, AJTE 99-6421. 32. Ohta, H. (1997). Experiments on Microgravity Boiling Heat Transfer by Using Transparent Heaters. Nuclear Engineering and Design, Vol. 175, pp. 167–180. 33. Fujita,Y., Ohta,H.,andUchida,S.(1987).‘‘NucleateBoilingHeatTransferin VerticalNarrow Space’’. Proc. 2nd ASME-JSME Thermal Engineering Joint Conference, Vol. 5, pp. 469–476. 34. Kirk, K. M. and Merte, H., Jr. (1992). ‘‘Low Velocity Nucleate Flow Boiling at Various Orientations’’. Fluid Mechanics Phenomena in Microgravity, ASME AMD-Vol. 154/FEDVol. 142, pp. 1–10. 35. Ohta, H., Inoue, K., Yoshida, S., and Morita, T. S. (1997). ‘‘Nucleate Pool Boiling Heat Transfer in Microgravity’’. Physics of Heat Transfer in Boiling and Condensation, Institute for High Temperature, Russian Academy of Sciences, pp. 539–544.
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36. Merte, H., Jr. (1990) Nucleate Pool Boiling in Variable Gravity. Progress in Astronautics and Aeronautics 130, 15–69. 37. Oka, T., Abe, Y., Tanaka, K., Mori, Y. H., and Nagashima, A. (1992). Observation Study of Pool Boiling under Microgravity. JSME Int. J., Ser.II, Vol. 35, 280–286. 38. Straub, J. (2000). Pool Boiling in Microgravity. In ‘‘Microgravity Fluid Physics and Heat Transfer’’ (Dhir, V. K., ed.), Begell House, pp. 114–125. 39. Qiu, D. M., Dhir, V. K., Hasan, M. N., Chao, D., Neumann, E., Yee, G., and Birchenough, A. (2000). Single Bubble Dynamics During Nucleate Boiling under Low Gravity Conditions. In ‘‘Microgravity Fluid Physics and Heat Transfer’’ (Dhir, V. K., ed.), Begell House, pp. 62–71. 40. Steinbichler, M., Micko, S., and Straub, J. (1998). ‘‘Nucleate Boiling Heat Transfer on a Small Hemispherical Heater and a Wire Under Microgravity’’. Heat Transfer 1998, Proc. 11th Int. Heat Transfer Conf., Vol. 2, pp. 539–544. 41. Lee, H. S., and Merte, H. Jr. (2000). Pool Boiling mechanism in Microgravity. In ‘‘Microgravity Fluid Physics and Heat Transfer’’ (Dhir, V. K., ed.), Begell House, pp. 126–135. 42. Suzuki, K., Saito, H., and Matsumoto, K. (2001). ‘‘Heat Flux Cooling by Microbubble Emission Boiling’’. Proc. Microgravity Transport Process in Fluid, Thermal, Biological and Materials Sciences II, UEF, pp. 325–331. 43. Abe, Y. and Iwasaki, A. (2000). Single and Dual Vapor Bubble Experiments in microgravity. In ‘‘Microgravity Fluid Physics and Heat Transfer’’ (Dhir, V. K., ed.), Begell House, pp. 55–61.
ADVANCES IN HEAT TRANSFER VOL. 37
Fluid Mechanics and Heat Transfer with Non-Newtonian Liquids in Mechanically Agitated Vessels
R. P. CHHABRA Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India
I. Introduction Mixing is perhaps one of the oldest and the commonest unit operation encountered in the chemical, biochemical, polymer, food, agriculture, ceramic, paper, pharmaceutical and allied industries and in natural settings in every day life [1]. Almost all manufacturing processes entail some sort of mixing to varying extents, and this step may constitute a considerable proportion of the overall process time. Therefore, the financial investment in terms of both fixed and operating costs of mixing operations represents a significant fraction of the overall costs. For instance, improper and inadequate mixing is believed to add an estimated amount of US $ 1–10 billion per annum to the cost of process industries in the United States alone [2]. Consequently, there is a strong motivation to develop sound and reliable strategies for the design of mixing equipment which in turn requires a thorough understanding of mixing itself. The term mixing is applied to the operations or processes which are aimed to reduce the degree of non-uniformity or inhomogenity, or the gradient of a physical property such as colour, concentration, temperature, viscosity, electric charge, and so on, or to achieve a random distribution of one constituent into another medium. Obviously, mixing can be achieved by moving (convection) material from one region to another within the body of the fluid thereby reducing the overall degree of non-homogeneity. Of course, the ultimate homogenization occurs only by molecular motion. There are instances when the objective of mixing is to produce a desired level of homogeneity but mixing is also used to promote the rates of heat and mass transfer, often where a system is also undergoing a chemical reaction. At the Advances in Heat Transfer Volume 37 ISSN 0065-2717
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Copyright ß 2003 Elsevier Inc., All rights reserved
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outset, it is therefore instructive and useful to consider some common examples of problems encountered in industrial mixing operations, since this will not only reveal the ubiquitous nature of the phenomenon of mixing, but will also provide a glimpse of and an appreciation of some of the difficulties associated with this seemingly simple process. One can classify mixing applications in a variety of ways, such as the quality of the final product (mixture), flowability of the product in the mixing of powders, etc., bubble size in gas–liquid systems, Newtonian or non-Newtonian liquids, but it is perhaps most satisfactory (albeit quite arbitrary) to base such a classification on the phases present in the system such as gas–liquid, liquid–solid, solid– solid, liquid–liquid, etc. This scheme of classification not only brings out a degree of commonality but also facilitates the development of an unified framework to deal with the mixing problems cutting across diverse industrial settings ranging from food, polymer to the processing of agricultural products and waste streams. Table I provides a selection of representative examples for various kinds of mixing applications, together with some key references, as encountered in a range of technological settings. An inspection of Table I clearly reveals the near all pervasive nature of mixing. Furthermore, there are situations wherein the mixing equipment may be designed not only to achieve a pre-determined level of uniformity but also to enhance heat transfer. For example, if the rotational speed of an impeller in a stirred vessel is selected so as to achieve a required rate of heat transfer, the agitation may then be more than adequate for the mixing duty. Excessive or overmixing should be avoided. For example, in biological systems, excessively high impeller speeds or power input are believed by many to give rise to shear rates which may damage micro-organisms. In addition, there are very sensitive cells made by genetic engineering techniques that have more diverse requirements for cultivation. For instance, many cell cultures require anywhere from 2 to 6 months for processing rather than the usual 5 to 7 days in antibiotic fermentation processing. This requires the development of mixing equipment that can be maintained aseptic for long periods of time. Similarly, in the pulp and paper industry, there is a growing recognition of many advantages of carrying out chemical processes at high pulp concentrations ( 12–25%) as opposed to the concentrations of 6–7% in traditional processes. The desire to process high-consistency pulps (which invariably exhibit complex non-Newtonian behaviour) puts one into a new arena where both macro- and micro-scale mixing are exceedingly difficult. Similarly, where the desirable flow (rheological) properties of some polymer solutions may be attributable to the chain length and the other architectural aspects of polymer molecules, excessive impeller speeds or agitation over prolonged periods may adversely
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TABLE I EXAMPLES Type of mixing
OF
DIFFERENT TYPES
Examples
OF
MIXING Remarks
Selected references
Single-phase liquid
Blending of miscible petroleum products and silicone oils; use of agitation to promote rates of heat and mass transfer and chemical reactions; homogenization of liquid metals [8].
More difficult to mix/agitate [3–7] highly viscous Newtonian and non-Newtonian systems, and when the density and viscosity of the two components differ vastly (water/honey or glucose syrup).
Liquid–liquid
Immiscible liquids as encountered in liquid–liquid extraction, formulations of emulsions in brewing, food processing, personal-care products and pharmaceutical processes; production of polymeric alloys.
[3–7] Main objective here is to produce large interfacial area, or a desired droplet size (or size distribution). The degree of the difficulty of mixing increases with the decreasing size of droplets and/or with severe interfacial tension effects.
Liquid–solid
Suspensions of particles in low viscosity systems by mechanical agitation; incorporation of carbon black powder and other fillers into a viscous non-Newtonian matrix (such as rubber) to produce composites [14].
Power input depends strongly [3,4,6,7, upon the size and density of 10,12,13] particles and viscosity in low viscosity systems. However, in case of fine particles, surface forces play an important role.
Gas–liquid
Aerobic fermentation, wastewater treatment, oxidation and chlorination of hydrocarbons, production of xanthan gums, batters, bread mixes, etc.
Main objective is to produce [3,6,15] large interfacial area to promote mass transfer and chemical reaction. Dispersion becomes increasingly difficult with the increasing levels of viscosity and non-Newtonian behaviour.
Gas–liquid– solid
Slurry and sparged reactors; three phase fluidized bed reactors.
Intimate mixing required for [3,6,15] efficient operation and product quality control.
Solid–solid
Formation of concrete by blending sand, cement and aggregates; production of gun powder, food mixtures [21], and condiments and spices, fertilizers, animal feeds, insecticides, solid– solid chemical reactions [22], etc.
Strongly dependent upon the [3,16– 20,375] size, shape and surface properties of the solid components.
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affect the quality of the final product. It is thus vital to appreciate that both ‘‘over-mixing’’ as well as ‘‘under-mixing’’ are equally undesirable for different reasons. In view of the diversity of mixing problems, it is clearly neither possible to consider the whole spectrum of mixing problems here nor it is fair to expect that a single framework (model) will work for all kinds of mixing situations. Indeed, highly specialized mixing techniques and equipment have been developed over the years. For instance, the mixing needs [23–25] of polymer-related processing is almost solely met by extruders whereas for highly viscous liquids, in-line mixers are gaining grounds, e.g., see Refs. [3,26]. Hence, in this chapter, consideration will be given primarily to batch mixing of liquids, followed by heat transfer in mechanically agitated systems. Over the years, considerable research effort has been directed at exploring and understanding the underlying physics of mixing in low viscosity systems when the fluid exhibits simple Newtonian flow behaviour such as that exhibited by water and other low molecular weight systems including molten metals and electrolytes. Indeed, scores of books, research monographs and review papers providing critically reasoned comprehensive accounts of developments in this area are available [3–7,15,27,28,28a] and the frequent publication of special issue of periodicals (e.g., see Refs. [29,30]) and regular conferences reporting significant advances in this field testify to the overwhelming theoretical and pragmatic importance of mixing even with Newtonian liquids. Unfortunately, many materials of pragmatic significance and encountered in a large number of industrial settings do not adhere to the simple Newtonian fluid behaviour and accordingly such substances are called nonNewtonian or rheologically complex fluids [31–34]. One particular sub-class of fluids of considerable interest is that in which the effective (or apparent) viscosity depends on shear rate or, crudely speaking, on the rate of flow. Most particulate slurries, emulsions, sewage sludges, gas–liquid dispersions (foams, froths, batters) exhibit varying degrees of non-Newtonian behaviour, as done by the melts and solutions of high molecular weight polymers or other large molecules such as soap or protein. Further examples of substances exhibiting non-Newtonian characteristics include foodstuffs (soup, jam, jelly, marmalade, meat extract, etc.) [35], paints, personal care products [36], propellants, synthetic lubricants and biological fluids (blood, saliva, synovial fluid, etc.). Evidently, non-Newtonian fluid behaviour is so widespread that it would be no exaggeration to say that the Newtonian fluid behaviour might be regarded as an exception rather than the rule! Although the earliest reference to non-Newtonian fluid behaviour dates back to 700 BC [37], the importance of non-Newtonian characteristics and their impact on process design and operations have been recognized only during the past 40–50 years or so, especially as far as the mixing in agitated vessels is
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concerned. Consequently, undoubtedly considerable research effort has been devoted to what one might call engineering analysis of non-Newtonian fluids as is evidenced by the number of books available on this subject [31–34,38–40]. Yet it is remarkable that, inspite of the overwhelming pragmatic significance of liquid mixing, most of the existing books on batch mixing concentrate on low viscosity Newtonian fluids, albeit some of the recent books have attempted to alleviate this situation by including short sections on the mixing of viscous Newtonian and non-Newtonian systems must. It is interesting to note that even in a recent extensive compilation of the literature on mixing, only about 15% of the papers relate to the mixing of non-Newtonian systems [40a].
II. Scope In general, the scope of this chapter is thus not only to summarize, in a comprehensive manner, the existing literature on the behaviour of nonNewtonian liquids in mechanically agitated vessels but also to attempt a reconciliation of the available information which is scattered in a diverse selection of journals. In particular, consideration is given to the effect of non-Newtonian characteristics of the liquid on the flow patterns, mixing times, rate of mixing, power input, heat transfer, scale-up and equipment selection for mechanically agitated systems. Although the major thrust of this chapter is on the mixing and agitation of non-Newtonian systems, some background information for Newtonian fluids is also included as it not only facilitates qualitative comparisons but also highlights the fact that there are situations where a great degree of similarity exists between the behaviour of Newtonian and non-Newtonian liquids in mixing vessels, at least at a macroscopic level. However, we begin with a brief discussion of thermophysical and rheological characteristics of non-Newtonian fluids relevant to the flow and heat transfer in non-Newtonian fluids.
III. Rheological and Thermo-physical Properties A. RHEOLOGICAL PROPERTIES As indicated previously, it is readily acknowledged that many fluids encountered in industrial practice exhibit flow characteristics which are not normally experienced when handling simple Newtonian fluids. It is neither possible nor the intent of this chapter to provide a detailed exposition to the mysterious world of non-Newtonian fluids. Besides, excellent books are now
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available on this subject, e.g., see Refs. [31,32,34,38,39]. It is, however, useful to introduce the simple rheological models here which have been used in correlating the results for flow and heat transfer with non-Newtonian media in mechanically agitated vessels. As will be seen in Section III.B, most of the work on the mixing and agitation and heat transfer with nonNewtonian fluids relates to the so-called shearthinning, shearthickening, viscoplastic, and viscoelastic liquids, with an occasional reference to the mixing of thixotropic substances [41]. For shearthinning and shearthickening fluids, the simple power-law model, Eq. (1), has been used extensively to correlate the data from mixing vessels: ¼ mð _ Þn
ð1Þ
where n<1 denotes shearthinning behaviour, n>1 predicts shearthickening behaviour and n ¼ 1, of course, represents the standard Newtonian fluid behaviour. The viscoplastic fluid behaviour is characterised by the existence of a yield stress, and for the externally applied stresses beyond the yield stress levels, the flow curve may be linear—the so-called Bingham plastic or it may be non-linear in which case the fluid is called the yield-pseudoplastic. The Bingham plastic model is written as: ¼ oB þ B _
for jj > oB
ð2aÞ
_ ¼ 0
for jj < oB
ð2bÞ
The yield-pseudoplastic fluid behaviour is invariably approximated by the well known Herschel–Bulkley model given below: ¼ oH þ mð _ Þn _ ¼ 0
for jj > oH for jj < oH
ð3aÞ ð3bÞ
It is appropriate to add here that considerable confusion exists in the literature whether a true yield stress exists or not [41a], but in practice the notion of an (apparent) yield stress is convenient since the behaviour of many industrially important materials notably suspensions and emulsions closely approximates to that predicted by Eq. (2) or (3). Furthermore, it is appropriate to make three observations at this juncture. Firstly, it is often possible to fit either Eq. (1) or (2) or any other such fluid model equally well to a given set of shear stress/shear rate data and this fact does not imply and should not be used to infer the nature of the material as being shearthinning
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or Bingham plastic or Herschel–Bulkley fluid. Secondly, oB in Eq. (2) and oH in (3) must only be seen as disposable parameters and as such their values must not be confused with the true yield stress if any [42]. Indeed, excellent independent techniques are now available to ascertain/measure the value of yield stress without invoking a rheological model [43]. In the case of viscoelastic fluids, the rheological models need to be much more elaborate than Eqs. (1)–(3), and frequently the importance of viscoelastic effects in a flow situation including that in mechanically agitated vessels is quantified in terms of a dimensionless Deborah, De, or a Weissenberg, Wi, number, which are defined as follows: Wi ¼
Characteristic time of fluid
f ¼ Characteristic time of process p
ð4Þ
There does not appear to be an unique way of evaluating f for a viscoelastic fluid. Some time its value is inferred from first normal stress difference data, N1, as: 0 m0 1=ðn nÞ
f ¼ 2m
ð5Þ
where N1 ¼ m0 ð _ Þn
0
ð6Þ
On occasions, the value of f is also deduced from shear stress–shear rate data itself when such data extend to sufficiently small shear rates to embrace the so-called zero shear region. Similarly, the commonest choice of the characteristic process time is typically taken to be of the order of O(N1) where N is the rotational speed of the impeller [31,32]. While considering heat transfer applications in mechanically agitated vessels, it is imperative to account for the effect of temperature on rheological properties. This is frequently achieved by evaluating the rheoloical constants (like m, n, oB , , oH , m0 , n0 , etc.) as functions of temperature within the temperature interval envisaged in the application under consideration. As expected, the decrease in apparent viscosity at a constant shear rate is well represented by the usual Arrhenius type expressions, with both the pre-exponential factor and the activation energy being shear rate dependent. Thus, for the commonly employed power-law model, Eq. (1), it is now reasonably well established that the flow behaviour index, n, for particulate suspensions, polymer solutions and melts is nearly independent of temperature at least over a 40–50 temperature interval [44]
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whereas the consistency coefficient, m, conforms to the usual exponential temperature dependence, i.e., m ¼ mo expðE=RTÞ
ð7Þ
where mo and E are evaluated using experimental results in the temperature range of interest. Similarly, for Bingham plastic fluids, both oB and B decrease with an increase in temperature, but obviously the values of the pre-exponential factor and the activation energy vary from one substance to another and unfortunately no predictive formulae are available for use in a new application [45–47]. Lastly, and probably most significantly, each non-Newtonian material is unique in itself in the sense that the reliable information about its rheological characteristics comes only from the direct rheometrical tests performed on it.
B. THERMO-PHYSICAL PROPERTIES In addition to the viscous and viscoelastic rheological characteristics, the other important physical characteristics required in heat transfer applications include thermal conductivity (k), density (), heat capacity (cp), surface tension ( ), and the coefficient of thermal expansion ( ). While the first three of these, namely, k, and cp enter into virtually all heat transfer calculations, surface tension ( ) exerts a strong influence on boiling heat transfer and bubble dynamics in non-Newtonian fluids. Similarly, the isobaric coefficient of thermal expansion is important in heat transfer by free or natural convection. Additionally, the values of molecular diffusivity and solubility are similarly needed when dealing with mass transfer processes such as that encountered in gas–liquid reactors [48] involving Newtonian liquids, and scores of biotechnological applications such as production of xanthan gum, fermentation, aeration, wastewater treatment, in all of which the liquid phase exhibits complex non-Newtonian behaviour [49–68]. Admittedly, very few experimental measurements of thermo-physical properties have been reported in the literature, and the available data for commonly used polymer solutions (dilute to moderate concentrations) of carboxymethyl cellulose (Hercules), polyethylene oxide (Dow), carbopol (Hercules), polyacrylamide (Allied colloids), etc., density, specific heat, coefficient of thermal expansion differ from the corresponding values for water by no more than 5–10% [69–76]. The thermal conductivity and molecular diffusivity may be expected to be shear rate dependent, as both of these are related to the viscosity which, as seen above, shows strong dependence on shear rate and the structure of polymer molecules, flocs, etc.
FLUID MECHANICS AND HEAT TRANSFER
85
While limited available measurements [77] confirm this expectation for the thermal conductivity of aqueous carbopol solutions, the effect, however, is rather small [78,78a]. For process engineering design calculations, there will thus be a little error incurred in using the values of these properties for that of water at the relevant temperature conditions. For the sake of completeness, the available body of knowledge on molecular diffusion in non-Newtonian liquids is incoherent and inconclusive, e.g., see Ref. [42] and the literature cited therein. For industrially important particulate slurries and pastes displaying shearthinning, shearthickening and viscoplastic characteristics, thermophysical properties, namely, density, specific heat and thermal conductivity can deviate significantly from that of its constituents. Early measurements [79] on aqueous suspensions of powdered copper, graphite, aluminium and glass beads suggest a linear variation of density and specific heat between the pure component values, i.e., sus ¼ s þ ð1 ÞL
ð8Þ
Cpsus ¼ Cps þ ð1 ÞCpL
ð9Þ
where is the fractional (by volume) concentration of the solids; subscripts sus, s, L correspond to the values for the suspension, the solid and the liquid phase, respectively. The thermal conductivity, ksus, of such systems, on the other hand, shows much more involved functional relationship than the linear dependence embodied in Eq. (8) or (9). The following expression seems to consolidate most of the data available in literature [37,79,80]: ksus 1 þ 0:5" ð1 "Þ ¼ 1 þ 0:5" þ ð1 "Þ kL
ð10Þ
where " ¼ ks/kL. Thermal conductivities of suspensions up to 60% (by weight) in water and other suspending media are well correlated by Eq. (10). For suspensions of highly conducting particles (" ! 1), the maximum value of the ratio (ksus/kL) approaches [(1þ2 )/(12 )]. Besides, the corresponding increase in the viscosity from such high values of would more than offset the effects of increase in thermal conductivity on the value of the convective heat transfer coefficient (and hence on heat transfer), as seen in the case of flow in a pipe [80a]. For slurries/suspensions of mixed size particles, the following
86
R. P. CHHABRA
expression due to Bruggemann [81] seems to correlate the results satisfactorily: ðksus =ks Þ 1 ¼ ðksus =kL Þ1=3 ð1 Þ ðkL =ks Þ 1
ð11Þ
Equation (11) seems to correlate rather well the data on alumina-in-paraffin oil suspensions in the range 0 0.3 [82]. An extensive review on the correlation and prediction of thermal conductivity of structured fluids including polymer solutions, filled and unfilled polymer melts, suspensions and food-suspensions is available in the literature [83].
IV. Non-Newtonian Effects in Agitated Vessels Over the years, considerable research effort has been directed at exploring the different aspects of mechanical agitation of non-Newtonian media in stirred vessels and research findings have been published in a wide ranging literature. Table II provides an extensive sampling of such studies and it also testifies to the overwhelming pragmatic relevance of this subject. A detailed inspection of this table shows the diverse variety of impellers used to agitate/mix an equally rich variety of non-Newtonian materials ranging from polymer solutions, to fermentation broths, to livestock manure slurries, to cookies dough, to chocolates, to clay and other particulates suspensions and polymerization reactive systems [150,151,260], though most of the literature relates to the use of laboratory scale of equipment. As mentioned previously, this diverse selection of experimental test fluids also reflects in a range of non-Newtonian characteristics including shearthinning, shearthickening, yield stress, thixotropy, viscoelasticity and combinations thereof. Furthermore, as each non-Newtonian substance is unique in itself, so is an agitator/tank assembly owing to a wide variation in the values of geometric parameters like the shape of tanks, type of bottoms (flat/ dished/conical/contoured), details of impeller, internals of the tank like number, type, thickness and width of baffles, cooling coils, draft tubes, etc. Such geometrical complexities alone preclude the possibility of detailed comparisons of results from different studies unless the equipments used are geometrically similar. However, in spite of this intrinsic difficulty, it is possible to discern and establish some overall generic trends. It is perhaps appropriate to add here that the bulk of the effort has been directed at the elucidation of the effect of non-Newtonian properties on the following aspects of the overall agitation process: (i) average shear rate, power consumption and scale up, (ii) flow patterns and flow field, (iii) rate and time of mixing, (iv) CFD modelling, (v) coil and jacket heat transfer, and
FLUID MECHANICS AND HEAT TRANSFER
87
(vi) design and selection of equipment. Accordingly, consideration will now be given to each of these topics in the ensuing sections arranged broadly under three major sections, namely, Fluid Mechanics (VI), Heat Transfer (VII) and Equipment Selection (VIII), respectively. However, prior to embarking upon a detailed treatment of each of these issues, it is worthwhile to recall that one is often confronted with two types of key engineering problems in mixing applications: how to ascertain and quantify the suitability of an available equipment for an envisaged application, and secondly, how to design and/or select an optimum system and optimum operating conditions for a new application. In either event, however, a thorough understanding of the underlying mixing mechanism is required to appreciate the problems of selection and optimum utilization of mixing equipment. Therefore, we begin with a description of different mechanisms of mixing in Section V. V. Mechanisms of Mixing In order to carry out mixing to produce a uniform product (or to distribute one phase into another in a random fashion), it is imperative to understand how mixtures of liquids move and approach uniformity or fully mixed state. Intuitively, for liquid mixing devices, it is essential to meet two requirements: firstly, there must be a bulk (convective) flow so that there are no stagnant regions, and secondly, there must be a zone of intensive or highshear mixing in which the inhomogeneities are progressively broken down by high levels of prevailing shear and elongational stresses [261–265]. Obviously, both these processes are energy-consuming and eventually the mechanical energy is dissipated as heat; the proportion of energy attributable to each of these steps obviously varies from one application to another and is also somewhat dependent upon the type of mixer itself, but is mainly governed by the flow field in the tank and the physical properties of the liquid phase. Similarly, depending upon the fluid properties (mainly viscosity), the main flow in a mixing vessel may be laminar or turbulent, with a substantial transition zone in between, and frequently, both types of flow conditions (laminar and turbulent) occur in different parts of the vessel. Since the laminar and turbulent flows arise from different physical mechanisms and as such represent intrinsically different types of flow, it is convenient to treat them separately. A. LAMINAR MIXING While there is no simple way to predict a priori whether laminar or turbulent mixing will occur in a given situation, large-scale laminar mixing
88
R. P. CHHABRA
in a mechanically stirred vessel is associated with low Reynolds number (<10) or with high viscosity (typically > 10 Pa s) which may exhibit either Newtonian or non-Newtonian flow behaviour. Under such conditions, inertial forces therefore tend to die out quickly, and the rotating impeller must sweep through a significant proportion of the cross-section of the vessel to induce adequate bulk flow. Due to the rather large velocity gradients close to the rotating impeller, the fluid elements in this region repeatedly deform, stretch (elongate) as shown schematically in Fig. 1. Each time a fluid element passes through this high shear zone, it becomes thinner and thinner and ultimately breaks down into smaller elements. Obviously, the flow field in a mixing vessel is three-dimensional, and shear and elongation occur simultaneously. As shown schematically in Fig. 2, for an incompressible liquid, elongational flow also results in the thinning or flattening of a fluid element. Thus, both these mechanisms (shear and elongation) give rise to forces or stresses in the liquid which in turn effect a reduction in droplet size and an increase in interfacial area, by which means the desired degree of homogeneity is obtained. However, the relative importance of the prevailing shear and elongational stresses is believed to be strongly dependent on the geometry of the impeller [265a]. In the context of viscoelastic liquids, it is appropriate to mention here that while Newtonian
FIG. 1. Schematics of the thinning of a fluid element due to shear flow.
FIG. 2. Schematics of the thinning of a fluid element due to elongation flow.
FLUID MECHANICS AND HEAT TRANSFER
89
and inelastic liquids have elongational viscosities which are typically three times the corresponding shear viscosity, this ratio—Trouton ratio—can be as large as 103–104 in the case of viscoelastic liquids thereby indicating much greater resistance to extension than that to shear deformation offered by viscoelastic liquids [31,32,34]. This factor alone, as will be seen later, makes the mixing and agitation of viscoelastic liquids not only a difficult task but much greater power is needed for mixing such fluids. Under laminar flow conditions, a similar mixing process also occurs when a liquid is sheared between the two rotating cylinders. In this device, during each revolution, the thickness of an initially radial fluid element (dispersed phase) is reduced, as shown schematically in Fig. 3, and eventually when the fluid elements become sufficiently thin, molecular diffusion comes into play. Obviously, if an annular tracer element is introduced to begin with, then no mixing would occur (Fig. 3). This emphasises the importance of the orientation of the fluid elements relative to the direction of shear produced by the mixer. Another model configuration which has been used extensively to explore the underlying mechanism of mixing is a two-dimensional cavity whose walls are subject to a periodic motion. The resulting shear forces in the cavity can stretch and fold the tracer [1,261–264]. Based on the notion of a point transformation coupled with elegant visualization experiments, Ottino and co-workers [1,261–264] have argued that viscous fluids flowing in simple and periodic patterns in two-dimensions can result in chaos that, in turn, induce efficient mixing, similar to that encountered in microchannels [266]. Finally, mixing can also be induced by physically ‘‘splicing’’ the fluid into successively smaller units and then re-distributing them. In-line (static) mixers for viscous fluids operating under laminar flow conditions rely primarily on this mechanism (Fig. 4). Thus, mixing in viscous liquids is achieved by a combination of some or all of the aforementioned mechanisms which reduce the size or scale of the inhomogeneity and then re-distribute them by bulk flow.
FIG. 3. Laminar shear mixing in a concentric cylinder configuration.
90
R. P. CHHABRA
FIG. 4. Schematics of mixing by cutting and folding of fluid elements.
B. TURBULENT MIXING In contrast, in low viscosity systems (< 10 m Pa s), the main flow generated in mixing vessels with a rotating impeller is usually turbulent, albeit laminar flow conditions also exist away from the impeller. The inertia imparted to the liquid by the impeller is sufficient to result in the circulation of the liquid throughout the vessel. Turbulence may occur throughout the vessel but clearly will be the greatest in the impeller region. Mixing by eddy diffusion is much faster than that by molecular diffusion, and thus, turbulent mixing occurs much more rapidly than the laminar mixing. Mixing is fastest in the impeller region owing to the high shear rates and the associated Reynolds stresses in vortices formed at the tips of the impeller blades; also, a high proportion of the energy is dissipated here. From a theoretical standpoint, turbulent flow is inherently complex and has defied predictions from first principles. Consequently, the flow fields prevailing in a mixing tank are not amenable to a theoretical treatment. At sufficiently high values of the Reynolds number of the main flow, some qualitative insights can be gained by using the theory of locally isotropic turbulence. Under these conditions, it is reasonable to postulate that the flow contains a spectrum of velocity fluctuations in which eddies of different sizes are superimposed on an overall time-averaged steady flow. In a mixing vessel, intuitively it appears reasonable to postulate that the large (primary) eddies, of a size of the order of the impeller diameter, would give rise to large velocity fluctuations of low frequency. Such eddies are anisotropic, and account for much of the kinetic energy present in the system. Interactions between these primary eddies and the slow moving fluid streams produce smaller eddies of higher frequency which undergo further disintegration until finally, their energy is dissipated as heat via viscous forces. This is how mixing occurs under these conditions. Admittedly, the foregoing description is a gross over-simplification, but nonetheless it does afford some qualitative insights about turbulent mixing. Qualitatively, this process is similar to that of the turbulent flow of a fluid close to a boundary surface. Although some quantitative results-both
FLUID MECHANICS AND HEAT TRANSFER
91
experimental and numerical for the scale size of eddies in Newtonian liquids (mostly water) and shearthinning fluids are available in the literature [15,27,269–275,275a], but it is not at all obvious that how this information can be integrated into the existing design procedures and practices for mixing equipment to improve their performance. Furthermore, owing to their generally high viscosities, non-Newtonian substances are processed (mixed) frequently in laminar flow conditions and hence, most of the aforementioned studies relating to turbulence in stirred vessels are only of marginal interest in the present context. In addition to these flow configuration, many investigators, e.g., Refs. [277–279] have employed simple but novel configurations including mixing in shaker table containers and in cavity flows to gain useful physical insights into the mechanisms of dispersion of one phase into another. Finally, in addition to such convection effects, molecular diffusion always acts in such a way as to reduce the scale and intensity of inhomogeneities, but its contribution is insignificant until the fluid elements have been sufficiently reduced in size for their specific areas to become large. It is appropriate to recall that the ultimate homogenization of miscible systems is brought about only by molecular diffusion. In the case of high viscosity liquids, this is also a slow process. VI. Fluid Mechanics As noted earlier and also revealed by an inspection of the extensive literature summarized in Table II, the bulk of the research effort has been devoted to the elucidation of the interplay between the physical properties of the liquid (Newtonian and non-Newtonian) and the geometry of the system (tank–impeller assembly) on scale up, power input and average shear rate, flow patterns, and the rate and time of mixing. Some preliminary studies on the numerical modelling (CFD) are also available which provide further physical insights. Furthermore, most work relates to laboratory scale equipment (i.e., cylindrical vessels of diameter rarely exceeding 500 mm or so) and with model but well characterised non-Newtonian fluids (aqueous polymer solutions and suspensions, etc.), and only scant literature is available with industrial scale equipment. We begin with the scale-up of stirred vessels in Section IV.A. A. SCALE UP Undoubtedly, one of the key problems confronting the designers of mixing equipment is that of deducing the most satisfactory configuration for a large unit from experiments carried out at small scale (laboratory units).
92
TABLE II SUMMARY
OF
EXPERIMENTAL STUDIES
ON
NON-NEWTONIAN LIQUIDS
IN
MECHANICALLY AGITATED SYSTEMS
Type of impeller and dimensions
Systems
Main objectives
Metzner and co-workers [84–87]
Six-bladed flat turbine 0.182 (D/T) 0.77 0.15 T 0.55 m
Pseudoplastic solutions of CMC, carbopol and suspensions of Attasol
Flow patterns, mixing times, average shear rate and power input for single phase systems
Calderbank and Moo-Young [88]
Six-bladed turbine and twobladed paddle T ¼ 0.25 m 0.34 (D/T) 0.67
Aqueous solutions of CMC, clays in water, Average shear rate and power input paint, paper pulp suspension, etc.
Nagata and co-workers [89–98]
Helical ribbon, halfellipsoidal, paddle, turbine, anchor 0.2 T 0.4 m 0.3 (D/T) 0.95
Pseudoplastic and viscoplastic suspensions of CaCO3, MgCO3, TiO2, kaolin and solutions of CMC, PVA
General studies on mixing of viscous Newtonian and non-Newtonian liquids, wall and coil heat transfer for single phase liquids under aerated and unaerated conditions
Godleski and Smith [99]
Turbine 0.14 T 0.44 m
Solutions of Natrosol
Power input and blending/mixing times for pseudoplastic liquids
Chapman and Holland [100]
Helical screw, turbine T ¼ 0.14, 0.178, 0.24 and 0.29 m D/T ¼ 0.26–0.55
High viscosity Newtonian liquids
Power input for centered and off-centered agitators
R. P. CHHABRA
Investigator
Gluz and Pavlushenko [101– 103]
Turbine, blade screw anchor, Solutions of sodium CMC bell-spiral, propeller
Beckner and Smith [104]
Flat- and pitch-bladed anchor T ¼ 0.23 m 0.68 (D/T) 0.96
Solutions of CMC and of polybutradiene in Power input for single phase pseudoplastic ethylbenzene liquids
Peters and Smith [105,106]
Anchor T ¼ 0.15, 0.23, 0.3 m (D/T) 0.85
Polyacrylamide solutions in water
Flow pattern and mixing time for single phase systems
Mizushina et al. [107– Anchor, paddle and 110] propeller T ¼ 0.3 m 0.6 (D/T) 0.8
Aqueous solutions of CMC; polystyrenein-toluene; cement slurry
Wall and cooling coil heat transfer under turbulent conditions; homogenization of temperature
Hagedorn and Salamone [111]
Paddle, propeller, anchor turbines T ¼ 0.35 m 0.30 (D/T) 0.65
Aqueous solutions of carbopol (0.36 n 0.69)
Wall heat transfer correlations
Hall and Godfrey [112,113]
Sigma-blade and helical ribbon 0.04 T 0.56 D/T 0.90
Solutions of hydroxypropyl methyl cellulose Blending times and power input
Bourne and Butler [114,115]
Helical ribbon Six and 160 gallon tanks 0.89 (D/T) 0.98
Aqueous solutions of CMC and HPMC
Dimensional analysis, power input, homogenization and heat transfer in single phase systems
FLUID MECHANICS AND HEAT TRANSFER
Flow patterns and power input
93
(Continued)
94
TABLE II (CONTINUED) Investigator
Type of impeller and dimensions
Systems
Main objectives
Paddle, propeller, anchor and Solutions of carbopol (0.36 n 0.69) Rushton turbine T ¼ 0.35 m
Jacket heat transfer
Skelland and Dimmick [117]
Three-bladed propellers T ¼ 0.45 m D/T ¼ 0.16–0.25
Solutions of carbopol (0.53 n 0.91)
Coil heat transfer in the range 340 Re 260,000
Mitsuishi and co-workers [118,119]
Two-, six-bladed turbines T ¼ 0.1, 0.3 m 0.5 (D/T) 0.8
Solutions of CMC, PVA and clay suspensions
Power input and heat transfer studies
Coyle et al. [120]
Helical ribbon T ¼ 0.35 and 0.76 m D/T ¼ 0.93, 0.97
Pseudoplastic systems (0.2 n 1)
Wall heat transfer results
O’Shima and Yuge [121]
Helical, anchor and helical screw impellers
Newtonian systems
Circulation times for highly viscous liquids
Sandall and co-workers [122–124]
Anchor and turbines T ¼ 0.18 m D/T ¼ 0.33, 0.98
Chalk–water slurries, solutions of carbopol Jacket heat transfer results for single phase liquids and slurries, and on gas absorption in polymer solutions
Rieger and Novak [125–128]
Solutions of CMC and PAA (0.16 n 1) Helical screw with and without a draft tube (centered and off centered) (D/T) ¼ 0.61, 0.9 and 0.95 T ¼ 0.1–0.15 m
Power input, scale-up and homogenization studies
R. P. CHHABRA
Hoogendoorn and den Hertog [116]
Helical ribbon and screw impellers 0.45 (D/T) 0.64 T ¼ 0.46 m
Solutions of CMC, PAA and Natrosol
Theoretical and experimental results on power input, internal circulation, mixing and blending times in inelastic and viscoelastic systems
Edney and Edwards [137]
Six-bladed (flat) turbine T ¼ 1.22 m D/T ¼ 0.41
Solutions of CMC and PAA
Heat transfer studies under aerated and unaerated conditions
Ford and co-workers Helical screw with a draught-tube [138–140] T ¼ 0.46 m, D/T ¼ 0.4, 0.5
Solutions of CMC and PAA
Effects of rheological properties on mixing and blending times
Edwards et al. [41]
Anchor, helical ribbon and screw T ¼ 0.13 m 0.67 (D/T) 0.94
Aqueous solutions of CMC and carbopol; tomato sauce, salad cream, yoghurt, etc.
Power input for pseudoplastics and thixotropic materials
Kale et al. [141]
Circular discs and turbines
Aqueous solutions of CMC and PAA
Effect of elasticity on power input at high Reynolds numbers
Quarishi et al. [142,143]
Rushton turbine, paddle T ¼ 0.24, 0.3 m D/T ¼ 0.2–0.33
Solutions of PAA and PEO
Reduction in power input in drag reducing polymer solutions
Yagi and Yoshida [144]
Six-bladed turbine T ¼ 0.25 m (D/T) ¼ 0.40
Solutions of CMC and PAA
Mass transfer studies in polymer solutions
FLUID MECHANICS AND HEAT TRANSFER
Chavan and co-workers [129–136]
(Continued)
95
96
TABLE II (CONTINUED) Investigator
Type of impeller and dimensions
Systems
Main objectives
Anchor and helical ribbon 0.5 (D/T) 0.98
–
New power input data for Newtonian liquids for 30 Re 104, and re-analysis of literature data for power law fluids
Hiraoka et al. [146]
Paddle 0.3 (D/T) 0.9
–
Numerical (2-D) simulation of variation of viscosity for power-law fluids for Re 10
Hocker and co-workers [147,148]
Turbine T ¼ 0.4 m D/T ¼ 0.33
Aqueous solutions of CMC and PAA
Power input for single phase agitation and aerated conditions
Ranade and Ulbrecht Six-bladed turbine [149] T ¼ 0.3 m 0.25 (D/T) 0.35
Solutions of CMC and PAA
Effect of rheology (pseudoplasticity and elasticity) on mass transfer in gas–liquid systems
White and co-workers Turbine, screw and anchor [150,151]
Polybutadiene in polystyrene solutions
Flow pattern and stream line visualization
Rautenbach and Bollenrath [152]
Helical ribbon
Solutions of PAA
Heat transfer in high viscosity Newtonian and non-Newtonian media
De Maerteleire [153]
Four-bladed impeller T ¼ 0.18 m (D/T) ¼ 0.56
Newtonian liquids
Coil heat transfer in aerated systems (170 Re 2.6 105)
R. P. CHHABRA
Sawinsky et al. [145]
Helical ribbon and screw agitators with draft coil system 0.145 T 0.291 m 0.73 (D/T) 0.91
Solutions of Natrosol, CMC and PAA
Mixing time, flow patterns, power input and coil heat transfer in inelastic and viscoelastic media
Nishikawa et al. [163–165]
Six-bladed turbine T ¼ 0.3 m (D/T) ¼ 0.6
Solutions of CMC
Heat and mass transfer to viscous Newtonian and non-Newtonian media under aerated and unaerated conditions
Poggemann et al. [166]
Flat blade, pitched bladed turbine, propeller, paddle anchor and helical ribbon D/T ¼ 0.3–0.98
–
Extensive evaluation of data on coil and wall heat transfer studies with single phase Newtonian and non-Newtonian liquids
Solomon et al. [167]
Rushton disc turbine and pitched bladed turbine T ¼ 0.29 m
Aqueous solutions of CMC and carbopol
Power input and cavern sizes in viscoplastic fluids
Chen [9]
Propellers and turbines T ¼ 0.298 m 0.2 (D/T) 0.34
Live stock manure slurries (power law fluids) Power input
Prud’homme and co-workers [168,169]
Rushton turbine T ¼ 0.23 m D/T ¼ 0.33
Boger fluids
Ayazi Shamlou and Edwards [170,171]
Helical ribbon T ¼ 0.15, 0.4 m 0.75 (D/T) 0.925
Solutions of CMC and carbopol; chocolates Power input and heat transfer studies
Effect of elasticity on mixing times and power input
FLUID MECHANICS AND HEAT TRANSFER
Carreau and co-workers [154–162]
(Continued)
97
98
TABLE II (CONTINUED) Investigator
Bertrand and Couderc [172–174]
Type of impeller and dimensions
Systems
Two-bladed anchor and gate Carbopol solutions impellers D/T ¼ 0.5, 0.66, 0.78
Main objectives
Experimental and predicted power input results for pseudoplastic liquids
Xanthan gum solutions
X-ray flow visualization studies for viscoplastic and shearthickening fluids
Ismail et al. [177]
Disc turbines T ¼ 0.4 m D/T ¼ 0.28–0.50
Air/water
Power input and cavity formation dynamics under aerated conditions
Nienow et al. [178]
Rushton turbine T ¼ 0.29 m
Aqueous solutions of xanthan gum, CMC, and carbopol
Effect of rheological properties on power input under aerated and unaerated conditions
Kuboi and Nienow [179]
Dual impellers (angled bladed and turbine) T ¼ 0.29 m D/T ¼ 0.5
Solutions of CMC and carbopol
Flow patterns and mixing rates
Wichterle and co-workers [180,181]
Standard turbine and propellers T ¼ 0.18, 0.24, 0.3 m 0.3 (D/T) 0.6
Slurries of Bentonite and limestone
Flow patterns and power input for viscoplastic and pseudoplastic systems. Measurements of shear rate on turbine impeller tip
R. P. CHHABRA
Elson and co-workers Rushton turbine [175,176] D/T ¼ 0.25–0.6 T ¼ 0.071 m
Heat transfer and boiling of Newtonian and non-Newtonian solutions
Kuriyama et al. [185] Helical ribbon T ¼ 0.16 m D/T ¼ 0.9
Solutions of CMC (n ¼ 0.5, 0.7)
Heat transfer analysis for pseudoplastic liquids
Kamiwano et al. [186] Six-bladed flat turbine 0.1 T 0.4 m D/T ¼ 0.4–0.5
Solutions of hydroxyethyl cellulose
Flow pattern and velocity fields using an imaging method
Turbines T ¼ 0.29, 0.63 and 1 m D/T ¼ 0.80–0.81
Etchells et al. [187]
Several impellers including TiO2 slurries radial, axial, Lightnin A 310 T ¼ 0.305 m D/T ¼ 0.25–0.33
Power input for mixing of Bingham plastics at industrial scale
Kai and Shengyao [188]
MIG, disc turbine, plate paddles, anchors, etc.
Aqueous solutions of CMC
Power input and coil heat transfer for single phase systems in the range of 7 Re 18,400
Zeppenfeld and Mersmann [189]
Rushton turbine D/T ¼ 0.33
Aqueous solutions of CMC and xanthan gum
Power input calculation for Newtonian and power law liquids in the intermediate range of Reynolds numbers
Koloni et al. [190]
Six-bladed turbine D/T 0.33–0.4 T ¼ 0.3 and 0.7 m
Power law slurries of CaCO3 and Ca(OH)2 Power input, gas holdup and interfacial area in aerated systems in square vessels
Galindo and co-workers [51– 55,63,191]
Rushton turbine, SCABA-6 RGT, Intermig, Lightnin T ¼ 0.21 m, D/T ¼ 0.47, 0.53
Xanthan gum broths; aqueous solutions of carbopol
FLUID MECHANICS AND HEAT TRANSFER
Aqueous solutions of carbopol (0.40 n 0.65)
Desplanches et al. [182–184]
Power input and cavern formation in yield stress fluids
(Continued)
99
100
TABLE II (CONTINUED) Investigator
Type of impeller and dimensions
Systems
Main objectives
Anchor impeller, six-bladed turbine and paddle
Bingham plastic and Ellis model fluids
Numerical analysis (velocity field) of Bingham plastic fluids in a vessel stirred by an anchor
Jomha et al. [194]
Helical ribbon and anchor T ¼ 0.152 m D/T ¼ 0.71–0.93
Suspensions of superclay in water
Power input for shearthickening fluids
Sestak et al. [195]
Anchor impeller T ¼ 01, 0.15, 0.43 m D/T ¼ 0.9
Solutions of CMC, PAA, Polyox and kaolin; Power input for pseudoplastic and thixotropic wall paper paint and laponite suspensions fluids
Wang and Yu [196]
Disc turbine, plate paddles, Solutions of CMC (0.49 n 0.92) MIG impeller, anchors, semiellipsoidal impeller – Solutions of CMC and carbopol
Sinevic et al. [197]
Coil and wall heat transfer correlations
Power input and secondary flows in coaxial flow of non-Newtonian systems
Oliver et al. [198]
Six-bladed Rushton turbine T ¼ 0.22 m D/T ¼ 0.45
Non-shearthinning elastic liquids
Effect of viscoelasticity on power consumption
Shervin et al. [199]
Rushton turbine, helical ribbon and double helix T ¼ 0.15 m D/T ¼ 0.5, 0.7, 0.83
Acrylic polymer in a mineral oil
Flow visualization, blending time and scale up for viscoelastic liquids
R. P. CHHABRA
Kaminoyama et al. [192,193]
Skelland and Kanel [200]
Flat-curved and pitch-blade turbines, propeller T ¼ 0.214 m D/T ¼ 0.36, 0.475
Dispersion of di-isobutyl ketone in aqueous Dispersion of a Newtonian liquid in polymer solutions carbopol solutions
Anchor and helical ribbon T ¼ 0.10, 0.13 m D/T ¼ 0.82–0.96
Aqueous solutions of hydroxy-ethyl cellulose
Effects of geometric parameters on power consumption with pseudoplastic liquids
Pandit et al. [11]
Helical ribbon T ¼ 0.38 m D/T ¼ 0.895
Suspensions of turmeric and pepper
Power input and mixing times for thixotropic suspensions
Tran et al. [207]
Pitched- and six-bladed turbines T ¼ 0.2, 0.3, 0.367 m D/T ¼ 0.20–0.25
Clay slurries with viscosity modifiers
Power input for viscoplastic slurries
Boger fluids, solutions of Gellan, xanthan Helical ribbon, combined geometries, double planetary, gum, CMC kenics and SMX static mixers T ¼ 0.21, 0.44 m D/T ¼ 0.88
Numerical and experimental studies on power input for inelastic and viscoelastic liquids with a variety of impellers. Comparative performance of static mixers
Amanullah et al. [226,227]
SCABA and axial flow impellers
Aqueous solutions of carbopol
Correlations of cavern sizes in shearthinning fluids
Hjorth [228]
SCABA impeller
Aqueous solutions of CMC and carbopol
Velocity profiles in laminar and transitional flow
Newtonian systems
Blending of liquids of different densities and viscosities
Tanguy and co-workers [208–225]
Bouwmans et al. [229] Pitched- and six-bladed turbine T ¼ 0.29, 0.64 m
101
(Continued)
FLUID MECHANICS AND HEAT TRANSFER
Takahashi and co-workers [201–206]
102
TABLE II (CONTINUED) Investigator
Type of impeller and dimensions
Systems
Main objectives
Anchor T ¼ 0.225 m D/T ¼ 0.98
Solutions of CMC
Boiling heat transfer in stirred vessels
Delaplace and co-workers [231–235]
Helical ribbon T ¼ 0.346 m D/T ¼ 0.925
Aqueous solutions of CMC, alginate, guargum and adragante gum
Power input, mixing time for shearthinning and shearthickening fluids
Ozcan-Taskin and Nienow [236]
Pitched- and six-bladed Boger fluids turbine MaxfloT, Chemineer HE3 T ¼ 0.22 m 0.35 (D/T) 0.53
Flow field and power input for viscoelastic liquids
Ruan et al. [237]
–
Cookie dough
Evaluation of rheological properties from mixing power curves
Reilly and Burmster [238]
Rushton turbine T ¼ 0.13 and 0.29 m
Solutions of CMC
Homogenization of liquids of different viscosities and densities
Jaworski et al. [239]
Pitched-blade turbine T ¼ 0.05 m
Aqueous carbopol solutions
Flow field around caverns in viscoplastic liquids
Masiuk and Lacki [240]
Helical ribbons T ¼ 0.345 m D/T ¼ 0.97
Aqueous solutions of CMC
Effect of ribbon geometry on power input and mixing time
R. P. CHHABRA
Foroquet-Murh and Midoux [230]
Moore and co-workers [241,242]
Six-bladed turbine T ¼ 0.15 m D/T ¼ 0.3
Mavros et al. [243]
Rushton turbine, Lightnin A Aqueous solutions of CMC 310 and Mixel TT agitator D/T ¼ 0.5
Effect of impeller geometry and nonNewtonian properties on flow patterns using LDV
Velasco et al. [65]
Dual impeller, Rushton turbine, six-bladed turbine T ¼ 0.20 m D/T ¼ 0.53
Rifamycin production
Power input in an industrial fermenter
Youcefi et al. [244]
Two-bladed impeller T ¼ 0.3 m D/T ¼ 0.5
Aqueous solutions of CMC, carbopol and PAA
Velocity field and power input in inelastic and viscoelastic liquids
Shimizu et al. [245]
Rushton turbine T ¼ 0.09 m D/T ¼ 0.54
Solutions of CMC and xanthan gum
Drop breakage in stirred non-Newtonian liquids
Torrez and Andre [246,247]
Rushton turbine T ¼ 0.3 m D/T ¼ 0.15
Bingham plastic and Herschel–Bulkley model fluids
Numerical and experimental results on power input in shearthinning and viscoplastic liquids
Wang et al. [248]
Composite (inner–outer) helical ribbon T ¼ 0.24 m D/T ¼ 0.9–0.97
Aqueous solutions of CMC (0.46 n 1)
Power input is reduced in pseudoplastic liquids
Aqueous polymer (carboxy-vinyl) solutions Velocity distribution for viscoplastic liquids
FLUID MECHANICS AND HEAT TRANSFER
(Continued)
103
104
TABLE II (CONTINUED) Investigator
Type of impeller and dimensions
Systems
Main objectives
Rai et al. [249]
Helical ribbon T ¼ 0.115 m D/T ¼ 0.7
Aqueous solutions of CMC and PAA
Nouri and Hockey [250]
Rushton turbine T ¼ 0.294 D/T ¼ 0.33
Aqueous solutions of CMC (0.56 n 0.9) Power curves at high Reynolds numbers, 100– 105
Mishra et al. [251]
Disc turbine T ¼ 0.3 m D/T ¼ 0.33
Aqueous solutions of PAA
Flow field for viscoelastic liquids
Fangary et al. [252]
Lightnin A 320 and A 410 agitators T ¼ 0.29 m D/T ¼ 0.60
Aqueous solution of CMC
Fluid trajectories using positron emission particle tracking method
Curran et al. [254]
Bohme and Stenger [255]
Rushton 45 turbine T ¼ 0.2, 0.4 m D/T ¼ 0.33
Solutions of CMC, PAA and xanthan gum Regime maps for dispersion of a gas into liquids
Single- and double flight ribbon impellers T ¼ 0.208 m D/T ¼ 0.89
Aqueous solutions of carbopol
Circulation times and power input data for viscoplastic fluids
Aqueous solutions of PAA
Power input and scale up
Turbines T ¼ 0.094, 0.144, 0.288 m D/T ¼ 0.5
R. P. CHHABRA
Vlaev et al. [253]
Coil heat transfer under aerated and unaerated conditions
Anchor and helical ribbon T ¼ 0.3 m D/T ¼ 0.90–0.97
Aqueous solutions of CMC
Power input and wall heat transfer results
Chowdhury and Tiwari [257]
Helical ribbon screw T ¼ 0.38, 0.57 and 1 m D/T ¼ 0.89–0.95
Aqueous solutions of CMC and guar gum (0.27 n 1)
Power input data and correlation (Re 5000)
Ducla et al. [258]
Turbine impellers
Aqueous solutions of CMC and PAA
Calculations of average shear rate from power input data
Pollard and Kantyka Anchor [259] T ¼ 0.3, 0.6, 92 m D/T ¼ 0.90–0.91
Chalk–water slurries and polymer solutions Jacket and coil heat transfer (0.38 n 1) ( 200 Re 106)
Pandey et al. [267]
Marine type impellers D ¼ 75, 127, 184 mm
CMC solutions (0.7 n 1)
Jacket/wall and coil heat transfer (400 Re 107)
Blasinski and Kuncewicz [268]
Ribbon agitators T ¼ 0.3 m D/T ¼ 0.93
Solutions of CMC (0.75 n 0.93)
Wall heat transfer (30 Re 7000)
Delaplace et al. [276]
Helical ribbon impeller T ¼ 0.34 m D/T ¼ 0.93
Solutions of guargum and carbopol (shearthinning and viscoplastic fluids)
Numerical and experimental results on wall heat transfer
FLUID MECHANICS AND HEAT TRANSFER
Heim [256]
CMC, carboxymethyl cellulose; PAA, polyacrylamide; PEO, polyethylene oxide; PVA, polyvinyl alcohol. In most cases, tap water has been used as solvent.
105
106
R. P. CHHABRA
An even more common problem is the design of small scale experiments to optimize the use of existing (out of use) large scale equipment for new applications. From a theoretical standpoint, scale-up demands geometrical, kinematic and dynamic (also chemical for reactive system, thermal similarity for heat transfer applications) similarity between the two units. Furthermore, the boundary conditions must also be identical in the two equipments. However, owing to process (material) and/or operational constraints, it is seldom possible to achieve the complete similarity and therefore considerable experience (intuition) is required for a successful scale up of laboratory data [3,6,15,27,31,32,103,280–282]. Since detailed descriptions of the scale up method and the associated difficulties are available in a number of excellent sources [3,6,15,27,281], only the salient points are re-capitulated here. It is customary and perhaps convenient to relate the power input to the agitator to the geometrical and mechanical arrangement of the mixer and thus to obtain a direct measure of the change in power consumption due to the alteration of any of the factors relating to the mixer. A representative mixer arrangement is shown schematically in Fig. 5. The mathematical description of the complete similarity between the two systems is usually expressed in terms of the dimensionless ratios of geometric dimensions and of the forces present in the fluid in a mixing vessel. For a Newtonian fluid and in the absence of heat and mass transfer and chemical reactions, for geometrically similar systems, the resulting dimensionless ratios of the forces are the familiar Reynolds (Re), Froude (Fr) and Weber (We) numbers defined as follows:
Re ¼
D2 N
FIG. 5. Typical tank (with a jacket)–impeller assembly.
ð12Þ
FLUID MECHANICS AND HEAT TRANSFER
Fr ¼
N 2D g
107
ð13Þ
N 2 D3 ð14Þ It is customary to use the impeller diameter, D, as the characteristic linear dimension in this field. For applications involving heat and mass transfer, additional dimensionless groups include Prandtl number (Pr), Schmidt number (Sc), Nusselt number (Nu), Sherwood number (Sh) and Grashof numbers (Gr). Besides, additional factors come into play depending upon whether the heat transfer is occurring between a fluid in a jacket and that in the mixer (wall or jacket heat transfer) or via a cooling coil and/or draft tube (coil heat transfer). It is also important to recognize the role of the two commonly used boundary conditions, namely, the constant temperature (e.g., if condensing steam is used to heat up a batch of liquid) or the constant heat flux (if the vessel is wound with an electric heating coil). Irrespective of all these features, free convection, how so ever small, is always present in heat and mass transfer studies which ought to be accounted for in the interpretation/correlation of such results. In the case of inelastic (purely viscous) non-Newtonian fluids, an appropriate value of the apparent viscosity must be identified for use in Eq. (12) and in other dimensionless groups such as Prandtl, Schmidt and Grashof numbers. Over and above this, it may also be necessary to introduce further non-dimensional parameters indicative of the other nonNewtonian effects such as a Bingham number, Bi (¼ oB =B N) for viscoplastic fluids and a Weissenberg number, Wi (¼ f N) for viscoelastic liquids. It is thus imperative that the complete similarity between the two systems implies the equal values of all such dimensionless groups for the two systems. As noted previously, owing to the conflicting requirements imposed by the complete similarity, in practice only partial or distorted similarity is possible. In general, it is thus necessary to identify the one or two key features that must be matched at the expense of the secondary level factors. Thus, for instance, for purely shearthinning power type materials, both the Bingham (Bi) and Weissenberg numbers (Wi) are irrelevant as is the Froude number in the absence of significant vortex formation, such as in highly viscous systems. The fact that each non-Newtonian substance is unique in terms of its rheology adds further to the complexity of scale-up, especially for viscoelastic systems. Aside from the above-noted theoretical considerations, additional difficulties can also result from the choice of scale-up criteria, and these We ¼
108
R. P. CHHABRA
again vary from one application to another including the type and the main goal of mixing. Thus, for geometrically similar systems, the size of the system is determined by the scale-up factor. For power consumption, one commonly used criterion is to maintain the power input per unit volume of liquid constant in two mixers. In blending operations, it may be necessary to keep dimensionless blend time constant. Similarly, heat transfer processes in agitated vessels are scaled up either on the basis of equal heat transfer per unit volume of liquid batch or by maintaining a constant value of heat transfer coefficient. Yet in instances when chemical reactions are carried out in agitated vessels, it may be necessary to ensure the same regime (e.g., mass transfer or kinetic controlled) and/or to have equal residence time in the two systems. Using geometric similarity, it is useful to make the following observations [344]: (i) Circulation and mixing times in a large vessel will be considerably longer than that in a small tank. (ii) The maximum shear rate (in the impeller zone) will be higher in the larger tank, but the average shear rate will be lower thereby giving rise to a much wider variation in shear rates in a full-scale equipment than that in a laboratory scale equipment. (iii) The Reynolds numbers in the large tank are typically 5–25 times larger than that in a small vessel. Thus, if the small scale equipment operates in the laminar regime, the corresponding full scale plant is likely to operate in transitional region. (iv) Heat transfer is usually much more demanding on a large scale than that in the laboratory scale equipment. In summary, while the equality of the Reynolds number ensures the complete similarity of flow in geometrically similar systems, the actual scaleup criterion varies from one application to another as seen above, e.g., see Refs. [125,187,255,280,282–284,311,344]. B. POWER INPUT From a practical standpoint, it is readily conceded that the power input (or power consumption) is the most important design parameter in mixing processes. Unfortunately, power input depends upon a large number of process variables, the geometrical arrangement and the physical characteristics of the liquids; this dependence is far too complex to be established from first principles. Therefore, most of the developments in this field are based on dimensional considerations aided by experimental observations. Owing to the inherently different mechanisms of mixing in low and high viscosity liquids and the way these influence the power input, it is convenient
FLUID MECHANICS AND HEAT TRANSFER
109
to consider the issue of power consumption in low and high viscosity systems separately. 1. Low Viscosity Liquids Typically, low viscosity liquids are mixed (agitated) in a vertical cylindrical tank fitted with baffles, with a height-to-diameter ratio of 1.5–2, and fitted with an agitator (shown schematically in Fig. 5). Under such conditions, high speed impellers (typically, 0.33 (D/T ) 0.5) are suitable, running with tip speeds of ca. 1–3 m/s. The main flow pattern in the vessel is usually turbulent. Admittedly, such studies on the single phase mixing of low viscosity Newtonian liquids are of limited industrial interest, it does, however, serve as a useful starting point for the subsequent treatment of high viscosity systems. With reference to the schematic arrangement shown in Fig. 5, simple dimensional considerations lead to the following functional relationship: P ¼ f ðRe; Fr; geometric ratiosÞ N 3 D5
ð15Þ
where the Power number, Po, is defined as, Po ¼
P N 3 D5
ð16Þ
Since we are dealing with the agitation of single phase liquids and the surface tension effects are assumed to be negligible. Therefore, the Weber number is redundant here. Furthermore, for geometrically similar systems, geometrical ratios are all fixed and thus Eq. (15) reduces to: Po ¼ f ðRe; FrÞ
ð17Þ
Similarly, the Froude number is generally important only when severe vortex formation occurs, and in single phase mixing it can be neglected for Re< 300 or so, as shown in Fig. 6 for a propeller agitator [285]. Due to the detrimental effect of vortex formation on the quality of mixing, in practice it is minimised and/or avoided by either installing baffles in the tank (Fig. 5) or by installing the agitator in an off-center position. Hence in most situations involving low viscosity Newtonian liquids, the Power number is a unique function of the Reynolds number and the mixer geometry only. Furthermore, as the viscosity of the liquid progressively increases, the tendency for vortex formation progressively diminishes and so does the necessity of baffles in the mixing tanks for liquids of > 5 Pa s. The effect of the type of impeller and the geometry on power input is shown in Fig. 7
110
R. P. CHHABRA
FIG. 6. Effect of Froude number on the power curve for a propeller impeller (re-plotted from Ref. [285]).
FIG. 7. Effect of impeller design on Power number–Reynolds number relationship for Newtonian liquids (re-plotted from Ref. [286]).
for scores of liquids encompassing a five orders of magnitude variation in Reynolds number [286]. For a fixed geometrical arrangement and a single phase liquid, in the absence of vortex formation, experimental results on power input can be represented by a unique power curve, in accordance with Eq. (17). The dependence in Fig. 7 is seen to be similar to the Moody diagram for friction in pipes. Three distinct regions can be discerned in the power curve: at low Reynolds numbers (< 10 or so), the Power number varies inversely with the Reynolds number, i.e., a slope of 1 on log–log coordinates which is typical of viscosity dominated flows. This region, which is characterised by slow mixing both at macro- and molecular level, is where
FLUID MECHANICS AND HEAT TRANSFER
111
unfortunately the majority of highly viscous (Newtonian and nonNewtonian) liquids are processed. The actual value of the Reynolds number marking the end of this region is strongly dependent upon the type of fluid and the configuration of the mixer. As will be seen later, the smaller the value of the power-law index, the larger is the value of the Reynolds number at which the transition occurs. At very high Reynolds numbers (> 104), the main flow in the tank is fully turbulent and inertia dominated, resulting in fast mixing. In this region, the Power number is almost independent of the Reynolds number and is nearly constant, as can be seen in Fig. 7 for Newtonian fluids. This type of limiting behaviour is also displayed by shearthinning inelastic liquids, as demonstrated recently by Nouri and Hockey [250]. However, once again, the asymptotic value of the Power number is solely dependent upon the geometrical configuration of the impeller/tank combination. Gas–liquid, liquid–liquid and liquid–solid contacting operations are typically carried out in this region. In between the laminar and turbulent regimes, there exists a substantial transition zone in which the both viscous and inertial forces are of comparable magnitudes. No simple mathematical relationship exists between the Power number and the Reynolds number, and for a given value of Re, the corresponding value of Po must be read off the appropriate power curve. Little is known about the critical value of Reynolds number marking the onset of the fully turbulent flow conditions. The following correlation [287] for Newtonian fluids is available to predict the value of the Reynolds number corresponding to the onset of the fully turbulent conditions: Rec ¼ 6370 Pot0:33
ð18Þ
where Pot is the constant value of the Power number under fully turbulent conditions which is strongly geometry dependent and hence the effect of the impeller/tank geometry is implicitly included in Eq. (18). The changes in the flow patterns associated with laminar–turbulent transitions have been studied by Hjorth [228]. Similar power curves for many different impeller geometries including dual, composite and proprietary designs, baffle arrangements, shapes of tanks, etc., are available in the literature [3,15,27,40a,52,53,62,148,172, 174,215,288–302], but it must be remembered that though the power curve approach is applicable to the mixing of any single phase liquid, at any impeller speed, each such curve is valid only for an unique impeller–tank combination. Unfortunately, little is known about the influence of variation in geometric parameters such as non-standard baffles, impeller-to-bottom clearance, etc., and this makes it almost inevitable to perform experiments
112
R. P. CHHABRA
on an envisaged system to deduce useful information about the large scale system. Notwithstanding this intrinsic limitation, adequate information is now available on the calculation of power consumption for the agitation of low viscosity Newtonian liquids under most conditions of practical interest, though little is available in terms of the standardization of equipment configurations [305]. 2. High Viscosity Newtonian and Inelastic Non-Newtonian liquids It is useful to recall that mixing in high viscosity liquids is slow both at the molecular scale (owing to low values of molecular diffusivity) as well as at the macroscopic level, due to poor bulk flow. In high viscosity liquids, only the fluid in the immediate vicinity of the agitator is influenced by the agitator and the flow is usually laminar. Therefore, efficient mixing of viscous fluids requires specially designed impellers with close clearances at the side and bottom walls of the vessel and which sweep rather large volumes of the liquid in the tank. High speed stirring with small impellers (Rushton turbine, propeller, etc.) merely wastefully dissipates energy in the impeller region of the vessel, particularly when the liquid is highly shearthinning or possesses a yield stress. Although highly viscous Newtonian fluids include lubricating oils, glycerol, sugar syrups, most of the high viscous fluids of interest in chemical, food, pharmaceutical, and allied processing industries exhibit non-Newtonian flow characteristics, notably shearthinning and viscoplastic characteristics, albeit a few substances also display shearthickening and time dependent thixotropic behaviour. However, in spite of such complexities, the power curve approach is generally applicable in such cases as detailed here. Since most non-Newtonian materials exhibit an apparent viscosity which is a function of the rate of shear. The flow in a mixing tank is three-dimensional and rather complex. Furthermore, the rate of shear shows a great degree of variation, being maximum in the impeller region and it may even approach zero in parts of the tank where the fluid is virtually stagnant. Furthermore, in view of the non-viscometric flow conditions in the tank, the shear rate may depend upon the rheology itself. Thus, it is not at all possible to estimate the rate of shear (and its distribution) in a stirred tank from first principles. In view of this, and from a practical point, the notion of an average shear rate is perhaps convenient and useful, at least for the purpose of calculating power input in a new application provided a power curve has been established for Newtonian liquids in a geometrically similar system. Perhaps the simplest way to develop an expression for the average shear rate is to force the power input data for non-Newtonian liquids on to the corresponding power curve established with Newtonian fluids for a fixed geometry, as discussed below.
FLUID MECHANICS AND HEAT TRANSFER
113
3. Average Shear Rate A remarkably simple relationship has been shown to exist between the power consumption for time-independent non-Newtonian substances and for Newtonian liquids under laminar and geometrically similar conditions. This link, which was first established by Metzner and Otto [84] for shearthinning polymer solutions and slurries, hinges on the fact that there appears to be a characteristic average shear rate _avg for a mixer which determines power consumption, and which is directly proportional to the rotational speed of the impeller, i.e., _avg ¼ Ks N
ð19Þ
while initially Ks was postulated to be a function of the impeller–vessel configuration only, albeit some of the subsequent investigators have questioned the validity of this assumption [88,104,118]. If the apparent viscosity corresponding to the average shear rate defined by Eq. (19) is used in the equation for a Newtonian liquid, the power consumption for laminar conditions is satisfactorily predicted for most inelastic shearthinning and viscoplastic liquids. Subsequently, the linear relationship embodied in Eq. (19) was confirmed by the flow visualization experiments of Metzner and Taylor [85]. A compilation of the literature values of Ks reported up to 1983 has been given by Skelland [288] and is shown here in Table III in a slightly modified form. An inspection of Table III suggests that for shearthinning fluids, Ks lies approximately in the range 10–13 for most configurations studied thus far (and presumably of practical interest) with propellers, turbine type impellers, while slightly large values of 25–30 have been reported for close clearance impellers like anchors and helical ribbons [160,206,303]. Skelland [288] also reconciled most of the power consumption data in the form of power curves as shown in Fig. 8. The range of the impeller–vessel configuration used by different investigators is reflected in the diversity of power curves, albeit all are of qualitatively similar form. Irrespective of the mixer configuration, Re 10 seems to mark the end of the laminar region. In contrast, the transition to the fully turbulent conditions (characterised by a constant value of the Power number, Pot) seems to occur at different values of the Reynolds number ranging from 100 (curve BB) to Re>1000 (see curves D–D2, D–D3 in Fig. 8). Clearly, the critical value of the Reynolds number is strongly dependent on the impeller–vessel geometry. Furthermore, the available scant data shown in Fig. 8 and other available in the literature [250] on this transition boundary does not seem to conform to the behaviour predicted by Eq. (18) developed for Newtonian fluids.
114
TABLE III VALUES
OF
0.05–1.5 0.18–0.54 0.14–0.72
11.5 1.5 11.5 1.4 11.5 1.4
4, WB/T ¼ 0.1 or none
–
0.85–0.98
0.14–0.72
11.5 1.4
0.10–0.20 0.10–0.30 0.13
0.33–0.75 0.21–0.45
0.21–0.26 1.0–1.42 0.16–0.40
13 2 13 2 10 0.9
0.13
0.21–0.45
0.16–0.40
10 0.9
with
4, WB/T ¼ 0.1 or none 4, WB/T ¼ 0.1 or none None, (i) shaft vertical at vessel axis, (ii) shaft 10 from vertical, displaced D/6 from centre None, (i) shaft vertical at vessel axis, (ii) shaft 10 from vertical, displaced D/6 from centre None, position (ii) None, position (i) 4, WB/T
0.30 0.30 0.15
0.5–0.53 0.5–0.53 0.6
0.16–0.40 0.16–0.40 0.16–0.60
10 0.9 10 0.9 10
with
None, position (ii)
–
0.33–0.71
0.16–0.40
10 0.9
with
None, position (i)
–
0.33–0.71
0.16–0.40
10 0.9
with
4, WB/T ¼ 0.1
0.12
0.47
0.05–0.61
10
with
4, WB/T ¼ 0.1
0.12
0.47
1.28–1.68
–
4, WB/T ¼ 0.1 None 0 or 4, WB/T ¼ 0.08
0.09–0.13 0.28 0.10–0.15
0.33–0.5 0.98 0.35–0.52
0.16–1.68 0.34–1.0 0.34–1.0
10 11 5 11 5
Same as for D–D Same as for D–D Square-pitch marine propeller three blades Double-pitch marine propeller three blades (downthrusting) Double-pitch marine propeller three blades (downthrusting) Square-pitch marine propeller four blades Square-pitch marine propeller four blades Two-bladed paddle Anchor Cone impellers
R. P. CHHABRA
0.182–0.77 0.18–0.77 0.286
D–D2 D–D3 E–E
H–H – –
FIG. 8
0.051–0.20 0.051–0.20 –
Same as for D–D—but upthrusting
G–G1
TO
4, WB/T ¼ 0.1 None 4, WB/T ¼ 0.1
Number and size of baffles
D–D1
G–G
KEY
Ks (n<1)
Single turbine with six flat blades Single turbine with six flat blades Two turbines, each with six flat blades and T/2 apart Two turbines, each with six flat blades and T/2 apart Fan turbine with six blades at 45 Fan turbine with six blades at 45 Square-pitch marine propellers with three blades (downthrusting)
F–F1
OF IMPELLERS AND
N (Hz)
A–A A–A1 B–B
F–F
VARIOUS TYPES
D/T
Impeller type
C–C C–C1 D–D
FOR
D (m)
Curve
B–B1
Ks
FLUID MECHANICS AND HEAT TRANSFER
115
FIG. 8. Power curves for time-independent fluids. Key to the curves is given in Table III (based on Ref. [288]).
However, this transition is perhaps of minor relevance here, for most nonNewtonian fluids are usually processed in laminar flow conditions and the flow conditions seldom approach turbulent regime. Over the years, it is increasingly being recognized that the high speed agitators and close clearance anchor or gate type impellers are not very effective in mixing highly viscous Newtonian and non-Newtonian liquids for different reasons. While the high speed impellers wastefully dissipate most of the energy in a relatively small body of the liquid in the impeller zone, close clearance impellers create very little circulation and pumping action away from the wall. Therefore, impellers with high pumping capacity are preferred for highly viscous systems. The two designs, shown schematically in Fig. 9, have gained wide acceptance are helical screw and helical ribbon impellers [306–309] and the modifications thereof such as a having a draft tube to improve its pumping capacity. Using the analogy with the couette flow and replacing the rotating impeller by a rotating cylinder of an equivalent diameter, some analytical predictions of the average shear rate and power consumption are available in the literature which are in line with experimental results for such impellers [114,115,129,131,135,136]. Subsequently, this flow has also been modelled as a drag flow and analogous analytical expressions for Ks and power consumption for power law fluids are available in the literature [156,157]. However, all these analyses implicitly endeavour to collapse the data for non-Newtonian systems on to the power curve for Newtonian fluids for a specific arrangement of the mixer. Admittedly, the approach of Metzner and Otto [84] has enjoyed a great degree of success in correlating much of the power input data for a variety of
116
R. P. CHHABRA
FIG. 9. Schematics of Helical ribbon, Helical screw and Helical screw with a draft tube.
FIG. 10. Effect of power law index on Ks for a helical ribbon impeller.
configurations [304], it is not completely satisfactory and has thus also come under severe criticism. For instance, this approach does not always lead to a unique power curve for a given geometrical configuration if the value of the power-law index, n, varies widely [38,118,250]. This implies that the constant Ks in Eq. (19) is not truly a constant, and its value depends upon the values of rheological parameters, such as n [88,104,195,235], time-dependency [194,195], etc. Fig. 10 shows this effect clearly for the agitation of scores of power law fluids by helical ribbon impellers [213].
FLUID MECHANICS AND HEAT TRANSFER
117
However, this problem seems to be much more acute for anchors, helical ribbons and the other close clearance impellers than that for turbines, propellers, etc. [206,213,233]. Owing to the intense shearing of relatively thin sheets of liquids at the wall, one would expect the value of Ks to be much higher for anchors and other similar impellers than propellers, turbines, etc. Indeed, the available experimental results suggest values of Ks ranging from 10 to as high as 80 [213]. It appears that the value of Ks for anchors not only depends upon rheology, but also is extremely sensitive to the geometrical details of an impeller–tank combination. For instance, Calderbank and Moo-Young [88] put forward and following correlation for Ks for the agitation of shearthinning liquids by anchors: n=ð1nÞ 9ðT=DÞ2 4n Ks ¼ 9:5 þ ðT=DÞ2 1 3n þ 1
ð20Þ
Equation (20) is applicable for T/D<1.4. Similarly, Beckner and Smith [104] suggested the following expression for Ks (0.27 n 0.77): Ks ¼ að1 nÞ
ð21Þ
The empirical constant a appearing in Eq. (21), in turn, is a function of geometry alone, particularly of the side and bottom clearances. Some analytical efforts [129,135,160] have also been made to replace the impeller by a rotating cylinder of an equivalent diameter such that the values of the torque (hence power) in two cases are the same. This approach, though highly idealised, but clearly does bring out the effect of rheology and geometry on the factor Ks [31,160]. For instance, Yap et al. [157] presented the following expression for Ks for helical ribbons used to mix shearthinning and mildly viscoelastic systems: Ks ¼ 41=ð1nÞ ðD=TÞ2 ðl=DÞ
ð22Þ
Equation (22) predicts a rather strong dependence of Ks on the flow behaviour index n and showing more than an order as magnitude variation as n changes from 0.5 to 0.8, eventually becoming indeterminate at n ¼ 1. Perhaps the most reliable expression for Ks for helical ribbon agitators used for pseudoplastic fluids is that due to Delaplace and Leuliet [233]: "ðn þ 1Þ n=ðn1Þ Kp D Ks ¼ ð" þ 1Þn p2 l
ð23Þ
118
R. P. CHHABRA
where 1 2ðT=DÞ2 lnðT=DÞ " ¼ 1 þ 1 þ ðT=DÞ2 1
ð24Þ
Kp is the power constant under laminar flow conditions and is given by (Po Re) for Newtonian liquids. Thus, the use of Eq. (23) necessitates some experimental tests with Newtonian liquids to ascertain the value of Kp for a given mixing system. This, in turn, then can be combined with viscometric data (value of n) to calculate the value of Ks and thus to find the average shear rate via Eq. (19). Similarly, admittedly while the value of Ks is geometry dependent but fortunately it is nearly independent of the equipment size and thus there are no scale-up problems with this approach. Data on power consumption in viscoplastic [51–55,63,89–98,175,176,180, 181,191,207], dilatant or shearthickening fluids [194] and time-dependent thixotropic suspensions [41,195] have been similarly correlated using this approach. In summary, the prediction of power consumption for the agitation of a given time-independent fluid of known rheology in a specific impeller–tank assembly, at a desired impeller speed, proceeds as follows: (i) The average shear rate is estimated using Eq. (19). The relevant value of Ks thus must be known either from small scale experiments or from expressions like Eq. (20) or Eq. (21) or Eq. (22), etc. (ii) The relevant apparent viscosity is evaluated either from shear stress– shear rate plot (rheogram), or by means of an appropriate viscosity model such as power-law, Bingham plastic, or Herschel–Bulkley models. However, extrapolation of such data beyond the experimental range of conditions must be avoided as far as possible. (iii) The value of the apparent viscosity calculated in step (ii) is used to evaluate the Reynolds number of flow via Eq. (12), and then the value of the Power number (hence of P) is obtained from the appropriate power curve such as shown in Fig. 7 or Fig. 8 where such power curves are not available for the specific impeller–vessel combination, power consumption data with Newtonian liquids must be obtained to produce a power curve over as wide ranges of conditions as anticipated with time-independent non-Newtonian fluids. Despite the uncertainty associated with this approach, in terms of the value of Ks, this scheme yields the values of power consumption under laminar flow conditions with an accuracy of the order of 25–30%. However, beyond the laminar flow conditions, Eq. (19) is not applicable and indeed
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FIG. 11. Dependence of average shear rate on the rotational speed for an anchor in laminar range.
FIG. 12. Dependence of average shear rate on the rotational speed for an Intermig impeller in laminar and transitional region.
the dependence of the average shear rate on the rotational speed of the impeller gradually changes for being proportional to N to that N2 under fully turbulent conditions [310], as shown in Fig. 11 for an anchor impeller and in Fig. 12 for a composite impeller of the INTERMIG turbine type [311]. In concluding this section on power input, it is worthwhile to add here that there is sufficient evidence in the literature [147,160,310] to suggest that an extended laminar region exists with an increasing degree of shear thinning value, i.e., decreasing value of n. Indeed, the slope of power curves continues to be 1 almost up to Re ¼ 100 or so, as can be seen in Fig. 13 for a helical ribbon impeller. It is, however, not yet possible to predict
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FIG. 13. Typical power curves for inelastic shearthinning fluids stirred by helical ribbon screw impellers (re-drawn from Ref. [160]).
a priori the value of the Reynolds number up to which such laminar flow conditions will exist in an envisaged application. This is solely due to the complex interplay between the flow patterns and the geometry of the system. Therefore, it is worthwhile to re-iterate that the uncertainty surrounding the prediction of Ks in the intermediate region is perhaps not all that relevant in the context of non-Newtonian fluids. In other words, it is thus possible to estimate the power consumption for the agitation of purely viscous nonNewtonian systems with about as much as accuracy as that of the power curve developed based on data for Newtonian liquids. Before leaving this section, it is also useful to mention that Eq. (19) has also been used in the reverse sense, that is, using impeller power data as a means of characterising rheology [237,312–316]. Indeed, the well-established vane technique for measuring the yield stress is based on the analysis of the power–time response curves for viscoplastic materials [312]. Aside from this justifiable applications, many other investigators [237,313–316,316a–d] have attempted to infer the values of rheological parameters (such as n, m) from power input data. Attention must be drawn to the fact that due to the indeterminate and complex 3-D nature of the flow field produced by an impeller, such an approach, in principle, is suspect, albeit it can be used as an useful tool for quality control purposes. However, it must be borne in mind that this approach of inferring rheological properties from power input data is fraught with danger. 4. Effect of Viscoelasticity In contrast, much less is known and therefore considerable confusion exists about the role of viscoelasticity on power input to the impeller.
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Some studies suggest that since the primary flow pattern in mixing tanks is unaffected by viscoelasticity, and hence the power input is little influenced by fluid viscoelasticity, at least in the laminar flow regime. Indeed, the early experimental studies [136,317–320] lend support to this notion, though this is not necessarily so [236]. On the other hand, Nienow et al. [178] documented a slight increase in power input for viscoelastic liquids, whereas Ducla et al. [258] reported a slight decrease in power consumption with turbine impellers. However, in practice, most polymer solutions (used as test fluids) in experimental studies exhibit simultaneously both shear dependent viscosity and viscoelasticity, and it is therefore not possible to isolate the contributions of these two rheological characteristics on power input. However, this dilemma can be resolved by using the so-called Boger fluids [319a], which in steady shear display varying levels of viscoelasticity in the absence of shearthinning. The available experimental studies with these fluids [168,169,198,236] reveal a strong interplay between the geometry and fluid viscoelasticity on one hand and the fluid viscoelasticity and the kinematic conditions (laminar, transitional flow) on the other. The effect of viscoelasticity may even depend upon the size of the equipment [168]. For the standard Rushton turbine and modifications thereof, Oliver et al. [198], Prud’homme and Shaqfeh [168] and Collias and Prud’homme [169] concluded that the power input may increase or decrease below the corresponding value for Newtonian fluids (same geometry and the Reynolds number) depending upon the values of the Reynolds and Weissenberg numbers. For instance at low Reynolds numbers, viscoelasticity seems to increase the power consumption [168,169,198]. On the other hand, in the case of axial flow impellers, Ozcan-Taskin and Nienow [236] concluded that even at low values of the Elasticity number (defined as Wi/Re), the power consumption was significantly higher in viscoelastic media than that in Newtonian liquids otherwise under identical conditions. For instance, even when El 0.02 and 5 Re 500 in these experiments they documented up to 50–60% increase in power input at low Reynolds number and it appears to taper off as the value of the Reynolds number progressively increases. Qualitatively similar increases in power input have been also reported by Youcefi et al. [244,325] for a two-bladed impeller. On the other hand, no such increase in power input has been reported for double planetary type mixers [223]. Representative analogous results for a helical ribbon [160] are shown in Fig. 14 where the results for both the usual polymer solutions (showing both shearthinning and viscoelasticity) and the Boger fluids (purely elastic liquids) are included. In addition to an increase in power input, data for viscoelastic fluids tend to veer away from the Newtonian line at small values of the Reynolds number. For instance, for the 0.35% PIB/PB/Kerosene solution, power input data begin to deviate from the
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FIG. 14. Effect of viscoelasticity on power input to helical ribbon screw impellers (re-plotted from Ref. [160]).
Newtonian line at as small values of the Reynolds number as 1 whereas in other cases this departure occurs at Re 30–40 thereby indicating a complex interplay between rheology and flow parameters. This behaviour is in stark contrast to that for inelastic shearthinning fluids wherein the laminar region is extended. However, this departure from the Newtonian line is mainly due to the fluid elasticity because the Reynolds number is still too small for the inertial effects to be significant. This is believed to be due to the rather high values of extensional viscosities for these fluids. At high Reynolds numbers, viscoelasticity seems to suppress the secondary motion, thereby resulting in a reduction in power consumption [141,310] and as seen also in Fig. 14. Collias and Prud’homme [169] have further explored the role of viscoelasticity on power consumption and mixing times. Utilizing Eq. (19) and the equality of the Reynolds number for two geometrically similar systems, one can readily show that: 1n0 El1 N2 ¼ ð25Þ El2 N1 where the subscripts ‘‘1’’ and ‘‘2’’ refer to the two geometrically similar systems; n0 is the power-law type index for the first normal stress difference 0 ( _ n ). For the same fluid, the equality of the two Reynolds numbers yields, 2 N1 D2 ð26Þ ¼ N2 D1
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and this in turn yields, 2ð1n0 Þ El1 D1 ¼ El2 D2
ð27Þ
Note that the same fluid, i.e., constant values of m, n, n0 , , etc., has been used in the two different size equipments. Eq. (27) clearly brings out the complex interplay between the rheology (n0 ), size of equipment (D) and kinematics (because El ¼ Wi/Re). Indeed, this well may be the basic reason for much of the confusion in the literature regarding the role of viscoelasticity [169]. Clearly the value of n0 whether greater than 1 (as in Ref. [178]) or smaller than 1, as in the study of Bartels and Janssen [321] is crucial to the role of viscoelasticity in an envisaged application. Consequently, no satisfactory and universally applicable correlations are available for the prediction of power input in viscoelastic systems. Before concluding this section on the effect of viscoelasticity on power consumption, it is also of interest to draw attention to the scant work available with the drag reducing systems (containing only few ppm of polymers). The reduction in power input accompanied by concomitant reduction in gas– liquid mass transfer occurs only under turbulent conditions in agitated systems [142,143,149]. While usually such a reduction is attributed to the viscoelastic behaviour, but this statement has seldom been backed up by appropriate rheological measurements. Finally, it is important to note here that the whole of the preceding section pertaining to the prediction of power input relies on the fact that the value of the rotational speed N is known. It is a much more difficult question to answer that what is the optimum or an appropriate value of N for a new application. Since a full understanding of the underlying physical phenomena is not yet available, the choice of the optimum operating speed remains (and will continue to be) primarily a matter of experience. Though some guidelines in the literature are available but unfortunately all such recommendations vary from one impeller to another, and even from one fluid to another depending upon the severity of non-Newtonian effects. For the simplest type of fluid behaviour, namely, the Newtonian fluids being stirred by a disc turbine, Hicks et al. [322] introduced a scale (degree) of agitation, SA, which ranges from 1 to 10 with 1 being mildly mixed and 10 being intensely mixed. The scale of agitation in turn is defined as [282,303,323]: SA ¼ 32:8
Nq ND3 VL
ð28Þ
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where Nq is the dimensionless pumping number ( ¼ Q/ND3), Q being the bulk flow produced by the impeller and these characteristics are provided by the manufacturers of mixing equipment; VL is the volume of the liquid batch. Most chemical and processing applications are characterised by scales of agitation in the range 3–6, while values of 7 SA 10 are typical of chemical reactors, fermentors and of mixing of high viscosity liquids. Some guidelines are also available for choosing a suitable value of SA for specific applications employing turbine agitators [324] which unfortunately are known to be not at all suitable for the mixing of rheologically complex systems. Furthermore, the use of this methodology involving SA is not recommended for viscoelastic and viscoplastic fluids [303]. C. FLOW PATTERNS
AND
FLOW FIELDS
Further physical insights into mixing can be gained from the analysis of flow fields created by an impeller in a mixing tank. Such studies also facilitate the identification of stagnant or dead zones in the vessel. Flow patterns are strongly dependent upon the type of the impeller and geometry of the system, and these are further influenced by the rheological properties of the liquid. It is customary and convenient to classify the agitators used with non-Newtonian liquids into three types: (i) The first type or class I impellers operate at relatively high (rotational) speeds, generating high deformation rates in the vicinity of the impeller, as well as relying on the favourable momentum imparted to the whole volume of liquid. Turbine impellers and propellers (shown in Fig. 29) exemplify this class of impellers, with impeller-to-tank diameter ratio somewhere in the range 0.16– 0.6. Usually, the flow conditions relate to intermediate to high values of the Reynolds number. (ii) The second type or class II impellers are characterised by close clearance at the wall which extend over the whole diameter of the tank and their operation relies on intense shearing of the liquid in the small gaps at the walls with D/T> 0.8–0.9. Gates and anchors (also shown in Fig. 29) are representative of. (iii) Finally, there are slow rotating class III impellers which do not result in high shear rate, but rely on their excellent pumping capacity to ensure that an adequate momentum is imparted to the liquid in all parts of the vessel. Helical screw and helical ribbon impellers (shown in Fig. 9) are representative of this category. Due to the intrinsically different type of flow features created by these impellers, it is appropriate to deal with each of these categories separately.
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1. Class I Impellers The flow patterns for single phase Newtonian and inelastic nonNewtonian fluids in tanks agitated by this class of impellers have been reported by many investigators [85,86,167,175,176,180,227,236,239,251, 252,269,326–329]. The experimental techniques used include the introduction of tracer liquids, neutrally buoyant particles or hydrogen bubbles, Positron emission, X-ray visualization method; and measurement of local velocities by means of pitot tubes, laser doppler velocimeters and so on. The salient features of the flow patterns produced by propellers and disc turbines are shown schematically in Fig. 15. Basically, the propeller creates an axial flow through the impeller, which may be upwards or downwards depending upon the direction of rotation. Strictly speaking the flow field is threedimensional and unsteady; the study of circulation patterns such as that shown in Fig. 15 are helpful in delineating the presence of dead zones. If the propeller is mounted centrally and in a tank without baffles, severe vortex formation can occur (especially in low viscosity systems) which can be circumvented by installing baffles and/or by mounting the agitator in an offcentered position. In either event, the resulting flow patterns are much more complex than that shown in Fig. 15 and the power consumption also increases [33,295]. The flat-bladed turbine impeller creates a strong radial flow outwards from the impeller, thereby establishing circulation zones in the top and bottom of the vessel (Fig. 15). The flow pattern can be changed by altering the impeller geometry and, for instance, if the turbine blades are angled to the vertical, a strong axial flow component is also produced. Such a flow pattern may be advantageous in applications where it is necessary to suspend solids. However, as the Reynolds number decreases, the flow is primarily in the radial direction. Similarly, a flat paddle produces a flow field with significant tangential components of velocity, which does not promote
FIG. 15. Qualitative flow patterns for propellers and disc turbine impellers.
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FIG. 16. Qualitative flow patterns for a dual (two turbines mounted on a single shaft) impeller.
mixing. Propellers, turbines and paddles are commonly used impellers for low viscosity Newtonians and inelastic shearthinning liquids, usually operating in the transitional and turbulent flow regions. For tall vessels employing liquid depth-to-tank diameter ratio (Z/T ) larger than 1, it is common to use multiple impellers mounted on a single shaft. Clearly, the resulting flow patterns will be more complex than that seen in Fig. 15. Using two axial flow turbines mounted on the single shaft gives rise to two ‘‘zones of action’’ as shown in Fig. 16 [344]. For a shearthinning substance, the apparent viscosity is lowest in the impeller region and the fluid motion decreases rapidly away from the impeller. This decay in velocity is much more rapid in pseudoplastic fluids than that in Newtonian liquids. Viscoplastic fluids possessing a yield stress also display qualitatively similar behaviour in the sense that the shear induced by the rotating impeller is restricted to a small cavity (cavern) and there is no (or little) mixing outside this cavity. Intuitively, the shearthickening fluids would display exactly the opposite behaviour which is counter-intuitive, i.e., poor mixing in the impeller region! In a pioneering study using tanks of square cross-section, Wichterle and Wein [180] delineated the regions of motion/no-motion in shearthinning fluids being stirred by disc-turbine and propeller-type impellers, as shown in Fig. 17a and b, respectively. While at low Reynolds number, the size of the wellmixed region Dc is of the order of D, but as the Reynolds number gradually increases, the value of Dc/D increases thereby the well-mixed zone progressively grows in size. Wichterle and Wein [180] also put forward the following expressions for Dc: Dc ¼1 D
for a2 Re < 1
ð29aÞ
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FIG. 17. Flow patterns in highly shearthinning liquids, with a disc turbine and a propeller [180].
Dc ¼ aðReÞ0:5 D
for a2 Re > 1
ð29bÞ
where a is a constant which is 0.3 for propellers, 0.6 for turbines and approximately equal to 0.375(Pot)1/3 for other types of impellers; Pot is the constant value of the Power number under fully turbulent conditions. In Eq. 29, the Reynolds number is defined by setting Ks ¼ 1, i.e., Re ¼ N2nDn/m. For viscoplastic materials, a direct link between the flow pattern and the corresponding power input is illustrated by the study of Nagata et al. [330]. They reported a cyclic increase and decrease in power input which can be explained qualitatively as follows. Initially, the power input is high due to the high (apparent) viscosity of the solid-like structure; however, once the yield stress is overcome and the material begins to yield and to exhibit fluidlike characteristics, the power consumption decreases and the stress level drops. The structure is then re-established and the solid-like behaviour results leading to an increase in power input and hence the cycle repeats itself. Also, there was a propensity for a vortex to form at the liquid free surface during this cyclic behaviour. This tendency was considerably diminished or almost eliminated by using class II impellers. In this case, the solid-like behaviour can occur in the center of the vessel. More detailed and quantitative information on flow patterns in viscoplastic materials stirred by the standard Rushton disc turbine has been gleaned using X-rays [175,176], hot-wire anemometry [167] and laser doppler anemometry [241,242]. Many attempts have been made to develop predictive relations for the size of caverns seen in viscoplastic liquids [53,167,226,227,239]. For instance, Solomon et al. [167] put forward the following relation for Dc: 1=3 Dc 4Po N 2 D2 ¼ oB D p3
ð30Þ
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Equation (30) was stated to apply in the following ranges of conditions: 2 2 3 N 2 D2 4 N D D 3 ; B B p Po T o o
i:e:; when D Dc T:
Subsequently, based on a consideration of the fluid velocity at the cavern boundary, and assuming the cavern to be of spherical shape, Amanullah et al. [227] put forward the following expression for cavern diameter in power law shear thinning fluids: " # ðn2Þ=n ðn2Þ=n Dc 2 n 4pm 1=n T ¼ Vo þ n F 2 2
ð31Þ
Similar expressions for toroidal shaped caverns as observed with radial flow SCABA 6SRGT and Lightnin A 315 axial flow impellers [52,53] are also available in the literature [227]. However, unlike Eq. (29) or (30), this approach necessitates a knowledge of the total force, F, acting at the cavern boundary and the fluid velocity Vo which are neither generally known [239] nor amenable to a priori prediction for a new application. Despite these limitations, Amanullah et al. [227] reported a good match between the predictions of Eq. (31) and their own experiments. Similar results on cavern sizes and shapes generated by dual impellers like Intermig are also available in the literature [191]. Overall, it can be concluded that these laboratory scale studies on cavern characteristics are also of considerable significance is industrial settings [187,283]. In contrast, the influence of liquid viscoelasticity is both more striking and difficult to assess [236,251]. An early photographic study [331] of turbine and propeller type impellers rotating in viscoelastic fluids suggests two distinct flow patterns. In a small region near the impeller the flow is outwards, whereas elsewhere the flow is inwards towards the impeller in the equatorial plane and outwards from the rotating impeller along the axis of rotation. These two regions are separated by a close streamline thereby allowing no convective transport between the two regions. A more quantitative study [332] reveals that, irrespective of the nature of the secondary flow pattern, the primary flow pattern around a rotating body is virtually unaffected by the viscoelastic behaviour of the liquid. Indeed, a variety of flow patterns may be observed depending upon the type of the impeller and the kinematic conditions, i.e., the values of the Reynolds and Weissenberg numbers or Elasticity numbers [236,251].
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2. Class II Impellers While gate and anchor-type close clearance impellers produce poor axial circulation of the liquid in the tank, it appears that the liquid viscoelasticity promotes axial flow [105,106]. Thus, Peters and Smith [105] reported the axial flow to be almost 15 times greater in a viscoelastic medium than that in a comparable Newtonian fluid. Figure 18 displays the resulting shear rate distribution obtained in a viscoelastic fluid being stirred by an anchor where it is seen that the liquid in the tank is virtually unaffected (shear rate 0) by the passage of the anchor impeller, except close to the wall of the vessel. Broadly speaking, both anchor and gate impellers promote fluid motion near the wall, but leave the body of the liquid near the shaft relatively stagnant, as can be inferred from the typical streamline pattern shown in Fig. 19. Besides owing to the poor top to bottom turnover, significant vertical concentration gradients usually exist, which can be minimised by using a helical ribbon or a helical screw twisted in the opposite sense, pumping the fluid downward near the shaft. Typical qualitative flow patterns for an anchor impeller are sketched in Fig. 20. In these systems, the flow pattern changes with the impeller speed and thus neither the notion of an average shear rate, nor its linear variation with N implicit in Eq. (19), is strictly valid. Furthermore, any rotational motion induced within the tank wall will also produce a secondary flow in the vertical direction; the liquid near the tank bottom is virtually stationary while that at higher levels is rotating and hence will experience centrifugal forces. Consequently,
FIG. 18. Shear rate profiles for an anchor rotating in a viscoelastic medium [105,106].
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FIG. 19. Streamline patterns (relative to the arm of the impeller) for a viscoelastic liquid in a tank agitated by a gate impeller.
FIG. 20. Secondary flow pattern in an anchor agitated tank.
the unbalanced forces present within the liquid lead to the formation of a toroidal vortex. Depending upon the viscosity level and type (Newtonian, inelastic or viscoelastic) of fluid, the secondary flow pattern may be single(Fig. 20) or double-celled as shown schematically in Fig. 21. Indeed, such flow patterns are also borne out by numerical predictions and experimental observations for inelastic shearthinning media [332]. 3. Class III Impellers Apart from the qualitative results for a composite impeller (anchor fitted with a ribbon or screw) mentioned in the preceding section, only scant results are available on the flow patterns created by helical ribbon and helical screw impellers. The first study of the flow pattern produced by a helical ribbon impeller is that of Nagata et al. [333] and Fig. 22 displays the
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FIG. 21. Schematic representation of a twin-celled secondary flow pattern.
FIG. 22. Flow pattern produced by a helical ribbon impeller [333].
complex flow pattern induced by a helical ribbon impeller. The primary topto-bottom circulation, mainly responsible for mixing, is principally due to the axial pumping action of the ribbon. The shear produced by the helical ribbon is confined in the regions inside and outside of blade, whereas the shear between the bulk of the liquid and the wall is cyclic in nature. Notwithstanding the degree of scatter present in Fig. 23, from Bourne and Butler [114], the velocity data appear to be scale independent and the type of fluid, i.e., inelastic shearthinning or viscoelastic. Furthermore, Bourne and Butler [114] concluded that there was virtually no radial flow except in the top and bottom regions of the vessel, and the vertical velocity inside the ribbon helix varied only from 4 to 18% of the ribbon speed. In addition to the aforementioned primary flow pattern, secondary flows also develop with the increasing rotational speed of the impeller, similar to those observed with an anchor and shown in Fig. 21. Carreau et al. [155]
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FIG. 23. Variation of axial (liquid) velocity in the core region of helical ribbon impellers pumping down in 27 and 730 l tanks. The solid lines indicate the upper and lower bounds of data [114,115]. þ: D/T ¼ 0.89; : D/T ¼ 0.952 (small tank); s : D/T ¼ 0.954 (large tank).
also studied flow patterns for a helical ribbon impeller in viscoelastic systems. They also reported significant reduction in axial circulation as can be seen in Fig. 24 where the non-dimensional axial velocity is plotted for an inelastic (2% CMC solution) and a viscoelastic PAA solution at N ¼ 0.67 Hz. The values of the axial velocities in the inelastic CMC solution were comparable to that in Newtonian liquids. On the otherhand, the tangential velocities were so high in viscoelastic liquids that the entire contents of the tank, except for a thin layer at the wall, rotated as a solid body with the impeller. Even less is known about the flow patterns produced by a helical screw. In a preliminary study, Chapman and Holland [100] presented photographs of dye-flow patterns for an off-centered helical screw agitator, pumping upwards without a draft tube. There appeared to be a dispersive flow present between the flights of the screw, the dispersion being completed at the top of the screw. The flow into the screw impeller was from the other
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FIG. 24. Axial velocity profile in an inelastic (2% CMC) and in a viscoelastic (1% PAA) solution with a helical ribbon impeller rotating at N ¼ 0.67 Hz [155].
side of the tank, whereas the liquid in the remaining parts of the tank appeared to be virtually stagnant. Preliminary three-dimensional numerical predictions for the flow pattern produced by a helical screw appear to be in line with experimental results for Newtonian liquids [208]. Aside from the aforementioned results for different class of impellers, limited results are also available for other types of mixing devices used for thick pastes with complex rheological behaviour [112,334]. One common geometry used for the mixing of thick pastes is that of sigma-blade mixer (Fig. 33), with thick S- or Z-shaped blades, which look like high pitch helical ribbons. Usually, two units are placed horizontally in separate troughs inside a mixing chamber and the blades rotate in opposite directions at different speeds. Preliminary results obtained using a positive displacement mixer point to their potential advantages over helical ribbon and sigma blade mixers for thick pastes and extremely viscous materials [335]. From the aforementioned description, it is abundantly clear that the flow patterns developed in a mixing tank are strongly dependent on the tank– impeller configuration, rheology of the liquid and the operating conditions. Needless to emphasize here that in selecting suitable equipment, extreme care is needed to ensure that the resulting flow pattern is suitable for the envisaged application. D. MIXING
AND
CIRCULATION TIMES
Before addressing the question of circulation and mixing times, and the related issue of the rate of mixing, one must deal with the methods of
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FIG. 25. Qualitative representation of the relationship between the ‘‘scale’’ and ‘‘intensity’’ of mixing.
assessing and quantifying the quality of a mixture. Due to the wide scope and spectrum of mixing problems and the objectives of mixing, it is not possible to develop a single criterion for all possible mixing applications. Aside from such practical difficulties, even from a theoretical standpoint, mixing is poorly understood. This is so primarily due to the fact that perfect mixing implies three-dimensional randomisation of materials, and unfortunately three-dimensional processes are not yet readily amenable to mathematical treatment. Often times, the quality of a mixture is qualitatively judged by visual criteria. Another intuitive and convenient, but perhaps unscientific, criterion is whether or not the product (mixture) meets the required specifications. For many applications this criterion may be acceptable, but many high quality products require more stringent and definitive criteria for assessing the quality of a mixture. Figure 25 illustrates the intrinsic problem in defining the quality of a mixture. This figure shows a matrix of two materials mixed to different degrees by two different mechanisms, namely, size of inhomogeneity (‘‘scale of segregation’’), and diffusion (‘‘scale of intensity’’). In order to appreciate the problem associated with qualifying mixtures, a third parameter is needed, namely, the scale of examination. This denotes the smallest area or volume that can be resolved by whatever technique is used to assess the quality of the mixture. If the scale of examination equals the area of one of the dark or light squares in the right hand column of Fig. 25, then clearly all mixtures in this column will be judged as poorly mixed. Keeping the scale of examination at this level, as one moves to the left on the ‘‘scale’’ axis, the quality of mixing improves. Therefore, a homogeneous
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mixture can be produced by reducing the size of each component to some level below the smallest scale of observation and distributing these components throughout the system in a random manner. More detailed discussions on the assessment of the quality of mixtures are available in the literature [3,136,334]. Irrespective of the criteria used, mixing time is defined as the time needed to produce a mixture or a product of pre-determined quality, and the rate of mixing is the rate at which the mixing progresses towards the final state. When a tracer is added to a single-phase liquid in a mixing tank, the mixing time is measured as the time interval between the introduction of tracer and the time when the contents of the vessel have attained the required degree of uniformity. If the tracer used is completely miscible and has the same density and viscosity as the process liquid in the tank, the tracer concentration may be measured as a function of time at any point in the tank by an appropriate detector, such as by way of refractive index, or by electrical conductivity. For a given amount of tracer, the equilibrium concentration C1 is readily calculated and this value will be approached asymptotically at any point (Fig. 26). In practice, it is, however, customary to define the mixing time m as that required for the mixture composition to come within a specified (1 or 5%) deviation from the equilibrium value C1. Unfortunately, this value is strongly dependent on the way the tracer is added and the location of the detector, etc. It is thus not uncommon to record the tracer concentration at several points in the tank, and to define the variance of concentration, 2, about the equilibrium value as: 2 ¼
i¼p 1 X ðCi C1 Þ2 p 1 i¼1
FIG. 26. Qualitative representation of mixing-time measurement curve.
ð32Þ
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FIG. 27. Reduction in variance of concentration of tracer with time.
where Ci is the response of the ith detector at time t. Fig. 27 schematically shows a typical variance curve. Over the years, many experimental methods have been developed and used to measure mixing times in stirred vessels. Typical examples include acid–base titrations, measurement of electrical conductivity, temperature, refractive index, and pH, light absorption, etc. However, in each case, it is important to specify the manner of tracer addition, the position and the number of points of detection, the sampling volume of the detector (scale of examination), and the criterion used for locating the end point. Each of these factors exerts varying levels of influence on the experimental value of mixing time, and therefore, extreme care must be exercised in comparing results from different investigations [334,336]. Inspite of all such inherent limitations, the notion of a single (average) mixing time is convenient in practice, albeit extrapolations from one system to another must be treated with reserve. Furthermore, irrespective of the technique and the criterion used, the response curve may show periodic behaviour. This may be due to the repeated passage of a fluid element with a locally high concentration of tracer. The time interval between any two successive peaks is known as the circulation time, c. For a given geometrical configuration, dimensional arguments suggest the dimensionless mixing and circulation times to be functions of the Reynolds number, Froude number, Weber number and Weissenberg number, i.e., ¼ Nm ¼ f1 ðRe; Fr; We; WiÞ m
ð33aÞ
FLUID MECHANICS AND HEAT TRANSFER
c ¼ Nc ¼ f2 ðRe; Fr; We; WiÞ
137 ð33bÞ
For geometrically similar systems and in the absence of vortex formation and surface tension effects, Eq. (33) simplify to: m ¼ f3 ðRe; WiÞ
ð34aÞ
c ¼ f4 ðRe; WiÞ
ð34bÞ
Evidently for inelastic liquids, the Weissenberg number is also redundant. In general terms, m is constant both in laminar and in fully turbulent conditions, with a substantial transition zone in between these two limits. Undoubtedly, the functional relationship between m and Re is strongly dependent on the tank–impeller geometry and the type of the impeller, namely, class I, or II, or III. Little information is available about the mixing times for class I impellers in non-Newtonian systems. The scant experimental results [86] for turbine impellers in baffled tanks suggest that the correlations developed for Newtonian fluids can also be used for inelastic systems via the notion of an effective viscosity corresponding to the shear rate given by Eq. (19). The results of Godleski and Smith [99] point to much larger mixing times than those predicted by Norwood and Metzner [86], thereby implying severe segregation between the high shear (impeller region) and low shear (wall region) zones of the tank. On the other hand, Bourne and Butler [114,115] concluded that the rate of mixing and the mixing times are not very sensitive to the rheological properties of the liquids. Furthermore, in highly shearthinning systems with a yield stress, a cavern of turbulent flow engulfs the fast rotating agitator, whereas the rest of the liquid may be at rest. Under such conditions, the utility of mixing and circulation times is severely limited. Intuitively, one would expect similar, or perhaps even more severe, deterioration in mixing (hence mixing times) in viscoelastic liquids, especially when the secondary flow and flow reversal occur. However, the lack of information on mixing times with class I impellers is not too serious, for these impellers are rarely used for the agitation of viscous nonNewtonian fluids. The only study pertaining to the use of class II impellers for nonNewtonian materials is that of Peters and Smith [105,106] which seems to suggest a reduction in both mixing and circulation times for viscoelastic polymer solutions agitated by an anchor impeller. The decrease in mixing time is primarily due to the increased axial circulation as noted in the preceding section.
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In contrast to the meagre information pertaining to class I and II impellers, class III impellers have received much more attention. It is readily agreed that the shearthinning behaviour does not exert any great influence on the pumping capacity of helical impellers, whence the circulation times are little influenced [130,134,155,159,310]. Thus, the dimensionless circulation time c is constant in the laminar regime (Re< 10) and it decreases with the increasing Reynolds number and the decreasing value of power-law index n in the transition zone, eventually becoming independent of the rheology [310]. In this regard, this finding is in line with that of Bourne and Butler [114,115] for class I impellers. Another study [158] with a helical ribbon impeller shows that even though the average circulation times are not influenced significantly by shearthinning behaviour, their distribution becomes narrower with the decreasing value of power-law index n, thereby suggesting poor mixing between the high shear and low shear regions in the tank for highly shearthinning fluids. in inelastic Qualitatively similar observations can also be made about m fluids, that is, m is independent of the Reynolds number in the laminar region (Re< 10) and it decreases with the Reynolds number in the intermediate regime. Figure 28 confirms this expectation for a Newtonian fluid, an inelastic CMC solution and a Boger fluid being mixed by a helical ribbon. While in each case the constant mixing time limit is seen to be reached at low Reynolds numbers, but the cessation of the so-called laminar flow conditions is seen to occur at different values of the Reynolds number. The mixing time seems to increase progressively as the fluid behaviour changes from the Newtonian to inelastic shear thinning to viscoelastic
FIG. 28. Representative results on mixing times for a Newtonian [u], an inelastic [s] and a viscoelastic [m, j] fluids stirred by a helical ribbon impeller [310].
FLUID MECHANICS AND HEAT TRANSFER
139
behaviour thereby implying the inherent difficulty of homogenization of non-Newtonian systems in general and viscoelastic fluids in particular. Extensive reviews on the efficacy of class III impellers in homogenizing highly viscous Newtonian and non-Newtonian systems are available in the literature [206,234]. Based on an extensive evaluation of the literature data for power input and mixing time for Newtonian liquids, Delaplace et al. [234] suggest that the laminar flow conditions can exist up to about Re 60 for helical ribbon impellers which is also borne out by the results shown in Fig. 28. They also alluded to the possible difficulties in linking the performance of a helical ribbon impeller to the geometric configuration such as the wall clearance, and the pitch ratio, etc. It is abundantly clear from the foregoing discussion that the available body of knowledge about the mixing and circulation times is much less extensive and is also somewhat incoherent as compared with that for power input. Indeed, significant advances are still being made in this area even for Newtonian systems and/or in novel impeller systems such as jet mixers [337]. Also, notwithstanding the inherent drawback of using the single mixing time, alternative suggestions based on the production of inter-material surface area and energetic considerations have also been made to quantitatively describe the efficiency of mixing [338]. Thus, Ottino and Macosko [338] defined an efficiency parameter for laminar mixing as the ratio of energy expended in the creation of inter-material area and the total energy dissipated. This criterion can also be used to rank various mixing devices to ascertain their suitability for an anticipated application. E. NUMERICAL
AND
CFD MODELLING
In recent years, considerable research effort has been expended in the numerical and/or CFD modelling of the batch mixing of liquids in mechanically agitated systems, e.g., see [146,173,193,208,210,217,246,247, 265,273,275,276,325,328,329,332,339–347,347a]. There is no doubt that such modelling can potentially define many of the fluid mechanical parameters for an overall mixing system [340]. Many of the models, particularly for turbulent flow, divide the whole tank into many small microcells. However, all such efforts tend to be very computation intensive. The main impediment which has hindered the widespread use of CFD modelling in mixing processes is the very elusive nature of the complex phenomena (fluid rheology and geometry) of any practical mixing process. The fluid mechanics (kinematics) required to achieve a process result is generally not known. Notwithstanding these inherent formidable conceptual difficulties, some successful attempts have been made at the numerical modelling of flows in stirred tanks, albeit most of these relate to turbulent
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flow conditions for Newtonian fluids [275,276,327–329,346]. Indeed, scores of methods including finite element, boundary element methods and commercially available CFD packages like FLUENT and POLYFLOW etc., as well as a variety of tank–impeller configurations have been used to get detailed structure of the flow field and integral parameters like circulation times and torque, etc., for specific geometric configuration. However, the work with turbulent flow conditions is of little interest in the context of non-Newtonian fluids, albeit one may encounter turbulent conditions in some applications such as in industrial paper pulp processing wherein a chest fitted with a side entering impeller is used [339]. Bakker and Fasano [339] used FLUENT to predict the velocity profiles using the k–" model for a paper pulp modelled as a Bingham plastic in such a geometry. Both laminar and turbulent flow conditions were encountered in different parts of the tank. However, they reported qualitative agreement between the predicted and experimental flow patterns. Similarly, Venneker and van den Akker [273] simulated the flow patterns for the turbulent flow of a power law liquid (n ¼ 0.77) in a tank fitted with a Rushton turbine. The numerical predictions were substantiated by LDA measurements and a good match was reported. Since for viscous materials, the laminar flow conditions are encountered much more frequently than the turbulent conditions, there have been some modelling efforts under these conditions. Ottino et al. [345] introduced a theoretical framework for describing the phenomenon of mixing. They suggested the use of deformation of contact interfaces between materials in case of multi-phase systems or of the originally designated material surface as means of mathematically describing mixing. Similarly, Khayat et al. [342] have developed some general ideas about threedimensional mixing flow of Newtonian and viscoelastic fluids, which in principle can provide some clues about the batch mixing. In the context of laminar mixing, even the response of viscous Newtonian liquids can provide useful insights into the mixing of at least inelastic liquids. Thus, Abid et al. [332] investigated the laminar flow of Newtonian fluids induced by an anchor impeller fitted in a tank. Based on a detailed analysis of the tangential velocity distributions, the flow appeared to be planar and hence they concluded that under such conditions 2-D modelling would be adequate. Similarly, Tanguy et al. [208] employed a 3-D finite element scheme to predict circulation times and power input in laminar region for Newtonian fluid being agitated by a helical ribbon screw impeller. While the predicted and experimental values of circulation times are within 5% of each other, the discrepancy for power input was up to 50% which increased with the increasing speed of impeller. Subsequently, this work has been extended to the agitation of a second-order model fluid under laminar conditions [217]. Within the range of applicability, viscoelasticity was seen to exert only a
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141
minor influence on gross parameters. A similar study of viscous nonNewtonian fluids stirred by a paddle impeller has also been reported [146]. The scant studies to elucidate the role of viscoelasticity [217,325] seem to suggest very little effect on power input which is clearly at odds with the available experimental results. Lafon and Bertrand [343] have predicted the flow fields for power law fluids (n ¼ 0.174) agitated by an anchor which are qualitatively consistent with the pattern shown in Fig. 20. Similarly, Kaminoyama et al. [193] presented three-dimensional simulations for a Bingham plastic fluid agitated by an anchor (idealised as a cylinder) and predicted that the fluid deformation ceased at radial position (nondimensional using tank radius) of 0.4. Some preliminary studies with highly idealised impeller geometries such as an elliptic cylinder [347] as well as with novel impellers such as hydrofoil [341] with Newtonian and nonNewtonian liquids are also available. Similarly, the relative performance of static mixers can also be evaluated via CFD modelling [224] whereas Wunsch and Bohme [348] have numerically analysed three-dimensional fluid flow and convective mixing in a static mixer. Similarly, the three-dimensional flow in a Banbury mixer (shown in Fig. 29) has been numerically studied by Yang and Manas-Zloczower [349]. All in all, it is perhaps fair to say that undoubtedly the CFD is a powerful tool for developing physical understanding as well as in optimizing the performance of a mixing device. Such studies may also facilitate the development of new designs of impellers. The only major drawback is that the complex rheology coupled with the complicated geometrical aspects does not permit extrapolation of results from one system to another thereby requiring each situation to be dealt with as a new problem.
VII. Heat Transfer It is an established practice to enhance the rate of heat transfer to process fluids by externally applied motion, both within the bulk of the material and at the proximity of heat transfer surfaces. In most applications, fluid motion is promoted either by pumping through straight and coiled tubes or by mechanical agitation as in stirred tanks. A simple jacketed vessel (Fig. 5) is frequently used in chemical, food, biotechnological and pharmaceutical process engineering applications to heat/cool process streams to control the rate of reaction, or to bring it to completion. This is usually accomplished by using condensing steam or cooling water in a jacket fitted outside the mixing tank or in an immersed cooling coil in the tank contents. As is the case with power input, mixing time and flow patterns, etc., the rate of heat transfer (wall or coil) is strongly dependent on the tank–impeller configuration, type
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and number of baffles, fluid behaviour, kinematic conditions and the type of heat transfer surface, for example jacket or coil. Since voluminous literature is available on heat transfer to low viscosity Newtonian fluids in mixing tanks [5,40a,350–356], it is thus possible to predict the value of the convective heat transfer coefficient in such systems under most conditions of practical interest. In contrast, much less is known about the analogous situations involving viscous Newtonian, non-Newtonian and viscoelastic systems as can be seen from the listing in Table II. On account of generally high viscosities, overall heat transfer tends to be poor in non-Newtonian fluids, and additional complications from viscous dissipation may also arise under certain circumstances. Most of the progress in this area has also been made through the application of dimensional analysis supplemented by experimental results. It is often not justifiable to make cross-comparisons between different studies unless the two systems exhibit complete similarity, i.e., geometric, kinematic and thermal. A simple dimensional analysis of the pertinent variables suggests the following functional relationship: Nu ¼ f ðRe; Pr; Gr; Fr; We; Wi; geometric ratiosÞ
ð35Þ
As mentioned previously, usually the Froude and Weber numbers are not very important in mixing of viscous single phase liquids. Furthermore, for geometrically similar systems and inelastic fluid behaviour, Eq. (35) simplifies to: Nu ¼ f ðRe; Pr; GrÞ
ð36Þ
The new dimensionless groups appearing in Eq. (36) are: hLc k
ð37Þ
Cp k
ð38Þ
ð tÞgL3c 2 2
ð39Þ
Nusselt number; Nu ¼ Prandtl number; Pr ¼ and the Grashof number; Gr ¼
where Lc is a characteristic linear dimension of the system. Thus, for instance, for the purpose of correlating power input, mixing time data, it is
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143
customary to use the impeller diameter, D, as the characteristic linear dimension. For heat transfer applications, an unambiguous choice of Lc is far from obvious and indeed different choices for Lc further add to the complexity of the situation. The Grashof number clearly is a measure of the importance of natural convection effects, which are generally small in low viscosity liquids, due to high values of the Reynolds number. It becomes increasingly significant in highly viscous liquids agitated by low speed close clearance anchors, gates and helical ribbon or screw impellers [161]. The effective viscosity term appearing in the Reynolds, Prandtl and Grashof numbers is evaluated via Eq. (19) with an appropriate value of Ks. An examination of Table II shows that most of the heat transfer studies have attempted to establish the functional relationship embodied in Eq. (36) for a given tank–impeller configuration and under the conditions when natural convection is negligible in comparison with the forced convection. It is convenient to present the pertinent information separately for each class of impellers.
A. CLASS I IMPELLERS These impellers operate at relatively high rotational speeds and are effective only in low to medium viscosity liquids. In most cases, the flow conditions in the tank correspond to transitional and/or turbulent and therefore the natural convection effects are assumed to be negligible. For shearthinning polymer solutions and slurries stirred by paddles turbines and propellers, many correlations of varying forms and complexity are available in the literature [101,166,182,355]. Most such expressions are of the following general form: Nu ¼ AReb Prc ðViÞd
ð40Þ
where Vi is the viscosity number and accounts for the temperature dependence of viscosity. It is usually defined as the ratio of the fluid viscosity evaluated at the wall and that at the bulk temperature. Obviously, the values of the constants A, b, c, d are strongly dependent on the tank– impeller configuration and the type of heat transfer surface, namely, jacket or coil. For jacketed vessels, it is a well established practice to use either D or T as the characteristic linear dimension in the definition of the Nusselt number. In most cases, the effective viscosity has been calculated using Eq. (19) with Ks ¼ 4p ¼ 12.56, as proposed by Gluz and Pavlushenko [101–103]. By way of illustration, Gluz and Pavlushenko [101] put forward the following
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correlation for heat transfer: 0:18 hD 0:67 0:33 mw ¼ 0:215Re Pr Nu ¼ k mb
ð41Þ
where mw and mb are the values of the power-law consistency coefficient at the wall and bulk temperature, respectively. Equation (41) is based on experimental data encompassing the following ranges of variables: 0.6 n 1; 5 Re 2 105, Pr 2.5 104, T ¼ 300 mm and Ks ¼ 4p. Understandably, the natural convection effects are likely to be unimportant at such high values of Reynolds number, as also reflected by the absence of the Grashof number in Eq. (41). The literature abounds with such correlations, but their utility is severely limited by the fact that each of them applies to a specific tank–impeller configuration [353]. Some of these correlations, however, explicitly contain geometric ratios. In contrast, the analogous expressions for heat transfer coefficient for coil heat transfer with class I impellers tend to be more complex than Eq. (41) and involve additional geometric parameters. The effective viscosity is still estimated via Eq. (19) with a suitable value of Ks. One such correlation, which covers fairly wide ranges of conditions, is due to Edney and Edwards [137]: 0:375 hdc T eff;b 0:2 ¼ 0:036Re0:64 Pr0:35 Nu ¼ ð42Þ Dc k w where dc is the coil tube diameter, Dc is the mean helix diameter. Edney and Edwards [137] used Ks ¼ 11.5 for a six-blade turbine and were able to obtain a unified representation of their data for both Newtonian and nonNewtonian fluids over the following ranges of conditions: 400 Re 106; 4 Pr 1900 and 0.65 eff 280 m Pa s. Preliminary results also suggest that moderate levels of aeration did not influence the heat transfer characteristics appreciably. B. CLASS II IMPELLERS These impellers, such as the gates and anchors, reach the far corners of the tank directly rather than relying on momentum transport, and operate at relatively low rotational speeds. For heat transfer applications, it thus becomes even more important to induce fluid motion close to the heat transfer surface, i.e., wall and/or coil. The bulk of the literature relating to heat transfer for anchors rotating in Newtonian and inelastic liquids has been reviewed by Ayazi Shamlou and Edwards [171] and others [350,355].
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FLUID MECHANICS AND HEAT TRANSFER
In jacketed vessels, the bulk of the resistance to heat transfer lies in the thin liquid film between the impeller and the tank wall. Some analytical efforts have also been made to model this process. The simplest approach hinges on the fact that in view of the poor bulk flow, heat transfer occurs mainly by conduction across the thin liquid film [120]. As expected, this gross-over simplification severely under-estimates the value of the Nusselt number by up to a factor of 4 [171]. Heim [256], on the other hand, invoked the boundary layer flow approximation and developed a closed form expression for the Nusselt number (at the wall) as a function of Re, Pr and (D/T). The impeller-to-tank diameter ratio was found to be a more significant variable under laminar flow conditions than under turbulent condition. Subsequent experimental results for Newtonian liquids agitated by an anchor and screw seem to lend a general support to the qualitative trends predicted by this approach. Other approaches include the penetration model which essentially treats the process as an unsteady, one-dimensional heat conduction problem in an semi-infinite domain, in between the two successive passages of the impeller. This approach has been shown to over-estimate the value of Nusselt number almost by an order of magnitude [152]. One plausible explanation for such a large discrepancy is perhaps due to the fact that the impeller does not completely wipe the liquid off the wall of the tank thereby leaving a static liquid film adhering to the wall. Thus, Rautenbach and Bollenrath [152] put forward the following modified expression for Nu:
Nu ¼ 0:568 NðT DÞn2b
0:23
1 1 ðD=TÞ
ð43Þ
where is the thermal diffusivity, and nb is the number of impeller blades. Pollard and Kantyka [259] reported an extensive experimental study on heat transfer from a coil to chalk-in-water slurries (0.3 n 1) in vessels up to 1.1 m in diameter fitted with anchor agitators; they correlated their results on Nusselt number as follows: 0:14 0:48 0:27 hT T T 0:667 0:33 eff;b ¼ 0:077Re Nu ¼ Pr k D dc eff;w
ð44Þ
Equation (44) applies over the following ranges of conditions: 200 Re 105. The effective viscosity appearing in Eq. (44) is evaluated using the value of Ks given by Eq. (20). Similarly for jacketed vessels (fitted with baffles), Hagedorn and Salamone [111] measured the rates of heat transfer to water, glycerine and aqueous carbopol solutions over wide ranges of conditions (0.36 n 1; Re 7 105; Pr 24,000) and for a range of class I and II impellers. Based
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R. P. CHHABRA
TABLE IV VALUES
OF
CONSTANTS
IN
EQ. (45)
Impeller
a
b
C
d
e
f
g
i
T/D
Paddle Propeller Turbine Anchor
0.96 1.28 1.25 1.43
0.15 0 0 0
2.51 0.55 3.57 0.56
0.26 0.30 0.24 0.30
0.31 0.32 0.30 0.34
0.46 0.40 0 –
0.46 – 0 –
0.56 1.32 0.78 0.54
1.75–3.5 2.33–3.41 2–3.50 1.56
on the measurement of temperatures at various locations in the tank, they developed the following generic form of heat transfer correlation: e f g hT T W a=ððnþ1ÞþbÞ d mb ¼ C Re Pr ni Nu ¼ k D D mw
ð45Þ
where the effective viscosity is evaluated via Eq. (19) using Ks ¼ 11. The values of the empirical constants appearing in Eq. (45) vary from one impeller to another and are listed in Table IV. Note the inverse dependence of the Nusselt number on the scale of the equipment (T/D) which is obviously due to the large stagnant zones present beyond the impeller region. The predictions of Eq. (45) are believed to be reliable to within 15% for moderately shearthinning behaviour (n0.69) and these progressively deteriorate ( 20%) as the value of n drops further. Similarly, Sandall and Patel [124] and Martone and Sandall [122] have presented empirical expressions for the heating of pseudoplastic (carbopol in water) solutions and viscoplastic chalk-in-water slurries in a steam-jacketed tank fitted a turbine impeller and baffles or with an anchor agitator. Based on only one value of tank diameter, T, their correlation is of the following form: Nu ¼
hT eff;b d ¼ C Rea Prb k eff;w
ð46Þ
In this case also, the effective viscosity appearing in the Reynolds and Prandtl numbers and in the viscosity ratio term is evaluated using Eqs. (19) and (20). Equation (46) encompasses over the following ranges of experimental conditions: 0.35 n 1; 80 Re 105, and 2 Pr 700. The values of the constants are as follows: a ¼ 2/3, b ¼ 1/3, d ¼ 0.12. The remaining constant C showed some dependence on the type of the impeller; thus for instance, C ¼ 0.48 for turbine and C ¼ 0.32 for anchors. Attention is drawn
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147
to the fact that the aforementioned values of a, b and d also coincide with the corresponding values for Newtonian fluids and hence the effect of nonNewtonian behaviour is reflected by the values of C.
C. CLASS III IMPELLERS These impellers are characterised by relatively low shear rates, excellent pumping capacity and considerably improved mixing efficiency for highly viscous Newtonian and non-Newtonian media. Hence, considerable attention has been accorded to heat transfer aspects in these systems. While the theoretical ideas of Heim [256] and Coyle et al. [120] mentioned in the preceding section are also applicable here to a certain extent, most of the progress in this area has been made by means of dimensional and scaling considerations. Several excellent experimental studies with jacketed vessels [96,120,161,171,185,188,256] and coil heat transfer [96] have been reported in the literature. Once again, owing to geometrically different configurations employed in different studies, it is strictly not possible to make meaningful cross-comparisons. Therefore, only a representative selection of widely used correlations is presented here to give the reader an idea of what is involved in attempting to develop universal correlations in this field. Carreau et al. [161] studied heat transfer between a coil (also acting as a draft tube) and viscous Newtonian, shearthinning and viscoelastic polymer solutions agitated by a helical screw in a flat-bottomed tank. Experiments were performed in both heating and cooling mode to avoid any spurious effects. The flow rate of water inside the coil was sufficiently high to ensure the high values of heat transfer coefficient on the inside. The value of Ks ¼ 16, a value deduced from their previous study [159], was used to evaluate the effective viscosity. They developed the following correlation:
Nu ¼
0:59 hco dco dco ¼ 0:387Re0:51 Pr0:33 k D
ð47Þ
where the subscript ‘‘o’’ refers to the outside of the coil. All physical properties were evaluated at the mean film temperature (twþtb)/2. Carreau et al. [161] noted that Eq. (47) predicts the values of hco in viscoelastic systems with lower accuracy than that for Newtonian and inelastic fluids. The strong influence of (dco/D) on Nu in Eq. (47) is in line with the model predictions [256]. Carreau et al. [161] also proposed alternative correlations in terms of the liquid circulation velocity rather than the impeller tip velocity used in the conventional definition of the Reynolds number.
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R. P. CHHABRA
The mean circulation velocity of the liquid Vc, is defined as: Vc ¼
lc tc
ð48Þ
where lc and tc are the mean circulation length and time, respectively. The value of lc is strongly influenced by the flow pattern and the geometry and needs to be inferred from experimental results. In their study [161], lc ¼ 1.08 m, and the circulation time tc was calculated from the following correlation [159]: V ¼ ½0:124 þ 0:265ð1 expð0:00836 ReÞÞ ð1 0:811Wi0:25 Þ D3 Ntc
ð49Þ
where V is the volume of the tank, and Wi ¼ N1/2eff _ e. The effective shear rate _e for heat transfer is calculated as: _e ¼
Vc dco
ð50Þ
They introduced the following modified definitions of the Reynolds, Prandtl and Nusselt numbers: Rec ¼
Vc2n ðns dco Þn m
ð51Þ
Prc ¼
Cp mVcn1 n1 kdco
ð52Þ
Nuc ¼
hco ðns dco Þ k
ð53Þ
and
The factor (nsdco) accounts for the number of loops in the coil. In terms of these new groups, Carreau et al. [161] re-correlated their results as follows: Nuc ¼ 2:82 Rec0:385 Prc0:33
ð54Þ
All thermo-physical properties are evaluated at the mean film temperature. While the general form of Eq. (54) is similar to that of Eq. (47), but Eq. (54) does incorporate some description of the flow patterns via the use of lc and tc.
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As stated at the outset, only a selection of widely used correlation is presented here and the reference must be made to the extensive compilations due to Edwards and Wilkinson [355], Poggemann et al. [166] and Dream [354]. It should be emphasized again that the currently available information on heat transfer to non-Newtonian fluids in agitated vessels relates to specific vessel–impeller configurations. Few experimental data are available for independent validation of the individual correlations available in the literature. However, another comment is also in order at this juncture. Although the methods used for the estimation of the effective viscosity via Eq. (19) vary from one correlation to another, especially in terms of the value of Ks, this appears to exert only a moderate influence on the value of h, at least for shearthinning fluids. For instance for n ¼ 0.3, a twofold variation in the value of Ks will give rise to a 40% reduction in the value of the effective viscosity and its effect on the value of h is further diminished . Thus, an error of 100% in the estimation of eff because h varies as 0:3–0:7 eff will result in an error of only 25–60% in the value of h which is not at all bad in view of the complexity of the flow in an agitated vessel. It is worthwhile to re-iterate here that for a given fluid rheology and impeller–tank geometry, there is a little point in attempting to augment the value of the heat transfer coefficient by increasing the speed of rotation, as the small increase in the value of h is more than off-set by the corresponding increase in power input to the system. Thus, unfortunately, not much can be done about improving heat transfer in agitated tanks and one must live with what one gets! Some preliminary results are also available on the nucleate boiling of non-Newtonian polymer solutions [230,378–380] in stirred tanks. GastonBonhomme et al. [378–380] studied boiling heat transfer in a mechanically stirred tank fitted with a helicoidal heating coil for moderate degrees of shearthinning behaviour and relatively thin fluids. Subsequently, FloquetMuhr and Midoux [230] have examined the effect of power law consistency coefficient on the convective heat transfer coefficients in a tank fitted with an anchor. Most such correlations are of the following form: qsfcnb ¼ qsfc þ qsnb
ð55Þ
where s ¼ 2 for Newtonian fluids and s ¼ 1 for power law shearthinning fluids; qfcnb is the heat flux under forced convection with nucleate boiling, qfc is the heat flux under forced convection conditions only and qnb under the nucleate boiling conditions only. These heat fluxes, in turn, are estimated using appropriate correlations available in the literature [230,378–380].
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R. P. CHHABRA
VIII. Mixing Equipment and its Selection The wide range of mixing equipment available commercially reveals the great variety of mixing duties encountered in the processing and allied industries. It is reasonable to expect therefore that no single piece of mixing equipment will be suitable for carrying out such wide ranging duties. Furthermore, most of the equipment available commercially is based on the assumption of Newtonian fluid behaviour and only a few manufactures have taken into account non-Newtonian characteristics of the material. However, broadly speaking, the mixers suitable for highly viscous Newtonian materials are also likely to be acceptable at least for shearthinning inelastic fluids. In general, the higher the viscosity, smaller are the clearances between the moving and fixed parts, and these devices operate at low rotational speeds. Such considerations have led to the development of a number of distinct and proprietary types of mixers over the years. Unfortunately, very little has been done, however, by way of standardisation of equipment [6,303]. As noted previously, the lack of such standards not only makes it necessary to conduct some experimental tests for each application but it also makes it virtually impossible to formulate universally applicable design methods. The choice of a mixer type, its design and optimum operating conditions is therefore primarily a matter of experience. In the following sections, an attempt has been made to present the main mechanical features of commonly used equipment, their range of applications, etc. Extensive discussions of design and selection of different types of mixers, and the effects of various physical and operational parameters on the performance of the equipment are available in the literature [3,6,27,357–372]. Most equipment manufacturers also provide performance profiles and recommended range of applications of their products, and also offer some guidelines for the selection of most suitable configuration and operating conditions for a specific application. Finally, it is also worthwhile to mention here that the fixed and operating costs for the mixing equipment also vary significantly from one type to another type of mixer, and this must not be overlooked while selecting the most appropriate configuration for an envisaged application [373]. The equipment used for batch mixing of viscous liquids by mechanical agitation (impeller) has three main elements: a tank or vessel, baffles and an impeller. A. TANK
OR
VESSEL
These are often vertically-mounted cylindrical tanks, up to 10 m in diameter, and height-to-tank diameter ratio of at least 1.5, and typically
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151
filled to a depth equal to about one tank diameter. In some applications, especially in gas–liquid and liquid–liquid mixing applications, tall vessels are used and the liquid depth is then up to three tank diameters; multiple impellers fitted on to a single shaft are frequently used [179,301]. The vessels of square cross-section have also been used in some applications [298]; similarly, the vessels may be closed or open at the top [297–299]. The base of the tanks may be flat, dished, conical, or specially contoured, depending upon a variety of considerations such as the ease of emptying/draining, or the need to suspend solid particles, etc. For the batch mixing of thick pastes and doughs using helical screw and ribbon impellers, Z- or sigma-blade mixers, the vessels may be installed horizontally. In such applications, the working volume of thick pastes and doughs is often relatively small, and the mixing blades (impeller) are massive in construction.
B. BAFFLES In order to prevent (or to minimise the tendency for) gross vortexing, which has deleterious effect on the quality and efficiency of mixing, particularly in low viscosity liquids, baffles are often fitted to the wall of the tank. These take the form of thin strips, about 0.1T in width, and typically four equi-spaced baffles may be used. The baffles may be mounted flush with the wall or a small clearance may be left between the wall and the baffle to facilitate fluid motion in the wall region. Minor variations in the length of baffles usually have only a small influence on power input [290]. Baffles are, however, generally not required for high viscosity liquids (> 5 Pa s) because in these fluids viscous stresses are sufficiently large to damp out the tendency for the rotary motion. In some cases, the problem of vortexing is obviated by mounting impellers off-centre or horizontally.
C. IMPELLERS This is perhaps the most important component of a batch mixer and a wide variety of impellers have evolved over the years to meet ever increasing requirements for the mixing of rheologically complex materials. Figure 29 shows a selection of the commonly used impellers or agitators. Propellers, turbines, paddles, gates, anchors, helical ribbon and screws are usually mounted on a central vertical shaft in a cylindrical vessel, and they are selected for a specific duty, largely on the basis of liquid viscosity or non-Newtonian characteristics [358]. As the viscosity of the liquid progressively increases, it becomes generally necessary to move from a
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FIG. 29. Some of the commonly used impeller designs.
propeller to a turbine and then, in order, to a paddle, to a gate or to an anchor, and then to a helical ribbon, and finally to a screw. The speed of rotation or agitation is gradually reduced as the medium viscosity increases. Propellers, turbines and paddles are typically used with low viscosity liquids and operate in the transitional/turbulent regime. A typical velocity (ND) for a tip of the blades of a turbine is 3 m/s, with a propeller being a little faster and a paddle little slower. These agitators are also known as remote-clearance impellers because of the significant gap between the wall
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FIG. 30. A selection of variation in turbine impeller designs.
and the impeller, 0.13 (D/T ) 0.67. For each of the impeller shown in Fig. 29, minor design variations are available which have been introduced by individual equipment manufacturers. In the case of the so-called standard six-bladed Rushton turbine, possible variations available are shown in Fig. 30. Thus, it is possible to have angled-blades, retreating blades, hollow bladed turbines, wide blade hydro-foils, etc. Figure 31 shows some further novel designs of this class of impellers. For tall mixing vessels (such as that used in fermentation applications), it is quite common to mount two or more disc turbines (T/2 distance apart) on the same shaft to improve mixing over the whole depth of the liquid. Gates, anchors, helical ribbons and screws (also see Fig. 29) are usually employed for the mixing of highly viscous Newtonian and non-Newtonian media. The gate, anchor and ribbon type impellers are usually arranged with a close clearance at the vessel wall, whereas the helical screw has a smaller diameter and is often used inside a draft tube (Fig. 29) to promote liquid motion throughout the vessel. Helical ribbons or interrupted ribbons are often used in horizontally installed cylindrical tanks. A variation of the simple helix mixer is the helicone, shown schematically in Fig. 32, which has
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FIG. 31. Some novel designs of impellers.
FIG. 32. A double helicone impeller.
the added advantage that the gap between the blade and the vessel wall is easily adjusted by a small axial shift of the impeller. In some applications involving dispersion of particles in high viscosity liquids, the shear stresses generated by an anchor may not be adequate for the breakup and dispersion of agglomerates, and it may be necessary to use an anchor to promote general flow in the vessel together with a high shear mixing impeller mounted on a separate off-centred, inclined shaft and operating at high speed.
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FIG. 33. A sigma-blade mixer.
Kneaders, Z-blade (Fig. 29) and sigma-blade (Fig. 33), and Banbury mixers (Fig. 29) are required for the mixing of highly viscous materials like pastes, rubbers, doughs, and so on, many of which exhibit non-Newtonian flow characteristics. The tanks are usually mounted horizontally, and two impellers are used. The impellers are massive and clearances between blades, as well as between the vessel wall and the blade, are very small thereby ensuring the entire mass of liquid is subjected to intense shearing. While mixing heavy pastes and doughs using a sigma blade mixer, it is not uncommon for the two blades to rotate at different speeds in the ratio of 3:2. The blade design differs from that of the helical ribbons due to the fact that the much higher viscosities, of the order of 10 kPa s, require a more solid construction; the blades consequently tend to sweep a greater quantity of the fluid in front of them, and the main small-scale mixing occurs by extrusion between the blade and the wall. Partly for this reason, the mixers of this type are operated only partially full, though the Banbury mixer used in the rubber industry is filled completely and pressurized as well. The pitch of the blades produces the necessary motion along the channel, and this gives the large scale blending needed to limit the batch blending times to reasonable levels. Figure 34 shows the various designs of impellers of Banbury type mixers which are used extensively in rubber and polymer industries. In addition to the aforementioned selection, many other varieties such as double planetary, two intersecting cylinders [373a], composite and dual impellers are also available. In view of such a wide variety of impeller designs coupled with the diversity of mixing problems, it is virtually impossible to offer guidelines for the selection of the most appropriate equipment for a given duty. This choice is further made difficult depending upon the main objective of mixing, that is, whether it is to achieve homogenization or to
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Fig. 34. Some variations in the design of Banbury mixer internals.
add/remove heat or to promote chemical reactions, etc. In spite of all these uncertainties, it is readily conceded that the choice of an impeller is primarily governed by the viscosity of the medium. Therefore, some attempts have been made to devise selection criteria for impellers solely based on viscosity. Figure 35 shows one such selection chart. In general, the discussion herein has been restricted to the batch mixing of liquids. It is, however, appropriate to direct the reader to some lead references to other types of mixing problems. Gas–liquid mixing has been thoroughly treated by Tatterson [15,27] for low viscosity systems. The corresponding literature on high viscosity liquids is summarized in various sources [3,6,7,59,60,374]. The contemporary literature on the mixing of solids is reviewed in a series of papers by Lindley and co-workers [16–20] and more recently by Ottino and Khakhar [375].
IX. Concluding Summary In this chapter, the voluminous literature available on the batch mixing of single phase liquids by mechanical agitation has been critically and thoroughly reviewed. Starting with the diversity of industrial settings where mixing is encountered in process engineering applications, various mechanisms of mixing in laminar and turbulent flow conditions have been examined. Following this are addressed the issues of scale-up, power input,
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FIG. 35. Suggested ranges of operations of various impellers (based on the values of viscosity) (modified after Niranjan et al. [19]).
flow pattern and mixing times, numerical modelling, wall and coil heat transfer, and finally the selection of equipment. It is instructive to recall here that adequate information is available on all these aspects of mixing for Newtonian liquid media, albeit most developments are based on dimensional considerations supplemented by experimental results. In contrast, the corresponding body of information is neither as extensive nor as coherent in the case of highly viscous Newtonian and non-Newtonian fluids. Furthermore, most of the information relates to inelastic (or time independent) shearthinning and viscoplastic media. Much less is known about the agitation of shearthickening, time-dependent (thixotropic and rheopectic) and viscoelastic liquids. A bulk of the research effort has been expended in elucidating the effect of non-Newtonian characteristics on scale-up, power input, mixing time, flow patterns and on heat transfer. In each case, it is endeavoured to define an average shear rate for a given geometrical configuration so that the results for non-Newtonian fluids collapse on to the corresponding relationship for Newtonian media for the same geometry. Under laminar flow conditions in the tank, the average shear rate has been found to be proportional to the rotational velocity of the impeller and this dependence becomes stronger in the transitional region. Under laminar conditions, the constant of proportionality, Ks, is a function of geometry only, though in some cases it has been found to depend upon the rheology of the liquid also. Some analytical efforts have also been made to predict the value of this
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constant, especially for close clearance impellers. Thus for a given geometry, it is imperative to establish the power curve with Newtonian liquids and few tests are then needed with non-Newtonian fluids to calculate the pertinent value of the Ks for the specific configuration. This chart can then be used for geometrically similar systems to calculate the power input for large scale equipment. In general terms, this approach is able to predict power input with an accuracy of 25–30% for shearthinning and viscoplastic fluids, and to a lesser extent for mildly viscoelastic systems. In general, the increasing levels of shearthinning conditions extend the so-called laminar region whereas on the other hand, results for viscoelastic media begin to veer away from the power curve at much lower values of the Reynolds number. Such deviations are not due to turbulence in the conventional sense (i.e., due to inertial effects), but these may well be due to the so-called elastic turbulence and/or elongational stresses [376,377]. In spite of all these limitations, the role of non-Newtonian rheology on power input is probably the most widely aspect of liquid mixing. Much less is known about the mixing times. While, in principle, the notion of an average shear rate has also proved useful in interpreting mixing and circulation times, little experimental data is available owing to the experimental difficulties inherent in such measurements. Qualitatively speaking, the dimensionless mixing (and circulation) time is independent of the Reynolds number, in both the laminar and the fully turbulent conditions. Thus, it decreases with increasing Reynolds number from the upper asymptotic value to the lower one. While shearthinning does not appear to exert much influence, mixing times tend to be much higher for viscoplastic and even larger for viscoelastic liquids than that for Newtonian media otherwise under identical conditions. Similarly, considerable segregation occurs in highly shearthinning and viscoplastic fluids as the momentum imparted by the rotating member is confined to a small cavity (cavern) surrounding the rotating impeller, with very little motion outside this cavity. At low Reynolds numbers, the size of the cavity is of the order of the impeller diameter, and it, however, grows with the increasing Reynolds number. Little is known about the effect of viscoelasticity on flow patterns. Similarly, much less is known about the heat transfer to/from non-Newtonian fluids stirred by an impeller. Irrespective of the type of heat transfer, namely, through the wall jacket or from a coil, the approach has been to reconcile the data for nonNewtonian fluids (at least for inelastic systems) with that for Newtonian liquids using the notion of an average shear rate deduced from power input data. This approach has been quite successful in reconciling experimental data which is rather surprising. The value of the Ks factor inferred from power input data implicitly reflects the gross fluid mechanical phenomena in the impeller region whereas the bulk of the thermal resistance to heat
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transfer lies outside this region! The fact that even this approach works is presumably due to the fact that in high viscosity systems, convection does not contribute to the overall heat transfer as much as in low viscosity Newtonian fluids. Besides, one must learn to accept poor heat transfer characteristics in rheologically complex liquids, as any attempt to enhance heat transfer by increasing the speed of rotation is self-defeating because the power input depends upon the rotational speed much more strongly than the Nusselt number. The chapter is concluded by presenting a short overview of the mechanical equipment available to cope with a variety of single phase liquid mixing duties. Unfortunately, no design codes are available, but some guidelines are presented for the selection of an appropriate system for a new application. From the foregoing treatment, it is abundantly clear that even the simplest type of mixing involving single phase non-Newtonian liquids has not been studied as systematically and thoroughly as that for Newtonian media and this area merits much more attention than it has received in the past. In particular, the following is a (partial) list of the related topics which need further systematic exploration: (i) For a given geometry, the effect of viscoelasticity on flow patterns, mixing times, and efficiency of mixing needs to be examined to deal with the mixing of viscoelastic systems. (ii) There seems to be a complex interplay between the geometry, rheology and kinematics and until this relationship is established, extrapolation/scale up cannot be carried out with a great degree of confidence. (iii) CFD and/or numerical modelling has just begun to provide some insights into the underlying physical processes and the full potential of CFD needs to be realized to inspire confidence in the optimal design and operation of mixing equipment. (iv) In view of the changing patterns in the processing in biotechnology, pulp and paper and other process engineering applications, new designs of equipment are needed, and CFD studies can provide some hints in this direction. (v) More effort needs to be directed at detailed kinematical studies involving flow visualization so that the underlying fluid mechanics can be understood better than that through gross measurements of power input and mixing time, etc. (vi) Certainly, much more experimental work is needed on heat transfer characteristics in these systems to cover as wide range of conditions as possible.
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Nomenclature a c Cp D Dc dc E El Fr Gr g h k Kp Ks l m m0 N1 n n0 nb Nu Nq P Po Pot Pr Q Re R
constant in Eq. (29) (–) height of impeller from tank to bottom (m) heat capacity (J/kg K) impeller diameter (m) coil helix diameter (m) coil tube diameter (m) energy of activation of flow (J/ mol K) Elasticity number, Wi/Re, (–) Froude number (–) Grashof number (–) acceleration due to gravity (m/s2) heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) ¼ Po Re for Newtonian fluids in laminar region constant in Eq. (19) (–) length (m) power law consistency coefficient for shear stress (Pa sn) power law coefficient for first normal 0 stress difference (Pa sn ) first normal stress difference (Pa) flow behaviour index (–) power law index for first normal stress difference (–) number of blades in the impeller (–) Nusselt number (–) Pumping number (–) power (W) Power number (–) constant-value of Power number under turbulent conditions (–) Prandtl number (–) circulation flow rate (m3/s) Reynolds number (–) universal gas constant (J/mol K)
Sc Sh SA t t T Vi VL We Wi Z
Schmidt number (–) Sherwood number (–) intensity of agitation (–) temperature difference (K) temperature (K) tank diameter (m) ratio of viscosity at wall and in the bulk (–) volume of liquid batch (m3) Weber number (–) Weissenberg number (–) depth of liquid (m)
GREEK LETTERS _ c m
f
p B oB oH
thermal diffusivity (m2/s) isobaric coefficient of expansion (1/ K) shear rate (1/s) circulation time (s) mixing time (s) fluid characteristic time (s) process characteristic time (s) viscosity (Pa s) Bingham plastic viscosity (Pa s) density (kg/m3) surface/interfacial tension (N/m) shear stress (Pa) Bingham model parameter (Pa) Herschel–Bulkley model parameter (Pa)
SUBSCRIPTS avg b eff w
average bulk effective wall
References 1. Ottino, J. M. (1989). The mixing of fluids. Sci. Am. 260(1), 56–67. 2. Tatterson, G. and Calabrese, R. (1989). ‘‘Industrial Mixing Research Needs’’. AIChE, New York. 3. Harnby, N., Edwards, M. F., and Nienow, A. W. (1992). ‘‘Mixing in the Process Industries’’, 2nd edn. Butterworth-Heinemann, Oxford.
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4. McDonough, R. J. (1992). ‘‘Mixing for the Process Industries’’. Van Nostrand Reinhold, New York. 5. Nagata, S. (1975). ‘‘Mixing: Principles and Applications’’. Wiley, New York. 6. Oldshue, J. Y. (1983). ‘‘Fluid Mixing Technology’’. McGraw Hill, New York. 7. Ulbrecht, J. J. and Patterson, G. K. (1985). ‘‘Mixing of Liquids by Mechanical Agitation’’. Gordon and Breach, New York. 8. Beishon, D. S. and Robertson, T. (1987). Problems Associated with the Homogenisation of Liquid Metals. I. Chem. E. Symp. Ser. (No. 108): Fluid Mixing III, pp. 1–31. 9. Chen, Y. R. (1981). Impeller power consumption in mixing livestock manure slurries. Trans. ASAE 24, 187–192. 10. Dorris, G. M. (2000). Effects of impeller type and mixing power on settling and filtering of lime mud. J. Pulp Paper Sci. 26(2), 47–53. 11. Pandit, A. B., Tenefrancia, N., Joshi, J. B., and Tiwari, K. K. (1984). Power consumption and homogenization time for Oleoresin suspensions. Chem. Eng. J. 28, 25–38. 12. Bourne, J. R., Buerli, M., and Regenass, W. (1981). Power and heat transfer to agitated suspensions: Use of heat flow calorimetry. Chem. Eng. Sci. 36, 782–784. 13. Weisman, J. and Efferding, L. E. (1960). Suspension of slurries by mechanical mixers. AIChE J. 6, 419–426. 14. Coran, A. Y. and Donnet, J.-B. (1992). The dispersion of carbon black in rubber Part I. Rapid method for assessing quality of dispersion. Rubber Chem. Technol. 65, 973–997. 15. Tatterson, G. B. (1991). ‘‘Fluid Mixing and Gas Dispersion in Agitated Tanks’’. McGraw Hill, New York. 16. Lindley, J. A. (1991). Mixing processes for agricultural and food materials: 1. Fundamentals of mixing. J. Agric. Eng. Res. 48, 153–170. 17. Lindley, J. A. (1991). Mixing processes for agricultural and food materials: Part 2, Highly viscous liquids and cohesive materials. J. Agric. Eng. Res. 48, 229–247. 18. Lindley, J. A. (1991). Mixing processes for agricultural and food materials: 3. Powders and particulates. J. Agric. Eng. Res. 49, 1–19. 19. Niranjan, K., Smith, D. L. O., Reilly, C. D., Lindley, J. A., and Phillips, V. R. (1994). Mixing processes for agricultural and food materials: Part 5, Review of mixer types. J. Agric. Eng. Res. 59, 145–161. 20. Reilly, C. D., Smith, D. L. O., Lindley, J. A., Niranjan, K., and Phillips, V. R. (1994). Mixing processes for agricultural and food materials: Part 4, Assessment and monitoring of mixing systems. J. Agric. Eng. Res. 59, 1–18. 21. Aarons, B. L. and Hepner, L. (1975). Mixing and blending in the food processing industries. Food Trade Rev. 45(1), 7–11. 22. Tamhankar, S. S. and Doraiswamy, L. K. (1979). Analysis of solid–solid reactions: A review. AIChE J. 25, 561–582. 23. Schenkel, G. (1966). ‘‘Plastics Extrusion Technology’’. Cliffe Books, London. 24. Janssen, L. P. B. M. (1978). ‘‘Twin Screw Extrusion’’. Elsevier, Amsterdam. 25. Rauendaal, R. C. (1992). ‘‘Mixing in Polymer Processing’’. Hanser, Munich. 26. Heywood, N. I., Viney, L. J., and Stewart, I. W. (1984). I. Chem. E. Sym. Ser. No.89, Fluid Mixing II, p. 147. 27. Tatterson, G. B. (1994). ‘‘Scaleup and Design of Industrial Mixing Processes’’. McGraw Hill, New York. 28. Uhl, V. W. and Gray, J. B. (1967). ‘‘Mixing: Theory and Practice’’. Academic, New York. 28a. Zlokarnik, M. (2001). ‘‘Stirring: Theory and Practice’’. Wiley-VCH, Weinheim. 29. Papers in J. Chem. Eng. Jpn. (2001), 34(5). 30. Special issue on Mixing: Chem. Eng. Res. Des. (2001), 79A(8).
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31. Carreau, P. J., DeKee, D., and Chhabra, R. P. (1997). ‘‘Rheology of Polymeric Systems: Principles and Applications’’. Hanser, Munich. 32. Chhabra, R. P. and Richardson, J. F. (1999). ‘‘Non-Newtonian Flow in the Process Industries’’. Butterworth-Heinemann, Oxford. 33. Coulson, J. M. and Richardson, J. F. (1999). In ‘‘Chemical Engineering’’, Vol. 1, 6th edn., Butterworth-Heinemann, Oxford. 34. Bird, R. B., Armstrong, R. C., and Hassager, O. (1987). Dynamics of Polymeric Liquids Vol. 1: Fluid Dynamics, 2nd edn. Wiley, New York. 35. Steffe, J. (1992). ‘‘Rheological Methods in Food Process Engineering’’. Freeman Press, East Lansing, MI. 36. Laba, D. (1993). ‘‘Rheological Properties of Cosmetics and Toiletries’’. Marcel-Dekker, New York. 37. Slattery, J. C. (1972). ‘‘Momentum, Energy and Mass Transfer in Continua’’. McGraw Hill, New York. 38. Skelland, A. H. P. (1967). ‘‘Non-Newtonian Flow and Heat Transfer’’. Wiley, New York. 39. Siginer, D., DeKee, D., and Chhabra, R. P. (1999). ‘‘Advances in the Flow and Rheology of Non-Newtonian Fluids’’. Elsevier, Amsterdam. 40. Tanner, R. I. (2000). ‘‘Engineering Rheology’’, 2nd edn. Oxford University Press, New York. 40a. Mezaki, R., Mochizuki, M., and Ogawa, K. (1999). ‘‘Engineering Data on Mixing’’. Elsevier, Amsterdam. 41. Edwards, M. F., Godfrey, J. C., and Kashani, M. M. (1976). Power requirement for the mixing of thixotropic liquids. J. Non-Newt. Fluid Mech. 1, 309–322. 41a. Barnes, H. A. (1999). The yield stress—A review. J. Non-Newt. Fluid Mech. 81, 133–167. 42. Chhabra, R. P. (1993). In ‘‘Bubbles, Drops and Particles in Non-Newtonian Fluids’’ pp. 112–113. CRC Press, Boca Raton, FL. 43. Nguyen, Q. D. and Boger, D. V. (1992). Measuring the flow properties of yield stress fluids. Annu. Rev. Fluid Mech. 24, 47–72. 44. Lin, C.-X. and Ko, S.-Y. (1995). Effects of temperature and concentration on the steady shear properties of aqueous solutions of carbopol and carboxymethyl cellulose. Int. Comm. Heat Mass Transfer 22, 157–169. 45. Chhabra, R. P. (1999). Heat and mass transfer in rheologically complex systems. In ‘‘Advances in the Flow and Rheology of Non-Newtonian Fluids’’ (D. Siginer, D. DeKee, and R. P. Chhabra, eds.), Part B, 1435–1488. Elsevier, Amsterdam. 46. Manoliadis, O. and Bishop, P. L. (1984). Temperature effect on rheology of sludges. J. Environ. Eng. (ASCE) 110, 286–290. 47. DeKee, D. and Chhabra, R. P. (1994). Elastic modulus and yield stress of suspensions. Rheol. Acta 33, 238–240. 48. Charpentier, J.-C. (1981). Mass-transfer rates in gas–liquid absorbers and reactors. Adv. Chem. Eng. 11, 1–133. 49. Aiba, S. and Okamoto, R. (1965). Power requirements of agitation and capacity coefficient of mass transfer in bubble aeration. Chem. Eng. (Japan) 43(8), 609–616. 50. Aunins, J. G., Woodson, B. A., Jr., Hale, T. K., and Wang, D. I. C. (1989). Effects of paddle impeller geometry on power input and mass transfer in small-scale animal cell culture vessels. Biotechnol. Bioeng. 34, 1127–1132. 51. Galindo, E. (1994). Aspects of the process for xanthan production. Trans. IChemE 72C, 227–237. 52. Galindo, E. and Nienow, A. W. (1992). Mixing of highly viscous simulated xanthan fermentation broths with the Lightnin A-315 impeller. Biotechnol. Prog. 8, 233–239. 53. Galindo, E. and Nienow, A. W. (1993). Performance of the scaba 6SRGT agitator in mixing of simulated Xanthan gum broths. Chem. Eng. Technol. 16, 102–108.
FLUID MECHANICS AND HEAT TRANSFER
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54. Galindo, E., Nienow, A. W., and Badhan, R. S. (1988). Mixing of simulated xanthan gum broths. In ‘‘Bioreactor Fluid Dynamics’’ (R. King, ed.), pp. 65–78. Elsevier, Amsterdam. 55. Hannote, M., Flores, F., Torres, L., and Galindo, E. (1991). Apparent yield stress estimation in xanthan gum solutions and fermentation broths using a low cost viscometer. Chem. Eng. J. 45, B49–B56. 56. Herbst, H., Schumpe, A., and Deckwer, W.-D. (1992). Xanthan production in stirred tank fermenters: Oxygen transfer and scale up. Chem. Eng. Technol. 15, 425–434. 57. Lee, Y. H. (1983). Aeration. In ‘‘Annual Reports on Fermentation Processes’’ (G. T. Tsao, ed.), Vol. 6, pp. 101–147. Academic Press, San Diego. 58. Moo-Young, M. and Blanch, H. W. (1981). Design of biochemical reactors: Mass transfer criteria for simple and complex systems. Adv. Biochem. Eng. 19, 1–69. 59. Nienow, A. W. (1998). Hydrodynamics of stirred bioreactors. Appl. Mech. Rev. 51(1), 3–32. 60. Oldshue, J. Y. (1983). Mixing in fermentation processes. In ‘‘Annual Reports on Fermentation Processes’’ (G. T. Tsao, ed.), Vol. 6, pp. 75–99. Academic Press, San Diego. 61. Reusss, M. (1994). Stirred tank bioreactors. In ‘‘Bioreactor System Design’’ (J. A. Asenjo and J. C. Merchuk, eds.), pp. 207–255. Marcel Dekker, New York. 62. Sanchez, A., Martinez, A., Torres, L., and Galindo, E. (1992). Power consumption of three impeller combinations in mixing xanthan fermentation broths. Process Biochem. 27, 351–365. 63. Serrano-Carreon, L. and Galindo, E. (1997). Studies on cavern development in mixing a yield stress fluid in a pilot-scale proto-fermenter. Recent Progres en Genie des Procedes 11(51), 161–168. 64. Tecante, A., Brito de la Fuente, E., Choplin, L., and Tanguy, P. A. (1996). Oxygen transfer in non-Newtonian fluids stirred with a helical ribbon screw impeller. In ‘‘Advances in Engineering Fluid Mechanics: Multiphase Reactor Polymerisation System Hydrodynamics’’ (N. P. Cheremisinoff, ed.), pp. 431–453. Gulf, Houston. 65. Velasco, D., Martinez, A., and Galindo, E. (1994). Hydrodynamics of industrial fermentations: Rheology and power draw in Rifamycin production. AIChE Symp. Ser. 90(299), 157–165. 66. Vlaev, D., Mann, R., Lossev, V., Vlaev, S. D., Zahradnik, J., and Seichter, P. (2000). Macro-mixing and streptomyces fradiae. Trans. IChemE 78A, 354–362. 67. Wachsen, O., Lange, C., and Reichert, J.-H. (1995). Zur on-line-bestimmung der viskositat von nicht-newtonschen biosuspensionen in geruhrten fermentern. Chem. Ing. Tech. 68, 268–272. 68. Yoshida, F. (1982). Aeration and mixing in fermentation. In ‘‘Annual Reports on Fermentation Processes’’ (G. T. Tsao, ed.), Vol. 5, pp. 1–34. Academic Press, San Diego. 69. Porter, J. E. (1971). Heat transfer at low Reynolds numbers (highly viscous liquids in laminar flow). Trans. IChemE 49, 1–29. 70. Cho, Y. I. and Hartnett, J. P. (1982). Non-Newtonian fluids in circular pipe flows. Adv. Heat Transfer 15, 59–143. 71. Paul, D. D. and Abdel-Khalik, S. I. (1983). Nucleate boiling in drag reducing polymer solutions. J. Rheol. 27, 59–76. 72. Prud’homme, R. K. and Long, R. E. (1983). Surface tensions of concentrated xanthan and polyacrylamide solutions with added surfactants. J. Colloid Interfacial Sci. 93, 274–276. 73. Irvine, T. F., Jr., Kim, I., Cho, K., and Gori, F. (1987). Experimental measurements of isobaric thermal expansion coefficients of non-Newtonian fluids. Exp. Heat Transfer 1, 155–162. 74. Hu, R. Y. Z., Wang, A. T. A., and Hartnett, J. P. (1991). Surface tension measurement of aqueous polymer solutions. Exp. Thermal Fluid Sci. 4, 723–729. 75. Ishiguro, S. and Hartnett, J. P. (1992). Surface tension of aqueous polymer solutions. Int. Comm. Heat Mass Transfer 19, 285–295.
164
R. P. CHHABRA
76. Miyairi, Y. and Mitsuishi, N. (1976). Measurements of thermal properties of aqueous polymer solutions. Heat Transfer-Japanese Res. 5(4), 26–29. 77. Loulou, T., Peerhossaini, H., and Bardon, J. P. (1992). Etude experimentale de la conductivite thermique de fluides non-Newtoniens sous cisaillement application aux solutions de carbopol 940. Int. J. Heat Mass Transfer 35, 2557–2562. 78. Lee, D.-L. and Irvine, T. F., Jr. (1997). Shear rate dependent thermal conductivity measurements of non-Newtonian fluids. Exp. Thermal Fluid Sci. 15, 16–24. 78a. Lee, W. Y., Cho, Y. I., and Hartnett, J. P. (1981). Thermal conductivity of non-Newtonian fluids. Lett. Heat Mass Transfer 8, 255–259. 79. Orr, C., Jr. and Dallavalle, J. M. (1954). Heat transfer properties of liquid–solid suspensions. Chem. Eng. Prog. Sym. Ser. 59(9), 29–45. 80. Tareef, B. M. (1940). Colloid J. (USSR) 6, 545. 80a. Shin, S. (1996). The effect of the shear rate-dependent thermal conductivity of nonNewtonian fluids on the heat transfer in a pipe flow. Int. Comm. Heat Mass Transfer 23, 665–678. 81. Bruggemann, D. A. G. (1935). Berechnung verschiederner physikalischer konstanten von heterogenen substanzen. Ann. Physik 24, 636–679. 82. Rajaiah, J., Andrews, G., Ruckenstein, E., and Gupta, R. K. (1992). Thermal conductivity of concentrated, sterically stabilised suspensions. Chem. Eng. Sci. 47, 3863–3868. 83. Dutta, A. and Mashelkar, R. A. (1987). Thermal conductivity of structured liquids. Adv. Heat Transfer 18, 161–239. 84. Metzner, A. B. and Otto, R. E. (1957). Agitation of non-Newtonian fluids. AIChE J. 3, 3–10. 85. Metzner, A. B. and Taylor, J. S. (1960). Flow patterns in agitated vessels. AIChE J. 6, 109–114. 86. Norwood, K. W. and Metzner, A. B. (1960). Flow pattern and mixing rates in agitated vessels. AIChE J. 6, 432–437. 87. Metzner, A. B., Feehs, R. H., Ramos, H. L., Otto, R. E., and Tuthill, J. D. (1961). Agitation of viscous Newtonian and non-Newtonian fluids. AIChE J. 7, 3–9. 88. Calderbank, P. H. and Moo-Young, M. B. (1959). The prediction of power consumption in the agitation of non-Newtonian fluids. Trans. IChemE 37, 26–33. 89. Nagata, S., Yanagimoto, M., and Yokoyama, T. (1957). A study on the mixing of highviscosity liquid. Kagaku Kogaku 21, 278–286. 90. Nagata, S., Nishikawa, M., Gotoh, S., and Nakajima, M. (1970). Power consumption of helical mixer for the mixing of highly viscous liquid. Kagaku Kogaku 34, 1115–1117. 91. Nagata, S., Nishikawa, M., Tada, H., and Gotoh, S. (1971). Power consumption of mixing impellers in pseudo-plastic liquids. J. Chem. Eng. Jpn. 4, 72–76. 92. Nagata, S., Nishikawa, M., Takimoto, T., Kida, F., and Kayama, T. (1971). Jacket side heat transfer coefficient in mixing vessel. Kagaku Kogaku 35, 924. 93. Nagata, S., Nishikawa, M., and Takimoto, T. (1971). Heat transfer to helical coils immersed in mixing vessel. Kagaku Kogaku 35, 1028. 94. Nagata, S., Nishikawa, M., and Katsube, T. (1971). Mixing of highly viscous nonNewtonian liquids. Kagaku Kogaku 35(7), 794–800. 95. Nagata, S., Nishikawa, M., Kayama, T., and Nakajima, M. (1972). Heat transfer to cooling coil acting as rotating impeller in highly viscous liquids. J. Chem. Eng. Jpn. 5, 187–192. 96. Nagata, S., Nishikawa, M., and Kayama, T. (1972). Heat transfer to vessel wall by helical ribbon impeller in highly viscous liquids. J. Chem. Eng. Jpn. 5, 83–85. 97. Nagata, S., Nishikawa, M., Itaya, M., and Ashiwake, K. (1975). Study of heat transfer for aerated mixing vessel and aerated tower. Kagaku Kogaku Ronbunshu 1, 460. 98. Nagata, S., Nishikawa, M., Hattori, M., and Kayama, T. (1976). Heat transfer to halfellipsoidal impeller in highly viscous non-Newtonian liquids. Heat Transfer-Japanese Res. 5(1), 15–25.
FLUID MECHANICS AND HEAT TRANSFER
165
99. Godleski, E. S. and Smith, J. C. (1962). Power requirements and blends times in the agitation of pseudoplastic fluids. AIChE J. 6, 617–620. 100. Chapman, F. S. and Holland, F. A. (1965). A study of turbine and helical-screw agitators in liquid mixing. Trans. IChemE 43, 131–140. 101. Gluz, M. D. and Pavlushenko, I. S. (1966). Heat transfer during mixing of non-Newtonian liquids. Zh. Prikl. Khim 39(11), 2475–2483. 102. Gluz, M. D. and Pavlushenko, I. S. (1966). Homogenizing time during mixing of nonNewtonian liquids. Zh. Prikl. Khim 39(12), 2719–2724. 103. Pavlushenko, I. S. and Gluz, M. D. (1966). Dimensionless equation of transfer processes in mixing of non-Newtonian liquids. Zh. Prikl. Khim 39(10), 2288–2295. 104. Beckner, J. L. and Smith, J. M. (1966). Anchor-agitated: Power input with Newtonian and pseudo-plastic fluids. Trans. IChemE 44, 224–236. 105. Peters, D. C. and Smith, J. M. (1967). Fluid flow in the region of anchor agitator blades. Trans. IChemE 45, 360–366. 106. Peters, D. C. and Smith, J. M. (1969). Mixing in anchor agitated vessels. Can. J. Chem. Eng. 47, 268–273. 107. Mizushina, T., Ito, R., Murakami, Y., and Kiri, Y. (1966). Experimental study of the heat transfer to the cooling coil in an agitated vessel. Kagaku Kogaku 30, 827. 108. Mizushina, T., Murakami, Y., and Tanaka, S. (1966). Experimental study of the heat transfer of non-Newtonian fluids to the wall of an agitated vessel. Kagaku Kogaku 30(9), 819–826. 109. Mizushina, T., Ito, R., Hiraoka, S., Ikuta, Y., Konaka, Y., and Watanabe, J. (1967). Transport phenomena at the wall of agitated vessel—Heat Transfer. Kagaku Kogaku 31, 1208–1212. 110. Mizushina, T., Ito, R., Hiraoka, S., and Watanabe, J. (1970). Uniformalization of temperature field in agitated reactors. Kagaku Kogaku 34(11), 1205–1212. 111. Hagedorn, D. and Salamone, J. J. (1967). Batch heat transfer coefficients for pseudoplastic fluids in agitated vessels. Ind. Eng. Chem. Process Des. Dev. 6, 469–475. 112. Hall, K. R. and Godfrey, J. C. (1968). The mixing rates of highly viscous Newtonian and non-Newtonian fluids in a laboratory sigma-blade mixer. Trans. IChemE 46, 205–212. 113. Hall, K. R. and Godfrey, J. C. (1970). Power consumption by helical ribbon impellers. Trans. IChemE 48, 201. 114. Bourne, J. R. and Bulter, H. (1969). An analysis of the flow produced by helical ribbon impellers. Trans. IChemE 47, 11–17. 115. Bourne, J. R. and Butler, H. (1969). Power consumption of helical ribbon impellers in viscous liquids. Trans. IChemE 47, 263–270. 116. Hoogendoorn, C. J. and den Hertog, A. P. (1967). Model studies on mixers in the viscous flow region. Chem. Eng. Sci. 22, 1689–1699. 117. Skelland, A. H. P. and Dimmick, G. R. (1969). Heat transfer between coils and nonNewtonian fluids with propeller agitation. Ind. Eng. Chem. Process Des. Dev. 8, 267–274. 118. Mitsuishi, N. and Hirai, N. (1969). Power requirements in the agitation of non-Newtonian fluids. J. Chem. Eng. Jpn. 2, 217–224. 119. Mitsuishi, N., Miyairi, Y., and Katamine, T. (1973). Heat transfer to Newtonian liquids in an agitated vessel. J. Chem. Eng. Jpn. 6, 409. 120. Coyle, C. K., Hirschland, H. E., Michel, B. J., and Oldshue, J. Y. (1970). Heat transfer to jackets with close clearance impellers in viscous materials. Can. J. Chem. Eng. 48, 275–278. 121. O’Shima, E. and Yuge, K. (1970). Circulation time distribution of viscous liquids in a mixing vessel with helical ribbon type, anchor type and helical screw type impellers. Kagaku Kogaku 34, 779–785.
166
R. P. CHHABRA
122. Martone, J. A. and Sandall, O. C. (1971). Heat transfer to Bingham plastic slurries in agitated vessels. Ind. Eng. Chem. Process Des. Dev. 10, 86–91. 123. Perez, J. E. and Sandall, O. C. (1974). Gas absorption by non-Newtonian fluids in agitated vessels. AIChE J. 20, 770–775. 124. Sandall, O. C. and Patel, K. G. (1970). Heat transfer to non-Newtonian pseudoplastic fluids in agitated vessels. Ind. Eng. Chem. Process Des. Dev. 9, 139–144. 125. Rieger, F. and Novak, V. (1972). Scale-up method for power consumption of agitators in the creeping flow regime. Chem. Eng. Sci. 27, 39. 126. Rieger, F. and Novak, V. (1973). Power consumption of agitators in highly viscous nonNewtonian liquids. Trans. IChemE 51, 105–111. 127. Novak, V. and Rieger, F. (1975). Homogenization efficiency of helical ribbon and anchor agitators. Chem. Eng. J. 9, 63–70. 128. Netusil, J. and Rieger, F. (1993). Power consumption of screw and helical ribbon agitators in highly viscous pseudoplastic fluids. Chem. Eng. J. 52, 9–12. 129. Chavan, V. V. and Ulbrecht, J. (1972). Power correlation for helical ribbon impellers in inelastic non-Newtonian fluids. Chem. Eng. J. 3, 308–311. 130. Chavan, V. V. and Ulbrecht, J. (1973). Internal circulation in vessels agitated by screw impellers. Chem. Eng. J. 6, 213–223. 131. Chavan, V. V. and Ulbrecht, J. (1973). Power correlation for close clearance helical impellers in non-Newtonian liquids. Ind. Eng. Chem. Process Des. Dev. 12, 472. 132. Chavan, V. V. and Ulbrecht, J. (1973). Power correlation for off-centered helical screw impellers in highly viscous Newtonian and non-Newtonian liquids. Trans. IChemE 51, 349. 133. Chavan, V. V., Jhaveri, A. S., and Ulbrecht, J. (1972). Power consumption for mixing of inelastic non-Newtonian fluids by helical screw agitators. Trans. IChemE 50, 147–155. 134. Chavan, V. V., Ford, D. E., and Arumugam, M. (1975). Influence of fluid rheology on circulation, mixing and blending. Can. J. Chem. Eng. 53, 628–635. 135. Chavan, V. V. (1983). Close clearance helical impellers: A physical model for Newtonian liquids at low Reynolds number. AIChE J. 29, 177–186. 136. Chavan, V. V. and Mashelkar, R. A. (1980). Mixing of viscous Newtonian and nonNewtonian fluids. Adv. Transport Process 1, 210–252. 137. Edney, H. G. S. and Edwards, M. F. (1976). Heat transfer to non-Newtonian and aerated fluids in stirred tanks. Trans. IChemE 54, 160–166. 138. Ford, D. E. and Ulbrecht, J. J. (1975). Blending of polymer solutions with different rheological properties. AIChE J. 21, 1230–1233. 139. Ford, D. E. and Ulbrecht, J. J. (1976). Influence of rheological properties of polymer solutions upon mixing and circulation times. Ind. Eng. Chem. Process Des. Dev. 15, 321–326. 140. Ford, D. E., Mashelkar, R. A., and Ulbrecht, J. J. (1976). Mixing times in Newtonian and non-Newtonian fluids. Process Technol. Int. 7, 803–807. 141. Kale, D. D., Mashelkar, R. A., and Ulbrecht, J. (1974). High speed agitation of nonNewtonian fluids: Influence of elasticity and fluid inertia. Chem. Ing. Tech. 46(1), 69. 142. Quaraishi, A. Q., Mashelkar, R. A., and Ulbrecht, J. (1976). Torque suppression in mechanically stirred liquids and multiphase liquid systems. J. Non-Newt. Fluid Mech. 1, 223–245. 143. Quaraishi, A. Q., Mashelkar, R. A., and Ulbrecht, J. (1977). Influence of drag reducing additives on mixing and dispersing in agitated vessels. AIChE J. 23, 487–492. 144. Yagi, H. and Yoshida, F. (1975). Gas absorption by Newtonian and non-Newtonian fluids in sparged agitated vessels. Ind. Eng. Chem. Process Des. Dev. 14, 488–492. 145. Sawinsky, J., Havas, G., and Deak, A. (1976). Power requirement of anchor and helical ribbon impellers for the case of agitating Newtonian and pseudo-plastic liquids. Chem. Eng. Sci. 31, 507–509.
FLUID MECHANICS AND HEAT TRANSFER
167
146. Hiraoka, S., Yamada, I., and Mizoguchi, K. (1979). Two-dimensional model analysis of flow behaviour of highly viscous non-Newtonian fluid in agitated vessel with paddle impeller. J. Chem. Eng. Jpn. 12, 56. 147. Hocker, H. and Langer, G. (1977). Zum Leistungsverhalten begaster Ruhrer in Newtonschen und nicht-newtonschen flussigkeiten. Rheol. Acta 16, 400–412. 148. Hocker, H., Langer, G., and Werner, U. (1981). Power consumption of stirrers in nonNewtonian liquids. Ger. Chem. Eng. 4, 113–123. 149. Ranade, V. R. and Ulbrecht, J. (1978). Influence of polymer additives on the gas–liquid mass transfer in stirred tanks. AIChE J. 24, 796–802. 150. White, J. L., Chankraiphon, S., and Ide, Y. (1977). Rheological behaviour and flow patterns around agitators in polymer solutions. J. Appl. Polym. Sci. 21, 1–18. 151. Ide, Y. and White, J. L. (1974). Rheological phenomena in polymerization reactors: Rheological properties and flow patterns around agitators in polystyrene–styrene solutions. J. Appl. Polym. Sci. 18, 2997–3018. Also see ibid 19, 719–727. 152. Rautenbach, R. and Bollenrath, F. M. (1979). Heat transfer in stirred vessels to highviscosity Newtonian and non-Newtonian substances. Ger. Chem. Eng. 2, 18–24. 153. De Maerteleire, E. (1978). Heat transfer to a helical cooling coil in mechanically agitated gas–liquid dispersions. Chem. Eng. Sci. 33, 1107–1113. 154. Carreau, P. J., Guy, C., and Corneille, J. L. (1966). Heat transfer to agitated nonNewtonian fluids. Can. J. Chem. Eng. 44, 3–8. 155. Carreau, P. J., Patterson, W. I., and Yap, C. Y. (1976). Mixing of viscoelastic fluids with helical-ribbon agitators I—Mixing time and flow pattern. Can. J. Chem. Eng. 54, 135–142. 156. Patterson, W. I., Carreau, P. J., and Yap, C. Y. (1979). Mixing with helical ribbon agitators Part II: Newtonian fluids. AIChE J. 25, 508–516. 157. Yap, C. Y., Patterson, W. I., and Carreau, P. J. (1979). Mixing with helical ribbon agitators Part III. Non-Newtonian fluids. AIChE J. 25, 516–521. 158. Guerin, P., Carreau, P. J., Patterson, W. I., and Paris, J. (1984). Characterization of helical impellers by circulation times. Can. J. Chem. Eng. 62, 301–309. 159. Carreau, P. J., Paris, J., and Guerin, P. (1992). Mixing of Newtonian and non-Newtonian liquids: Screw agitator and draft coil system. Can. J. Chem. Eng. 70, 1071–1082. 160. Carreau, P. J., Chhabra, R. P., and Cheng, J. (1993). Effect of rheological properties on power consumption with helical ribbon agitators. AIChE J. 39, 1421–1430. 161. Carreau, P. J., Paris, J., and Guerin, P. (1994). Heat transfer to Newtonian and nonNewtonian liquids in a screw agitator and draft coil system. Can. J. Chem. Eng. 72, 966–974. 162. Cheng, J., Carreau, P. J., and Chhabra, R. P. (1995). On the effect of wall and bottom clearance on mixing viscoelastic fluids. AIChE Sym. Ser. 91(304), 115–122. 163. Nishikawa, M., Kamata, N., and Nagata, S. (1976). Heat transfer for highly viscous liquids in mixing vessel. Heat Transfer-Japanese Res. 5(4), 84–92. 164. Nishikawa, M., Kunioka, S., Fujieda, S., and Hashimoto, K. (1982). Heat transfer to nonNewtonian liquid in aerated mixing vessel. Kagaku Kogaku Ronbunshu 8, 494. 165. Nishikawa, M., Nakamura, M., and Hashimoto, K. (1981). Gas absorption in aerated mixing vessels with non-Newtonian liquid. J. Chem. Eng. Jpn. 14, 227. 166. Poggemann, R., Steiff, A., and Weinspach, P.-M. (1980). Heat transfer in agitated vessels with single-phase liquids. Ger. Chem. Eng. 3, 163–174. 167. Solomon, J., Elson, T. P., Nienow, A. W., and Pace, G. W. (1981). Cavern sizes in agitated fluids with a yield stress. Chem. Eng. Commun. 11, 143–164. 168. Prud’homme, R. K. and Shaqfeh, E. (1984). Effect of elasticity on mixing torque requirements for Rushton turbine impellers. AIChE J. 30, 485–486. 169. Collias, D. J. and Prud’homme, R. K. (1985). The effect of fluid elasticity on power consumption and mixing times in stirred tanks. Chem. Eng. Sci. 40, 1495–1505.
168
R. P. CHHABRA
170. Ayazi Shamlou, P. and Edwards, M. F. (1985). Power consumption of helical ribbon mixers in viscous Newtonian and non-Newtonian fluids. Chem. Eng. Sci. 40, 1773–1781. 171. Ayazi Shamlou, P. and Edwards, M. F. (1986). Heat transfer to viscous Newtonian and non-Newtonian fluids for helical ribbon mixers. Chem. Eng. Sci. 41, 1957–1967. 172. Bertrand, J. and Couderc, J. P. (1982). Agitation de fluides pseudoplastiques par un agitateur bipale. Can. J. Chem. Eng. 60, 738–747. 173. Bertrand, J. and Couderc, J. P. (1985). Etude numerique des ecoulements generes par une ancre dans le cas de fluides visqueux, Newtoniens ou pseudoplastiques. Entropie 125/126, 48–59. 174. Bertrand, J. and Couderc, J. P. (1985). Evaluation of the power consumption in agitation of viscous Newtonian or pseudoplastic liquids by two bladed, anchor or gate agitators. Trans. IChemE 63, 259–263. 175. Elson, T. P. (1990). X-ray flow visualisation of flow patterns during the mixing of yield stress, Newtonian and dilatant fluids. Chem. Eng. Commun. 94, 143–158. 176. Elson, T. P., Cheesman, D. J., and Nienow, A. W. (1986). X-ray studies of cavern sizes and mixing performance with fluids possessing of yield stress. Chem. Eng. Sci. 41, 2555–2562. 177. Ismail, A. F., Nagase, Y., and Imon, J. (1984). Power characteristics and cavity formation in aerated agitations. AIChE J. 30, 487–489. 178. Nienow, A. W., Wisdom, D. J., Solomon, J., Machon, V., and Vlcek, J. (1983). The effect of rheological complexities on power consumption in an aerated, agitated vessel. Chem. Eng. Commun. 19, 273–293. 179. Kuboi, R. and Nienow, A. W. (1986). Intervortex mixing rates in high-viscosity liquids agitated by high speed dual impellers. Chem. Eng. Sci. 41, 123. 180. Wichterle, K. and Wein, O. (1981). Threshold of mixing of non-Newtonian liquids. Int. Chem. Eng. 21, 116–120. 181. Wichterle, K., Kadlec, M., Zak, L., and Mitschka, P. (1984). Shear rates on turbine impeller blades. Chem. Eng. Commun. 26, 25–32. 182. Desplanches, H., Llinas, J. R., and Chevalier, J.-L. (1980). Transferts thermiques entre un serpentin et des fluides newtoniens ou non-newtoniens agites par une turbine a pales inclines dans une cuve. Can. J. Chem. Eng. 58, 160–170. 183. Desplanches, H., Bruxelmane, M., and Chevalier, J.-L. (1990). Transfert de chaleur entre un serpentin et une solution agitee Newtonienne en regime intermediate—Grandeurs caracteristiques, methodes d’estimation et criteres d’extrapolation. Entropie 26, 3–13. 184. Desplanches, H., Gaston-Bonhomme, Y., and Chevalier, J. L. (1994). Boiling of viscous liquids in a stirred tank. Part 1. Newtonian liquids. Int. Chem. Eng. 34, 225–233. Also see ibid 234–239. 185. Kuriyama, M., Arai, K., and Saito, S. (1983). Mechanism of heat transfer to pseudoplastic fluids in an agitated tank with helical ribbon impeller. J. Chem. Eng. Jpn. 16, 489–494. 186. Kamiwano, M., Saito, E., and Kaminoyama, M. (1988). Flow pattern and apparent viscosity of pseudoplastic liquid in a stirred vessel. Kagaku Kogaku Ronbunshu 14, 316. 187. Etchells, A. W., III, Ford, W. N., and Short, D. G. R. (1987). Mixing of Bingham Plastics on an Industrial Scale. I. Chem. E. Symp. Ser.: Fluid Mixing III, No. 108, 1–10. 188. Kai, W. and Shengyao, Y. (1989). Heat transfer and power consumption of nonNewtonian fluids in agitated vessels. Chem. Eng. Sci. 44, 33–40. 189. Zeppenfeld, R. and Mersmann, A. B. (1988). A new fluid dynamic model for mixing of Newtonian and power-law liquids in the transient regime. Chem. Eng. Technol. 11, 162–170. 190. Koloni, T., Plazl, I., and Zumer, M. (1989). Power consumption, gas holdup and interfacial area in aerated non-Newtonian suspensions in stirred tanks of square crosssection. Chem. Eng. Res. Des. 67, 526.
FLUID MECHANICS AND HEAT TRANSFER
169
191. Galindo, E., Arguello, M. A., Velasco, D., Albiter, V., and Martinez, A. (1996). A comparison of cavern development in mixing a yield stress fluid by Rushton and Intermig impellers. Chem. Eng.Technol. 19, 315–323. 192. Kaminoyama, M., Saito, E., and Kamiwano, M. (1990). Flow analogy of pseudoplastic liquid in geometrically similar stirred vessels based on numerical analysis. J. Chem. Eng. Jpn. 23, 214. 193. Kaminoyama, M., Akabane, K., Arai, K., Saito, F., and Kamiwano, M. (1994). Numerical analysis of flow of a Bingham fluid in an anchor impeller. Int. Chem. Eng. 34, 263–269. 194. Jomha, A. I., Edwards, M. F., and Woodcock, L. V. (1990). New method for predicting the power requirement for mixing shearthickening suspensions. Chem. Eng. Sci. 45, 1389–1396. 195. Sestak, J., Zitny, R., and Houska, M. (1986). Anchor agitated systems: Power input correlation for pseudoplastic and thixotropic fluids in equilibrium. AIChE J. 32, 155–158. 196. Wang, K. and Yu, S. (1989). Heat transfer and power consumption of non-Newtonian fluids in agitated vessels. Chem. Eng. Sci. 44, 33. 197. Sinevic, V., Kuboi, R., and Nienow, A. W. (1986). Power numbers, Taylor numbers and Taylor vortices in viscous Newtonian and non-Newtonian fluids. Chem. Eng. Sci. 41, 2915–2923. 198. Oliver, D. R., Nienow, A. W., Mitson, R. J., and Terry, K. (1984). Power consumption in the mixing of Boger fluids. Trans. IChemE 62, 123–127. 199. Shervin, C. R., Raughley, D. A., and Romaszewski, R. A. (1991). Flow visualization and scale up studies for the mixing of viscoelastic fluids. Chem. Eng. Sci. 46, 2867–2873. 200. Skelland, A. H. P. and Kanel, J. S. (1990). Minimum impeller speeds for complete dispersion of non-Newtonian liquid–liquid systems in baffled vessels. Ind. Eng. Chem. Res. 29, 1300–1306. 201. Takahashi, K., Arai, K., and Saito, S. (1980). Power correlation for anchor and helical ribbon impellers in highly viscous liquids. J. Chem. Eng. Jpn. 13, 147–150. 202. Takahashi, K., Arai, K., and Saito, S. (1982). An extended power correlation for anchor and helical ribbon impellers. J. Chem. Eng. Jpn. 15, 77–79. 203. Takahashi, K., Sasaki, M., Arai, K., and Saito, S. (1982). Effects of geometrical variables of helical ribbon impellers on mixing of highly viscous Newtonian liquids. J. Chem. Eng. Jpn. 15, 217–224. 204. Takahashi, K., Yokota, T., and Konno, H. (1984). Power consumption of helical ribbon agitators in highly viscous pseudoplastic liquids. J. Chem. Eng. Jpn. 17, 657–659. 205. Takahashi, K., Yokota, T., and Konno, H. (1988). Mixing of pseudoplastic liquid in an agitated vessel equipped with a variety of helical ribbon impellers. J. Chem. Eng. Jpn. 21, 63–68. 206. Takahashi, K. (1988). Mixing of non-Newtonian liquids with helical ribbon impellers. In ‘‘Encyclopedia of Fluid Mechanics’’ (N. P. Cheremisinoff, ed.), Vol. 7, pp. 869–886. Gulf, Houston. 207. Tran, Q. K., Horsley, R. R., Reizes, J. A., and Ang, H. M. (1992). The mixing of pseudoplastic yield stress slurries. Proc. 11th Int. Cong. Rheol., Brussels, 631–633. 208. Tanguy, P. A., Lacroix, R., Bertrand, F., Choplin, L., and Brito de la Fuente, E. (1992). Finite element analysis of viscous mixing with a helical ribbon-screw impeller. AIChE J. 38, 939–944. 209. Tanguy, P. A., Bertrand, F., and Brito de la Fuente, E. (1994). Mixing of Viscoplastic Fluids with Anchor Impellers. I. Chem. E. Symp. Ser. No. 136, 525–532. 210. Tanguy, P. A., Bertrand, F., Labrie, R., and Brito de la Fuente, E. (1996). Numerical modelling of the mixing of viscoplastic slurries in a twin-blade planetary mixer. Trans. IChemE 74A, 499–504.
170
R. P. CHHABRA
211. Tanguy, P. A., Thibault, F., and Birto de la Fuente, E. (1996). A new investigation of the Metzner-Otto concept of anchor mixing impellers. Can. J. Chem. Eng. 74, 222–228. 212. Tanguy, P. A., Thibault, F., Brito-De La Fuente, E., Espionosa-solares, T., and Tecante, A. (1997). Mixing performance induced by coaxial flat blade-helical ribbon impellers rotating at different speeds. Chem. Eng. Sci. 52, 1733–1741. 213. Tanguy, P. A. and Brito de la Fuente, E. (1999). Non-Newtonian mixing with helical ribbon impellers and planetary mixers. In ‘‘Advances in the Flow and Rheology of NonNewtonian Fluids’’ (D. A. Siginer, D. DeKee, and R. P. Chhabra, eds.), Part A, pp. 301– 330. Elsevier, Amsterdam. 214. Tanguy, P. A., Thibault, F., Dubois, C., and Ait-Kadi, A. (1999). Mixing hydrodynamics in a double planetary mixer. Trans. IChemE 77A, 318–324. 215. Espinosa-Solares, T., Brito de la Fuente, E., Tecante, A., and Tanguy, P. A. (1997). Power consumption of a dual turbine-helical ribbon impeller mixer in ungassed conditions. Chem. Eng. J. 67, 215–219. 216. Bertrand, F., Tanguy, P. A., and Brito-de la Feunte, E. (1996). A new perspective for the mixing of yield stress fluids with anchor impellers. J. Chem. Eng. Jpn. 29, 51–58. 217. Bertrand, F., Tanguy, P. A., Brito de la Fuente, E., and Carreau, P. (1999). Numerical modeling of the mixing flow of second order fluids with helical ribbon impellers. Comput. Methods Appl. Mech. Eng. 180, 267–280. 218. Brito de la Fuente, E., Choplin, L., and Tanguy, P. A. (1997). Mixing with helical ribbon impellers. Trans. IChemE 75A, 45–52. 219. Brito de la Fuente, E., Espinosa, T., Tecante, A., Thibault, F., and Tanguy, P. A. (1997). Mixing non-Newtonian fluids under gassed and ungassed conditions with a double geometry impeller. Recente Progres en Genie des Procedes 11(51), 153–160. 220. Brito de la Fuente, E., Leuliet, J. C., Choplin, L., and Tanguy, P. A. (1992). On the effect of shearthinning behaviour on mixing with a helical ribbon impeller. AIChE Symp. Ser. 88(286), 28–32. 221. Brito de la Fuente, E., Nunez, M. C., and Tanguy, P. A. (1997). Non-isothermal mixing of rheologically complex fluids with close-clearance impellers: Effect of natural convection. Chem. Eng. Technol. 20, 203–207. 222. Dubois, C., Thibault, F., Tanguy, P. A., and Ait-Kadi, A. (1996). Characterization of viscous mixing in a twin intermeshing conical helical mixer. Fluid Mixing 5, 249–258. 223. Zhou, G., Tanguy, P. A., and Dubois, C. (2000). Power consumption in a double planetary mixer with non-Newtonian and viscoelastic materials. Trans. IChemE 78A, 445–453. 224. Rauline, D., Le Blevec, J.-M., Bousquet, J., and Tanguy, P. A. (2000). A comparative assessment of the performance of the kenics and SMX static mixers. Trans. IChemE 78A, 389–396. 225. Leuliet, J. C., Brito de la Fuente, E., and Choplin, L. (1992). Melanges de Fluides elastiques avec un agitateur a ruban helicoidal. Entropie 28(171), 53–58. 226. Amanullah, A., Hjorth, S. A., and Nienow, A. W. (1997). Cavern sizes generated in highly shearthinning viscous fluids by SCABA 3SHP1 impellers. Trans. IChemE 75C, 232–238. 227. Amanullah, A., Hjorth, S., and Nienow, A. W. (1998). A new mathematical model to predict cavern diameters in highly shearthinning power law fluids using axial flow impellers. Chem. Eng. Sci. 53, 455–469. 228. Hjorth, S. (1997). On the laminar to turbulent transition in mixing vessels. Recente Progres en Genie des Procedes 11(51), 43–50. 229. Bouwmans, I., Bakker, A., and van den Bakker, H. E. A. (1997). Blending liquids of differing viscosities and densities in stirred vessels. Trans. IChemE 75A, 777–783. 230. Foroquet-Murh, L. and Midoux, N. (1994). Nucleate boiling of non-Newtonian fluids in a stirred tank reactor. Chem. Eng. Process. 33, 459–464.
FLUID MECHANICS AND HEAT TRANSFER
171
231. Delaplace, G., Dieulot, J.-Y., Brienne, J.-P., and Leuliet, J.-C. (1999). Determination experimentale et prediction des temps de circulation et de melange pour un systeme d’agitation helicoidal. Can. J. Chem. Eng. 77, 447–457. 232. Delaplace, G., Torrez, C., Andre, C., Leuliet, J.-C., and Fillaudeau, L. (2000). ‘‘CFD Simulation of Foodstuff Flows in an Agitated Vessel’’. Proc. Ist Int. Conf. Simulation in Food & Bioindustries, published by the Society for Computer Simulation International, pp. 179–186. 233. Delaplace, G. and Leuliet, J.-C. (2000). A new expression of the Ks factor for helical ribbon agitators. Can. J. Chem. Eng. 78, 393–394. 234. Delaplace, G., Leuliet, J. C., and Relandeau, V. (2000). Circulation and mixing times for helical ribbon impellers. Review and experiments. Exp. Fluids 28, 170–182. 235. Delaplace, G., Leuliet, J.-C., and Ronse, G. (2000). Power requirement when mixng a shearthickening fluid with a helical ribbon impeller type. Chem. Eng. Technol. 23, 329–336. 236. Ozcan-Taskin, N. G. and Nienow, A. W. (1995). Mixing viscoelastic fluids with axial flow impellers: Flow fields and power consumption. Trans. IChemE 73C, 49–56. 237. Ruan, R., Almaer, S., Zou, C., and Chen, P. L. (1997). Spectrum analysis of mixing power curves for neural network prediction of dough rheological properties. Trans. ASAE 40, 677–681. 238. Reilly, C. D. and Burmester, S. S. H. (1994). Homogenization of liquids with different densities and viscosities. AIChE Symp. Ser. 90(299), 175–180. 239. Jaworski, Z., Pacek, A. W., and Nienow, A. W. (1994). On flow close to cavern boundaries in yield stress fluids. Chem. Eng. Sci. 49, 3321–3324. 240. Masiuk, S. and Lacki, H. (1993). Power consumption and mixing time for Newtonian and non-Newtonian liquids mixing in a ribbon mixer. Chem. Eng. J. 52, 13–17. 241. Moore, I. P. T., Cossor, G., and Baker, M. R. (1995). Velocity distribution in a stirred tank containing a yield stress fluid. Chem. Eng. Sci. 50, 2467. 242. Hirata, Y., Nienow, A. W., and Moore, I. P. T. (1991). LDA studies of velocity distributions and cavern sizes in a yield stress fluid agitated by a Rushton turbine. In ‘‘Proc. 7th European Mixing Conf.’’ (M. Bruxelmane and G. Froment, eds.), pp. 167–172. 243. Mavros, P., Xuereb, C., and Bertrand, J. (1996). Determination of 3-D flow fields in agitated vessels by laser-doppler velocimetry: Effect of impeller type and liquid viscosity on liquid flow pattern. Trans. IChemE 74A, 658. 244. Youcefi, A., Anne-Archard, D., Boisson, H. C., and Sengelin, M. (1997). On the influence of liquid elasticity on mixing in a vessel agitated by a two-bladed impeller. J. Fluids Eng. (ASME) 119, 616–622. 245. Shimizu, K., Minekawa, K., Hirose, T., and Kawase, Y. (1999). Drop breakage in stirred tanks with Newtonian and non-Newtonian fluid systems. Chem. Eng. J. 72, 117–124. 246. Torrez, C. and Andre, C. (1998). Power consumption of a Rushton turbine mixing Newtonian and shear-thinning fluids: Comparison between experimental and numerical results. Chem. Eng. Technol. 21, 599–604. 247. Torrez, C. and Andre, C. (1999). Simulation of a Rushton turbine mixing yield stress fluids: Application of the Metzner-Otto concept. Chem. Eng. Technol. 22, 701–706. 248. Wang, J.-J., Feng, L.-F., Gu, X.-P., Wang, K., and Hu, C.-H. (2000). Power consumption of inner–outer helical ribbon impellers in viscous Newtonian and non-Newtonian fluids. Chem. Eng. Sci. 55, 2339–2342. 249. Rai, C. L., Devotta, I., and Rao, P. G. (2000). Heat transfer to viscous Newtonian and non-Newtonian fluids using helical ribbon agitator. Chem. Eng. J. 79, 73–77. 250. Nouri, J. M. and Hockey, R. M. (1998). Power number correlation between Newtonian and non-Newtonian fluids in a mixing vessel. J. Chem. Eng. Jpn. 31, 848–852.
172
R. P. CHHABRA
251. Mishra, V. P., Kumar, P., and Joshi, J. B. (1998). Flow generated by a disc turbine in aqueous solutions of polyacrylamide. Chem. Eng. J. 71, 11–21. 252. Fangary, Y. S., Barigou, M., Seville, J. P. K., and Parker, D. J. (2000). Fluid trajectories in a stirred vessel of non-Newtonian liquid using positron emission particle tracking. Chem. Eng. Sci. 55, 5969–5979. 253. Vlaev, S. D., Gyurov, P., Vlaev, D., and Mann, R. (1997). Regime maps and a network of zones regime selection algorithm for gas–liquid mixing in complex-rheology fluids. Recent Progres en Genie des Procedes 11(51), 193–200. 254. Curran, S. J., Hayer, R. E., Afacan, A., Williams, M. C., and Tanguy, P. A. (2000). Experimental mixing study of a yield stress fluid in a laminar stirred tank. Ind. Eng. Chem. Res. 39, 195–202. 255. Bohme, G. and Stenger, M. (1988). Consistent scale up procedure for the power consumption in agitated non-Newtonian fluids. Chem. Eng. Technol. 11, 199–205. 256. Heim, A. (1980). Model of momentum and heat transfer in mixers with close-clearance agitators. Int. Chem. Eng. 20, 271–289. 257. Chowdhury, R. and Tiwari, K. K. (1979). Power consumption studies of helical ribbon screw mixers. Ind. Eng. Chem. Process Des. Dev. 18, 227–231. 258. Ducla, J. M., Desplanches, H., and Chevalier, J. L. (1983). Effective viscosity of nonNewtonian fluids in a mechanically stirred tank. Chem. Eng. Commun. 21, 29–36. 259. Pollard, J. and Kantyka, T. A. (1969). Heat transfer to agitated non-Newtonian fluids. Trans. IChemE 47, 21–27. 260. Ryan, D. F., Janssen, L. P. B. M., and van Dierendonck, L. L. (1988). Circulation time prediction in the scale up of polymerization reactors with helical ribbon agitators. Chem. Eng. Sci. 43, 1961–1966. 261. Ottino, J. M. (1990). Mixing, chaotic advection and turbulence. Annu. Rev. Fluid Mech. 22, 207–253. 262. Ottino, J. M. (1990). ‘‘The Kinematics of Mixing’’. Cambridge University Press, London. 263. Ottino, J. M. (2000). The art of mixing with an admixture of art: Viewing creativity through P.V. Danckwerts’s early work. Chem. Eng. Sci. 55, 2749–2765. 264. Ottino, J. M., DeRoussel, P., Hansen, S., and Khakhar, D. V. (2000). Mixing and dispersion of viscous liquids and powdered solids. Adv. Chem. Eng. 25, 105–204. 265. De Roussel, P., Khakhar, D. V., and Ottino, J. M. (2001). Mixing of viscous immiscible liquids. Part 1: Computational models for strong,weak and continuous flow systems. Chem. Eng. Sci. 56, 5511–5529. Also see ibid 5531–5537. 265a. Wille, M., Langer, G., and Werner, U. (2001). The influence of macroscopic elongational flow on dispersion processes in agitated tanks. Chem. Eng. Technol. 24, 119–127. 266. Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezic, I., Stone, H. A., and Whitesides, G. M. (2002). Chaotic mixer for microchannels. Science 295, 647–651. 267. Pandey, D. K., Prakash, O., and Gupta, S. N. (1977). Differential viscosity correlation for heat transfer to non-Newtonian fluids in a jacketed agitated vesseel. Indian J. Technol. 15, 223–228. 268. Blasinski, H. and Kuneewicz, C. (1981). Heat transfer during the mixing of pseudoplstic fluids with ribbon agitators. Int. Chem. Eng. 21, 679–683. 269. Calabrese, R. V. and Stoots, C. M. (1989). Flow in the impeller region of a stirred tank. Chem. Eng. Prog. 85(5), 43–50. 270. Kovacs, T., Tragardh, C., and Fuchs, L. (2001). Fourier spectrum to recover deterministic and stochastic behaviour in stirred tanks. AIChE J. 47, 2167–2176. 271. Mavros, P., Xuereb, C., and Bertrand, J. (1997). Effect of radial or axial impeller and liquid viscosity on energy dissipation in agitated vessels. Recente Progres en Genie des Procedes 11(51), 51–58.
FLUID MECHANICS AND HEAT TRANSFER
173
272. van der Molen, K. and van Maanen, H. E. A. (1978). Laser–Doppler measurements of the turbulent flow in stirred vessels to establish scaling rules. Chem. Eng. Sci. 33, 1161–1168. 273. Venneker, B. C. H. and van den Akker, H. E. A. (1997). CFD calculations of the turbulent flow of shear-thinning fluids in agitated tanks. Recent Progres en Genie des Procedes 11(51), 179–186. 274. Zalc, J. M., Alvarez, M. M., Muzzio, F. J., and Arik, B. E. (2001). Extensive validation of computed laminar flow in a stirred tank with three Rushton turbines. AIChE J. 47, 2144– 2154. 275. Derksen, J. (2001). Assessment of large eddy simulations for agitated flows. Chem. Eng. Res. Des. 79A, 824–830. 275a. Brodkey, R. S. (1985). Fundamentals of turbulent mixing and kinetics. In ‘‘Mixing of Liquids by Mechanical Agitation’’ (J. J. Ulbrecht and G. K. Patterson, eds.), pp. 29–58. Gordon and Breach, New York. 276. Delaplace, G., Torrez, C., Leuliet, J.-C., Belaubre, N., and Andre, C. (2001). Experimental and CFD simulation of heat transfer to highly viscous fluids in an agitated vessel equipped with a non-standard helical ribbon impeller. Chem. Eng. Res. Des. 79A, 927–937. 277. Ali, A. M., Yuan, H. H. S., Dickey, D. S., and Tatterson, G. B. (1981). Liquid dispersion mechanisms in agitated tanks: Part 1. Pitched blade turbine. Chem. Eng. Commun. 10, 205–213. Also see ibid 215–222. 278. Gardner, J. and Tatterson, G. (1992). Characterization of mixing in shaker table containers. Biotechnol. Bioeng. 39, 794–797. 279. Tatterson, G. B. and Stanford, T. G. (1981). Liquid dispersion mechanisms in agitated tanks. Part III. Low viscosity discrete phase into high viscosity continuous phase. Chem. Eng. Commun. 6, 371–376. 280. Garrison, C. M. (1983). How to design and scale mixing pilot-plants. Chem. Eng. 90(2), 63–70. 281. Zlokarnik, M. (1998). Problems in the application of dimensional analysis and scale-up of mixing operations. Chem. Eng. Sci. 53, 3023–3030. 282. Bowen, R. L. (1985). Agitation intensity: Key to scaling up flow-sensitive liquid systems. Chem. Eng. 92(3), 195. 283. Carpenter, K. J., Lines, P. C., Aldington, R. J., Keron, A., and Hindson, W. A. J. (1994). Two examples of designing full scale reactors for multi-stage synthesis involving nonNewtonian mixtures. AIChE Sym. Ser. 90(299), 101–105. 284. Kelkar, J. V., Mashelkar, R. A., and Ulbrecht, J. (1973). Scale-up method for the power consumption of agitators in the creeping flow regime. Chem. Eng. Sci. 28, 664–666. 285. Rushton, J. H., Costich, E. W., and Everett, H. J. (1950). Power characteristics of mixing impellers. Chem. Eng. Prog. 46, 395–404. Also see ibid 467–476. 286. Bates, R. L., Fondy, P. L., and Corpstein, R. R. (1963). An examination of some geometric parameters of impeller power. Ind. Eng. Chem. Process Des. Dev. 2, 310–314. 287. Grenville, R. K. et al. (1995). Paper presented at Mixing XV 15th Biennial North American Mixing Conference, Banff, Canada (June 18–23). 288. Skelland, A. H. P. (1983). Mixing and agitation of non-Newtonian fluids. In ‘‘Handbook of Fluids in Motion’’ (N. P. Cheremisinoff and R. Gupta, eds.), pp. 179–209. Ann Arbor Science, Ann Arbor, MI. 289. Ibrahim, S. and Nienow, A. W. (1995). Power curves and flow patterns for a range of impellers in Newtonian fluids: 40 < Re < 5 105. Trans. IChemE 73A, 485–491. 290. Karcz, J. and Major, M. (1998). An effect of baffle length on the power consumption in an agitated vessel. Chem. Eng. Process. 37, 249–256. 291. Pahl, M. H., Brenke, A., and Luo, Y. (1996). Power consumption of multiple shaft agitators. Chem. Eng. Technol. 19, 503–509.
174
R. P. CHHABRA
292. Papastefanos, N. and Stamatoudis, M. (1989). Effect of impeller and vessel size on impeller power number in closed vessels for Reynolds numbers between 40 and 65,000. Chem. Eng. Commun. 80, 69–79. 293. Papastefanos, N. and Stamatoudis, M. (1989). Effect of vessel and impeller geometry on impeller power number in closed vessels for Reynolds numbers between 40 and 65,000. Trans. IChemE 67, 169–174. 294. Seichter, P., Kognal, J., and Rieger, F. (1981). Process characteristics of screw impellers with a draught tube for Newtonian liquids—The power input. Collect. Czech. Chem. Commun. 46, 2007–2020. 295. Strek, F. and Karcz, J. (1993). Experimental studies of power consumption for agitated vessels equipped with non-standard baffles and high speed agitator. Chem. Eng. Process. 32, 349–357. 296. Sumi, Y. and Kamiwano, M. (2001). Development and mixing characteristics of a multistage impeller for agitating highly viscous fluids. J. Chem. Eng. Jpn. 34, 485–492. 297. Theliander, H. (1995). Mixing in closed vessels—Suspension of solid particles and power demand. Chem. Eng. Commun. 132, 169–186. 298. Xanthopoulos, C. and Stamatoudis, M. (1986). Turbulent range impeller power numbers in closed cylindrical and square vessels. Chem. Eng. Commun. 46, 123–128. 299. Xanthopoulos, C. and Stamatoudis, M. (1988). Effect of impeller and vessel size on impeller power number in closed vessels for Reynolds numbers above 75,000. Chem. Ing. Technik. 60(7), 560–562. 300. Hiraoka, S., Kato, Y., Ozaki, N., Murakami, Y., and Lee, Y. S. (2001). Power consumption and mixing time in an agitated vessel with double impeller. Chem. Eng. Res. Des. 79A, 805–810. 301. Jahoda, M. and Machon, V. (1994). Homogenization of liquids in tanks stirred by multiple impellers. Chem. Eng. Technol. 17, 95–101. 302. Hentrich, P., ed. (2000). ‘‘Handbook of Mixing Technology’’. Ekato, Schopfheim, Germany. 303. Bakker, A. and Gates, L. E. (1995). Viscous mixing. Chem. Eng. Prog. 101(12), 25–34. 304. Doraiswamy, D., Grenville, R. K., and Etchells, A. W., III (1994). Two-score years of the Metzner-Otto correlation. Ind. Eng. Chem. Res. 33, 2253–2258. 305. Kraume, M. and Zehner, P. (2001). Experience with experimental standards for measurements of various parameters in stirred tanks: A comparative test. Chem. Eng. Res. Des. 79A, 811–818. 306. Seichter, P. (1971). Efficiency of the screw mixers with a draft tube. Trans. IChemE 49, 117–123. 307. Seichter, P. (1981). Process characteristics of screw impellers with a draught tube for Newtonian liquids—Pumping capacity of the impeller. Collect. Czech. Chem. Commun. 46, 2021–2031. 308. Seichter, P. (1981). Process characteristics of screw impellers with a draught tube for Newtonian liquids—Time of homogenization. Collect. Czech. Chem. Commun. 46, 2032– 2042. 309. Rehr, E. O. and Bohm, R. (1970). Ruhren nicht-Newtonscher flussigkeiten-II. Mitteilung: Der leistungsbedarf furden bandschecken-und ander-rurhen. Chem. Technol. 22, 136–146. 310. Cheng, J. and Carreau, P. J. (1994). Mixing in the transition flow regime with helical ribbon agitators. Can. J. Chem. Eng. 12, 418–430. 311. Forschner, P., Krebs, R., and Schneider, T. (1991). ‘‘Scale up Procedures for Power Consumption of Mixing in Non-Newtonian Fluids’’. Proc. 7th European Conference on Mixing, Brugge, Belgium, pp. 161–165. 312. Barnes, H. A. and Nguyen, Q. D. (2001). Rotating vane rheometry—A review. J. NonNewt. Fluid Mech. 98, 1–14.
FLUID MECHANICS AND HEAT TRANSFER
175
313. Costell-Perez, M. E., Steffe, J. F., and Moreira, R. G. (1991). Simple determination of power law flow curves using a paddle type mixer viscometer. J. Texture Stud. 22, 303–316. 314. Keentok, M., Milthorpe, J. F., and O’Donovan, E. (1985). On the shearing zone around rotating vanes in plastic liquids: Theory and experiment. J. Non-Newt. Fluid Mech. 17, 23–35. 315. Kemblowski, Z. and Kristiansen, B. (1986). Rheometry of fermentation liquids. Biotechnol. Bioeng. 28, 1474–1483. 316. Leong-Poi, L. and Allen, D. G. (1992). Direct measurement of the yield stress of filamentous fermentation broth with the rotating vane technique. Biotechnol. Bioeng. 40, 403–412. 316a. Furling, O., Choplin, L., and Tanguy, P. A. (2001). On-line (in situ) viscosity follow-up of concentrated slurries using during make down. Chem. Eng. Res. Des. 79A, 915–920. 316b. Lai, K., Steffe, J. F., and Ng, P. (2000). Average shear rates in the rapid viscoanalyser (RVA) mixing system. Cereal Chem. 77, 714–716. 316c. Cantu-Lozano, D., Rao, M. A., and Gasparetto, C. (2000). Rheological properties of non-cohesive apple dispersion with helical and vane impellers: Effect of concentration and particle size. J. Food Process Eng. 23, 373–385. 316d. Aerts, L. and Verspaille, M. (2001). Absolute rheometry in the brabender viscograph by mixer viscometry calibration. Starch-Starke 53, 59–63. 317. Knoch, A. (1999). Influence of non-Newtonian flow behaviour on mixing process characteristics. Chem. Eng. Technol. 22, 112–118. 318. Prokopec, L. and Ulbrecht, J. J. (1970). Ruhrleistung eines schraubenruhrers mit leitrohr beim mischen nicht-Newtonschen flussigkeiten. Chem. Ing. Tech. 42, 530–534. 319. Ulbrecht, J. J. (1974). Mixing of viscoelastic liquids by mechanical agitation. Chem. Eng. (June), 347–353. 319a. Boger, D. V. (1977). A highly elastic constant-viscosity fluid. J. Non-Newt. Fluid Mech. 3, 87–91. 320. Kelkar, J. V., Mashelkar, R. A., and Ulbrecht, J. (1972). On the rotational viscoelastic flows around simple bodies and agitators. Trans. IChemE 50, 343–352. 321. Bartels, P. V. and Janssen, L. P. (1983). ‘‘Effect of Viscoelasticity on Mixing Flows in Stirred Tanks’’. Proc. IX Mixing Conference, Henniker, NH. 322. Hicks, R. W., Morton, J. R., and Fenic, J. G. (1976). How to design agitators for desired process response. Chem. Eng. 83(April 26), 102–110. 323. Fasano, J. B., Bakker, A., and Penney, W. R. (1994). Advanced impeller geometry boosts liquid agitation. Chem. Eng. 101(8), 110–116. 324. Gates, L. E., Hicks, R. W., and Dickey, D. S. (1976). Application guidelines for turbine agitators. Chem. Eng. 83(Dec. 6), 165–170. 325. Anne-Archard, D. and Boisson, H. C. (1997). Numerical simulation of Newtonian and viscoelastic 2-D laminar mixing in an agitated vessel. Recent Progres en Genie des Procedes 11(51), 145–152. 326. Myers, K. J., Ward, R. W., and Bakker, A. (1997). A digital particle image velocimetry investigation of flow field instabilities of axial flow impellers. J. Fluids Eng. (ASME) 119, 623–632. 327. Aubin, J., Mavros, P., Fletcher, D. F., Bertrand, J., and Xuereb, C. (2001). Effect of axial agitator configuration (up-pumping, down-pumping, reverse rotation) on flow patterns generated in stirred vessels. Chem. Eng. Res. Des. 79A, 845–856. 328. Mavros, P., Mann, R., Vlaev, S. D., and Bertrand, J. (2001). Experimental visualization and CFD simulation of flow patterns induced by a novel energy-saving dual-configuration impeller in stirred vessels. Chem. Eng. Res. Des. 79A, 857–866.
176
R. P. CHHABRA
329. Jaworski, Z., Dyster, K. N., and Nienow, A. W. (2001). The effect of size, location and pumping direction of pitched blade turbine impellers on flow patterns. Chem. Eng. Res. Des. 79A, 887–894. 330. Nagata, S., Nishikawa, M., Tada, H., Hirabayashi, H., and Gotoh, S. (1970). Power consumption of mixing impellers in Bingham plastic liquids. J. Chem. Eng. Jpn. 3, 237–243. 331. Giesekus, H. (1965). Some secondary flow phenomena in general viscoelastic fluids. Proc. 4th Int. Conf. Rheol. 1, 249–266. 332. Abid, M., Xuereb, C., and Bertrand, J. (1992). Hydrodynamics in vessels stirred with anchors and gate agitators: Necessity of a 3-D modelling. Chem. Eng. Res. Des. 70A, 377– 384. 333. Nagata, S., Yanagimoto, M., and Yokoyama, T. (1956). Studies on the mixing of highly viscous liquids. Memoirs Fac. Eng. Kyoto Univ. (Japan) 18, 444–460. 334. Kappel, M. (1979). Development and application of a method for measuring the mixture quality of miscible liquids. Part I, II, III. Int. Chem. Eng. 19, 196–215; 431–444 and 571–590. 335. Cheng, D. C.-H., Schofield, C., and Jane, R. J. (1974). Proc. Ist Engineering Conf. Mixing & Centrifugal Separation, BHRA Fluids Eng., Cranfield, Paper # C2-15. 336. Manna, L. (1997). Comparison between physical and chemical methods for the measurement of mixing times. Chem. Eng. J. 67, 167–173. 337. Grenville, R. K. and Tilton, J. N. (1996). A new theory improves the correlation of blends time data from turbulent jet mixed vessels. Trans. IChemE 74A, 390–396. 338. Ottino, J. M. and Macosko, C. W. (1980). An efficiency parameter for batch mixing of viscous fluids. Chem. Eng. Sci. 35, 1454–1457. 339. Bakker, A. and Fasano, J. B. (1994). A computational study of the flow pattern in an industrial paper pulp chest with a side entering impeller. AIChE Sym. Ser. 89(293), 118–124. 340. Bakker, A., Fasano, J. B., and Leng, D. E. (1994). Pinpoint mixing problems with lasers and simulation software. Chem. Eng. 101(1), 94–100. 341. Kelly, W. J. and Humphrey, A. E. (1998). Computational fluid dynamics model for predicting flow of viscous fluids in a large fermentor with hydrofoil flow impellers and internal cooling coils. Biotechnol. Prog. 14, 248–258. 342. Khayat, R. E., Derdouri, A., and Frayce, D. (1998). Boundary element analysis of threedimensional mixing flow of Newtonian and viscoelastic fluids. Ind. J. Numer. Methods Fluids 28, 815–840. 343. Lafon, P. and Bertrand, J. (1988). ‘‘A Numerical Model for the Prediction of Laminar Mixing’’. Proc. European Conference on Mixing (BHRA), pp. 493–500. 344. Oldshue, J. Y. (1989). Fluid mixing in 1989. Chem. Eng. Prog. 85(5), 33–42. 345. Ottino, J. M., Ranz, W. E., and Macosko, C. W. (1981). A framework for description of mechanical mixing of fluids. AIChE J. 27, 565–577. 346. Pericleous, K. A. and Patel, M. K. (1987). The modelling of tangential and axial agitators in chemical reactors. Physicochem. Hydrodyn. 8, 105–123. 347. Takigawa, T., Kataoka, K., Ema, H., Yoshimura, T., and Ohmura, N. (2000). Mixingeffective motion of high viscosity fluid around a rotating elliptic-cylinder. J. Chem. Eng. Jpn. 33, 420–426. 347a. Peixoto, S., Nunhez, J., and Duarte, G. (2000). Characterizing the flow of stirred vessels with anchor type impellors. Braz. J. Chem. Eng. 17, 925–935. 348. Wunsch, O. and Bohme, G. (2000). Numerical simulation of 3-D viscous fluid flow and convective mixing in a static mixer. Arch. Appl. Mech. 70, 91–102. 349. Yang, H.-H. and Manas-Zloczower, I. (1992). 3-D flow field analysis of a banbury mixer. Int. Polymer Process. 7, 195–203.
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350. Uhl, V. W. and Gray, J. B. (1966). ‘‘Mixing’’, Vol. 1, Chapter 5. Academic, New York. 351. Hewitt, G. F., Shires, G. L., and Bott, T. R. (1994). ‘‘Process Heat Transfer’’. CRC Press, Boca Raton, FL. 352. Rohsenow, W. M., Hartnett, J. P., and Cho, Y. I. (1998). ‘‘Handbook of Heat Transfer’’, 3rd edn. McGraw Hill, New York. 353. Balakrishna, M. and Murthy, M. S. (1980). Heat transfer studies in agitated vessels. Chem. Eng. Sci. 35, 1486–1494. 354. Dream, R. F. (1999). Heat transfer in agitated jacketed vessels. Chem. Eng. 106(1), 90–96. 355. Edwards, M. F. and Wilkinson, W. L. (1972). Heat transfer in agitated vessels Part I— Newtonian fluids. Chem. Eng. 8, 310–319. Also see ibid 9, 328–335. 356. Strek, F. and Karcz, J. (1997). Heat transfer to Newtonian fluid in a stirred tank—A comparative experimental study for vertical tubular coil and a jacket. Recent Progres en Genie des Procedes 11(51), 105–112. 357. Cohen, D. (1998). How to select rotor–stator mixers. Chem. Eng. 105(8), 76–79. 358. Dickey, D. S. (2000). Facing the challenge of mixing problem fluids. Chem. Eng. 107(5), 68–75. 359. Dietsche, W. (1998). Mix or match: Choose the best mixers everytime. Chem. Eng. 105(8), 70–75. 360. Einekel, W.-D. (1980). Influence of physical properties and equipment design on the homogenity of suspensions in agitated vessels. Ger. Chem. Eng. 3, 118–124. 361. Gladki, H. (1997). Keep solids in suspension. Chem. Eng. 104(10), 213–216. 362. Heywood, N. I. (1999). Stop your slurries from stirring up trouble. Chem. Eng. Prog. 95(9), 21–41. 363. Ho, F. C. and Kwong, A. (1973). A guide to designing special agitators. Chem. Eng. 80(July 23), 94–104. 364. Masucci, S. F. (1992). Effectively make emulsions and dispersions. Chem. Eng. 99(7), 112–115. 365. Munier, M. (1997). Performance of some agitators for gas–liquid dispersion. Recent Progres en Genie des Procedes 11(51), 271–278. 366. Myers, K., Reeder, M. F., Bakker, A., and Dickey, D. S. (1997). In ‘‘Recent Progres en Genie des Procedes’’, Vol. 11 (51), pp. 115–122. 367. Myers, K. J., Bakker, A., and Ryan, D. (1997). Avoid agitation by selecting static mixers. Chem. Eng. Prog. 93(6), 28–38. 368. Myers, K. J., Reeder, M. F., Ryan, D., and Daly, G. (1999). Get a fix on high-shear mixing. Chem. Eng. Prog. 95(11), 33–42. 369. Pasquali, G., Fajner, D., and Magelli, F. (1983). Effect of suspension viscosity on power consumption in the agitation of solid–liquid systems. Chem. Eng. Commun. 22, 371–375. 370. Shaw, J. A. (1994). Understand the effects of impeller type, diameter and power on mixing time. Chem. Eng. Prog. 100(2), 45–48. 371. von Essen, J. (1998). Gas–Liquid-mixer correlation. Chem. Eng. 105(8), 80–82. 372. Rzyski, E. (1993). Liquid homogenization in agitated tanks. Chem. Eng. Technol. 16, 229–233. 373. Muskett, M. J. and Nienow, A. W. (1987). Capital vs. Running costs: The economics of mixer selection. I. Chem. E. Symp. Ser. No. (108): Fluid Mixing III, pp. 33–48. 373a. Schaffer, M., Marchilden, E., McAuley, K., and Cunningham, M. (2001). Assessment of mixing performance and power consumption of a novel polymerisation reactor system. Chem. Eng. Technol. 24, 401–408. 374. Nienow, A. W. and Elson, T. P. (1988). Aspects of mixing in rheologically complex fluids. Chem. Eng. Res. Des. 66, 5–15.
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375. Ottino, J. M. and Khakhar, D. V. (2000). Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32, 55–91. 376. Larson, R. G. (2000). Turbulence without inertia. Nature 405(6782), 27–28. 377. Groisman, A. and Steinberg, V. (2000). Elastic turbulence in a polymer solution flow. Nature 405(6782), 53–55. 378. Gaston-Bonhomme, Y., Desplanches, H., and Chevalier, J. L. (1989). Ebullition nuclee de liquides newtoniens et non-newtoniens a partir d’un serpentin en cuve agitee. Recents Progres en Genie des Procedes 8a, 38. 379. Desplanches, H., Gaston-Bonhomme, Y., and Chevalier, J. L. (1990). Ebullition de liquides visqueux en cuve agitee-lere partie: Liquides Newtoniens. Entropie 157–158, 65. 380. Gaston-Bonhomme, Y., Bouvenot, A., Desplanches, H., and Chevalier, J. L. (1992). Ebullition de liquides visquex en cuve agitee-zone partie: Liquides non-Newtoniens. Entropie 167, 19.
ADVANCES IN HEAT TRANSFER VOL. 37
Optical and Thermal Radiative Properties of Semiconductors Related to Micro/Nanotechnology
Z. M. ZHANG, C. J. FU, and Q. Z. ZHU George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. E-mail:
[email protected]
Abstract Optical and radiative properties of semiconductor materials and structures are often critical to the functionality and performance of many devices, such as semiconductor lasers, radiation detectors, tunable optical filters, waveguides, solar cells, selective emitters and absorbers, etc. This chapter reviews the optical and thermal radiative properties of semiconductor materials related to the recent technological advancements that are playing a vital role in the integrated-circuit manufacturing, optoelectronics, and radiative energy conversion devices. Some fundamental aspects related to important micro/nanoscale processes in semiconductor optoelectronics are presented. These include the electronic band structure, energy gap, interband and intraband transitions, free-carrier absorption, optical and acoustic phonons, and the effects of impurities and temperature. Theoretical and experimental studies on the radiative properties of thin films and multilayer systems, rough surfaces, and nanostructured surfaces are summarized. Quantum confinement in nanomaterials is described, followed by a review of the radiative properties of photonic crystals and porous silicon. Potential applications and future developments are outlined. I. Introduction Semiconductor-based integrated circuits have widespread applications in industry and everyday life in the 21st century. Improvements of performance and shrinkage of device sizes in microelectronics have been a major driving force for scientific and economic progress over the past quarter of the Advances in Heat Transfer Volume 37 ISSN 0065-2717
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Copyright ß 2003 Elsevier Inc., All rights reserved
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century. Nanoelectronics is the closely watched next frontier, and the successful demonstration of circuits made of nanowires, nanotubes, and single molecules was selected by the Science magazine as the breakthrough-of-theyear in 2001 [1–3]. The ability to fabricate and control materials and devices with very small feature sizes is the hallmark of contemporary technologies. Figure 1 shows a scanning electron microscope (SEM) image of an array of InGaAs/GaAs quantum wires with a width of 40 nm [4]. The quantum wires were fabricated from multiple quantum wells using high-resolution electron beam lithography. There are 20 InGaAs wells, each having a thickness of 3 nm, separated by 60-nm-thick GaAs barriers. The etching depth is 1.4 mm. The final quantum wires have dimensions of 20 nm 40 nm 1.4 mm of InGaAs and 60 nm 40 nm 1.4 mm of GaAs. Figure 2 shows an atomic force microscope (AFM) image of InAs quantum dots grown on Si substrate using molecular beam epitaxy (MBE) [5]. Self-assembled high-density growth of quite uniform dot dimensions has been demonstrated, with 4–5 monolayers of coverage [5,6]. In addition to quantum-confined structures for electronic states, confinement of photons has been realized with photonic crystals. Figure 3
FIG. 1. An SEM image of InGaAs/GaAs quantum wires, after Ref. [4].
FIG. 2. An AFM image of InAs nanodots on Si substrate, after Ref. [5].
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FIG. 3. Two-dimensional hexagonal photonic crystal, after Ref. [7]. Copyright 2000 by the American Physical Society, with permission.
FIG. 4. Comparison of wavelength ranges with some characteristic dimensions.
shows an SEM image of a two-dimensional microporous silicon photonic crystal [7]. Photolithography and alkaline etching were used to define a triangular lattice of pore nuclei on an n-type Si wafer. The patterned wafer was anodically etched in HF to form cylindrical pores with a depth of 100 mm. The lattice constant was 1.5 mm, and the pore radius after thermal oxidation and wet etching was 0.644 mm. The transmission spectra demonstrated the potential use for highly birefringent, optically integrated devices [7]. Figure 4 compares the wavelength ranges with some characteristic dimensions. One can see that microelectromechanical systems (MEMS) generally produce micromachining capabilities from millimeters down to a few micrometers. Currently, the smallest feature of integrated circuits is about 150 nm. The layer thickness of thin films can be as small as a few nanometers. Nanomachining is an active area of research and development and some unique optical properties are associated with micro/nanostructures.
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The use of microstructures not only modifies the optical properties for optoelectronic applications and processing control, but also facilitates some important energy conversion devices, such as solar cells and thermophotovoltaic applications [8]. The key is to modify the absorption and emission spectra using one-, two-, and three-dimensional nanostructures. These include, for example, III–V semiconductor multiple quantum wells [9] and microstructured surfaces or bulk structures [10,11]. Optical techniques and radiative processes play important roles in current industry and daily life. Examples are advanced lighting and display, materials and surface characterization, real time processing monitoring and control, laser manufacturing, rapid thermal processing (RTP), communication, data storage and reading, radiation detection, biomedical imaging and treatment, ground and space solar energy utilization, direct energy conversion, etc. Optical and thermal radiative properties are fundamental physical properties that describe the interaction between electromagnetic waves and matter from deep ultraviolet to far-infrared spectral regions. In fact, optical property measurements and analysis offer powerful tools to our understanding of the physics of solids and other materials. Optical and radiative properties depend on a large number of variables, making them difficult to measure and to analyze the observed results. Numerous studies have been devoted to the measurements, analysis, modeling, and simulation of optical and radiative behaviors of materials in solid, liquid, gas, and plasma forms. This review is divided into six sections. After this introductory section, a summary of the physical foundations of semiconductor optical properties is provided in Section II. Sections III and IV discuss the theoretical and experimental studies on the radiative properties of thin films and multilayer systems, rough surfaces, and nanostructured surfaces. In Section V, quantum confinement in nanomaterials is briefly described, followed by an introduction to photonic crystals. Some concluding remarks with future opportunities are outlined in Section VI.
II. Fundamentals of Optical Properties of Semiconductors The nature of radiation may be understood either as electromagnetic waves or as a collection of particles, called photons. The propagation of electromagnetic waves is governed by a set of equations, i.e. Maxwell’s equations, and can be written as [12–14], rE ¼
@ðHÞ @t
ð1Þ
OPTICAL AND THERMAL RADIATIVE PROPERTIES
r H ¼ E þ
@ð"EÞ @t
183
ð2Þ
r ð"EÞ ¼
ð3Þ
r ðHÞ ¼ 0
ð4Þ
These equations are expressed in SI units. Here, H is the magnetic field vector, E is the electric field vector, " is the electric permittivity, is the magnetic permeability, is electric charge density, and is the electric conductivity. Ohm’s law gives the electric current density [A/m2] as J ¼ E
ð5Þ
In most materials, ¼ 0 because the number of electrons equals the number of protons in the nuclei. Except in non-linear processes or for inhomogeneous materials, " and are independent of time and position, although they are frequency dependent. For a dielectric material, ¼ 0, Eqs. (1) and (2) can be combined to yield, r2 E ¼ "
@2 E @t2
ð6Þ
using the relation r (r E) ¼ r(r E)r2E ¼ r2E. Eq. (6) is the wave equation. It can also be written in terms of the magnetic field. The solution of the wave equation for a monochromatic wave may be written as E ¼ E þ eið!tqrÞ
ð7Þ
where Eþ is a complex vector, ! is the angular frequency, and q is the wavevector, which points towards the direction of propagation. In order for Eq. (7) to be a solution of Eq. (6), the magnitude of q must be equal to pffiffiffiffiffiffi q ¼ ! ". At any time t, the field is a sinusoidal function of position. Moreover, in each plane with q r ¼ const, the field is a sinusoidal function of time. At any instant of time, the surface normal to the wavevector q has the same phase; this surface is called a wavefront. The wavefront travels in the direction of q with a speed, c¼
! 1 ¼ pffiffiffiffiffiffi " q
ð8Þ
This is the phase speed of the wave. The magnitude of the wavevector is related to the wavelength by q ¼ 2/ . Figure 5 illustrates a linearly-polarized ˆ electromagnetic wave propagating in the positive x-direction, i.e. q ¼ qx.
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FIG. 5. Illustration of an electromagnetic wave.
In free space (vacuum), 0 ¼ 4 107 N/A2 and "0 ¼ 8.854 1012 F/m. pffiffiffiffiffiffiffiffiffiffi The speed of electromagnetic wave in vacuum is given by c0 ¼ 1= "0 ¼ p0ffiffiffiffiffiffiffiffiffi 8 2:998 10 m=s. The refractive index of the medium is defined as n ¼ "="0 . For non-magnetic materials, ¼ 0. Hence, c ¼ c0/n and ¼ 0/n, where 0 is the wavelength in vacuum. Notice that n and " are functions of frequency (or wavelength) and are in general temperature dependent. The ratio "/"0 is called dielectric constant. For polychromatic light, the phase speed depends on wavelength in a dispersive medium. This gives rise to the phenomenon of dispersion in optics. An example is the dispersion of light by a prism. For polychromatic waves, the group velocity vg in one-dimension is defined as, vg ¼
d! dq
ð9Þ
In a conductive medium, 6¼ 0, similar to Eq. (6), one can derive the following equation: r2 E ¼
@E @2 E þ " 2 @t @t
ð10Þ
It can be shown that p Eq. ffiffiffi (7) is a solution of Eq. (10), with the complex wavevector q ¼ ð!=c0 Þ "~, where "~ is the complex dielectric function that is related to the complex refractive index (nþi) by " ¼ "1 þ i"2 ¼ ðn þ iÞ2 ð11Þ "~ ¼ þ i "0 "0 ! The imaginary part of the complex refractive index is called the extinction coefficient. From Eq. (11), "1 ¼ n22 and "2 ¼ 2n. To study the power flux [W/m2], the Poynting vector is used. The timeaverage power is one half of the real part of the Poynting vector, i.e., S¼
1 ReðE H Þ 2
where * denotes complex conjugate.
ð12Þ
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In an absorbing medium, the electric and magnetic fields will attenuate exponentially. As an example, consider a wave propagating in the positive x-direction with its electric field polarized in the y-direction: ˆ 0 eið!tqre xÞ eqim x E ¼ yE
ð13Þ
where E0 is a constant (the field amplitude at x ¼ 0 and t ¼ 0), and qre ¼ !n/c0 and qim ¼ !/c0 are the real and the imaginary parts of the wavevector, respectively. The magnetic field can be derived from Eq. (13) using Eq. (2), that is H ¼ zˆ
ðn þ iÞ E0 eið!tqre xÞ eqim x c0
ð14Þ
By substituting Eqs. (13) and (14) into Eq. (12), one obtains Sx ¼
n n E02 e2qim x ¼ E 2 ea x 2c0 2c0 0
ð15Þ
where a ¼ 4p/ 0 [m1] is called the absorption coefficient. The inverse of a
is called the radiation penetration depth: ¼
1
0 ¼ a 4p
ð16Þ
It is the distance through which the radiation power is attenuated by a factor 1/e ( 37%). The particle theory treats radiation as a collection of photons. The energy of an individual photon is proportional to the frequency, i.e., E ¼ h ¼ h!
ð17Þ
where h ¼ 6.626 1034 J s is Planck’s constant, is the frequency in Hz, and h ¼ h/2. The momentum of a photon is, p¼
h h ¼ c
ð18Þ
Photons obey Bose-Einstein statistics without requiring the total number to be conserved because the number of photons depends on temperature. In an isothermal enclosure (cavity) of temperature T with or without a medium, the electromagnetic wave energy per unit volume per unit frequency interval can be derived from quantum mechanics as [15–17] u ¼
8ph3 c3 ðeh=kB T 1Þ
ð19Þ
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TABLE I SPECTRAL REGIONS EXPRESSED IN DIFFERENT UNITS
Wavelength, (m) Wavenumber, (cm1) Frequency, (THz) Energy, E (eV)
UV From–To
VIS Up to
NIR Up to
MIR Up to
FIR Up to
0.01–0.4 1 M–25 k 30 k–750 124–3.1
0.78 12,820 384 1.59
2.5 4000 120 0.5
20 500 15 0.0625
1000 10 0.3 0.0012
where kB ¼ 1.381 1023 J/K is the Boltzmann constant. The radiant energy flux is related to the energy density and speed of light by q00rad; ¼ u c=4. If a blackbody is placed inside the enclosure, it will absorb all incoming radiant energy reaching its surface; under thermal equilibrium, it must emit the same amount of energy. Hence, the spectral emissive power (per unit area) of a blackbody in terms of wavelength is [17–20] eb; ð ; TÞ ¼
2phc2
5 ðehc=kB T
1Þ
ð20Þ
This is called Planck’s law due to Planck’s original derivation using the concept of energy quantization. Because of the broad spectral region of electromagnetic waves, alternative units are often used, such as wavelength, frequency, photon energy, wavenumber (1/ ), etc. Generally speaking, optical radiation covers the spectral region from ultraviolet (UV), visible (VIS), near-infrared (NIR), mid-infrared (MIR) through far-infrared (FIR). Table I outlines the subdivisions of the spectral region in different units. In order to understand the optical properties of semiconductors, it is important to become familiar with the physics of electrons and phonons and the interactions between them. The electron band structure and phonons will be discussed, followed by a discussion on the scattering, absorption, and emission mechanisms. The frequency-dependent dielectric function models will then be introduced. In the last part of this section, the quantum size effect on the optical behavior will be discussed.
A. ELECTRONIC BAND STRUCTURES A crystal is constructed by the continuous repetition in space of identical structural unit, called a lattice. Saying in other words, a crystal is a threedimensional periodic array of lattices. However, a lattice is only a mathematic abstraction, and the crystal structure is formed when a group of atoms, called a basis, is attached identically to each lattice point. The
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structure of any crystal can therefore be described in terms of a single lattice. In three dimensions, crystal lattices can be grouped into 14 different types as required by the point symmetry operations. These lattice types are then categorized into seven systems according to the seven types of conventional unit cells, namely, cubic, tetragonal, orthorhombic, monoclinic, triclinic, rhombohedral, and hexagonal. The physical properties of crystallized solids are controlled by the arrangement of atoms in a unit cell and the chemical bonds between the atoms. Hence, it is of great importance to know the structure of a crystal first in order to understand its electrical, optical, thermal, and other properties. This section introduces the semiconductor crystal structures and their electronic bands in the reciprocal lattice space. Emphases are given to the tetravalent semiconductors, that is, the diamond and zinc blende structure semiconductors, because these two types of materials are archetypical semiconductors and have enormous technological values. 1. Crystal Structures of Diamond and Zinc Blende Semiconductors The crystal structures of diamond and zinc blende semiconductors are derivatives of the cubic structure. The diamond lattice is formed from two face-centered cubic lattice, which are displaced along the body diagonal by one quarter the length of the diagonal. As shown in Fig. 6, each atom in the diamond lattice has a covalent bond with four adjacent atoms, which together form a tetrahedron by promoting an s-electron to a p-state to form sp3 hybrids [21–26]. This tetrahedral structure can be seen in the sub-cubic cell delineated by the dashed lines. In essence, the diamond lattice can be thought as a face-centered cubic lattice with a basis containing two identical atoms-one is on the edge, the other is on the body diagonal with a distance of one quarter the length of the diagonal between them. The lattice constant
FIG. 6. Diamond crystal structure indicating the tetrahedral coordination, after Ref. [25].
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a is defined as the lateral length of the unit cubic. The zinc blende lattice is similar to that of the diamond. The difference is that there are two different types of atoms distributed on the diamond lattice. Therefore, the zinc blende lattice can also be thought as a face-centered cubic lattice, but with a basis containing two atoms of different types. Table II presents the most commonly used diamond and zinc blende semiconductors with associated lattice constants. Notice that GaN crystal is wurtzite in its stable form with a hexagonal symmetry. This is also the case for AlN and InN. The III-nitride materials have a large energy gap and are important for UV-blue–green LEDs and lasers [27]. ZnS, ZnO, CdS, and
TABLE II PARAMETERS OF COMMON SEMICONDUCTOR MATERIALS, WHERE i AND d DENOTE INDIRECT AND DIRECT INTERBAND TRANSITIONS, RESPECTIVELY [21–25] Structure Diamond
Zinc blende
a
Crystal
Eg (eV)
m e =me
m h =me
0.2 0.98 a 0.19 b 1.64 a 0.08 b
0.25 0.16 c 0.49 d 0.04 c 0.28 d
C Si
3.567 5.431
5.47 1.11
(i) (i)
Ge a-Sn
5.657 6.491
0.66 0
(i) (d)
AlSb BN BP CdS CdSe CdTe GaAs GaN (w) GaP GaSb HgTe InAs InP InSb SiC ZnO ZnS ZnSe ZnTe
6.135 3.615 4.538 5.818 6.05 6.48 5.654 5.451 6.118 6.429 6.036 5.869 6.478 4.348 4.63 5.409 5.668 6.089
1.58 7.5 2 2.42 1.7 1.56 1.42 3.36 2.26 0.72 <0 0.36 1.35 0.17 2.36 3.35 3.68 2.58 2.26
(d) (i)
0.12
0.98
(d) (d) (d) (d) (d) (d) (d) (d) (d) (d) (d) (d) (d) (d)
0.21 0.13 0.07 0.067 0.19 0.82 0.042 0.023 0.077 0.015
0.8 0.45 0.64 0.082 0.6 0.6 0.4
Longitudinal effective mass. Transverse effective mass. c Light-hole effective mass. d Heavy-hole effective mass. b
a [A˚]
7 0.27
0.04 0.64 0.4
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189
CdSe can also be wurtzite. HgTe is a semimetal with a negative band gap and can be mixed with the wide-gap semiconductor CdTe to form the ternary compound of Hg1xCdxTe that has important applications as infrared MCT detectors [28]. It should be noted that there exist different ways to choose a unit cell of the crystal lattice. Another way of choosing a unit cell is to follow the two steps: (1) draw lines to connect a given lattice point to all nearby lattice points; (2) at the midpoint and normal to these lines, draw new lines or planes. The smallest volume enclosed in this way is called the Wigner-Seitz primitive cell [21]. The reciprocal lattice of a crystal structure is defined in the k-space (wavevector-space). Since a crystal is a periodic array of lattices in real space, the reciprocal lattice can be obtained by performing a spatial Fourier transform of the crystal. A Brillouin zone is defined as a WignerSeitz cell in the reciprocal lattice and the smallest of which is called the first Brillouin zone. The definition of the Brillouin zone gives a vivid geometric interpretation of the Bragg diffraction condition and thus is of importance in the study of electron and phonon states in crystals and their interactions with electromagnetic waves. Figure 7 shows the first Brillouin zone of a facecentered cubic lattice in the k-space. The directions kx, ky, and kz are called the h100i, h010i, and h001i directions, respectively. The center of the Brillouin zone is called the -point, and the intersection of the three axes with the zone edge is called the X-point. The body diagonal, or the h111i direction, meets the zone edge at the L-point. Other points of interest, such as the K-, U-, and W-points can also be defined, as illustrated in Fig. 7.
FIG. 7. The first Brillouin zone of a face-centered cubic crystal.
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2. Electronic Band Theory a. The Bloch Theorem for a Periodic Potential The total potential in a crystal includes the core–core, electron–electron, and electron–core Coulomb interactions. In order for solving electron wave functions subjected to such a potential, the problem will be a many-body problem, which indicates very difficult in mathematics. However, this problem can be simplified using the Hartree approximation that each electron moves in the average field created by the other electrons. This is the one-electron model and the appropriate electronic wavefunctions are the products of one-electron wavefunctions. Thus the Hamiltonian operator (H) for the one-electron model is H¼
p2e þ UðrÞ 2me
ð21Þ
where pe and me are the momentum and mass of the electron, respectively, and U(r) is a periodic potential function resulted from both the electron– electron and electron–core interactions. The one-electron Schro¨dinger equation is [21–23]:
2 h r2 þ UðrÞ ðrÞ ¼ E ðrÞ 2me
ð22Þ
where E is the electron energy and (r) is the electron wavefunction. The periodicity of lattice structure yields the boundary condition UðrÞ ¼ Uðr þ RÞ
ð23Þ
where R is the vector between two lattice points. The potential U(r) can be expanded as a Fourier series in terms of the reciprocal lattice vector G, that is UðrÞ ¼
X
UG eiGr
G
ð24Þ
where UG’s are complex Fourier expansion coefficients. According to the Bloch theorem, the wavefunction of an electron in a periodic potential must have the form: ðrÞ ¼ eikr uk ðrÞ
ð25Þ
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191
where uk(r) is a periodic function with the periodicity of the lattice. The wavefunction (r) can also be expressed as a Fourier series summed over all values of the permitted wavevector so that X ðrÞ ¼ Ck eikr ð26Þ k
This general expansion can be related to the Bloch form by substituting k with kG, which gives X CkG eiGr ð27Þ ðrÞ ¼ eikr G
Compared Eq. (25) with Eq. (27), one obtains X uk ðrÞ ¼ CkG eiGr G
ð28Þ
Substituting Eqs. (24), (26) and (27) into Eq. (22), the Schro¨dinger equation is rewritten in the form as ! X X h2 k 2 Ck þ UG CkG ECk eikr ¼ 0 ð29Þ 2m e k G Hence, 2 2 X h k E Ck þ UG CkG ¼ 0 2me G
ð30Þ
This is the most important equation in the electronic band theory of crystals. When U(r):0, Eq. (30) reduces to Ek0 ¼ h2 k2 =2me for the case of free electrons. For the energy levels near the first Brillouin zone boundaries, Eq. (30) reduces to the following two equations: ðE Ek0 ÞCk ¼ UG CkG
ð31aÞ
0 ÞCkG ¼ UG Ck ¼ UG Ck ðE EkG
ð31bÞ
and
where UG is the complex conjugate of UG. These equations have solutions only when E E0 UG k ¼0 ð32Þ U E EkG G
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The two roots are then obtained as 0 E 0 þ EkG E¼ k 2
" #1=2 2 0 Ek0 EkG þjUG j2 2
ð33Þ
Equation (33) gives the dominant effect of the periodic potential on the 0 , the result is energy near the zone edge. At the zone edge, Ek0 ¼ EkG particularly simple, i.e., E ¼ Ek0 jUG j
ð34Þ
As shown in Fig. 8, a band gap of 2jUGj is formed at the first Brillouin zone edge. The corresponding wavefunctions at the zone edge are two standing waves that have the following forms [21,25] 1 ðrÞ
/ cos ð0:5G rÞ
and
2 ðrÞ
/ sin ð0:5G rÞ
ð35Þ
The lower energy state Ek0 jUG j corresponds to 1 with probability density j 1j2 peaked at core sites. This state piles up charges at the core sites. The upper energy state Ek0 þ jUG j corresponds to 2 with probability density j 2j2 that localizes charges between the cores. The energy difference between these two states is the origin for the formation of the gap at the zone edge (see Fig. 9). When all the Brillouin zones and their associated Fourier components are included, the result is a set of curves as those shown in Fig. 10(a). This particular way of depicting the energy levels is known as the extended-zone scheme. It is useful to plot all the energy levels in the first Brillouin zone.
FIG. 8. Plot of the energy bands given by Eq. (33). The lower band corresponds to the choice of the minus sign, and the upper band to the plus sign. When k ¼ G/2, the two bands are separated by a gap of magnitude 2jUGj.
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193
FIG. 9. Probability density j j2 for bands near the Brillouin zone edge.
FIG. 10. Electronic band structure: (a) the extended-zone scheme; (b) the reduced-zone scheme.
This is done by translating (folding) the branches in Fig. 10(a), using the reciprocal lattice vector, as shown in Fig. 10(b). This representation of electronic bands is known as the reduced-zone scheme. b. Electronic Band Structures for Semiconductors Due to the complicated three-dimensional structure, the actual electronic band structures for semiconductors are much more complicated than that shown in Fig. 10. The electronic band structures of Si and GaAs in the first Brillouin zone are shown in Figs. 11 and 12, respectively, along reciprocal lattice directions L– –X–U (or K)– (see Fig. 7). Si and GaAs are chosen here because these two types of semiconductors have distinctly different energy gap features and are representative for a wide range of semiconductor materials. Degeneracy causes additional subbands within the conduction band and the valence band. The energy is relative to the top of the valence band (i.e. Ev is set to zero in the plot). The valence band is formed by the bonding orbitals of the valence electrons, whereas electrons in
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FIG. 11. Calculated energy band structure of silicon, after Refs. [25,26].
FIG. 12. Calculated energy band structure of GaAs, after Refs. [25,26].
the conduction band are dissociated from the atom and hence become free charges. The energy gap, Eg, is defined as the energy difference between the top of the valence band (Ev) and the bottom of the conduction band (Ec). For Si, as shown in Fig. 11, the bottom of the conduction band and the top of the valence band do not occur at the same value of k. This type of semiconductor is called an indirect gap semiconductor. For a direct gap semiconductor, such as GaAs, the bottom of the conduction band and the top of the valence band occur at the same value of k at the -point, as shown
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195
in Fig. 12. The mechanisms for electron transition between the valence band and the conduction band in a direct gap semiconductor are completely different from that in an indirect gap semiconductor. At absolute zero temperature, there are no electrons in the conduction band and the valence band is completely filled. When the temperature increases or there exist optical excitations, electrons in the valence band can transit to the conduction band, leaving behind some vacancies in the valence band. The vacancies left in the valence band are called holes, which have the same mass but opposite charge as electrons. Usually the electrons are found almost exclusively in levels near the conduction band minima, while holes are found in the neighborhood of the valence band maxima. Therefore, the energy versus wavevector relations for the carriers can generally be approximated by quadratic forms in the neighborhood of such extrema, i.e. Ee ðkÞ ¼ Ec þ
2 k2 h 2m e
and
Eh ðkÞ ¼ Ev
h2 k2 2m h
ð36Þ
where subscript e and h are for electrons and holes, respectively. The effective mass m* for electrons and holes are defines as, 1 m e ðor hÞ
¼
1 d2 Ee ðor hÞ h2 dk2
ð37Þ
The physical significance of the definition of the effective mass is that, when an electron or a hole is related to an external field E, the subjected forces (accelerations) can be expressed as: ae ¼ eE=m e
and
ah ¼ eE=m h
ð38Þ
where e is the electron charge. The values of the energy gap and the effective mass for some common semiconductors are listed in Table II. There are longitudinal effective mass and transverse effective mass of electrons in the conduction band for silicon. This is because the constant energy surface of silicon is not spherical but ellipsoidal near the minimum of the conduction band, which gives rise to different curvature along the cube axis and perpendicular to the axis. Thus the effective mass along the axis is called longitudinal effective mass while that perpendicular to the axis is called transverse effective mass. The valence band maximum occurs at k ¼ 0 ( point in Fig. 11), where two degenerate subbands with different curvatures meet, giving rise to the so-called ‘‘light holes’’ and ‘‘heavy holes’’. It should be noted there is another subband just below the valence band maximum with a very small energy difference between them, as shown in Fig. 13 [25].
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FIG. 13. Illustration of the subband structure for a direct gap semiconductor.
This is due to spin–orbit coupling and the holes in this band are called the ‘‘split-off holes’’. c. Number Density of Carriers in Semiconductors Because the number density of carriers in a semiconductor determines the electrical and thermal radiative properties of the material, the calculation of the number density of electrons and holes at any temperature T is thus a very important procedure in semiconductor materials. We know that electrons obey the Fermi-Dirac distribution function, which is given as [21]: fe ðEÞ ¼
1 eðEF Þ=kB T
ð39Þ
þ1
where F denotes the Fermi level and kB is Boltzmann’s constant. The distribution function fh for holes is related to fe by fh ¼ 1fe. The number density of carriers at temperature T will be given by ne ¼
Z
1
Ec
De ðEÞdE and ðE F Þ=kB T þ 1 e
nh ¼
Z
Ev 1
Dh ðEÞdE ð F e EÞ=kB T þ
1
ð40Þ
where De(E) and Dh(E) are the densities of states in the conduction and valence bands, respectively. With the approximated quadratic forms of the conduction and valence bands, Eq. (36), the densities of states can be written as
1=2 1 2m e 3=2 E Eg De ðEÞ ¼ 2 2 2 h
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197
1 2m h 3=2 Dh ðEÞ ¼ 2 ðE Þ1=2 h2 2p
ð41Þ
and
Except at very high temperatures, EcF kBT and FEv kBT are satisfied. In this case fe and fh can be simplified as, fe ðEÞ eðF EÞ=kB T
and
fh ðEÞ eðEF Þ=kB T
ð42Þ
The number density of carriers may be obtained by integration of Eq. (40) as m kB T ne ¼ 2 e 2 2p h
3=2
eðF Eg Þ=kB T
and nh ¼ 2
mh kB T 3=2 F =kB T e 2p h2
ð43Þ
The combination of the above equations gives
kB T ne nh ¼ 4 2p h2
3
ðm e m h Þ3=2 eEg =kB T
ð44Þ
This expression does not involve the Fermi level F. Therefore, it holds for both intrinsic and doped semiconductors. Note that ne ¼ nh in an intrinsic semiconductor and Eq. (44) gives the number density of electrons or holes. It can be seen that the number density increases with temperature and it is expected that electronic contributions become stronger at higher temperatures.
B. PHONONS In the above discussions of the electronic band structures of semiconductors, it is assumed that the cores of atoms are fixed. In real crystal, however, the cores of atoms are vibrating about their equilibrium positions and the vibration of atoms has important influence on energy storage and transport in crystals. Lattice vibration causes elastic waves to propagate in crystalline solids. Phonons are the energy quanta of lattice waves. For a given vibration frequency !ph, the energy of a phonon, given by h!ph, is the smallest discrete value of energy. Thermal vibrations in
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Z. M. ZHANG ET AL.
crystals are thermally excited phonons, like the thermally excited photons in a blackbody cavity. The phonon dispersion relation or the relationship between the vibration frequency and phonon wavevector can be obtained with a simple model of linear spring-mass arrays, as shown in Fig. 14. Spring represents chemical potential between ions; mass denotes individual ions of the lattice. In a onedimensional situation, one can identify the force imposed on individual ions using the relative displacements between them. Monatomic lattice vibrations exhibit a relationship of the following form [21–23]:
!ph ¼
1=2 4C sin kph a ; M 2
p p kph a a
ð45Þ
where !ph denotes vibration frequency, kph is the magnitude of the phonon wavevector, C is the spring constant, and M is the mass of an individual ion. The range of values of kph is restricted to the first Brillouin zone. Because wavevectors outside the first Brillouin zone merely reproduce lattice motions already described by values of kph in the first Brillouin zone. In fact, due to the periodicity of the mathematical solution in terms of kph, we may treat a value of kph outside the first Brillouin zone by subtracting the appropriate integral times of the reciprocal lattice constant, 2/a, to give a value of kph inside the limits of the first Brillouin zone. As a result, the wavelength is given by,
ph ¼
2p ; kph
2a ph < 1
FIG. 14. The spring-mass representation of lattice vibration.
ð46Þ
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199
At the boundaries k ¼ /a of the Brillouin zone the lattice vibrations form standing rather than propagating waves. When the lattice has more than one atom per primitive cell, the vibrational spectrum displays additional features. If there are two atoms per primitive cell, lattice vibrations for each polarization mode in a given propagation direction will have two possible solutions to !ph which is governed by the following equation [21] 2C M1 !2ph Cð1 þ eikph a Þ
Cð1 þ eikph a Þ ¼0 2C M2 !2ph
ð47Þ
where M1 and M2 represent the masses of the two atoms in the primitive cell. It is assumed that each plane interacts only with its adjacent planes and force constants are the same between any adjacent planes. For the limiting case when kpha < < 1, since cos ðkph aÞ ¼ 1 k2ph a2 þ , the two roots of Eq. (47) are 1 1 2 ; 2a< < ph < 1 ðoptical branchÞ !ph ffi 2C ð48Þ þ M1 M 2 !2ph ffi
C=2 k2 a2 ; 2a< < ph < 1 M1 þ M2 ph
ðacoustic branchÞ
ð49Þ
2C M2
ð50Þ
For kpha ¼ , the two roots to Eq. (47) are !2ph ¼
2C M1
and
!2ph ¼
The above discussion can be extended to three-dimensional systems. In three-dimensions, the lattice vibrations allow for both transverse and longitudinal modes. For the case of two atoms per primitive cell, there are one longitudinal branch and two transverse branches for both acoustic and optical vibration modes. In general, if there are p atoms in the primitive cell, there will be one longitudinal and two transverse acoustic branches; p1 longitudinal and 2p2 transverse optical branches. However, degeneracy of the transverse branches may occur due to symmetry [21–23]. The relationship of !ph and kph for diamond is shown in Fig. 15. The longitudinal optical (LO) and longitudinal acoustic (LA) branches in the X h100i direction illustrate the phonon dispersion relations described in Eqs. (47) to (50), which present the case of longitudinal vibrations in one dimension. Because M1 ¼ M2 for diamond, the two roots given in Eq. (50) are identical so that the LO and LA branches meet at the zone edge in the X direction.
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Z. M. ZHANG ET AL.
FIG. 15. Optical and acoustical branches of the dispersion relation for a three-dimensional diamond lattice.
If M1 6¼ M2, such as in the case of zinc blende lattice, the two roots in Eq. (50) will be different, resulting in a frequency gap between the LO and LA branches at the zone edge. The frequency gap is forbidden for propagating waves. The wavevector kph must be complex for real !ph, and the wave is damped in space. The group velocity is the velocity of energy transmission in the lattice, which is expressed as vph ¼
d!ph dkph
ð51Þ
It can be seen from Fig. 15 that the group velocity of the optical branch is very small; thus lattice vibrations according to the optical branch transmit only a small portion of energy. However, this type of vibration has significant influence on the radiative properties of the material and is therefore termed optical branches. According to the wave-particle duality, a phonon with energy h!ph will carry a momentum of pph which is given by pph ¼ hkph
ð52Þ
where kph is the wavevector and the value of which is determined by the relationship of !ph kph discussed above. In an energy transport process the energy of phonons absorbed or emitted is only an appropriate integral multiple of h!ph. Because a lattice can be taken as a collection of oscillators,
OPTICAL AND THERMAL RADIATIVE PROPERTIES
201
and the total vibrational energy of an oscillator for each vibration mode is given by 1 "ph ð!ph Þ ¼ nph þ h!ph 2
ð53Þ
where nph is the number of phonons for the particular vibration mode. The term 12 h!ph is the zero point energy. In order to find the total energy in the lattice, Eq. (53) has to be summed over all possible vibration modes. Because phonons are bosons, the thermal equilibrium distribution of the number of phonons of any given mode at energy h!ph and temperature T is given by Bose-Einstein distribution: nph ¼
1 expð h!ph =kB TÞ 1
ð54Þ
The total energy in thermal equilibrium of the collection of oscillators is then given by Eph ¼
X k
h!ph;k þ zero point energy expð h!ph;k =kB TÞ 1
ð55Þ
If the volume of the lattice is large enough, it is convenient to replace the summation in Eq. (55) by an integral. By introducing the phonon density of state D(!ph), i.e. there are D(!ph)d!ph modes of vibration in the frequency range !ph to !ph þ d!ph, the total energy is Eph ¼
Z
Dð!ph Þ h!ph d!ph þ zero point energy expð h!ph;k =kB TÞ 1
ð56Þ
The Debye model assumes that the phonons have a linear dispersion relation over all possible values of kph such that !ph ¼ vskph and thus Dð!ph Þ ¼ 3!2ph =ð2p2 v3s Þ, where vs is the sound velocity. In this linear approximation it is necessary to set an upper integration limit for Eq. (56) by a bounded frequency !D (Debye frequency) so that the number of atoms or oscillators Nph is fixed for a finite volume V. The total number of oscillators is given by Nph
V ¼ 3
Z
!D 0
Dð!ph Þd!ph ¼
!3D 2p2 v3s
ð57Þ
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Z. M. ZHANG ET AL.
The specific heat due to lattice vibration can be evaluated as follows: Z 1 @Eph 9nph kB T 3 D =T x4 ex dx cv ¼ ¼ 3 V @T V D ðex 1Þ2 0
ð58Þ
where nph ¼ Nph/V is the oscillator number density, and D is the Debye h!D/kB. It can be shown from Eq. (58) that temperature defined as D ¼ above the Debye temperature D, the specific heat cv approaches to a constant value equal to 3nphkB ¼ 3R, where R is Avogadro’s constant. When T is well below D, cv obeys the T 3 law as cv
3 12p4 T nph kB D 5
ð59Þ
Lattice vibrations dominate the specific heat of solids, and electronic contribution is important only at very low temperatures (below a few kelvins).
C. SCATTERING OF ELECTRONS AND PHONONS Electron scattering and phonon scattering are important for the transport properties and optical properties. Kinetic theory based on the Boltzmann transport equation (BTE) gives a complete description of the transport phenomena [15,24]. The BTE is an integro-differential equation of the distribution function in terms of space, velocity, and time. It takes into account changes in the distribution function caused by external forces and collisions between particles. The average scattering rate (number of collisions a particle experiences per unit time), mean free path (average distance between subsequent collisions), and average speed are important parameters for the first-order solution, i.e. the relaxation time approximation in solving the BTE. In this section, the phonon and electron scattering mechanisms will be briefly reviewed. More detailed discussion can be found in Refs. [21–24]. 1. Phonon Scattering a. Phonon–Phonon Scattering Phonon scattering governs the thermal transport properties of dielectric crystals. In a real insulator the thermal conductivity is of a finite value. From the kinetic theory, this means that the mean free path of phonons in a crystal is of a finite value. The anharmonic nature of the interatomic potential offers a coupling mechanism for phonon–phonon interactions,
OPTICAL AND THERMAL RADIATIVE PROPERTIES
203
which was not included in the above-discussed linear oscillator model. The phonon–phonon scattering is inelastic because the phonon frequency is different before and after the scattering event. There are two types of phonon–phonon interactions, namely, the normal (N) process and the umklapp (U) process. In an N process, either two phonons interact to form a third phonon or one phonon breaks up into two others. The energy and momentum of the phonons are conserved as given by [21–23] h! 1 þ h! 2 ¼ h! 3
or
h!1 ¼ h!2 þ h!3
ð60Þ
hk1 þ hk 2 ¼ hk 3
or
hk1 ¼ hk2 þ hk3
ð61Þ
In Eqs. (60) and (61), the left-handed terms are for phonon(s) before scattering and the right-handed terms are for phonon(s) after scattering. Since both the energy and momentum are conserved, N processes do not alter the direction of energy flow. Hence, N processes make no contribution to the thermal resistance (infinite thermal conductivity). In a U process, two phonons interact to form a third one, and the process satisfies Eq. (60) for energy conservation; however, the conservation of momentum puts the wavevector of the resultant phonon outside the first Brillouin zone. In order to bring the wavevector back into the first Brillouin zone, a reciprocal lattice wavevector G must be involved. The momentum relationship needs to be modified as (in terms of wavevectors) [21–23] k1 þ k2 ¼ k3 þ G
ð62Þ
Therefore, the net momentum is not conserved in the U processes, causing the thermal conductivity to be finite. Figure 16 schematically shows the relationships of wavevectors for an N process and a U process. b. Elastic Phonon Scattering In addition to the phonon–phonon interactions, phonons may also interact with defects (such as impurities, vacancies, or dislocations) and boundaries. These scattering processes may also influence the mean free path of phonons. Scattering of phonons by defects is elastic since the phonon frequency remains the same. At temperatures greater than the Debye temperature, phonon–phonon interactions are dominant. As the temperature drops, the wavelengths of phonons become comparable to the size of defects and the scattering of phonons by defects is important. When the bulk mean free path is comparable or greater than the characteristic dimension, such as the thickness of the film or diameter of the wire, scattering of phonons by boundaries becomes important. Boundary
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FIG. 16. Schematic illustrations of phonon–phonon scattering processes.
scattering is important for nanostructure materials and at low temperatures, when the phonon mean free path is large. Following Matthiessen’s rule, the scattering rate can be summed for all contributions, including phonon–phonon scattering, phonon–defect scattering, and phonon–boundary scattering. Quantitative discussions can be found from Refs. [23,24]. The effect of defect and boundary scattering of phonons on the thermal conductivity of thin films have been reviewed by Chen [29], Cahill [30], and Goodson [31]. 2. Charge Carrier Scattering The scattering of charge carriers controls the electric conduction in solids and dominates the thermal conduction in metals. It is also important for optical absorption by carriers, especially in the mid-infrared region. The scattering processes include carrier–carrier scattering, carrier–phonon scattering, carrier–defect scattering, and carrier–boundary scattering, as illustrated in Fig. 17. Carrier–carrier inelastic scattering is negligible except for highly conductive materials, such as in a high-temperature superconductor [32,33]. Therefore, carrier–carrier scattering will not be discussed further. Since lattice vibrations are enhanced with increasing temperature, electron–phonon scattering usually dominates the scattering process at high temperatures, while at low temperatures lattice vibrations are weak and defect scattering becomes important. Both the acoustic branch and optical branch can scatter electrons. A brief summary of the electron scattering processes is given below. More detailed discussions can be found from Refs. [23,24,34,35].
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205
FIG. 17. Schematic illustration of carrier scattering by phonon, defect, and boundary.
a. Carrier Scattering by Phonons The vibration of lattice ions causes deviations from the perfect periodic lattice and distorts the carrier wavefunction. This is more easily visualized as the scattering of electrons by phonons. But for phonon scattering, the mean free path of carriers would be infinite in a defect-free crystal. Carrier scattering is responsible for the electrical resistance. The energy and momentum conservations for carrier–phonon scattering can be written as Ef ¼ Ei h!ph
and
kf þ G ¼ ki kph
ð63Þ
where subscripts i and f indicate the initial and final states of the carrier, and the minus sign corresponds to phonon emission and plus sign corresponds to phonon absorption. If G is set to zero, the process is an N process, otherwise, it is a U process as in phonon–phonon scattering. In semiconductors at low temperatures, only N processes are energized. Typical electron–phonon scattering rate at room temperature ranges from 1012 to 1013 Hz. There are two types of phonons, the acoustic phonon and the optical phonon, that can scatter electrons but with different mechanisms. Usually, the energy of acoustic phonons can be neglected compared with the electronic energy. Therefore, scattering by acoustic phonons is elastic. Scattering by optical phonons is inelastic because the exchange of energy between the carriers and phonons can be significant. This process facilitates the energy transfer between electrons and phonons, which is associated with Joule heating. For materials with two different atoms per primitive cell, the asymmetric charge distribution in the chemical bond forms a dipole. Scattering by optical phonons in these materials is called polar scattering, which can effectively scatter electrons or holes.
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Z. M. ZHANG ET AL.
b. Carrier Scattering by Defects An electron or hole in a periodic lattice does not really collide with the ions. The transport of free carriers can be viewed as the propagation of a wave in a periodic potential created by the ions. In addition to lattice vibrations, defects or impurity may break the periodicity of the potential or alter its amplitude. Kinetic theory gives the defect scattering rate d as d ¼
1 ¼ nd d v e d
ð64Þ
where d is the relaxation time, nd and d are the defect number density and cross-sectional area, and ve is the average carrier velocity. All these parameters are functions of the wavevector. For metals, the electron velocity is the Fermi velocity, on the order of 106 m/s. For semiconductors, the velocity of thermally excited electrons or holes can be calculated by ve ¼ ð3kB T=m Þ1=2
ð65Þ
which is on the order of 105 m/s. The calculation of the scattering crosssectional area is more complex and detailed discussion can be found from Ferry [35]. c. Carrier Scattering by the Boundary Scattering at the interface or grain boundaries in a polycrystalline solid may be important when the characteristic dimension is comparable to or less than the mean free path of carriers. Extensive studies have been devoted to the size effect on electrical, thermal, optical and mechanical properties, mostly for metals and superconductors [36–38].
D. ABSORPTION AND EMISSION PROCESSES 1. Absorption Processes The presence of energy band gap in semiconductors gives rise to very different absorption spectra compared to other materials. Figures 18 and 19 show the absorption spectra of a lightly-doped Si and a pure GaAs, based on the extinction coefficient values of Palik [39]. The absorption spectra are consequence of several absorption mechanisms [40,41]. At short wavelengths, the photon energy is large enough to excite electrons from the valence band to the conduction band and leave behind holes in the valence band. This type of absorption is called interband transition and is very strong, resulting in a very large absorption coefficient. Because only photon
OPTICAL AND THERMAL RADIATIVE PROPERTIES
207
FIG. 18. The absorption spectrum of lightly doped Si at room temperature, data from Refs. [39,40].
FIG. 19. The absorption spectrum of pure GaAs at room temperature, data from Ref. [39].
energies greater than the band gap can be absorbed and the quantum efficiency increases with photon energy, a sharp absorption edge is formed. Therefore, the presence of the absorption edge in the absorption spectrum of a semiconductor provides a direct and perhaps the simplest method for
208
Z. M. ZHANG ET AL.
measuring the energy gap of a semiconductor. As the wavelength further increases beyond the absorption edge, the absorption coefficient could be affected by the existence of impurities and defects, free-carriers absorption, intraband or intersubband transitions by electrons and holes, and absorption by lattice vibrations, etc. For GaAs, which is a direct band gap semiconductor, the absorption edge is even sharper. The free-carrier absorption tends to increase gradually with wavelength, as seen from Fig. 18. Carrier contributions are negligible for a pure semiconductor unless the temperature is very high. The absorption mechanisms can be well explained using quantum mechanical theory [42]. In the following the major absorption mechanisms of semiconductors are described without complicated derivations [41]. a. Interband Transitions Interband transition is also called the fundamental absorption process, and it can be further classified into two types. One is direct interband transition; the other is indirect interband transition. As shown in Fig. 20(a), the lowest point of the conduction band occurs at the same value of k as the highest point of the valence for a direct gap semiconductor. An electron can be excited from the top of the valence band into the bottom of the conduction band by absorbing a photon of energy at least equal to the gap energy, Eg. In this transition process, the energy and momentum conservation can be expressed as [21] h! and Ef ¼ Ei þ
k f ¼ ki þ q
ð66Þ
where ! and q are the frequency and wavevector of the photon; i and f denote the initial and final states of the electron, respectively. Because the speed of light is much greater than the Fermi velocity of an electron, we can expect that the momentum of the photon to be negligible compared to that
FIG. 20. (a) Direct interband transition occurs in direct gap semiconductors. (b) Indirect interband transition involves the emission (process 1) or absorption (process 2) of a phonon.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
209
of the electron. In other words, the momentum of the electron is almost unchanged after the transition, i.e. kf ki. This kind of transition is called direct interband transition. When the valence band and the conduction band are parabola-like, the absorption coefficient due to band gap absorption can be expressed as [41]: aBG ¼ Að h! Eg Þ1=2
ð67Þ
where A is a parameter that depends on the effective masses of the electrons and holes and the refractive index of the material. It should be noted that Eg is temperature dependent and decreases as temperature increases. When a transition requires a change in both energy and momentum as in the case of an indirect gap semiconductor, see Fig. 20(b), a phonon is either emitted or absorbed for momentum conservation because the photon itself cannot provide a change in momentum. The energy and momentum conservation equations are [21] h! ¼ Ef Ei h!ph
and
q ¼ kf ki kph
ð68Þ
where the minus and plus signs correspond to phonon absorption and emission, respectively. This kind of transition is called indirect interband transition. With the involvement of phonons, the absorption coefficient is given as [41,42] aBG ð!Þ ¼ aa ð!Þ þ ae ð!Þ
ð69Þ
where aa and ae are the absorption coefficients for transitions with phonon absorption and emission, respectively, and are expressed below: aa ð!Þ ¼
Að h! E g þ h!ph Þ2 ; expð h!ph =kB TÞ 1
h! > Eg h!ph
ð70Þ
ae ð!Þ ¼
Að h! E g h!ph Þ2 ; 1 expð h!ph =kB TÞ
h! > Eg þ h!ph
ð71Þ
and
Clearly, aa (or ae) is nonzero only when the photon energy is greater than the band gap subtracted (or added) by the phonon energy. There may be several types of phonon-assistant interband transitions, and their effects on the absorption coefficient can be superimposed. Interband transition produces an electron–hole pair when a photon of energy greater than the energy gap Eg is absorbed in a crystal. The electron
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Z. M. ZHANG ET AL.
FIG. 21. The observed exciton absorption in GaAs for as the photon energy is very near the gap energy Eg [21,25].
and hole produced in this way are free and may move independently through the crystal. However, the attractive coulomb interaction between an electron–hole pair makes it possible to form a stable bound state of the two particles. This bound electron–hole pair is known as an exciton. The formation of excitons usually appears as narrow peaks in the absorption edge of direct gap semiconductors, or as steps in the absorption edge of indirect gap semiconductors. In direct gap semiconductors the exciton occurs when the energy is less than the gap energy Eg by the binding energy of the exciton Ex, i.e., h! ¼ Eg Ex . On the other hand, in indirect gap semiconductors phonon participation is needed to conserve momentum. Thus the free exciton occurs when the photon energy satisfies the following relation [21,25] h!ph h! ¼ Eg Ex
ð72Þ
Here again, the minus and plus signs denote the transition with phonon absorption and emission, respectively. An example of exciton absorption can be seen in Fig. 21. It is clear that the presence of exciton absorption contributes a component to the absorption coefficient in the region of bandto-band transitions. b. Transition between a Band and an Impurity Level When a semiconductor material is not pure or is doped with other elements, the impurities will make energy levels possible to locate inside the
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energy gap such as the acceptor level of a p-type semiconductor or the donor level of an n-type semiconductor. Thus the transition between the valence band and an ionized donor level or between an ionized acceptor and the conduction band may occur when a photon of energy h! > Eg Ei is absorbed, where Ei represents the ionization energy of the impurity [25,41]. Transitions between an impurity level and a band include the whole band levels, causing a shoulder in the absorption edge at the threshold lower than the energy gap by a value of energy Ei. In practice, the density of impurity states is much lower than the density of states in the bands, the absorption coefficient involving transitions between a band and an impurity level covers a much smaller range than the fundamental transitions. It should be noted that transitions between a neutral donor level and the conduction band or between the valence band and a neutral acceptor level might occur by the absorption of a low-energy photon. But the involved photon energy corresponds to the far-infrared region where the absorption by free carriers and by lattice vibrations is dominant. c. Intraband Transitions Intraband transitions include intersubband transitions by holes in the valence band and intersubband transitions by electrons in the conduction band. By spin–orbit interaction the valence band of most semiconductors is split into three subbands that are called heavy-hole band, light-hole band and split-off band, respectively, as illustrated in Fig. 13. In p-type semiconductors when the top of the valence band is populated with holes, three types of photon-absorbing transitions by holes are possible: (a) from the heavy-hole band to the light-hole band; (b) from the heavy-hole band to the split-off band; and (c) from the light-hole band to the split-off band. In general, the absorption is proportional to the hole density. A detailed discussion of intersubband transition in p-type semiconductors and comparisons of calculated and experimental reflectance spectra can be found in Ref. [43]. As for an n-type semiconductor, intraband transitions are by electron transitions between the conduction subbands. The general feature of an intraband transition in the absorption spectrum is represented by a peak with low-energy threshold for the case of direct transition and by a bump for the case of indirect transition. Both the peak and bump are adjacent to the rapid rise of the absorption curve due to free-carrier absorption [41]. d. Free-Carrier Absorption Free-carrier absorption occurs when the photon energy is absorbed by free electrons or holes. The absorption of a photon makes an electron or a
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hole transit to a higher energy state within the same band. The carrier must also collide with ionized impurities or phonons in order to conserve momentum. The collision with impurities acts as a damping force on the motion of the free carrier. The free-carrier absorption is broadband and generally increases with the wavelength. The Drude model, which is based on oscillation of an electron driven by a harmonic field and subject to a damping, predicts that the absorption coefficient is proportional to the square of the wavelength. However, the wavelength dependence of absorption is different for different scatterers. Pankove [41] summarized that scattering by acoustical phonons leads to a wavelength dependence of the absorption coefficient as 1.5, scattering by optical phonons gives a dependence of 2.5, and scattering by ionized impurities gives rise to a dependence of up to 3.5. When all the three scattering modes exist, the resultant free-carrier absorption coefficient af is the summation of the three processes, i.e., f ¼ A 1:5 þ B 2:5 þ C 3:5
ð73Þ
where A, B, and C are constants. The dominant mode of scattering depends on the impurity concentration. If the wavelength dependence of the absorption coefficient is expressed as p, the exponent p will range from 1.5 to 3.5 and increases with doping. e. Absorption by Lattice Vibrations Absorption by lattice vibrations is due to the existence of electric dipoles formed by the atoms of different species in compound semiconductors. A maximum absorption is achieved when the frequency equals the vibrational mode of the dipole, which is usually in the mid- to far-infrared region of the spectrum. Therefore, the momentum of a photon h/ is negligible compared with that of a phonon, which can be as large as h/a, where a is the lattice constant. Two or more phonons must be emitted to ensure the momentum conservation (multiphonon emission) in the absorption process [41]. In general, the number of lattice vibrational modes of semiconductors is six (since each transverse wave has two polarization states), the complex structure of the lattice absorption spectrum as seen in Figs. 18 and 19 can be explained by the multiplicity of the possible combinations of all the six modes. In some semiconductors such as silicon, the bonding is purely covalent; the observed lattice vibrational spectrum is caused by second order process: the perturbation of the lattice by thermal vibrations can produce a dipole moment which is capable of interacting with radiation and producing more phonons.
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The oscillator model predicts the phonon contribution to the dielectric function as follows. "~ð!Þ ¼ ðn þ iÞ2 ¼ "1 1 þ
!2LO !2TO !2TO !2 i !
ð74Þ
For GaAs, the longitudinal and transverse phonon frequencies are ! LO ¼ 292 cm1 and ! TO ¼ 269 cm1, the damping coefficient ¼ 2.4 cm1, and the high-frequency term "1 ¼ 11 [39]. These parameters can well describe the phonon absorption peak at 37 mm (269 cm1) shown in Fig. 18. Some impurities can form chemical bond with the semiconductor atoms. This is the case of oxygen in Si. The Si–O bond has a molecular vibration mode at about 9 mm, causing observable absorption depending on the concentration of interstitial oxygen, which is more prominent at low temperatures [44,45]. f. Temperature Effect There have been relatively few studies on the optical properties of semiconductors at high temperatures. The radiative properties at high temperatures are important for real time monitoring during materials processing, such as RTP, MBE, epitaxial chemical vapor deposition (CVD), etc. [46–56]. Temperature has a strong influence on the absorption processes. The band structure changes with temperature. Specifically, the energy gap becomes narrow with increasing temperature, causing a shift in the fundamental absorption edge towards longer wavelength. The temperature dependence of the energy gap can be expressed as [40,57] Eg ðTÞ ¼ Eg ð0Þ AT 2 =ðT þ BÞ
ð75Þ
where A and B are positive constants, Eg is expressed in eV, and T is in K. For Si, Eg(0) ¼ 1.155 eV, A ¼ 4.73 104, and B ¼ 635 K [58]; For GaAs, Eg(0) ¼ 1.52 eV, A ¼ 8.87 104, and B ¼ 572 K [59]. At high temperatures, thermally excited electrons and holes can greatly increase the carrier concentration; see Eq. (43). Free-carrier absorption is enhanced at elevated temperatures, even for lightly doped semiconductors. This can be seen from the increased opacity at wavelengths beyond the absorption edge and the increased spectral emissivity at high temperatures. Lattice vibrations are also enhanced as temperature rises, due to the increased phonon population. The phonon frequency may also change, causing the absorption peak to shift. The effects of impurities and defects are also temperature dependent. In most cases, the enhanced carrier absorption
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tends to dominate the infrared behavior and screen the phonon and impurity effects. Measurements of the radiative properties at high temperatures are difficult to perform and the data are rather limited. Some references on the radiative properties of semiconductors and related materials at elevated temperatures can be found from Refs. [40,58–71]. Some researchers showed that the photons emitted by the lamp during RTP could cause the carrier concentration to deviate from the thermal equilibrium value; and hence, induce a change in the emissivity and absorptivity of lightly doped Si wafers [72–74]. 2. Radiative and Non-radiative Recombinations Most of the transitions discussed above that cause absorption can occur in the opposite direction and give off photons. In essence, radiation emission is the inverse process of absorption. At thermal equilibrium, the rate of photon emission is equal to the rate of photon absorption. The process of light emission is called luminescence. For a material to give off light (optical radiation) in certain spectral region, the system must not be at equilibrium. The deviation from equilibrium can be created by some sort of excitation. Excitation can be created by the absorption of optical radiation (resulting in photoluminescence) or by passing through an electrical current (resulting in electroluminescence). Fluorescence and phosphorescence refer to luminescence that occurs during or after an optical excitation. The emission spectrum is generally different from the incident spectrum. For direct interband transition, the absorption of a photon generates an electron–hole pair, which can recombine to create a photon. Electronic transitions from a higher energy state to a lower energy state within the conduction band or the valence band are often called relaxation. Not all the recombination or relaxation processes give off photons. The so-called non-radiative recombination often generates phonons that cause heat dissipation. There are two types of emission, spontaneous and stimulated emission. A photon of energy h! traversing a semiconductor can stimulate a transition between two energy levels E1 and E2, whose energy difference is E2E1 ¼ h!. Two types of transition can occur: 1 ! 2, the absorption of h!12 ¼ photon generates an electron–hole pair; and 2 ! 1, the pair recombines to emit a photon. When the probability of emission is greater than that of absorption, the traversing photon can induce the emission of another photon (transition 2 ! 1), as shown in Fig. 22a. It is also possible for the emission to occur without apparent provocation, as shown in Fig. 22b. There is a fundamental difference between stimulated and spontaneous emission. In a stimulated emission, the emitted electromagnetic wave is in
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215
FIG. 22. Illustration of stimulated transition and spontaneous transition.
phase with and travels in the same direction as the incident photon. On the other hand, the wavevector for spontaneous emission can be in any direction. Stimulated emission is an important process for lasing, which requires phase coherence. Some representative processes related to radiative or non-radiative recombination are summarized in the following. More detailed discussion can be found from Pankove [41] and Svelto [75]. a. Radiative Transitions In a sufficiently pure semiconductor, excitons are formed from paired electrons and holes, which may recombine to emit a narrow spectral line. The photon energy is given by, h! ¼ Eg Ex mEph
ð76Þ
where m is the number of optical phonon emitted per transition. Note that m can be zero for a direct transition and must be equal to or greater than one for an indirect transition. The larger the m is, the smaller the transition probability and the weaker the corresponding emission line. The interband transition refers to the electrons from conduction band to valence band. The physical mechanisms are the same as the absorption processes described in Eqs. (66) to (71). The result is a broadband emission for photon energies less than Eg (or EgEph in an indirect semiconductor). Generally speaking, the rate of transition is proportional to the product of the density of the carrier in the upper energy state, the density of empty lower states, and the probability for the carrier in the upper energy state to make a radiative transition to the lower energy state. Self-absorption is coupled to the emission process and modifies the emission spectrum. Intraband transitions happen at much longer wavelengths because the difference between the upper and lower energy states is small. In an earlier work, radiative intravalence band transitions were observed in Ge, by creating a non-equilibrium distribution between light and heavy holes with an electric field [41]. More recently, researchers have used multiple quantum
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FIG. 23. Transition without an intermediate state (left) and with an intermediate state (right).
wells to produce a new type of laser, i.e. quantum cascade laser, which is based on band structure engineering. The transitions occur between discrete electronic states within the conduction band. Discrete energy levels arise from the quantization of electron motion in the active region. Continuouswave mid-infrared lasers have been demonstrated at room temperature, with the laser wavelength ranging from 4 to 24 mm [76–79]. The transition from an upper energy state to a lower energy state may proceed through several intermediate states, which may or may not be radiative, as shown in Fig. 23. The efficiency of radiative transition may be significantly affected by the non-radiative transitions. b. Non-radiative Recombination Auger effect and multiphonon emission are two common non-radiative processes. In the Auger effect, the energy released by a recombining electron is absorbed by another electron, which subsequently relaxes to dissipate energy by the emission of phonons. The process is shown in Fig. 24(a). Non-radiative transition may occur by emitting a cascade of phonons, since the energy of a phonon is much smaller than the energy loss of an electron during recombination. The probability of creating a large number of phonons is very low in pure semiconductors. However, crystalline imperfections such as point defects and dislocations may alter the band structure and provide energy stages within the gap, as shown in Fig. 24(b). Hence, electrons and holes may recombine non-radiatively. Instead of giving off light, heat is produced in such a process, which is one of the failure mechanisms of semiconductor light-emitting devices. 3. Photon–Phonon Scattering In addition to the absorption and emission, photons may be scattered by phonons. In Section III, we will discuss electromagnetic wave reflection at the interface, where we will treat each material as a homogeneous medium and the reflection will be specular at a smooth interface and non-specular with rough surfaces, which will be discussed in Section IV. In addition to the
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FIG. 24. Non-radiative processes: (a) Auger effect and (b) recombination that emits multiple phonons.
FIG. 25. Illustration of the photon–phonon scattering process.
elastic scattering (specular or diffuse), there exists inelastic scattering when photons are scattered by phonons. Photons can interact with optical phonons, resulting in Raman scattering, or acoustic phonons, resulting in Brillouin scattering. In a photon–phonon scattering process, the creation (emission) and annihilation (absorption) of a phonon cause a shift in the frequency of the radiation, namely, Stokes and anti-Stokes shifts, as shown in Fig. 25. The energy conservation equations are [41,80,81]: h! 1 ¼ h! 2 þ h!ph
for a Stokes shift
h! 1 þ h!ph ¼ h! 2
for an anti-Stokes shift
and ð77Þ
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Because the interaction involved two photons and one phonon, the momentum of the phonon is restricted to small values. The intensity of the anti-Stokes shift is usually much weaker than that of the Stokes shift, and their ratio can be expresses as exp( h!ph/kBT). In certain cases, however, the phonons generated by the Stokes process can subsequently participate in the anti-Stokes process, causing a strong excitation to the anti-Stokes component. a. Raman Scattering Raman scattering refers to the interaction between photons and the optical phonons. Note that the resulting photon can interact with the phonon again, creating a cascade process that emits m phonons. The photon energy is reduced by m times the energy per phonon. The probability decreases as the order increases. Raman spectroscopy has become a major analytical instrument for the study of solids [80,81]. High-intensity lasers, high-resolution spectrometers, and sensitive detectors such as photomultiplier tubes (PMTs) are often employed to measure the narrow Raman lines. b. Brillouin Scattering The acoustic wave (acoustic phonon modes) can also scatter photons. Acoustic phonons can be produced by thermal or optical excitation. A longitudinal acoustic wave creates periodic regions of higher and lower density in the solid. The periodic structure acts as a grating moving with the velocity of sound. When light is incident on the grating, the scattering is analogous to the X-ray scattering by lattice. Therefore, Bragg reflection will occur and the following relation holds 2 a sin ¼ m i
ð78Þ
where is the angle between the incident radiation and the acoustic wavefront, m is the order of scattering, a is the acoustic wavelength, and i is the radiation wavelength inside the medium. The motion of the acoustic wave causes a Doppler shift in the optical frequency, which can also be viewed as the generation or annihilation of a phonon. The dispersion of acoustic phonons allows the frequency shift to be continuous (at least in a limited region). Lasers with high intensity can create stimulated Brillouin scattering, generating coherent acoustic waves. The phonon strength can keep growing along the path of interaction to cause structural damage to the crystal. More detailed discussions on the inelastic (or non-linear) scattering of photons by phonons in semiconductors can be found in Refs. [80,81].
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E. DIELECTRIC FUNCTIONS As seen from Eq. (11), the dielectric function is a characteristic function governing the propagation of electromagnetic waves inside the material. It is also a response function that described how incident radiation will be reflected from or absorbed by the material. The refractive index and extinction coefficient are a different set of functions that contain the same information as the real part and imaginary part of the dielectric function. Notice that n and are commonly called optical constants, though they are functions of frequency (or wavelength) and generally temperature dependent. 1. Kramers–Kronig Dispersion Relations The real and imaginary parts of an analytic function are related by the Hilbert transform relations [82]. Kramers [83] and Kronig [84] were first to show that the real part and imaginary part of the dielectric function are interrelated. These relations are called the Kramers–Kronig dispersion relations or K–K relations for simplicity. The K–K relations can be interpreted as the causality in the frequency domain and are very useful in obtaining optical constants from limited measurements. The principle of causality states that the effect cannot precede the cause, or no output before input. Some important relations are given here, detailed derivation and proofs can be found from the literature [85–87]. The real part "1 and imaginary part "2 of a dielectric function are related by [86,87]: 2 "1 ð!Þ 1 ¼ } p
Z
1
$"2 ð$Þ d$ $ 2 !2
Z
1
0
0 2! } "2 ð!Þ ¼ "0 ! p
0
"1 ð$Þ 1 d$ $ 2 !2
ð79aÞ ð79bÞ
where 0 is the dc conductivity, } denotes the principal value of the integral, and $ is a dummy frequency variable. These relations can be written in terms of n and as, 2 nð!Þ 1 ¼ } p ð!Þ ¼
2! } p
Z
Z
1
$ð$Þ d$ $ 2 !2
ð80aÞ
1
nð$Þ 1 d$ $ 2 !2
ð80bÞ
0
0
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Z. M. ZHANG ET AL.
Equations (79) and (80) are called K–K relations, which relate the real part of a causal function to an integral of its imaginary part over all frequencies and vice versa. A number of sum rules can be derived based on the above relations that are useful in obtaining or validating the dielectric function of a given material [87]. The K–K relations can be applied to reflectance spectroscopy to facilitate the determination of optical constants from the measured reflectance of a material in vacuum. For radiation incident from vacuum upon a material, at normal incidence, the field reflection coefficient is (see Section III for more discussion), r~ð!Þ ¼ rð!Þei ð!Þ ¼
1 nð!Þ ið!Þ 1 þ nð!Þ þ ið!Þ
ð81Þ
where r is the amplitude and the phase of the reflection coefficient. The power reflectance is 0 ð!Þ ¼ r~ r~ ¼ r2
ð82Þ
The amplitude and the phase are related and it can be shown that [86,87] !
ð!Þ ¼ } p
Z
1 0
lnð$Þ d$ $ 2 !2
ð83Þ
The refractive index and extinction coefficient can be calculated from [88] nð!Þ ¼ and
1 0 pffiffiffiffi 1 þ 0 þ 2 cos 0
pffiffiffiffi 2 sin 0 pffiffiffiffi ð!Þ ¼ 1 þ 0 þ 2 cos 0
ð84aÞ
ð84bÞ
2. The Drude Model for Free Carriers The Drude model describes frequency-dependent conductivity of metals [21], and can be extended to free carriers in semiconductors. In the absence of an electromagnetic field, free electrons move randomly. When an electromagnetic field is applied, free electrons acquire a non-zero average velocity, giving rise to an electric current that oscillates at the same frequency as the electromagnetic field. The collisions with the stationary atoms result in a damping force on the free electrons, which is
OPTICAL AND THERMAL RADIATIVE PROPERTIES
221
proportional to their velocity. The equation of motion for a single free electron is then me x¨ ¼ me x_ eE
ð85Þ
where denotes the strength of the damping due to collision, that is, the scattering rate or the inverse of the relaxation time . Assume the electron motion under a harmonic field E ¼ E0ei!t is of the form x ¼ x0ei!t. Eq. (85) can be rewritten as x_ ¼
i!e E me ð!2 þ i !Þ
ð86Þ
The electric current density is J ¼ ne ex_ ¼ ~ ð!ÞE; therefore, the frequencydependent conductivity is ~ ð!Þ ¼
ne e 2 0 ¼ me ð i!Þ 1 i!=
ð87Þ
where the dc conductivity is 0 ¼ ne e2 =me . Equation (87) is the Drude freeelectron model. The electric conductivity approaches the dc conductivity at very low frequencies (or very long wavelengths). The dielectric function is related to the conductivity by Eq. (11). Hence, "~ð!Þ ¼ "1
0 "0 !ð! þ i Þ
ð88Þ
where "1 is a real constant, which is the limiting value of the dielectric function at very high frequencies. For very low frequencies, ! < < , Eq. (88) approaches the Hagen–Ruben equation [18] rffiffiffiffiffiffiffiffiffiffi 0 n
2"0 !
ð89Þ
For semiconductors, free carriers contributions include two terms [40], i.e., "~f ð!Þ ¼
ne e2 =m e "0 nh e2 =m h "0 2 2 ! þ i!=e ! þ i!=h
ð90Þ
Note that ne and nh depend on the doping concentration and temperature; see Eq. (43). The scattering rate can be related to the mobility through
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FIG. 26. Refractive index of a lightly doped and a heavily doped Si, data from Refs. [39,89].
¼ m*/e for both electrons and holes. When ! 1, for an n-type semiconductor, Eq. (90) gives 2n
ne e 2 m e e "0 !3
ð91Þ
or af
ne e 2 2 2 4p c3 "0 m e ne
ð92Þ
The above equation, however, does not represent the actual wavelength dependence and the modified absorption coefficient was given in Eq. (73). Free carriers can also modify the refractive index, resulting in a drop in the refractive index as wavelength increases until ! 1 and an increase in the refractive index as wavelength further increases. Figure 26 compares the refractive index for a lightly doped and a heavily doped Si. 3. The Lorentz Oscillator Model for Phonons Vibrations of lattice ions and bound electrons contribute to the dielectric function in certain frequency region, often in the infrared. The refractive index can be calculated using the Lorentz oscillator model, which assumes that a bound charge q is accelerated by the local electric field E. In contrast to free electrons, a bound charge experiences a restoring force determined by
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223
FIG. 27. The classical oscillator model.
a spring constant Kj. The oscillator is further assumed to have a mass mj and a damping coefficient j, as shown in Fig. 27. The force balance yields the equation of motion for the oscillator: mj x¨ þ mj j x_ þ Kj x ¼ qE
ð93Þ
When E is a harmonic field, there exists a solution valid for times greater than the relaxation time x¼
!2j
q=mj E i j ! !2
ð94Þ
where !j ¼ (Kj/mj)1/2 is the resonance frequency of the jth oscillator. The motion of the single oscillator causes a dipole moment qx, and if the number density of the jth Poscillator is nj, the polarization or dipole moment per unit volume is P ¼ N j¼1 nj qx, where N is the total number of active phonon modes. The constitutive relation gives P ¼ ð"~ 1Þ"0 E. It can be shown that "~ð!Þ ¼ 1 þ
N X j¼1
Sj !2j !2j i j ! !2
ð95Þ
where Sj ¼ nj q2 =ðmj "0 !2j Þ is called the oscillator strength. The real and imaginary parts of the dielectric function and refractive index for a simple oscillator are illustrated in Fig. 28 near the resonance frequency. It can be seen from Eq. (95) and Fig. 28 that, for frequencies much lower and much higher than the resonance frequency, the extinction coefficient of the oscillator is negligible. Only within an interval of j around the resonance frequency is there appreciable absorption. Within the absorption band, the real part of the refractive index decreases with frequency; this is called anomalous dispersion.
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FIG. 28. The Lorentz model: (a) real and imaginary parts of the dielectric function; (b) refractive index and extinction coefficient.
By comparison of Eqs. (95) and (74), the resonance frequency is the transverse phonon frequency (!j ¼ !TO) and Sj ¼ "1 ð!2LO =!2TO 1Þ
ð96Þ
This model has been applied to a number of dielectric materials by fitting the reflectance spectrum to the three parameters for each oscillator [39,89–92]. A more complicated treatment based on quantum mechanics yields a fourparameter model [93]. The above classical oscillator model can be considered as the approximation when the relaxation time of the longitudinal and transverse optical phonons are the same. Intraband transitions can be modeled using the Lorentz model with a transition probability [43].
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225
4. Interband Transitions Modeling the interband transitions requires quantum theory and is more complicated. The absorption mechanisms for direct and indirect transitions have been discussed earlier, see Eq. (66) through Eq. (71). Semi-empirical equations, obtained by fitting the measurement data, are often used to describe the frequency-dependence of the absorption coefficient (or extinction coefficient) and refractive index in a limited spectral range [39,40,94–98]. Jellison and Modine [94] used spectroellipsometry to obtain the optical constants of Si at elevated temperatures, in the wavelength region from 240 to 840 nm. Following the work of MacFarlan et al. [95], Timans et al. [40,96] developed a model for the absorption coefficient in the interband region of Si to describe the infrared emissivity at elevated temperatures. The temperature and wavelength dependence of the refractive index of several semiconductors were discussed in Refs. [97,98]. Forouhi and Bloomer [99,100] developed a unified formulation of the optical properties of materials in the interband region. The absorption process is modeled based on the quantum mechanical theory for a oneelectron model that includes a finite lifetime of the excited electronic state. The extinction coefficient is expressed as a function of frequency using timedependent perturbation theory. The result is given in the following form [99,100],
ð!Þ ¼
Q X j¼1
Aj ð h! Eg Þ2 ð h!Þ Bj h! þ C j 2
ð97Þ
where Q is the number of distinguishable transitions, each corresponding to a peak in (!), and Aj, Bj, and Cj are related to the band structure and lifetime, which may be obtained by fitting to the measured data. The refractive index can be calculated using the K–K relation, Eq. (80a), which gives [99,100]
nð!Þ ¼ n1 þ
Q X j¼1
B j h! þ C j 2 ð h!Þ Bj h! þ C j
ð98Þ
where
Bj ¼
2Aj ðB2j =2 þ Bj Eg Eg2 þ Cj Þ ð4Cj B2j Þ1=2
ð99aÞ
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and C j ¼
2Aj ½ðEg2 þ Cj ÞBj =2 2Eg Cj Þ ð4Cj B2j Þ1=2
ð99bÞ
Equations (97) and (98) have been applied to fit a number of semiconductor crystals. It was found that better agreement is obtained with n1 ¼ n(1)>1 (for Si, the best fit values is n1 ¼ 1.95). Further analysis of the parameters obtained by fitting the experimental data would help understand the absorption and relaxation processes. It would also be interesting to extend this model to high-temperature cases. Recently, several groups have attempted to compute the optical constants from first principles, based on the electronic band structure. It was found that when electron–hole interaction is included, the predicted optical constants in the interband region agree well with the measured spectra for several semiconductors [101–103]. A challenging task is to develop an ab initio model of the optical constants in the far-infrared region, considering single and multiphonon absorption processes. This requires a better understanding of the nature of the anharmonic interatomic forces that governs the lattice dynamics.
III. Radiative Properties of Layered Structures Crystalline films, from a few nanometers to several micrometers thick, have been deposited (by physical vapor deposition, CVD, sputtering, laser ablation, MBE, RTP, and other techniques) onto suitable semiconductor substrates. These layered structures play important roles in contemporary technologies, such as integrated circuits, semiconductor lasers, quantumwell detectors, superconductor/semiconductor hybrid devices, optical filters, and spectrally selective coatings for solar thermal applications [104–111]. Radiative energy transport in thin films differs significantly from that at bulk solid surfaces and through thick windows because of multiple reflections and interference effects. The regime where interference effects are important is identified as the first microscale radiative heat transfer (MRHT) regime [112–117]. A number of textbooks deal with geometric optics (using ray-tracing method or net-radiation method) and wave optics or thin-film optics (based on the solution of Maxwell’s equations) [12,14,17– 20,118–122]. Several groups have investigated the intermediate regime, called partial-coherence regime [123–128]. For very thin films, the optical constants may be thickness dependent; this could be the result of boundary
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227
scattering (that reduces the electron mean free path) or a variation in crystalline structure of the material. This regime is named the second MRHT regime [113–115]. Quantum size effects may modify the optical constants of materials when the thickness is less than a lattice constant. This regime is recognized as the third MRHT regime [113–115]. Measurement data for solids and thin films before early 1970s have been compiled in the collection [61]. The radiative properties of smooth, parallel laminae are summarized in this section, with emphasis on different formulations for various applications. Surface roughness and microstructure effects on the radiative properties will be discussed in following sections.
A. REFLECTION AND REFRACTION AT AN INTERFACE Consider a plane wave incident from a dielectric medium, whose refractive index is n1, to another dielectric medium, whose refractive index is n2, as shown in Fig. 29. It is assumed that the interface is perfectly smooth. The incident, reflected, and transmitted wavevectors must lie in the same plane, which defines the plane of incidence. The angle of incidence 1 is the angle between the incident wavevector and the normal direction of the interface. We will discuss incident plane waves with either the electric field or the magnetic field polarized in the y direction, because other polarization can be treated as a combination of these two components. When the electric field is in the y direction, as shown in Fig. 29, it is called a transverse electric (TE) wave or s-wave since the electric vector is normal
FIG. 29. Reflection and transmission at an interface for a TE wave.
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to the plane of incidence (also called perpendicular polarization). The incident electric field, divided by ei!t, can be expressed as ˆ i eiq1;z zþiq1;x x yE
ð100Þ
At the interface, the tangential E and H must be continuous. This implies that the x-component of the wavevector qx must be the same for the incident, reflected, and transmitted waves. Therefore, the angle of reflection must be equal to the angle of incidence (mirror reflection). For the transmitted or refracted wave, sin 2 ¼
qx n1 sin 1 ¼ q2 n2
ð101Þ
which is called Snell’s law. Near the interface, the y component of the electric field and z component of the magnetic field are Ey ¼
½Ei eiq1;z z þ Er eiq1;z z eiqx x Et e
iq2;z z iqx x
e
for z < 0 for z > 0
ð102Þ
and
Hx ¼
8 q1;z iq1;z z > Er eiq1;z z eiqx x < ! Ei e
for z < 0
0
> :
q2;z Et eiq2;z z eiqx x !0
for z > 0
ð103Þ
The Fresnel reflection and transmission coefficients are defined as r12 ¼ Er/Ei and t12 ¼ Et/Ei, respectively. Boundary conditions require that the tangential components of E and H, i.e. Ey and Hx be continuous at z ¼ 0. After some manipulations, it can be shown that r12;s ¼
n1 cos 1 n2 cos 2 n1 cos 1 þ n2 cos 2
ð104aÞ
t12;s ¼
2n1 cos 1 n1 cos 1 þ n2 cos 2
ð104bÞ
and
OPTICAL AND THERMAL RADIATIVE PROPERTIES
229
Similarly, the reflection and transmission coefficients for the transverse magnetic (TM) wave, p-wave, or parallel polarization can be derived as r12;p ¼
n1 cos 2 n2 cos 1 n1 cos 2 þ n2 cos 1
ð105aÞ
t12;p ¼
2n1 cos 1 n1 cos 2 þ n2 cos 1
ð105bÞ
and
Since both Eqs. (104) and (105) are defined based on the ratio of the electric fields [118,121], at normal incidence, they give the same reflection coefficient r12;s ¼ r12;p ¼
n1 n2 n1 þ n2
ð106Þ
Upon reflection, if n1
n2, it is the magnetic field that will experience a phase reversal. The directional-hemispherical spectral reflectivity, or simply reflectivity, 0l is given by the ratio of the reflected to the incident Poynting vector, and the absorptivity 0l is the ratio of the transmitted (since all transmitted energy will be absorbed inside the second medium) to the incident Poynting vector. Therefore, for either polarization [118], 0 ð1 Þ ¼ r212
ð107aÞ
and 0 ð1 Þ ¼
n2 cos 2 2 t n1 cos 1 12
ð107bÞ
It can be shown that 0l ð1 Þ þ 0l ð1 Þ ¼ 1, which is required by energy conservation. If the incident wave is randomly or circularly polarized, the reflectivity can be calculated by averaging the values for p- and s-wave, i.e., 0 ¼
0 ;p þ 0 ;s 2
ð108Þ
The reflectivity for radiation incident from air to a dielectric medium (n2 ¼ 3.4) is shown in Fig. 30. For TE wave, the reflectivity increases monotonically with the angle of incidence and reaches 1 at the grazing angle (90 ). The reflectivity for TM wave, in contrast, goes through a minimum,
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FIG. 30. Reflectivity vs. the angle of incidence for a non-absorbing medium.
which is 0. The angle at which 0l;p ¼ 0 is called the Brewster angle, B ¼ tan1(n2/n1). For parallel polarization, all the incident energy will be transmitted into medium 2 without reflection at the Brewster angle. When n1>n2, the reflectance will reach 1 at 1 ¼ sin1(n2/n1). This angle is called the critical angle and total reflection occurs at angles of incidence greater than the critical angle. This is the principle commonly used in optical fibers and waveguides, since light will be trapped in and propagate along the medium. The above discussion can be extended to absorbing media. The (complex) Fresnel coefficients r~12 and t~12 can be obtained by replacing the refractive index with the complex refractive index. The power or energy reflectivity and absorptivity are, respectively, 0 ¼ r~12 r~ 12
ð109aÞ
0 ¼ 1 0
ð109bÞ
and
When radiation from air (n1 1) or vacuum is reflected by an absorbing medium having a complex refractive index (nþi), the reflectance at normal incidence is 0 ¼
ðn 1Þ2 þ 2 ðn þ 1Þ2 þ 2
ð110Þ
OPTICAL AND THERMAL RADIATIVE PROPERTIES
231
This equation was used in the discussion of K–K relations; see Eq. (81) through (84). Since it is assumed that the second medium is semi-infinite, the transmitted energy must be absorbed inside it. The spectral-directional emissivity is equal to the spectral-directional absorptivity as prescribed by Kirchhoff’s law. Hence, the spectral-directional emissivity for an opaque surface is "0 ¼ 1 0
ð111Þ
The above equation can be integrated to obtain the hemispherical emissivity "h ¼
1 p
Z
2p 0
Z
p=2 0
"0 cos sin d d
ð112Þ
where and are the zenith and azimuthal angles, respectively. Since emission is not polarized, it can be inferred from Fig. 30 that the emissivity changes little below the Brewster angle and decreases to 0 as the incidence angle approaches 90 . The hemispherical emissivity for non-metallic surfaces is about 5–15% smaller than the normal emissivity. The totalhemispherical emissivity is evaluated using Planck’s distribution, Eq. (20), that is R1 h R1 h e ð Þeb; ð ; TÞ d
h 0 Re ð Þ eb; ð ; TÞ d
¼ 0
ð113Þ "tot ¼ 1 T 4 0 eb; ð ; TÞ d
B. RADIATIVE PROPERTIES OF A SINGLE LAYER A ‘‘thick’’ layer refers to the case when the interference between multiply reflected waves can be neglected. Saying in other words, the waves are incoherent. On the contrary, a ‘‘thin’’ film refers to the case when all multiply reflected waves are coherent and interfere with each other. The condition for being ‘‘thick’’ has often been commonly interpreted as that the layer thickness d is much greater than the wavelength. The correct definition is that the thickness is much greater than the coherent length, which could be much greater than the wavelength. Coherent length depends on the spectral width of the source and resolution of a spectrometer. In addition, beam divergence, surface roughness, and non-parallelism further reduce the degree of coherence. Generally speaking, when the thickness is comparable to the wavelength, interference effects are important. However, this does not guarantee complete coherence because of the nature of the source and imperfect surfaces. Interference may occur between the forward wave and
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backward wave at the second interface inside a highly absorbing medium, even though interferences between multiply reflected rays are negligible [129]. The radiative properties of a lamina under two limiting cases, incoherence or thick layer and complete coherence or thin film, are discussed first, followed by a brief discussion on the theory of partial coherence. 1. Formulation in either Coherent or Incoherent Limit a. Thick Slab In the incoherent case, ray tracing method or the net radiation method can be applied to find out the transmittance and reflectance of a thick layer. It is assumed that a slab of thickness d is placed in air or vacuum, as shown in Fig. 31. The refractive index and extinction coefficient of the material are n and , respectively. As mentioned earlier, it is generally required that the thickness be much greater than the wavelength to avoid interference effect. Because the intensity will attenuate exponentially inside an absorbing medium, the penetration depth ¼ 0/4p cannot be much smaller than the thickness of the layer in order to have appreciable transmission. Therefore, the extinction coefficient is much smaller than the refractive index, < < n. For given surface reflectivity 0l and internal transmissivity l0 , ray tracing yields the reflectance and transmittance as [18,130] R0
¼
0
ð1 0 Þ2 02 1þ 1 0 2 02
FIG. 31. Transmittance and reflectance of a lamina.
ð114Þ
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233
FIG. 32. Normal transmittance of several semiconductor materials (0.5 mm thick) at room temperature.
and T 0 ¼
ð1 0 Þ2 0 1 0 2 02
ð115Þ
respectively, and the absorptance of the layer is A0 ¼ 1 T 0 R0 ¼
ð1 0 Þð1 0 Þ 1 0 0
ð116Þ
The reflectivity 0l can be calculated from Eq. (108) as a function of the angle of incidence i and the refractive index since the influence of on 0l is often negligibly small. The internal transmissivity l0 is given by 0
4pd ¼ exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n2 sin2 i
!
ð117Þ
Figure 32 shows the calculated normal transmittance spectral for several semiconductor wafers for a thickness d ¼ 0.5 mm, using the roomtemperature optical constants from Palik [39]. b. Thin Non-absorbing Film To consider interference, the amplitude and the phase of the electric field (or magnetic field) must be traced inside the film. The method is usually
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Z. M. ZHANG ET AL.
referred to as thin film optics. Upon traversing the film, the wave acquires a phase shift given by 2pd b¼
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 sin2 i
ð118Þ
The reflected field is the sum of all the waves after multiple reflections that come out of the first interface, and the transmitted field is the sum of all the waves that come out of the second interface. The reflectance and transmittance are [120,121] R0 ¼
40 sin2 b ð1 0 Þ2 þ 40 sin2 b
ð119Þ
T 0 ¼
ð1 0 Þ2 ð1 0 Þ2 þ 40 sin2 b
ð120Þ
A change in the wavelength, thickness, or refractive index can cause the transmittance to oscillate. The transmittance spectrum will have a peak of 1 at ¼ mp and a valley at ¼ ðm þ ð1=2ÞÞp, where m is a non-negative integer [131]. Figure 33 shows the calculated normal transmittance for n ¼ 2, ¼ 0, and d ¼ 10 mm. The free spectral range is the interval between two peaks, which for normal incidence is given by ¼ ð2ndÞ1
FIG. 33. Calculated transmittance of a thin film with and without absorption.
ð121Þ
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235
Where d is in cm, is in cm1. The interference phenomenon can be applied to determine the optical constants as well as film thickness [39,131]. The effect of absorption will be discussed next. c. Thin Absorbing Film When the absorption in the film is not negligible, the phase shift given by Eq. (118) becomes complex, i.e., qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pd ~ ¼ ð122Þ ðn þ iÞ2 sin2 i
0 The reflectance and transmittance can be written as [118–122] r~ð1 e2i ~ Þ 2 R0 ¼ 1 r~2 e2i ~
and
T 0
ð1 r~2 Þei ~ 2 ¼ 1 r~2 e2i ~
ð123Þ
ð124Þ
where r~ is the complex Fresnel reflection coefficient at the air–film interface. It should be noted that 0l ¼ r~ r~ , which is in general not equal to r~2 . The absorptance is A0l ¼ 1 R0l Tl0 . The transmittance for a slightly absorbing film (<
ð1 0 Þ2 0 1 þ 0 2 0 2 20 0 cos ð2bÞ
ð125Þ
In Eq. (125), and 0l are calculated by neglecting , and l0 is from Eq. (117). The above discussion is applicable to parallel, perpendicular, or random polarization states, as long as the Fresnel coefficients are computed correspondingly. The effect of absorption on the transmittance spectrum is shown in Fig. 33 for ¼ 0.005 using Eq. (125). d. Total Properties and Spectral Averaging Taking the transmittance as an example, the total transmittance is defined as the fraction of the energy transmitted; therefore, for broadband radiation with a given spectral intensity (or radiance) Ll, the total transmittance is, Z 1 Z1 0 0 Ttot L d
ð126Þ L T d
¼ 0
0
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Z. M. ZHANG ET AL.
In some practice, one needs to integrate the transmittance over a narrow band. For example, the radiation comes through a filter or a spectrometer with a finite resolution. The intensity is nearly constant within the small bandwidth, the transmittance can be averaged over a spectral width
around to give T0
1 ¼
Z
þ =2
=2
T 0 d
ð127Þ
If the coherent formula, Eq. (125), is integrated over a free spectral range l ¼ =2 , the result is the same as the incoherent formula, Eq. (115). In the case when absorption is negligible, integrating Eqs. (119) and (120) correspondingly gives R0 ¼
20
1 þ 0 2
ð128Þ
T 0 ¼
1 0
1 þ 0
ð129Þ
and
which are exactly the same as the incoherent formula given by Eqs. (114) and (115) when l0 ¼ 0: Eq. (127) links the coherent formulas to the incoherent ones. The spectral averaging method is also a useful tool to obtain radiative properties in cases of partial coherence. e. Radiation Tunneling The above sections deal with the radiative properties of a layer in air or vacuum. The radiative transfer between two surfaces is important for many applications, such as cryogenic radiation shields or thermophotovoltaic energy conversion devices. The gap between the media may be transparent or absorbing. Interference effects become important when the gap width is comparable to or smaller than the important wavelengths. Another phenomenon called radiation tunneling may also become important to radiant energy transfer at small length scales. Evanescent waves arise from the total internal reflection that occurs when light is incident from an optically denser medium to another medium, which is often air or vacuum, at an incidence angle greater than the critical angle. Although no energy is transmitted to the second medium, there exists an electric field whose amplitude decays exponentially in the second medium. This exponentially decaying field is known as an evanescent wave, which
OPTICAL AND THERMAL RADIATIVE PROPERTIES
237
does not carry any energy (i.e. the real part of its Poynting vector normal to the surface is zero). If a third medium with a sufficiently large refractive index is placed very close to the first one, there will be a forward decaying evanescent wave and a backward decaying evanescent wave in the gap (medium 2) due to multiple reflections. The interference between these two waves gives rise to a net energy flow from the first to the third medium even at angles of incidence greater than the critical angle. This phenomenon is called photon tunneling or radiation tunneling [121,134]. The use of photon tunneling to enhance radiative energy transfer in microscale devices has been extensively studied [135–139]. Recently, some researchers proposed microscale thermophotovoltaic devices that use photon tunneling to increase the energy conversion efficiency [140,141]. Zhang and Fu [142,143] suggested the use of a negative-refractive-index layer to enhance photon tunneling. The practical realization of micro- and nano-structured materials with a negative refractive index is currently under intense investigation with controversial opinions [144–149]. 2. Partial Coherence Theory It should be noted that no source is perfectly coherent: even laser or atomic emission has a non-zero line width. Likewise, no source is completely incoherent: even the most chaotic blackbody radiation has small coherent length [150]. Although completely incoherent and coherent formulae can be applied to a variety of practical problems, there are situations that do not fall in either regime. An example is the measured transmittance spectra of a slab with a spectrometer, such as a grating spectrophotometer or Michelson interferometer. Due to the finite instrument resolution and the imperfections of the sample surfaces (not perfectly parallel or smooth), the fringe contrast (defined as relative difference of the transmittance extrema) is always less than that predicted by the coherent formula. Figure 34 shows the measured transmittance of a 267-mm-thick Si wafer near 5 mm (2000 cm1), measured with a Fourier transform spectrometer [126]. The absorption in the wafer is negligibly small. However, the maximum transmittance does not reach 1 even with an instrument resolution 1 cm1. The reason is believed to be the beam divergence. With a resolution of 16 cm1, no interference fringes can be seen and the transmittance changes little. The results calculated from the partial coherence theory and spectral averaging agreed well with the measurement but indicated that the coherence wavenumber interval is 2 cm1 when the instrument resolution is 1 cm1, as shown in Fig. 34. Some critical lengths that affect the coherence of the measured radiative properties of a layer are now discussed. In principle, the discussion is applicable to multilayer structures. The phase shift term in Eq. (125) dominates the degree of coherence. Increasing absorption causes a reduction
238
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FIG. 34. Transmittance spectra of a 267-mm-thick Si wafer [126].
in the internal transmissivity, which in turn, reduces the fringe contrast or coherence. From Eq. (118), ¼ 2pnd cos / 0, where is the refraction angle. If the spectral width or the source or instrument resolution is greater than the free spectral range l ¼ l20 =ð2nd cos Þ, it will fall in the incoherent regime. Taking n ¼ 3, d ¼ 200 mm, i ¼ 30 , and 0 ¼ 1.5 mm for example, the calculated free spectral range is l 1.9 nm and 8:4 cm1 . For a diode laser with a line width of 0.5 nm, interference effects are not negligible and the measurement will be temperature dependent because of thermal expansion and the change of refractive index with the thickness. Roughness and non-parallelism cause the thickness to vary. If the thickness variation is much greater than d ¼ l0/(2n cos), coherence will be destroyed. For the above example, d ¼ 0.25 mm, therefore, the thickness variation should be less than 0.1d ¼ 25 nm within the beam spot for it to be completely coherent. When the thickness variation is between 0.1d and 10d, or 25 nm and 2.5 mm in the above example, partial coherence is anticipated. Similar arguments can be made in regard to the refractive index change or the beam divergence. Although spectral averaging is one way to deal with the situation involving partial coherence for optical properties of thin films, the partial coherence theory may provide a direct solution and sometimes is easier to apply. Partial coherence theory was developed before 1960 [12], and has gone through significant advancement after the first laser in the 1960s, including the application to radiometry [151,152]. A comprehensive monograph on this topic was written by Mandel and Wolf [153] in 1995. A brief introduction is given here with a summary of the research works related to radiative properties of thin films. The electric field can be expressed in either frequency domain E() or time domain E(t), which are related by
OPTICAL AND THERMAL RADIATIVE PROPERTIES
239
Fourier transforms. The mutual coherence function of any two waves is defined as: Z1 Ej ðtÞEk ðtÞ ¼ 4 Gjk ðÞ d ð130Þ 0
where the angular brackets h i symbolize the time average operation, i.e., Z 1 Ej ðtÞEk ðtÞ ¼ lim Ej ðtÞEk ðtÞ dt ð131Þ ! 1 2 and Gjk() is the mutual spectral density, given by Gjk ¼ lim
! 1
1 Ej ðÞEk ðÞ 2
ð132Þ
where the long bar denotes ensemble averaging. The spectral density of a wave is defined by GðÞ ¼ lim
! 1
1 EðÞE ðÞ 2
ð133Þ
and the optical intensity, which is proportional to the radiant energy flux in a given medium, is Z1 EðtÞE ðtÞ ¼ 4 GðÞ d ð134Þ 0
The complex degree of coherence is defined as Ej ðtÞEk ðtÞ jk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Ej ðtÞEj ðtÞ Ek ðtÞEk ðtÞ
ð135Þ
Chen and Tien [123] employed the partial coherence theory to calculate the radiative properties of a layer, by taking the forward propagating field in the film as composed of two components: one is directly refracted from the incident field, and the other is the combination of multiple reflections inside the film. Richter et al. [124] extended this method to multilayer structures. Zhang [126] considered the degree of coherence between any two multiple reflected waves and expressed the radiative properties in terms of an infinite summation. This study also demonstrated how to apply the partial coherence formulae to a thin film on a substrate. A semi-empirical approach was used by Grossman and McDonald [127] to fit the degree of
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Z. M. ZHANG ET AL.
coherent with the experimentally obtained far-infrared transmittance spectrum. Anderson and Bayazitoglu [128] investigated the partial coherence effects on directional radiative properties at various incidence angles and hemispherical reflectance.
C. RADIATIVE PROPERTIES OF MULTILAYER STRUCTURES Since many applications involve a thin film on a substrate, expressions of the radiative properties are summarized in this section. The transfer matrix formulation for thin-film multilayer structure is then described. Method of generalizing the formulation to arbitrary numbers of thick and thin layers is outlined. 1. Radiative Properties of a Thin Film on a Substrate Radiative properties of a thin coating on a substrate are important for a large number of applications, such as a thermal oxide on a Si substrate, metallic coatings, superconducting films, etc. It is assumed that the coating thickness is not much greater than the radiation penetration depth in the film. Furthermore, the substrate is slightly absorbing (S < < nS). Three cases are studied here: first, the substrate thickness is much greater than the radiation penetration depth in the substrate. Therefore, the substrate can be treated as semi-infinite (opaque). Second, the substrate is thick enough to be considered incoherent, though semitransparent. Third, the substrate is coherent or thin. This is particularly important for high-resolution spectroscopy or far-infrared radiation, since the wavelength is long. When the substrate falls in the partial coherent regime, spectral integration or partial coherent formulation can be applied. Figure 35 illustrates various conditions just discussed. In this section, 0, F, and S will be used to indicate air (vacuum), film, and substrate, respectively. a. Opaque Substrate For a thin film on a semi-infinite substrate, the electric filed reflection coefficient (ra) and transmission coefficient (ta) can be obtained by tracing Fresnel’s coefficients considering the complex phase shift pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ~F ¼ ð2pdF =l0 Þ ðnF þ iF Þ2 sin2 0 inside the film. The reflectance R0a and transmittance Ta0 can be expressed as R0a ¼ ra r a
and
Ta0 ¼
nS cos S ta t a n0 cos 0
ð136Þ
OPTICAL AND THERMAL RADIATIVE PROPERTIES
241
FIG. 35. Illustration of the reflectance and transmittance for: (a) opaque substrate; (b) radiation from substrate to the film; (c) thick substrate, incident on the film; (d) thick substrate, incident on the substrate; (e) thin substrate, incident on the film; and (f ) thin substrate, incident on the substrate.
where ~
ra ¼
r0F þ rFS ei2 F
1 rF0 rFS ei2 ~F
ð137aÞ
and ~
ta ¼
t0F tFS ei F 1 rF0 rFS ei2 ~F
ð137bÞ
The absorptance of the film is 1 R0a Ta0 . 2. Thick Substrate When the substrate is thick but semi-transparent, the reflectance depends on whether the radiation is incident on the film side or the substrate side, see Fig. 35. The reflectance and transmittance of the film-substrate composite can be expressed as R0F ¼ R0a þ
0S S0 2 Ta2 1 0S S0 2 R0b
ð138Þ
242
Z. M. ZHANG ET AL.
R0S ¼ 0S þ
R0b S0 2 ð1 0S Þ2 1 0S S0 2 R0b
ð139Þ
and T0 ¼
S0 Ta0 ð1 0S Þ 1 0S S0 2 R0b
ð140Þ
where 0S ¼ r0S r 0S is the reflectivity at the air–substrate interface, S0 is the internal transmissivity of the substrate from Eq. (117), and R0b is the reflectance for radiation incident from within the substrate to the film, see Fig. 35(b), i.e., R0b
¼ rb
r b
r þ r ei2 ~F 2 SF F0 ¼ 1 rF0 rFS ei2 ~F
ð141Þ
The absorptance of the film substrate composite can be obtained by subtracting the reflectance and transmittance from unity. The absorptance also depends on which side the radiation is incident. a. Thin Substrate In this case, interference effects in both the film and the substrate must be taken into consideration. The resulting expressions are R0F
R0S and
r þ r ei2 ~S 2 a S0 ¼ 1 rb rS0 ei2 ~S
r þ r ei2 ~S 2 0S b ¼ 1 rb rS0 ei2 ~S
t t ei ~S 2 a S0 T0 ¼ 1 rb rS0 ei2 ~S
ð142Þ
ð143Þ
ð144Þ
where ~S is the complex phase shift inside the substrate. Notice that the reflection coefficients and transmission coefficients are generally complex in the above expressions. The absorptance depends on which side the radiation is incident. The above formulations were used to perform a design of
OPTICAL AND THERMAL RADIATIVE PROPERTIES
243
intensity modulators and infrared detectors made of high-temperature superconducting films on Si substrates [154–156]. 3. Transfer Matrix Method a. Thin Films Transfer matrix formulation is a convenient method to calculate the transmittance and reflectance of layered structures [121,157,158]. Consider an N-layer system shown in Fig. 36, where the first layer (layer 1) and the last layer (layer N) are semi-infinite. The (complex) refractive index of the lth layer is nl. For a monochromatic plane wave originating in medium 1, either a transverse electric (TE) or a transverse magnetic (TM) wave, the ˆ whose magnitude is 2pnl/l0. Phase matching wavevector is ql ¼ qlx xˆ þ qlz z, requires that qlx:qx ¼ (2p/l0)n1 sin 1, where 1 is the angle of incidence. The electric field in the lth layer can be expressed as El(z)eiqxxi!t with E1 ðzÞ ¼ A1 eiq1z z þ B1 eiq1z z and El ðzÞ ¼ Al eiqlz ðzzl1 Þ þ Bl eiqlz ðzzl1 Þ ;
l ¼ 2; 3; . . . ; N
ð145Þ
where Al and Bl are the amplitudes of the forward and backward waves at the interface, respectively, z1 ¼ 0, zl ¼ zl1þdl (l ¼ 2, 3, . . ., N) with dl as the layer thickness, and qlz ¼ ð2p=l0 Þnl cos l with l ¼ sin1(n1 sin 1/nl). The coefficients of adjacent layers are related by Alþ1 Al 1 ; l ¼ 1; 2; . . . ; N 1 ð146Þ ¼ Pl Dl Dlþ1 Blþ1 Bl
FIG. 36. Schematic of a multilayer structure.
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Z. M. ZHANG ET AL.
where Pl is the propagating matrix: Pl ¼ I ¼
1 0
0 1
and Pl ¼
eiqlz dl 0
iqlz dl ; 0
e
l ¼ 2; 3; . . . ; N 1
ð147Þ
Dl is the transfer matrix given below and D1 is its inverse. l 1 1 Dl ¼ for a TE wave; nl cos l nl cos l and Dl ¼
cos l nl
cos l nl
for a TM wave
ð148Þ
Hence,
A1 B1
ð149Þ
Pl D1 l Dlþ1
ð150Þ
¼M
AN BN
where M¼
N1 Y l¼1
The above formulation allows the determination of the field reflection and transmission coefficients. The field distribution in any layer can be solved from Eq. (146). When A1 is real and BN ¼ 0, the power reflectance and transmittance can be calculated from T 0 ¼
nN cos N AN A N n1 cos 1 A21
ð151Þ
B1 B 1 A21
ð152Þ
and R0 ¼
OPTICAL AND THERMAL RADIATIVE PROPERTIES
245
The absorptance of the composite layers can be calculated by subtracting the reflectance and transmittance from unity. The Poynting vector can be evaluated to obtain the radiant energy flux SðzÞ ¼ ð1=2ÞRe½EðzÞ H ðzÞ . The fraction of energy absorbed between z1 and z2 is given by z1 z2 ¼
Sðz1 Þ Sðz2 Þ Si
ð153Þ
where Si is the incident radiant energy flux. b. An Example Many millimeter-wave and far-infrared studies require high precision measurements in a very narrow band; this can be achieved with a FabryPerot resonator that consists of two reflecting mirrors facing each other separated by air or vacuum. HTS thin films have little absorption for frequencies up to at least 200 cm1. Hence, Fabry-Perot resonators built with HTS thin films may offer better performance than those with metallic meshes. Figure 37 shows the transmittance spectra of a Fabry-Perot resonator built from two 35-nm-thick YBa2Cu3O7 (YBCO) films on Si substrates [107]. The calculated transmittance accounts for partial coherence due to the non-parallelism of the two films and the limited spectral resolution. A theoretical analysis was carried out to study the effects of film thickness, substrate thickness and gap width on the performance of these
FIG. 37. Transmittance of a Fabry-Perot resonator made of YBCO films on Si substrates [107].
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Z. M. ZHANG ET AL.
resonators. The results demonstrated the feasibility of constructing FabryPerot resonators based on YBCO films on Si substrates [107]. c. Arbitrary Numbers of Thick and Thin Layers For incoherent multilayer structures, i.e., all the layers are much thicker than the wavelength and the corresponding coherence length, the intensity transfer matrix approach can be employed [159–162]. It is necessary to assume that every layer is either non-absorbing or slightly absorbing ð< <1Þ. Attention must be paid to the coherence length because cross-coherence can exist among the layers. An example is multiple parallel windows [18]. In the intensity transfer matrix, a forward propagating and attenuating intensity and a backward intensity are considered for each layer. By matching the boundary conditions, the intensity can be calculated and hence the transmittance and reflectance. Detailed expressions can be found from Refs. [159–162]. Radiative properties of arbitrary numbers of thick and thin layers can be derived theoretically. For each thin film stack, the electric field reflection and transmission coefficients are obtained first. The power transmittance and reflectance at the interfaces of each thick layer can then be obtained. The intensity transfer matrix can be carried out for all thick layers to obtain the reflectance and transmittance of the multilayer system [159]. Spectral averaging is another way of obtaining the transmittance and reflectance for systems involving thick and thin layers. In many practical applications, the surfaces of the layers are not perfectly parallel. In addition, surface roughness may be important. The coherence can degrade and specular properties can significantly deviate from those for perfect films. A number of studies deal with non-parallelism, surface roughness, and inhomogeneity by using modified Fresnel coefficients and/or phase average techniques [163–172]. A more detailed treatment of light scattering from rough surfaces is presented in the following section.
IV. Radiative Properties of Rough and Microstructured Surfaces Real surfaces contain irregularities or surface roughnesses that depend on the processing method. A surface appears to be smooth if the wavelength is much greater than the surface roughness height. A highly polished surface can have a roughness height on the order of nanometers. Some surfaces that look rough to human eyes may appear to be smooth for far-infrared radiation. The reflection of radiation by rough surfaces is more complicated. For randomly rough surfaces, the reflection often includes a peak around the direction of specular reflection, an off-specular lobe, and a diffuse
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247
component [173–177]. When the surface contains periodic structures, such as patterned surfaces or micromachined surfaces, diffraction effects may become important and several peaks could appear [178,179]. The directional dependence of the radiative properties due to surface roughness can cause a significant error in radiative heat transfer calculations that employ specular or diffuse surface simplifications [180–184]. The bidirectional reflectance distribution function (BRDF), which for a given wavelength is a function of the angles of incidence and reflection, fully describes the reflection from a rough surface. The effect of surface irregularities on the bidirectional reflectance distribution function (BRDF) has been applied to planet observations [185]. In the field of computer graphics, which deals with light reflection between objects, the current trends are to include the BRDF of real surfaces and to use physically based surface reflection models [186]. Furthermore, non-contact characterization of surfaces by scatterometry relies on BRDF measurements and relations between BRDF data and surface roughness parameters [187–190]. However, some researchers showed that it is necessary to incorporate BRDF into Monte Carlo simulations [181–183]. One of the applications is to determine the effective emissivity for accurate radiometric temperature measurement during RTP [184,191,192]. This section gives a general review of the methods of characterizing the surface topography and BRDF measurements and modeling.
A. SURFACE ROUGHNESS CHARACTERIZATION Roughness is a measure of the topographic relief of a surface. It describes features of irregularities on the surface. Surface roughness is referred to as microroughness in semiconductor manufacturing industry. The spatial wavelength of roughness ranges from a few nanometers to 100 mm and the vertical amplitude varies from 0.1 nm to a few micrometers. Surface roughness plays an important role in many industries. It can affect the performance of electrical circuits directly, or indirectly through influencing some process steps. Surface roughness also decreases the sensitivity for surface particle detection by laser scattering methods [193]. Another example is, in RTP systems, the emissivity and other radiative properties are significantly modified by the roughness on the unpolished side of a silicon wafer, causing error in lightpipe radiation thermometry. Roughness associated with optical components can cause scattering and stray light in optical systems and degrade the contrast and sharpness of optical images [194]. In mechanical application, surface roughness is an important factor in determining the satisfactory performance of the work piece [195]. Surface
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roughness also has important effects on the thermal contact conductance [196,197]. 1. Roughness Parameters and Functions Surfaces can be divided into two categories according to their topographic profiles. If the topographic (height) information in each point on the surface is known, the surface is called a deterministic surface; otherwise, it is called a random surface. The profile of a random surface is unknown but the height information can be modeled using the stochastic theory. Many surfaces in engineering applications are random surfaces and others may display a dominantly deterministic profile. Statistical descriptors are necessary to characterize surface roughness. Parameters, such as rootmean-square roughness, are defined to quantify some aspects of surface roughness with a single number. However, surface statistical functions, such as the power spectral density function, include many aspects of information about the surface. Statistical descriptors are also appropriate for deterministic surfaces but the meanings of some terms may be different. For the sake of simplicity, the definitions of the parameters and functions are expressed in one-dimensional form, which can be extended easily to twodimensional form. In the one-dimensional situation, the surface topographic profile can be described as a function of height variation at each point z(x). The mean surface is determined by z ¼ lim
L ! 1
1 L
Z
L
zðxÞ dx 0
ð154Þ
where L is the total sampling length. The arithmetic average roughness Ra is defined by Ra ¼ lim
L ! 1
1 L
Z
L 0
jzðxÞ zj dx
ð155Þ
and the root-mean-square (rms) roughness Rq is defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 L lim ½zðxÞ z 2 dx Rq ¼ L ! 1 L 0
ð156Þ
Ra and Rq are most commonly used roughness parameters but they only give the amplitude variation in the height direction and they do not tell the spatial extent and steepness of surface irregularities on the surface. Surface
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slope sðxÞ ¼ dz=dx can be estimated from the profile data and the rms slope sq is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 L ½sðxÞ s 2 dx ð157Þ sq ¼ lim L ! 1 L 0 Surface slope can be considered as a hybrid parameter, which combines the concepts of height deviation and lateral displacement [198]. The rms slope sq is an important parameter for modeling light scattering by a rough surface [175,176,199]. The surface profile can be considered as a superposition of many components whose amplitudes and lateral wavelengths are different. Power spectral density (PSD) function delineates the vertical and spatial extent of surface irregularities and is defined as follows [189]. Z 2 1 L i2pxfx zðxÞe dx L 0
PSDðfx Þ ¼ lim
L ! 1
ð158Þ
where fx is the spatial frequency in the x-direction. Since the PSD function includes information about surface roughness components at different lateral wavelengths, the rms roughness can be calculated from [189] ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z fmax
Rq ¼
2
fmin
PSDðfx Þ dfx
ð159Þ
In roughness measurement, the spatial frequency is limited between the minimum frequency fmin ¼ 1/L and the maximum frequency fmax ¼ 1/(2d), where d is the sampling interval. Figure 38 illustrates how the PSD function is sensitive to the different characteristics of surfaces. For an artificial surface shown in Fig. 38(a), there is a dominant spatial frequency of 0.16 mm1 and some higher-order harmonics in the power spectrum. This kind of features is typical for a surface machined by diamond tuning [198]. The profile and PSD for a Si surface are shown in Fig. 38(b). There is no dominant periodic roughness component. The PSD curve has a plateau at low frequency and it also displays a linear part (in log–log scale) at higher frequency, which is the characteristic of a fractal surface [200,201]. The autocovariance function ACV correlates deviations from the mean height with a translated version of itself [189]. ACVðÞ ¼ lim
L ! 1
1 L
Z
L 0
½zðxÞ z ½zðx þ Þ z dx
ð160Þ
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FIG. 38. Power spectral density for two rough surfaces: (a) an artificial periodic surface with several harmonic defects; (b) a silicon surface measured with an AFM.
where is a translation length. The two functions, ACV and PSD, form a Fourier transform pair; therefore, Z1 PSDðfx Þei2pfx dfx ð161aÞ ACVðÞ ¼ 1
By definition, the value for the autocovariance function at ¼ 0 is equal to the square of the rms roughness. The autocorrelation function ACF is the normalized autocovariance function by dividing the square of the rms roughness; that is [198] ACFðÞ ¼ ACVðÞ=R2q
ð161bÞ
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Autocorrelation function is very useful for visualizing the relative degrees of periodicity and randomness of a surface profile. The autocorrelation function of a periodic surface also shows high periodicity. A random profile exhibits a strongly decaying autocorrelation function. The autocorrelation length b is defined as the value of the translation length at which ACF() drops to 1/e of its value at ¼ 0; that is, ACFðbÞ ¼ ACFð0Þ=e. For a Gaussian random surface, the rms slope sq, rms roughness Rq, and the autocorrelation length b are related by [173] pffiffiffi ð162Þ sq ¼ 2Rq =b Due to practical limitations in the measurements, additional information needs to be given when reporting roughness measurement results [193]. Examples are the total sampling length and the lateral resolution, which determine the longest surface wavelength and the shortest surface wavelength, respectively. The PSD function is a promising way to describe the surface roughness and it has been used as a powerful tool to characterize roughness and compare roughness measurement results using different measurement techniques [202–206]. The PSD and ACF are related by the Fourier transform and they are useful in different ways. The PSD is useful for studying the strengths of various periodic components in the surface profiles while the ACF is useful for describing the lateral size of the random features on a surface [195,198]. 2. Surface Characterization Methods We can sense the surface roughness using the thumbnail and the eye. Both methods are completely subjective. Various types of instrumentations are being used to map the surface topography. Some instruments are following the tactile example of the nail, i.e. using a stylus probe. Some are mimicking the eye, i.e. using an optical method. The detailed information about these instrumentations can be found in Refs. [195,207,208]. a. Stylus Method As a stylus is dragged over a rough surface, it moves up and down when it rides over peaks and valleys on the surface. The deviation of the stylus to a reference can be transferred into an electrical signal and surface profile can be generated from the signal. The lateral resolution of the stylus profiler is limited by the radius of the tip. The radius of the tip can reach as small as 0.1 mm or less. The vertical resolution is in the range of 1 nm. One issue for the contacting stylus is the load of stylus on the surface. The force exerted by the stylus may be greater than 1 N so that it may scratch or even damage the scanned surface. Another issue is the scan time: if area profiling is
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required, the scan time will be very long since the stylus has to raster the whole area. Some recent developments in scanning probe techniques are discussed next. b. Scanning Probe Microscopy Scanning tunneling microscope (STM) and AFM both belong to the family of scanning probe microscope (SPM). The name of SPM comes from the probe, like stylus in phonograph, rastering a sample. The birth of SPM gives a marvelous way for scientists and engineers to view the fine feature in atomic scale. It has tremendous usage in microelectronic industry, biology, chemistry, and so on. Before the birth of SPM, fine features in atomic scale could only be seen by cumbersome, often destructive ways such as electron microscopy and X-ray diffraction. The first member of the SPM family is STM, which was invented by Binnig and Rohrer in 1981 [207,209]. The tip of the probe is grounded so fine that it may consist of only one atom. Piezoelectric controls enable the tip to be located a nanometer or two above the surface. At such short distance, the electron clouds of the atom at the tip overlap with those of nearest atoms on the surface. If there exists a voltage between the tip and the sample, electrons can tunnel through the small gap, generating a tunneling current. Piezoelectric controls also move the tip or the sample back and forth so that the microscope can generate an image for the scanned area. The tunneling current is extremely sensitive to the width of the gap. In order to keep a constant tunneling current, a feedback control changes the voltage applied to the piezoelectric component in charge of the movement normal to the sample. The variation of the applied voltage can be translated into an image of the surface topography. The STM can only be used for conducting surfaces. If a surface is not conductive, it must be coated with conductive layer so that it can be scanned by the STM. In 1985, Binnig and collaborators [207,210] came up with the first AFM that works both for conductors and insulators. When the tip is brought very close to the surface, there exists a repulsive force due to the overlap of electron clouds of atoms in the tip and those on the surface. Nowadays, the probe consists of two parts, the tip and the cantilever, and is usually made of silicon, silicon dioxide, or silicon nitride. The radius of the tip can be as small as 20 nm so that it can map the fine features on the scanned surface. Optical method is used to detect the deflection of the cantilever. Figure 39 illustrates the principle of an AFM using the optical readout scheme. The small mirror on the backside of the tip reflects the laser beam, and the reflected light is collected by a position sensitive photodiode. The output signal of the differential photodiode is proportional to the deflection of the cantilever. A feedback circuit maintains a constant distance between the tip and the surface. The image of surface topography
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FIG. 39. Schematic diagram of the optical readout method for AFM.
can be deducted from the applied voltage to the piezoelectric component controlling the vertical movement. The AFM has a vertical resolution about 0.1 nm and the lateral resolution is better than 10 nm. However, the AFM can only scan a small area and the scan rate is slow. Although the weight of the probe is very small (the force is about 1 nN), the pressure exerted by the probe on the surface may be high enough to destroy some surfaces, especially for biological applications. Usually, two operation modes are provided by the AFM, i.e. contact mode and tapping mode. In tapping mode, the oscillation of the tip is modulated and the tip contacts the surface intermittently. The load of the tip is extremely small and the tapping mode can even be used to map soft surfaces, such as biologic tissues and polymers [207]. c. Optical Interferometric Microscopy The optical interferometic microscope (OIM) exploits the wave behavior of light. A schematic diagram of an optical interferometric microscope is shown in Fig. 40 [211]. Light from the source is split into two parts by the beamsplitter in the interferometer. A part of light is reflected back by the rough surface and the other part is reflected by the reference surface. Because two lights are from the same source and the optical paths are different, they can generate an interferogram, which represents the topography of the surface. The interferogram is pictured by a CCD camera. Sophisticated hardware and software are combined to extract the surface height information. The most popular interferometric techniques are phase shift interferometry (PSI) and scanning white light interferometry (SWLI). The light source in PSI is a monochromatic light and the PSI is applicable to test
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FIG. 40. Schematic diagram of a three-dimensional interferometric microscope, after Ref. [211].
smooth surface. The vertical resolution is in the sub-nanometer range. If the surface is moderately rough (the height difference between the adjacent peak and valley is greater than a quarter of the selected wavelength), PSI cannot obtain a correct topographic image for the surface. For relatively rough surfaces, the SWLI is a better way to measure surface roughness. A basic principle is that the maximum intensity for white-light fringes occurs when the optical path difference is zero. When the optical path difference is varied, an intensity envelope is recorded for each point on the surface. The height information is deduced from the maxima in the intensity envelope. The vertical resolution of SWLI is about 1–2 nm. The OIM has a large field of view and the image can be obtained in a few seconds. However, the principle of the OIM is based on the light interference and it imposes some constrains on the scanned surface. For example, the surface must have a uniform reflectivity and the steep of surface irregularities must be smaller than a limit value. Furthermore, light reflected back by the subsurface of a sample with coating could deteriorate the measurement results. d. Light Scattering Method When light illuminates a rough surface, the reflected energy will be redistributed in the upper hemisphere, which can be measured by using a
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moving sensor, multiple sensors, or an integrating sphere. The distribution of scattered light can be described using physical optics. Exact solutions for light scattering on rough surfaces only exist for simple surface profiles. The Rayleigh-Rice vector perturbation theory relates the fraction of scattered power into an element solid angle to the power spectral density (PSD) function [189]. dP=d s 16p2 cos i cos2 s PSDð fx ; fy Þ ¼ p Pi
40
ð163Þ
where dP is the scattered energy within the solid angle d s in the direction ðs ; s Þ, Pi is the incident power, i is the incidence angle, and 0 is the radiation wavelength in vacuum. The two-dimensional PSD is a function of fx and fy. In Eq. (163), p is a reflectivity polarization factor, which depends on the optical constants, angles of incidence and reflection, and the polarization states of incident and reflected radiation. Fortunately, in many cases, its numerical value can be approximated by the sample reflectance. Spatial frequencies are calculated from fx ¼
sin s cos s sin i
0
ð164aÞ
sin s sin s
0
ð164bÞ
fy ¼
There are some restrictions for the Rayleigh-Rice vector perturbation theory. The surface must be smooth and free of contamination. The vertical amplitude limit in the visible spectrum is about 10 nm so as to satisfy the smoothness requirement. The Rayleigh-Rice vector perturbation provides a way to calculate surface statistics from scattering measurement results. Scatter measurement can be used as a fast, non-contact method of roughness characterization. If a surface does not meet the smoothness requirement, the Rayleigh-Rice vector perturbation theory is not applicable. However, the total integrated scatter (TIS) method is less restrictive and can be applied to a wider range of roughnesses. The measured scattered power other than the specular direction is normalized by the reflected specular power to give the TIS. Based on the Davies relation [173,174], the TIS can be calculated by [189]: Pscat 4pRq cos i 2
TIS ¼ Psp
0
ð165Þ
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where Pscat is the power scattered away the specular direction and Psp is the reflected power in the specular direction. It should be noted that Eq. (165) is only valid for a surface with a Gaussian height distribution. e. Other Techniques Scanning electron microscopy (SEM) uses electrons rather than light to visualize surface configuration. The short wavelength of electron beams provides very high resolution. The electron beam hits the surface, excites secondary electrons from the sample. A detector collects the secondary electrons and the signal is translated to the topography of the surface. Usually, the SEM does not provide quantitative parameters about surface roughness. The surface must be conductive for regular SEMs and the measurement is operated in vacuum. Ellipsometry is a common technique to measure film thickness and interfacial width and it can be extended to measure film roughness, defects, and particles [212–217]. Spectroellipsometry techniques have been applied to characterize microscopic roughness on specularly reflecting surfaces, i.e. roughness with amplitude much smaller than the wavelength of light [204]. Other techniques, such as the tunneling current oscillation [218], laser speckle [219], laser triangulation and phase shift interferometer [220], and Rutherford backscattering [221] have also been applied to characterize the surface roughness. Comparisons of different characterization techniques are listed in Table III. Some data are adopted from Ref. [195]. 3. Some Applications Related to the Semiconductor Industry The processes of making a Si wafer include an initial slicing process, followed by a lapping process that removes the wafer surface damage from the slicing process to give a nominally flat surface. The final polishing step is a combination of chemical etching and mechanical polishing. This process TABLE III COMPARISONS OF CHARACTERIZATION TECHNIQUES Lateral Resolution: x, y
Vertical Resolution: z
Stylus AFM STM OIM
0.1 mm < 2 nm < 2 nm 0.2 mm
TIS SEM
1 mm 5 nm
1 nm 0.1 nm 0.01 nm 1–2 nm for SWLI 0.1 nm for PSI 1 nm N/A
Comments Contact method Contact or tapping Conductive surface Sample must not be transparent Gaussian surface Vacuum needed
OPTICAL AND THERMAL RADIATIVE PROPERTIES
257
(a) Optically smooth surface, Rq ¼ 0.84 nm, OIM
(b) Optically rough surface, Rq ¼ 363 nm, AFM FIG. 41. Surface images: (a) A smooth surface measured with an OIM; (b) A rough surface measured with an AFM.
forms a mirror-like surface for further patterning processes. The final polishing is frequently done on one side of a wafer. The other side is left rough or etched after the lapping. For some device use, the rough side may receive a special process to induce crystal damage, called backside damage [222]. The typical methods are sand blasting and deposition of a polysilicon layer or a silicon nitride layer. Therefore, the rough sides of commercial wafers may have very different roughness levels and coated layers. Figure 41 shows the three-dimensional surface image of two silicon wafers. The image of the smooth side of a silicon wafer using OIM is plotted
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Z. M. ZHANG ET AL.
in Fig. 41(a) and that of the rough side using AFM is plotted in Fig. 41(b). AFM has been used to investigate the surface morphology of polysilicon deposits grown under two different temperatures [223]. The scanned area ranged from 0.4 to 655 mm2. For scanned area of 1 mm2, the rms roughnesses were 8 and 2.9 nm for two deposits, respectively. The PSD plots relative to the polysilicon were several orders of magnitude greater than the substrate PSD plots. Ion implantation allows the introduction of foreign species into host materials. The roughness created by ion implantation has been measured by AFM to study the effects of controlling parameters, such as dose and annealing [224]. The rms roughness was found to be nearly constant at 0.23 nm and insufficient to characterize those surfaces. PSD analysis showed that a low frequency roughness with feature size in the range of 200–600 nm was created above the critical dose. Researchers have done extensive studies to investigate the relation between the angle-resolved scatter and the power spectral density of surface roughness [189,225–227]. Bare silicon surface scatters almost exclusively from surface topography and scatter measurement can be used as a reliable source of surface roughness characterization [187]. Vorburger et al. [225] divided surface roughness measurable with light scattering into four regimes, based on their experiment results. They pointed out, for the smooth-surface regime, the angular distribution of the scatted light is closely related to the PSD of the roughness, ranging over 0 < Rq =l0 < 0:05. Measuring a sample using different techniques has been an interesting area for roughness characterization [193,202,203,206,227]. Most studies compared the power spectral densities measured by different instruments. Surface roughness measurements using AFM compared well with optical scatter data for a molybdenum mirror and glassy materials [202,227]. A number of techniques have been used to measure silicon wafers with different surface conditions. The measured rms roughness and PSD differ significantly from instrument to instrument, mainly due to the different frequency ranges in the measurements or coating on the surface [193,206].
B. BIDIRECTIONAL SCATTERING DISTRIBUTION FUNCTIONS The bidirectional reflectance distribution function (BRDF) and the bidirectional transmittance distribution function (BTDF) are important radiative properties related to the interaction of radiation with an imperfect surface. Sometimes the more general term, bidirectional scattering distribution function (BSDF) is used. The BRDFs and BTDFs of rough surfaces have been an attractive subject with considerable research devoted to the theory, numerical modeling, and measurement [173,189,228–232].
OPTICAL AND THERMAL RADIATIVE PROPERTIES
259
BRDF is a basic parameter for describing the nature of reflection from a surface element and is defined as [233] fr ð ; i ; i ; r ; r Þ ¼
dLr ðsr1 Þ Li cos i d i
ð166Þ
where ði ; i Þ and ðr ; r Þ denote the directions of incident and reflected beams, respectively, Li is the incident radiance, and dLr is the reflected radiance for radiation incident from an element solid angle d i . The denominator of Eq. (166) gives the incident irradiance (radiant flux). The geometry of the incident and reflected beams is shown in Fig. 42. All the radiative properties discussed in this section are spectral properties. In the experiment, the detector output signal is proportional to the radiant power reaching the detector. Hence, BRDF can be obtained from the following measurement equation: fr ¼
1 Pr Pi cos r r
ð167Þ
where Pi and Pr are the incident and reflected power reaching the detector. If a measured sample is semitransparent, the BTDF is defined as ft ¼
dLt ðsr1 Þ Li cos i d i
FIG. 42. Geometry of the incident and reflected beams in defining the BRDF.
ð168Þ
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Z. M. ZHANG ET AL.
The directional-hemispherical reflectance can be obtained by integrating the BRDF over the hemisphere [17–19]: 0
¼
Z
fr cos r d r 2p
ð169Þ
which is a function of the incident direction and wavelength. An important principle of the BRDF is reciprocity [18], which states symmetry of the BRDF with regard to reflection and incidence angles. In other words, the reflectance for energy incident from (i, i) and reflected to (r, r) is equal to that for energy incident from (r, r) and reflected to (i, i). That is fr ð ; i ; i ; r ; r Þ:fr ð ; r ; r ; i ; i Þ
ð170Þ
The reciprocity is a subject of several studies [234–237]. For a diffuse or Lambertian surface, the BRDF is independent of (r, r) and is related to the directional-hemispherical reflectance as fr;dif ¼ 0 =p
ð171Þ
On the other hand, the BRDF for an ideal specular, or mirror-like, surface can be represented as fr;mir ¼
0 ðr i Þð r i 180 Þ cos i
ð172Þ
where the Kronecker delta function (x) equals 1 when x ¼ 0 and equals 0 for all other values of x. The use of the BRDF is not necessary for such surfaces because the radiative properties can be calculated from the theories discussed in the previous section. 1. Theoretical Models of BSDF The complete BSDF solutions using electromagnetic theory are extremely computational intensive and unpractical in most cases [238–244]. Various approximation methods have been developed for different surface characteristics and spectral regions. The Fresnel approximation considers only specular components and is for perfectly smooth surfaces [245,246]. It is applicable to the polished wafer surface (Rq < 5 nm for visible light). The Rayleigh-Rice approximation is a perturbation theory that does not require matching boundary conditions [173]. It is based on a statistical Fourier analysis of the surface and predicts that fr / l4 PSD. It is commonly used
OPTICAL AND THERMAL RADIATIVE PROPERTIES
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for characterizing optical surfaces that are relatively smooth, i.e. Rq cos i =l0 < 0:05 [189,225,247]. From intermediate to very rough surfaces, physical optics method and geometrical optics method are usually applied. In the physical optics, Kirchhoff approximation and the Gaussian surface assumption are often used to model the surface scattering with wave characteristics, like wave diffraction. The Kirchhoff approximation is an extension of the Fresnel approximation that includes scattering but assumes that the radius of the surface curvature is much greater than the wavelength and that there is no multiple scattering. The conditions for this approximation to hold are: (i) Rq =ðb cos i Þ < 0:2 and (ii) Rq =l < 2 [240,248]. Hence, it is applicable to surfaces not too precipitous and when the incidence angles are not too large. The validity of the Kirchhoff approximation has been investigated by a number of authors and modifications have been suggested [249–253]. The geometric-optics approximation treats a rough surface as one with many small facets and employs the ray-tracing approach by neglecting the phase of the electromagnetic wave [175,254,255]. Multiple scattering can be incorporated into the geometric optics formulation. The applicable regions are approximately given by (i) Rq cos i =l0 > 0:2 and (ii) Rq =b < 2 [256,257]. The second criterion is satisfied for most surfaces without very deep valleys. A few analytical models are presented below. a. Physical-Optics Models Davies [174] assumed a Gaussian surface height distribution and employed the Kirchhoff approximation to derive expressions for the electromagnetic energy reflected from rough surfaces. For slightly rough surfaces (Rq <
0
ð173Þ
where 0sp is the directional-hemispherical reflectance of a smooth (specular) surface and is determined by the Fresnel equations. In essence, this equation relates the specular reflectance of a rough surface to an ideal smooth surface. If F is used to define this ratio, which is sometimes called specularity ratio, then " 2 # 4pRq F ¼ exp cos i
0
ð174Þ
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Z. M. ZHANG ET AL.
The off-specular component is given by 0sp ði Þp3 bRq 2 fr;off ði ; i ; r ; r Þ ¼ ðcos i þ cos r Þ4 cos i cos r 2 ( ) pb 2 2 2 sin i þ sin r þ 2 sin i sin r cos ð i r Þ exp
ð175Þ
Considering the actual measurement conditions, the specular component given by Eq. (173) exists in a finite solid angle around the specular direction [258]. The Davies model is also extended to moderately rough surfaces with Rq > l using the following expression, " # 2 0sp ði Þ 1 b b 2 2 fr ði ; i ; r ; r Þ ¼ exp tan 2Rq cos i cos r 16p Rq where
ð176Þ
is the half angle between the incident and reflected beams and tan2
¼
sin2 i þ sin2 r þ 2sin i sin r cos ð i r Þ ðcos i þ cos r Þ2
ð177Þ
As seen in Eq. (176), for very rough surfaces, the wavelength dependence comes from the Fresnel reflectance only. The Beckmann-Spizzichino model [173] is similar to Davies model but can be applied for a broader range of roughness. The result can be shown as [259]
F ¼e
g
"
# 2 X b 1 gm 1 þ r p cos i
m¼1 m!m
ð178Þ
where g ¼ ð4p cos i Rq =l0 Þ2 is the roughness factor in the above equation. This model also includes the scattering component around the specular direction within the detector solid angle. b. Geometric-Optics Model Torrance and Sparrow [175] assumed that the rough surface consists of small, randomly distributed, mirror-like facets. This model is based on the geometrical-optics approximation, which neglects the phase of the
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263
electromagnetic wave. The equations for calculating the BRDF are rearranged in the following form [260]: 0sp ðd ÞG tan2 d fr ði ; ’i ; r ; ’r Þ ¼ þ exp 2b 8p cos i cos r p
ð179Þ
where G is a masking and shading function, b is the rms slope of the microfacet, tan is the slope of the microfacet, d is the angle of incidence at the microfacet, and d is a diffuse reflectance component. These models may be used to fit the measured BRDF [260–262] and thus provide a model function that can be incorporated into Monte Carlo codes for radiative transfer analysis [180–184] and other applications [177,263–267]. The effects of surface roughness and patterning on the radiative properties during RTP have been investigated by several groups using areaaveraging method [268,269]. A large number of publications discussed how to use the numerical method [270–275] and statistical method [276,277] to obtain the BRDF from the optical constants of the surface material and the roughness parameters. Some works dealt with two-dimensional surfaces [251,257,275,278–285]. Fractal theory has also been applied to model the reflection from a rough surface [201,286–288]. Many surfaces contain a thin coating or oxide film; however, there are relatively few publications that deal with coating or thin film on the BRDF of rough surfaces [289–294]. Many groups used analytical, numerical, and experimental methods to investigate the enhanced backscattering and retroreflection [294–297]. Some researchers studied the effects of coherent scattering enhancement and non-Gaussian rough surface on the reflection [298–300]. 2. BRDF/BTDF Measurements The BRDF measurement instrument is usually called a bidirectional reflectometer or scatterometer. The main components are a light source (laser or spectrometer source), a goniometric stage, a detector or detectors, and a data acquisition system. Most measurement wavelengths are from visible to near-infrared range. Difficulty in obtaining appropriate sources, detectors, and low scatter optics complicates its feasibility at wavelengths less than approximately 0.25 mm. Diffraction effects that become important for wavelengths greater than 15 mm complicate its application at longer wavelengths. The light source with necessary optics provides a nearly collimated beam. There exist a variety of designs for the goniometric stage. It may involve the movements of detector, sample, or light source. In some experiments, the design can be rather simple when high angular resolution is not needed. In some cases, multiple turntable and translation stages are used. For example, Drolen [301] used four turntables and two
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translation stages with seven different lasers to study the wavelength and directional dependence of spacecraft themal control materials. In some facilities, the goniometric tables were placed in a vacuum environment [302,303]. Without considering the out-of-plane (out of incidence plane) measurements (i.e. capable of varying r ), the goniometric stage design is rather straightforward [304–308]. Among this kind of design the standard is the reference reflectometer at NIST [309,310]. If the out-of-plane measurements are needed, a more complicated hardware design is expected. Generally, there are two common designs: the gimbal system [311–314] and circulartrack system [182,315–317]. Both designs involve complicated components that usually need custom machining. The gimbal system also needs a complicated coordinate transformation to determine the incidence and scattering angles; see for example Germer and Asmail [314]. The polarization angle with respect to the sample may need another coordinate transformation to control. For the circular-track system, the stability and eccentricity may challenge its fabrication. Some instruments employed multiple detectors to improve the sampling speed [255,262,308,318]. Voss et al. [319] designed an instrument that uses multiple fibers to collect the reflected light and send to a CCD camera for fast detection of the BRDF distribution. Ford et al. [320] used a Fourier transform infrared spectrometer to measure the BRDF at a broad wavelength range from about 2.5 mm to 15 mm using an HgCdTe detector, with a limited angular resolution. The standard instrument to measure BRDF is the spectral tri-function automated reference reflectometer (STARR) in the Optical Technology Division at NIST [309,310]. Figure 43 shows the top view of the facility. The STARR instrument is capable of measuring spectral reflectance at angles of incidence from 0 to 80 , and angles of reflection greater than 5 from the incident beam. The instrument is automatic and controlled by a computer in a room separated from the light-tight room with black walls and ceiling. The source subsystem contains a monochromator, an optical chopper, and a polarizer. A xenon arc lamp is used from 200 to 400 nm, and a quartztungsten-halogen lamp is used from 400 to 2500 nm wavelengths. The spectral resolution is approximately 7 nm at 550 nm. The collection solid angle is 0.00177 sr and the corresponding half-cone angle is 1.35 . A Si photodiode is used for wavelengths from 200 to 1100 nm, and a thermoelectrically cooled InAs photodiode measures wavelength range from 900 to 2500 nm. The STARR is also capable of measuring the directionalhemispherical reflectance at different wavelengths and at an incidence angle of 6 , using an integrating sphere. Recently, a three-axis automated scatterometer (TAAS) was developed for measuring the BRDF of Si wafers with both in-plane and out-of-plane
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FIG. 43. Schematic of the STARR instrument at NIST [310].
FIG. 44. The three-axis automated scatterometer (TAAS) [261].
capability [261]. A picture of the instrument is shown in Fig. 44. The scatterometer consists of three major subsystems: goniometric table, light source, and detection subsystems. The goniometric table consists of three high-accuracy rotary stages, with computer-controlled step motors. The rotary stages, actuated by the step motors, have high angular resolution,
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repeatability, and accuracy. The collection solid angle is 0.000184 sr and the corresponding half-cone angle is approximately 0.45 . The rotary stages generally have an angular resolution better than 0.005 and an absolute accuracy within 0.001 . Different light sources are employed and can be conveniently interchanged. A fiber-coupled diode laser system is used to provide a coherent light source at wavelengths of 635, 790, and 1550 nm. A monochromator can be employed for broadband measurements. A lock-in amplifier sends an oscillating signal directly to the laser current controller and measures the modulated signals from the sample and reference photodiode detectors. The use of a phase-lock device effectively eliminates the influence of background radiation. Si and Ge photodiode detectors measure the radiant power in the wavelength ranges 350–1100 nm and 800– 1800 nm, respectively. Large dynamic range preamplifiers are used to maintain a near zero bias across the photodiodes and to have a large transimpedance range from 10 to 109 . A PC performs the data acquisition and automatic rotary-stage control in a LabView environment. A number of Si wafers have been characterized using the STARR and TAAS [261,262,321]. Figure 45 compares the BRDF of a rough Si surface (Rq ¼ 460 nm) at three wavelengths. The ratio Rq/l is generally considered as a parameter describing the optical smoothness. A surface would appear smoother at longer wavelengths. The peak value at the specular direction is 25% smaller for ¼ 785 nm than for l ¼ 635 nm, which could not be explained by the reduction of the specular reflectivity calculated from the Fresnel equations. The BRDF curve for l ¼ 1550 nm shows a prominent
FIG. 45. Measured BRDFs for a rough Si wafer at three wavelengths [321].
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FIG. 46. Comparison of the in-plane and out-of-plane BRDF [321].
specular spike. Since the silicon substrate becomes semi-transparent at l>1.1 mm [39], multiple reflections within the wafer may have played a role in the BRDF at 1550 nm. The out-of-plane distribution was measured within a small cone around the specular direction, and is plotted in Fig. 46, where 0 and 0 are the local polar angle and azimuthal angle, respectively. Because of the symmetry, the reported values are averaged over positive 0 and negative 0 . The difference between the BRDF values is small for 0 <5 , suggesting that the use of in-plane measurement results for out-ofplane BRDF is appropriate at very small cone angles. For large 0 , however, the BRDF appears to be anisotropic around the specular direction. For the same 0 , the largest values of BRDF occur at 0 ¼ 180 and the smallest values do not occur at 0 ¼ 0 but at 0 ¼ 45 . Measurements of coated Si wafers show additional features that could not be explained from conventional theory [321].
C. RADIATIVE PROPERTIES OF MICROSTRUCTURED SURFACES It has been known for a long time that radiative properties can be modified by surface microstructures, especially the directional properties [304,322,323]. Earlier studies, however, dealt with rather simple geometries and mostly for metallic surfaces. The emergence of microfabrication has led to more systematic investigation of the influence of microstructure on the radiative properties, especially thermal emission. Furthermore, it is now possible to design such structures to tailor the absorption and emission spectra for energy conversion devices. A brief survey is given below.
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Hesketh, Zemel, and Gebhart [324–327] published a series of studies on the thermal emission from periodically grooved micromachined silicon surfaces. The grooves were 45 mm deep and near square wave etched on heavily doped p-type Si wafers. The lateral periodicity was from 10 to 22 mm. The measured infrared wavelength ranged from 3 to 14 mm. Therefore, the ratio =l0 varied from 0.14 to 7.33, by nearly two orders of magnitude. Initially, the normal emittance was measured for both p- and spolarizations. Measurements showed a strong enhancement of thermal emission as compared with smooth Si wafers. Some resonance features were observed in the emission spectra but their dependence on was significantly different for p- and s-polarizations. The observed emissivity enhancement was explained by ‘‘organ pipe’’ radiant modes [324]. The directional emittance exhibited wavy features, associated with interference effects. It was shown that geometric optics models failed to predict the observed behavior, in both the normal and directional spectral emittance [325]. Zemel [328] discussed the ‘‘blackbody’’ radiation properties in microstructures when the geometric length of the cavities becomes comparable with the wavelength of the emitted radiation. It was shown that the photon density of states is dependent on the geometric length scale. When the length of the cavity is comparable with the emission wavelength, the radiation from the wall of the cavity may be coupled to the cavity itself. Wang and Zemel [329–331] continued the work by studying the spectral emittance of micromachined surfaces with undoped Si. Several theories were examined, including the Bloch-wave, coupled-mode, effective medium, and waveguide methods. It was found that the emission is spatially coherent from the microstructures [329]. For this reason, the effective medium theory cannot explain the observed directional and spectral variation of the measured emittance [330]. The large effect of doping was somewhat unclear. Greffet et al. [332] showed a rather strong coherent thermal emission, both directional and angular. Their work is based on a fundamental theoretical investigation on the near-field thermal emission due to surface excitations [333]. They concluded that the origin of coherent thermal emission lies in the diffraction of surface-phonon polaritons by the grating. Maruyama et al. [334] developed two-dimensional microcavities using Cr-coated Si surface, and demonstrated discrete thermal emission peaks from these structures. Sai et al. [335] compared the measured reflectance and emittance of Pt films on two-dimensional periodic microstructured Si surfaces with the rigorous coupled-wave analysis (RCWA) [178]. The agreement demonstrated again that thermal emission from periodic microcavities is spatially coherent. Kanamori et al. [336] fabricated subwavelength structures on a Si surface using a porous alumina membrane mask. Two-dimensional holes arranged in hexagon were formed with a
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feature size of 100 nm. The structure has been demonstrated to have antireflection effect in the visible region [336]. Surface modifications have great potential for selective emission and absorption, especially for space applications and thermophotovoltaic devices [8,10,11].
V. Quantum Confinement and Photonic Crystals Several phenomena can occur as the characteristic dimension is reduced to below some mechanistic length scale. As discussed earlier, boundary scattering of electrons and phonons can modify the transport and optical properties. In addition, the band gap can be tuned. Electron motion can be restricted in one-, two-, and three-dimensions, resulting in quantum wells, quantum wires, and quantum dots. An analog of electronic confinement is the photon confinement, resulting in photonic crystals. The study of photonic crystals and its potential applications is an area of great interest to the scientific community. In fact, photonic crystals are finding their way towards commercial applications.
A. QUANTUM CONFINEMENT Quantum confinement effects appear when the geometric dimensions of the nanostructures are comparable to the de Broglie wavelengths of electrons and holes of the material or reach the value of the bulk exciton Bohr radius aB [337–343], which is written in the form hn2 1 1 ð180Þ þ aB ¼ e m e m h where n is the refractive index of the semiconductor. The typical value of aB is about 4.3 nm for Si and about 2.8 nm for CdS. The dispersion relationship of the bottom conduction band or the top valence band in bulk semiconductors can be approximated as a parabola using the concept of effective mass of electrons or holes. Therefore, the corresponding density of states is proportional to the square root of energy. In the case of quantum well, the electron motion is confined in the x–y plane. If the potential walls of the quantum well are infinite, the quantized energy levels can be expressed as [338] El ¼
p2 h2 2 l ; 2m d 2
l ¼ 1; 2; 3; . . .
ð181Þ
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where m* is the effective mass of electrons or holes, and d is the characteristic dimension along the confinement direction (thickness of the quantum well). The energy levels are size dependent. The dispersion relation corresponding to the infinite motion in the x- and y-directions, but restricted in the z-direction, can be written in the form
EðkÞ ¼ El þ
2 ðk2x þ k2y Þ h 2m
ð182Þ
The density of electron or hole states obtained from Eq. (182) is a series of step functions, as shown in Fig. 47(a). If the quantum confinement is in twodimensions, the type of structure is referred to as a quantum wire. The calculation of the dispersion relation and the density of states for quantum wires can follow the same way as in the case of quantum wells. Figure 47(b) illustrates the density of states in a quantum wire as a function of energy. Finally, if the motion of electrons, holes, and excitons is restricted in all three dimensions, this quasi-zero-dimensional system is called a quantum dot. Because of the three-dimensional restrictions, a quantum dot is characterized by a discrete -function-like density of states, which is shown schematically in Fig. 47(c). A brief review discussion of the properties of quantum confinement is given below, based on the framework of effective
FIG. 47. Density of electron states: (a) Bulk materials (continuous curve) and quantum wells (step function); (b) quantum wires; and (c) quantum dots.
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mass approximation. Detailed theoretical and experimental studies of the quantum-confined structures can be found in Refs. [337–343]. Because of the difference in the density of states between bulk materials and quantum well structures, the optical properties of quantum-confined structures become size dependent. For instance, experimental and theoretical studies have confirmed that the band gap energy of HgTe-CdTe superlattices (multiple quantum wells) and the exciton binding energy of In0.135Ga0.865As/GaAs quantum wires are dependent on the sizes of the quantum structures [337,344]. In the last decade, with advancement of modern technique for nanocrystal growth, the quantum confinement effect in quantum dots has been extensively studied [339–343]. Quantum dots can be developed in inorganic glasses with diffusion-controlled method; in organic solutions or polymers; and on a crystal substrate by means of selforganized growth [339]. The simplest quasi zero-dimensional systems are spherical semiconductor nanocrystals embedded in glasses or organic matrices. This has proved to be the ideal model system for the study of basic questions of quantum dots. As the exciton Bohr radius aB is noticeably larger than the lattice constant for the most common semiconductors, the problem can be considered as a crystallite that has a very large number of atoms and can be treated as a macroscopic crystal compared with the lattice constant but at the same time should be considered as a quantum box for quasi-particles. Thus, it is reasonable to consider the quasi-particles featuring the properties inherent in an infinite crystal, but affected by the finite size of the crystallite and the potential jump at the boundaries. The effective mass approximation is based on this consideration under the assumption that the effective masses of electrons and holes are the same as in the bulk materials. Different regimes have been delineated in terms of the quantum dot radius d/2 and the exciton Bohr radius aB. 1. Weak Confinement This regime corresponds to the case when d/2 is a few times larger than the exciton Bohr radius aB. In this case the quantization of the exciton center-of-mass motion occurs. If the potential jump at the boundaries of the quantum dot is further assumed to be infinite, the ground state of an exciton energy can be obtained as [339,343] E ¼ Eg Ry þ
2 h2 p2 Md 2
ð183Þ
where Ry* is the exciton Rydberg energy and is given by Ry ¼ e2 =2n2 aB , and M ¼ m e þ m h is the exciton translation mass. Notice that Eq. (183) is
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also the energy of the first allowed optical transition. It is clear that the exciton absorption peak in the optical absorption spectra will have a blue shift with the decrease of the quantum dot radius. However, as long as d 2aB holds, the third term on the right-handed side is negligibly small compared with the exciton Rydberg energy Ry*. This is the reason for the term ‘‘weak confinement’’. 2. Strong Confinement The strong confinement regime corresponds to the condition d< <2aB , which means that the confined electron and hole have no bound state corresponding to the hydrogen-like exciton. However, since an electron and a hole are confined in space comparable with the extension of the exciton ground state for bulk material, the Hamiltonian of the electron–hole system should include the Coulomb potential and because the confinement potential does not allow the center-of-mass motion, the motion of a particle with reduced mass ð1=m e þ 1=m h Þ1 will be considered in this case. The problem has been treated successfully with the variational approach and the ground energy of electron–hole pair can be expressed as [345] " # 2 2aB 2 2aB 0:248 Ry E ¼ Eg þ p 1:786 ð184Þ d d in which the third term on the right-handed side accounts for the effective Coulomb electron–hole interaction. The strong monotonic blue-shift of the first allowed transition energy with the decrease of d is obvious from Eq. (184). Another case is when the ratio of the effective mass of a hole to that of an electron is very large, i.e. m h m e . If we define the electron and hole Bohr radii as ae ¼
n2 h2 me e2
and
ah ¼
n2 h2 m h e2
ð185Þ
Thus ae aB ah . In the case ah <
ð186Þ
The region of the hole motion around the dot center is much smaller than the dot radius and the size dependence of the exciton ground state can be described as the behavior of a donor localized at the center of the dot [339,343]. In this case the absorption spectrum is mainly determined by the quantization of the electron motion, but with the electron–hole Coulomb
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interaction, each electron level will split into several sublevels. The energy corresponding to the first absorption peak can be found to be in the form
d E ¼ Eg þ 2 aB
2
d Ry exp aB
ð187Þ
One should bear in mind that Eqs. (183) and (184) are obtained in the limits of weak and strong quantum confinement in the framework of effective mass approximation, and the confinement potential is assumed to be infinite which is not true in reality. Moreover, the effective mass approximation is not adequate in correctly describing the behaviors of electrons and holes since the conduction and valence bands may not be always expected to be parabolic-like; and the top of the valence band consists of four-fold degenerate subband. More accurate methods thus have been proposed, such as multiband effective mass approximation, tightbinding method with finite confinement potential, and pseudo-potential method [340,343,346]. However, the discussion of quantum confinement effect in terms of the weak and strong confinement limits provides a quantitative understanding of the physics of quantum confinement. Furthermore, Eqs. (183) and (184) have been found to provide a good fit to the exact solution in the range d > 8aB and d < 2aB , respectively [339,343]. The energy of the first allowed transition increases as the dot radius decreases. 3. Porous Silicon Promising applications of quantum dots include fast switching devices, quantum electronics, quantum computing, and quantum dot lasers [347– 350]. A brief discussion of a three-dimensional nanostructure, i.e. porous silicon (PS) is given here. There has been a surge of studies about the optical and radiative properties of Si nanocrystallites, shortly after the report of strong visible luminescence in porous silicon at room temperature in early 1990s [351–355]. As discussed above, crystalline Si is the dominant material in semiconductor industry but it is not an effective light emitter because of its indirect band gap. The feasibility of luminescence in PS renewed the hope of optoelectronic applications of Si-based materials. In addition to light emitting devices, applications of PS include solar cells, tunable optical filters, antireflection coatings, waveguides, chemical and biomedical sensors, micromachining applications, etc. The most striking quantum confinement effect in PS is the band gap widening, resulting in effective visible luminescence. Extensive theoretical investigations of the luminescent mechanisms, quantum size dependence, absorption, radiative recombination, and temperature dependence have been performed [356–364]. Optical absorption and refractive index of PS
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and oxidized PS have been measured and the theoretical analysis is often based on the effective medium theories [365–369]. The effective medium concept was developed in 1930s and used widely for inhomogeneous medium [370]. The Bruggeman effective medium approximation gives f1
"1 "eff "2 "eff þ f2 ¼0 "1 þ 2"eff "2 þ 2"eff
ð188Þ
where "1 and "2 are the dielectric function of medium 1 and 2, respectively, f1 and f2 are the volumetric fractions of each medium (f1 þ f2 ¼ 1), and "eff is the dielectric function of the composite. In the case of PS, f2 is the porosity and "2 ¼ 1. Eq. (188) can be used to study the effect of porosity on the refractive index and absorption coefficient. Several groups investigated the use of PS as the primary material for solar cells or as an emitter and antireflection layer [371–375]. Optical waveguides have been demonstrated with oxidized porous silicon [376,377]. Lammel et al. [378] proposed a microspectrometer based on tunable optical filter made of PS. Macroporous Si microstructures have been realized as photonic crystals for two-dimensional photon confinement [379]. More discussions about photonic crystals are given next.
B. PHOTONIC CRYSTALS Microstructure engineering can modify the density of states of a radiation field. The result is a change in the optical properties of atoms and molecules. Photonic crystals are periodic arrays of different refractive-index materials. The spatial periodicity is called the (photonic) lattice constant, based on the same concept for ordinary crystals. However, the lattice constant in photonic crystals is on the order of wavelength, which is usually much greater than those of ordinary crystals. For example, it is around 0.5 mm for visible light and 1 cm for microwave. Photon confinement may occur in one-, two-, or three-dimensions. Light absorption and emission can be strongly influenced by exciton-photon coupling, which is called a polariton. The appearance of a frequency range where no electromagnetic eigenmodes exist gives rise to the term: photonic band gaps, since they correspond to band gaps of electronic eigenstates in ordinary crystals [380]. It has been found that adding some defects to an otherwise perfect photonic crystal can enhance the functionality of a photonic crystal [381,382]. This could be done by putting extra materials or removing some materials, a concept equivalent to doping with donors or acceptors in semiconductors. Yablonovitch [383] gave a detailed comparison between the photonic band gap structures and the electronic band gap structures in ordinary crystals.
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Microfabrication techniques have enabled the manufacturing of photonic crystals in millimeter and microwave regions, because the lattice constant is quite large, from 100 mm to 1 cm [384–386]. On the other hand, threedimensional photonic crystals for the visible region are much more difficult to fabricate. Fleming and Lin [387] fabricated three-dimensional Si-based photonic crystals with a near-infrared stop band around 1.5 mm. The potential applications include high-Q resonant cavities [388], thresholdless single-mode light-emitting diode [383], waveguide and superprism [389–391], etc. Another application is the holey silica fibers [392]. In conventional optic fibers, light is guided inside the fiber based on total reflection at the interface between the high refractive index core and the lower refractive index cladding. A photonic band gap fiber restricts light radially and guides it along the fiber. Unlike conventional fiber, the core is hollow so that losses can be minimized. Detailed discussion and review can be found from Ref. [392]. There have been a large number of works dealing with the simulation of the transmission and reflection behavior of different structures. The transfer-matrix method developed by Pendry and MacKinnon [393] has demonstrated advantages in terms of speed and convenience. The finite difference time domain method has been used for general time-dependent solutions [394,395]. Sakoda and coworkers have done extensive analysis on the transmission spectra and band structure of two- and three-dimensional photonic crystals [396–399]. Figure 48 shows the band structure and transmittance spectra of a two-dimensional hexagonal lattice, see Fig. 3 [398,399]. The structure contains PbO glass with a refractive index of 1.65 and air (cylindrical holes). For the band structure, the solid curves represent symmetric modes and the dashed curves represent antisymmetric modes. The lattice constant is 1.15 mm and the hole radius is 0.486 mm. More detailed discussion on the transmittance and reflectance of photonic crystals can be found from Refs. [395,399]. Microcavities in photonic crystals have proven to be effective in enhancement and suppression of thermal emission, as well as transmission and reflection [400–402]. A thin slab of three-dimensional photonic crystal acts as a blackbody at the resonance wavelength, while the emission is suppressed at some other wavelengths [402].
VI. Concluding Remarks This article reviews the optical and thermal radiative properties of semiconductor-related materials: from bulk crystals to thin-film multilayer
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FIG. 48. Band structure and transmittance spectra of a hexagonal photonic crystal, after [398,399]: (a) TE wave; (b) TM wave. In (a) and (b), the left panel is the band structure and the right is the transmittance spectrum.
structures, and from rough and microstructured surfaces to quantum dots and photonic crystals. Only a very small portion of the enormously large number of publications relevant to this broad multidisciplinary topic is cited. Although the physics of bulk crystalline materials is well understood, further studies are needed to measure and model the spectral properties at elevated temperatures and with different doping concentrations. Theories for calculating the radiative properties of perfectly parallel multilayer systems of smooth interfaces have been well established. However, when the surface is rough or contains microstructures, the radiative property calculation is more complicated and often involves a number of simplifications. Nanostructure engineering holds great promise for novel optoelectronic and microscale energy conversion devices. There exist tremendous opportunities in fabrication, numerical simulation, and experimental investigation of micro and nanostructured materials.
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Acknowledgements The authors thank Professor Dave DeWitt, Professor David Tanner, and Dr. Paul Timans for their constant encouragement. This work was supported by the National Science Foundation (CTS-9875441, CTS0082969, and EEC-0080453) and the Optical Technology Division of the National Institute of Standards and Technology. A large portion of this work was completed when the authors were with the University of Florida. Thanks to former Ph.D. students, Yu-Jiun Shen and Yihui Zhou, and Master’s students, Jorge Garcia and Linxia Gu, for valuable contributions; and to current graduate students, Yu-Bin Chen, Bong Jae Lee, Hyunjin Lee, Vinh Khuu, and Keunhan Park, for carefully reviewing the manuscript. Nomenclature A0l a a aB al b c cv D d E E Ec Eg Ev e eb;l f G H H h i J k kB M m m* N n
absorptance of a single- or multilayered structure acceleration lattice constant Bohr radius absorption coefficient autocorrelation length speed of electromagnetic wave specific heat density of state diameter or thickness electric field vector energy energy at the bottom of the conduction band band gap energy ðEg ¼ Ec Ev Þ energy at the top of the valence band electron charge blackbody emissive power distribution function, spatial frequency reciprocal lattice vector magnetic field vector Hamiltonian operator Planck’s constantp (hffiffiffiffiffiffi ¼ffi 2ph) complex number 1 current density wavevector (for electron or phonon) Boltzmann’s constant mass of an ion or atom particle mass effective mass number of phonon oscillators refractive index; number density
vector between two lattice points in real space R0 reflectance r position vector r~ Fresnel reflection coefficient P radiant power } principal value of an integral p momentum q wavevector (for electromagnetic waves) q positive charge S Poynting vector Sj strength of the jth oscillator s surface slope T temperature T0 transmittance t time t~ Fresnel transmission coefficient U potential group velocity vg x, y, z Cartesian coordinates R
GREEK SYMBOLS 0l ~ l " "~ "0 Z
absorptivity complex phase shift through a layer scattering rate or damping coefficient radiation penetration depth electric permittivity complex dielectric function emissivity mobility zenith angle extinction coefficient (imaginary part of the refractive index)
278 l lc F 0 l0
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wavelength de Broglie wavelength magnetic permeability Fermi level electric charge density reflectivity electric conductivity scattering time internal transmissivity azimuthal angle wavefunction solid angle angular frequency
SUBSCRIPTS 0 1
2 e F f h i p ph q r S s sp t
medium 2 electron film free carrier hole incident TM wave, parallel polarization phonon root mean square reflected substrate TE wave or perpendicular polarization, scattered specular transmitted or refracted
vacuum or air medium 1
References 1. Service, R. F. (2001). Breakthrough of the year: molecules get wired. Science 294, 2442– 2443. 2. Bachtold, A., Hadley, P., Nakanishi, T., and Dekker, C. (2001). Logic circuits with carbon nanotube transistors. Science 294, 1317–1320. 3. Rao, A. M., Richter, E., Bandow, S., Chase, B., Eklund, P. C., Williams, K. A., Fang, S., Subbaswamy, K. R., Menon, M., Thess, A., Smalley, R. E., Dresselhaus, G., and Dresselhaus, M. S. (1997). Diameter-selection Raman scattering from vibrational modes in carbon nanotubes. Science 275, 187–191. 4. Bayer, M., Baars, T., Braun, W., and Forchel, A. (2001). Coherence spectroscopy on quantum wires. In ‘‘Ultrafast Phenomena in Semiconductors’’ (K.-T. Tsen, ed.), pp. 405– 442. Springer-Verlag, Berlin. 5. Wang, K. L. and Balandin, A. A. (2001). Quantum dots: physics and applications. In ‘‘Optics of Nanostructured Materials’’ (V. A. Markel and T. F. Geroge, eds.), pp. 515–549. Wiley, New York. 6. Sharma, P. C., Alt, K. W., Yeh, D. Y., Wang, D., and Wang, K. L. (1999). Formation of nanometer-scale InAs island on silicon. J. Electron Mater. 28, 432–436. 7. Genereux, F., Leonard, S. W., van Driel, H. M., Birner, A., and Gosele, U. (2001). Large birefringence in two-dimensional silicon photonic crystals. Phys. Rev. B 63, 161101/4. 8. Coutts, T. J. (1999). A review of progress in thermophotovoltaic generation of electricity. Renewable Sustainable Energy Rev. 3, 77–184. 9. Rohr, C. (2000). ‘‘InGaAsP Quantum Well Cells for Thermophotovoltaic Applications’’. Ph.D. dissertation, University of London. 10. Boueke, A., Kuhn, R., Fath, P., Willeke, G., and Bucher, E. (2001). Latest results on semitransparent POWER silicon solar cells. Solar Energy Mater. Solar Cells 65, 549–553. 11. Poruba, A., Fejfar, A., Remes, Z., Springer, J., Vanecek, M., Kocka, J., Meier, J., Torres, P., and Shah, A. (2000). Optical absorption and light scattering in microcrystalline silicon thin films and solar cells. J. Appl. Phys. 88, 148–160. 12. Born, L. and Wolf, E. (1999). ‘‘Principles of Optics’’, 7th edn. Cambridge University Press, Cambridge. 13. Jackson, J. D. (1999). ‘‘Classical Electrodynamics’’, 3rd edn. Wiley, New York.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
279
14. Kong, J. A. (1990). ‘‘Electromagnetic Wave Theory’’, 2nd edn. Wiley, New York. 15. Tien, C. L. and Lienhard, J. H. (1985). ‘‘Statistical Thermodynamics’’. Hemisphere, Washington, DC. 16. Kingston, R. H. (1978). ‘‘Detection of Optical and Infrared Radiation’’. Springer-Verlag, Berlin. 17. Mahan, J. R. (2002). ‘‘Radiation Heat Transfer: A Statistical Approach’’. Wiley, New York. 18. Siegel, R. and Howell, J. R. (1992). ‘‘Thermal Radiation Heat Transfer’’, 3rd edn. Hemisphere, Washington, DC. 19. Modest, M. F. (1993). ‘‘Radiative Heat Transfer’’. McGraw-Hill, New York. 20. Brewster, M. Q. (1992). ‘‘Thermal Radiative Transfer and Properties’’. Wiley, New York. 21. Kittel, C. (1996). ‘‘Introduction to Solid State Physics’’, 7th edn. Wiley, New York. 22. Ashcroft, N. W. and Mermin, N. D. (1976). ‘‘Solid State Physics’’. Harcourt College Publishers, Fort Worth, TX. 23. Ziman, J. M. (1960). ‘‘Electrons and Phonons’’. Oxford University Press, Oxford. 24. Majumdar, A. (1998). Microscale energy transport in solid. In ‘‘Microscale Energy Transport’’ (C. L. Tien, A. Majumdar, and F. M. Gerner, eds.), Chapter 1, pp. 3–93. Taylor & Francis, Washington, DC. 25. Cohen, M. L. and Chelikowsky, J. R. (1988). ‘‘Electronic Structure and Optical Properties of Semiconductors’’. Springer-Verlag, Berlin. 26. Hummel, R. E. (1993). ‘‘Electronic Properties of Materials’’. Springer-Verlag, Berlin. 27. Razeghi, M. (1998). GaN-based laser diodes. Int. J. High Speed Electron. Syst. 9, 1007–1080. 28. Willardson, R. K. and Beer, A. C. (eds.) (1981). Mercury cadmium telluride. In ‘‘Semiconductors and Semimetals’’, Vol. 18, Academic Press, New York. 29. Chen, G. (1996). Heat transfer in micro- and nanoscale photonic devices. Annu. Rev. Heat Transfer 7, 1–57. 30. Cahill, D. G. (1998). Heat transport in dielectric thin films and at solid–solid interfaces. In ‘‘Microscale Energy Transport’’ (C. L. Tien, A. Majumdar, and F. M. Gerner, eds.), Chapter 2, pp. 95–117. Taylor & Francis, Washington, DC. 31. Goodson, K. E. and Ju, Y. S. (1999). Heat Conduction in novel electronic films. Annu. Rev. Mater. Res. 29, 261–293. 32. Zhang, Z. M. and Frenkel, A. (1994). Thermal and nonequilibrium responses of superconductors for radiation detectors. J. Superconductivity 7, 871–884. 33. Gao, F., Carr, G. L., Porter, C. D., Tanner, D. B., Williams, G. P., Hirschmugl, C. J., Dutta, B., Wu, X. D., and Etemad, S. (1996). Quasiparticle damping and the coherence peak in YBa2Cu3O7-. Phys. Rev. B 54, 700–710. 34. Jacoboni, C. and Reggiani, L. (1983). The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials. Rev. Mod. Phys. 55, 645–705. 35. Ferry, D. K., Barker, J. R., and Jacoboni, C. (1980). ‘‘Physics of Nonlinear Transport in Semiconductors’’. Plenum, New York. 36. Tellier, C. R. and Tosser, A. J. (1982). ‘‘Size Effects in Thin Films’’. Elsevier, Amsterdam. 37. Goodson, K. E. and Flik, M. I. (1993). Electron and phonon thermal conduction in epitaxial high-Tc superconducting films. J. Heat Transfer 115, 17–25. 38. Flik, M. I., Zhang, Z. M., Goodson, K. E., Siegal, M. P., and Phillips, J. M. (1992). Electron scattering rate in epitaxial YBa2Cu3O7 superconducting films. Phys. Rev. B 46, 5606–5614. 39. E. D. Palik, ed. (1985). ‘‘Handbook of Optical Constants of Solids’’. Academic Press, New York. 40. Timans, P. J. (1996). Advances in rapid thermal and integrated processing. In ‘‘NATO ASI Series E: Applied Sciences’’ (F. Roozeboom, ed.), Vol. 318, pp. 1–67.
280
Z. M. ZHANG ET AL.
41. Pankove, J. I. (1971). ‘‘Optical Processes in Semiconductors’’. Dover Publications, New York. 42. Ridley, B. K. (1999). ‘‘Quantum Processes in Semiconductors’’, 4th edn. Oxford University Press, Oxford. 43. Hebb, J. P., Cravalho, E. G., and Flik, M. I. (1995). Thermal radiation absorption in doped semiconductors due to direct intersubband transitions. J. Heat Transfer 117, 948–954. 44. Ryoo, K., Kim, H. R., Koh, J. S., Seo, G., and Lee, J. H. (1992). Fine structure of oxygen absorption bands in Si at low temperatures. J. Appl. Phys. 72, 5393–5396. 45. Zhang, Z. M., Hanssen, L. M., Datla, R. U., and Drew, H. D. (1996). An apparatus for infrared transmittance and reflectance measurements at cryogenic temperatures. Int. J. Thermophys. 17, 1441–1454. 46. Bauer, G. and Richter, W., eds. (2000). ‘‘Optical Characterization of Epitaxial Semiconductor Layers’’. Springer-Verlag, Berlin. 47. Zhang, Z. M. (2000). Surface temperature measurement using optical techniques. Annu. Rev. Heat Transfer 11, 351–411. 48. Donnelly, V. M. and McCaulley, J. A. (1990). Infrared–laser interferometric thermometry: a nonintrusive technique for measuring semiconductor wafer temperatures. J. Vac. Sci. Technol. A 8, 84–92. 49. Dilhac, M., Ganibal, C., Nolhier, N., and Amat, L. (1990). In situ interferometric measurements in a rapid thermal processor. SPIE Proc. 1393, 349–353. 50. Boebel, F. G. and Mo¨ller, H. (1993). Simultaneous in situ measurement of film thickness and temperature by using multiple wavelengths pyrometric interferometry. IEEE Trans. Semicond. Manuf. 6, 112–118. 51. Ritter, G., Tillack, B., Weidner, M., Zaumseil, P., Bo¨bel, F. G., Hertel, B., and Mo¨ller, H. (1995). In-situ observation of chemical vapour growth of epitaxial SiGe thin films by reflexion supported pyrometric interferometry. J. Cryst. Growth 146, 119–124. 52. Boebel, F. G., Mo¨ller, H., Hertel, B., Grothe, H., Schraud, G., St. Schro¨der, and Chow, P. (1995). In situ film thickness and temperature control of molecular beam epitaxy growth by pyrometric interferometry. J. Cryst. Growth 150, 54–61. 53. Zhou, J. J., Li, Y., Thompson, P., Sato, D. L., Lee, H. P., and Kuo, J. M. (1996). Physical origins of temperature variation and background radiation associated with pyrometric interferometry measurement during III–V molecular-beam-epitaxy growth. Appl. Phys. Lett. 69, 2683–2685. 54. Sturm, J. C., Reaves, C. M., and Garone, P. M. (1990). Silicon temperature measurement by infrared transmission for rapid thermal processing applications. Appl. Phys. Lett. 56, 961–963. 55. Friberg, A. T. and Kajanto, I. (1989). Optical measurement of temperature with singlecrystal silicon. Rev. Sci. Instrum. 60, 2764–2767. 56. Malam, S. C., Pell, T., and Nix, R. M. (1994). Noninvasive infrared-based measurement of the temperature of GaAs semiconductor crystals. J. Vac. Sci. Technol. A 12, 2938–2939. 57. Thurmond, C. D. (1975). The standard thermodynamic functions for the formation of electrons and holes in Ge, Si, GaAs, and GaP. J. Electrochem. Soc. 122, 1133–1141. 58. Jellison, G. E., Jr. and Lowndes, D. H. (1982). Optical absorption coefficient of silicon at 1.152 m at elevated temperatures. Appl. Phys. Lett. 41, 594–596. 59. Pearsall, T. P., Saban, S. R., Booth, J., Beard, B. T., Jr., and Johnson, S. R. (1995). Precision of noninvasive temperature measurement by diffuse reflectance spectroscopy. Rev. Sci. Instrum. 66, 4977–4980. 60. Sato, T. (1967). Spectral emissivity of silicon. Jpn. J. Appl. Phys. 6, 339–347. 61. Touloukian, Y. S. and DeWitt, D. P. (eds.) (1970–1972). Thermal radiative properties. In ‘‘Thermophysical Properties of Matter’’ (Y. S. Touloukian and C. Y. Ho, eds.), Vols. 7, 8&9. IFI Plenum, New York.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
281
62. Weakliem, H. A. and Redfield, D. (1979). Temperature dependence of the optical properties of silicon. J. Appl. Phys. 50, 1491–1493. 63. Vandenabeele, P. and Maex, K. (1992). Influence of temperature and backside roughness on the emissivity of Si wafers during rapid thermal processing. J. Appl. Phys. 72, 5867– 5875. 64. Timans, P. J. (1993). Emissivity of silicon at elevated temperatures. J. Appl. Phys. 74, 6353– 6364. 65. Roy, S., Bang, S. Y., Modest, M. F., and Stubican, V. S. (1993). Measurement of spectral, directional reflectivities of solids at high temperatures between 9 and 11 mm. Appl. Opt. 32, 3550–3558. 66. Postlethwait, M. A., Sikka, K. K., Modest, M. F., and Hellmann, J. R. (1994). Hightemperature, normal spectral emittance of silicon carbide based materials. J. Thermophys. Heat Transfer 8, 412–418. 67. Jellison, G. E., Jr. and Modine, F. A. (1994). Optical functions of silicon at elevated temperatures. J. Appl. Phys. 76, 3758–3761. 68. Xu, H. and Sturm, J. C. (1995). Emissivity of rough silicon surfaces: measurement and calculations. Mat. Res. Soc. Symp. Proc. 387, 29–34. 69. Rogne, H., Timans, P. J., and Ahmed, H. (1996). Infrared absorption in silicon at elevated temperatures. Appl. Phys. Lett. 69, 2190–2192. 70. Ravindra, N. M., Abedrabbo, S., Chen, W., Tong, F. M., Nanda, A. K., and Speranza, A. C. (1998). Temperature-dependent emissivity of silicon-related materials and structures. IEEE Trans. Semicond. Manuf. 11, 30–39. 71. Ravindra, N. M., Sopori, B., Gokce, O. H., Cheng, S. X., Shenoy, A., Jin, L., Abedrabbo, S., Chen, W., and Zhang, Y. (2002). Emissivity measurements and modeling of siliconrelated materials: an overview. Int. J. Thermophys. 22, 1593–1611. 72. Chen, G., Borca-Tasciuc, T., and Fair, R. B. (1997). Photon effect on radiative properties of silicon during rapid thermal processing. J. Appl. Phys. 82, 830–835. 73. Boyd, I. W., Binnie, T. D., Wilson, J. I. B., and Colles, M. J. (1984). Absorption of infrared radiation in silicon. J. Appl. Phys. 55, 3061–3063. 74. Chen, G. (1996). Optical effect on thermal emission of semiconductors. Appl. Phys. Lett. 69, 512–513. 75. Svelto, O. (1998). In ‘‘Principles of Lasers’’, 4th edn. (Translated and edited by D. C. Hanna). Plenum Press, New York. 76. Faist, J., Capasso, F., Sivco, D. L., Sirtori, C., Hutchinson, A. L., and Cho, A. Y. (1994). Quantum cascade laser. Science 264, 553–556. 77. Paiella, R., Capasso, F., Gmachl, C., Sivco, D. L., Baillargeon, J. N., Hutchinson, A. L., Cho, A. Y., and Liu, H. C. (2000). Self-mode-locking of quantum cascade lasers with giant ultrafast optical nonlinearities. Science 290, 1739–1742. 78. Beck, M., Hofstetter, D., Aellen, T., Faist, J., Oesterle, U., Ilegems, M., Gini, E., and Melchior, H. (2002). Continuous wave operation of a mid-infrared semiconductor laser at room temperature. Science 295, 301–305. 79. Capasso, F., Gmachl, C., Sivco, D. L., and Cho, A. Y. (2002). Quantum cascade lasers. Phys. Today 55(5), 34–40. 80. M. Cardona, ed. (1975). ‘‘Light Scattering in Solids’’. Springer-Verlag, Berlin. 81. M. Cardona and G. Gu¨ntherodt, eds. (2000). ‘‘Light Scattering in Solids VIII: Fullerences, Semiconductor Surfaces, Coherent Phonons’’. Springer-Verlag, Berlin. 82. Nussenzveig, H. M. (1972). ‘‘Causality and Dispersion Relations’’. Academic Press, New York. 83. Kramers, H. A. (1956). ‘‘Collected Scientific Papers’’. North-Holland Publ, Amsterdam. 84. Kronig, R. de L. (1926). On the theory of dispersion of X-rays. J. Opt. Soc. Am. 12, 547–557.
282
Z. M. ZHANG ET AL.
85. Landau, L. D. and Lifshitz, E. M. (1960). ‘‘Electrodynamics of Continuous Media’’. Pergamon, Oxford. 86. Wooten, F. (1972). ‘‘Optical Properties of Solids’’. Academic Press, New York. 87. Smith, D. Y. (1985). Dispersion theory, sum rules, and their application to the analysis of optical data. In ‘‘Handbook of Optical Constants of Solids’’ (E. D. Palik, ed.), Chapter 3, pp. 35–68. Academic Press, New York. 88. Choi, B. I., Zhang, Z. M., Flik, M. I., and Siegrist, T. (1992). Radiative properties of Y– Ba–Cu–O films with variable oxygen content. J. Heat Transfer 114, 958–964. 89. Schumann, P. A., Jr., Keenan, W. A., Tong, A. H., Gegenwarth, H. H., and Schneider, C. P. (1971). Silicon optical constants in the infrared. J. Electrochem. Soc. 118, 145–148. 90. Spitzer, W. G. and Kleinman, D. A. (1961). Infrared lattice bands of quantz. Phys. Rev. 121, 1324–1335. 91. Spitzer, W. G., Miller, R. C., Kleinman, D. A., and Howarth, L. E. (1962). Far infrared dielectric dispersion in BaTiO3, SrTiO3, and TiO2. Phys. Rev. 126, 1710–1721. 92. Zhang, Z. M., Choi, B. I., Flik, M. I., and Anderson, A. C. (1994). Infrared refractive indices of LaAlO3, LaGaO3, and NdGaO3. J. Opt. Soc. Am. B 11, 2252–2257. 93. Gervais, F. and Piriou, B. (1974). Temperature dependence of transverse- and longitudinal-optic modes in TiO2 (rutile). Phys. Rev. B 10, 1642–1654. 94. Jellison, G. E., Jr. and Modine, F. A. (1994). Optical functions of silicon at elevated temperatures. J. Appl. Phys. 76, 3758–3761. 95. MacFarlane, G. G., McLean, T. P., Quarrington, J. E., and Roberts, V. (1958). Fine structure in the absorption-edge spectrum of Si. Phys. Rev. 111, 1245–1254. 96. Timans, P. J. (1993). Emissivity of silicon at elevated temperatures. J. Appl. Phys. 74, 6353–6364. 97. Li, H. H. (1980). Refractive index of silicon and germanium and its wavelength and temperature derivatives. J. Phys. Chem. Ref. Data 9, 561–601. 98. McCaulley, J. A., Donnelly, V. M., Vernon, M., and Taha, I. (1994). Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide. Phys. Rev. B 49, 7408–7417. 99. Forouhi, A. R. and Bloomer, I. (1991). Calculation of optical constants, n and k, in the interband region. In ‘‘Handbook of Optical Constants of Solids II’’ (E. D. Palik, ed.), Chapter 7, pp. 151–175. Academic Press, New York. 100. Forouhi, A. R. and Bloomer, I. (1988). Optical properties of crystalline semiconductors and dielectrics. Phys. Rev. B 38, 1865–1874. 101. Benedict, L. X., Shirley, E. L., and Bohn, R. B. (1998). Theory of optical absorption in diamond, Si, Ge, and GaAs. Phys. Rev. B 57, R9385–R9387. 102. Albrecht, S., Reining, L., Del Sole, R., and Onida, G. (1998). Ab Initio calculation of excitonic effects in the optical spectra of semiconductors. Phys. Rev. Lett. 80, 4510–4513. 103. Rohlfing, M. and Louie, S. G. (1998). Electron–hole excitations in semiconductors and insulators. Phys. Rev. Lett. 81, 2312–2315. 104. Timans, P. J., Sharangpani, R., and Thakur, R. P. S. (2000). Rapid thermal processing. In ‘‘Handbook of Semiconductor Manufacturing Technology’’, Chapter 9, pp. 201–286. Marcel Dekker, New York. 105. Zory, P. S., Jr., ed. (1993). ‘‘Quantum Well Lasers’’. Academic Press, New York. 106. Ivchenko, E. L. and Pikus, G. (1995). ‘‘Superlattices and Other Heterostructures’’. Springer-Verlag, Berlin. 107. Kumar, A. R., Boychev, V. A., Zhang, Z. M., and Tanner, D. B. (2000). Fabry-Perot resonators built with YBa2Cu3O7- films on Si substrates. J. Heat Transfer 122, 785–791. 108. Zhang, Z. M., Datla, R. U., and Hanssen, L. M. (1995). Development of neutral-density infrared filters using metallic thin films. Mat. Res. Soc. Symp. Proc. 374, 117–122.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
283
109. Grosse, P. (1991). FTIR-spectroscopy of layered structures: thin solid films coated substrates profiles multilayers. SPIE Proc. 1575, 169–179. 110. Zhang, Q.-C. (2000). Recent progress in high-temperature solar selective coatings. Solar Energy Mater. Solar Cells 62, 63–74. 111. Chen, J., Ge, X. S., and Hu, X. F. (1998). Single-layer heat mirror films and an improved method for evaluation of its optical and radiative properties in infrared. Solar Energy Mater. Solar Cells 55, 323–329. 112. Flik, M. I., Choi, B. I., and Goodson, K. E. (1992). Heat transfer regimes in microstructures. J. Heat Transfer 114, 666–674. 113. Tien, C. L. and Chen, G. (1994). Challenges in microscale conductive and radiative heat transfer. J. Heat Transfer 116, 799–807. 114. Tien, C. L., Qiu, T. Q., and Norris, P. M. (1994). Microscale thermal phenomena in contemporary technology. Thermal Sci. Eng. 2, 1–11. 115. Kumar, S. (1994). Thermal radiation transport regimes in micro-structures. Thermal Sci. Eng. 2, 149–157. 116. Kumar, S. and Mitra, K. (1999). Microscale aspects of thermal radiation transport and laser applications. Adv. Heat Transfer 33, 187–294. 117. Zhang, Z. M. (2001). ‘‘Microscale Thermal Radiation in Thin Solid Films’’. Proceedings of the International Conference on Energy Conversion and Applications (ICECA’01), pp. 281–286, Wuhan, China. 118. Heavens, O. S. (1965). ‘‘Optical Properties of Thin Solid Films’’. Dover Publications, New York. 119. Knittl, Z. (1976). ‘‘Optics of Thin Films’’. John Wiley & Sons, New York. 120. Bohren, C. F. and Huffman, D. R. (1983). ‘‘Absorption and Scattering of Light by Small Particles’’. Wiley, New York. 121. Yeh, P. (1988). ‘‘Optical Waves in Layered Media’’. Wiley, New York. 122. Haus, H. A. (1984). ‘‘Waves and Fields in Optoelectronics’’. Prentice-Hall, Englewood Cliffs. 123. Chen, G. and Tien, C. L. (1992). Partial coherence theory of thin film radiative properties. J. Heat Transfer 114, 636–643. 124. Richter, K., Chen, G., and Tien, C. L. (1993). Partial coherence theory of multilayer thinfilm optical properties. Opt. Eng. 32, 1897–1903. 125. Wong, P. Y., Trefethen, L. M., and Miaoulis, I. N. (1992). Cross correlation of optical properties of thin films under thermal radiation. J. Appl. Phys. 72, 4884–4887. 126. Zhang, Z. M. (1994). ‘‘Optical Properties of Layered Structures for Partially Coherent Radiation’’. Proceedings of the Tenth International Heat Transfer Conference, pp. 177– 182, Brighton, UK. 127. Grossman, E. N. and McDonald, D. G. (1995). Partially coherent transmittance of dielectric lamellae. Opt. Eng. 34, 1289–1295. 128. Anderson, C. F. and Bayazitoglu, Y. (1996). Radiative properties of films using partial coherence theory. J. Thermophys. Heat Transfer 10, 26–32. 129. Zhang, Z. M. (1997). Reexamination of the transmittance formulae of a lamina. J. Heat Transfer 119, 645–647. 130. Zhang, Z. M., Hanssen, L. M., and Datla, R. U. (1996). Polarization-dependent angular reflectance of silicon and germanium in the infrared. Infrared Phys. Technol. 37, 539–546. 131. Zhang, Z. M., Lefever-Button, G., and Powell, F. R. (1998). Infrared refractive index and extinction coefficient of polyimide films. Int. J. Thermophys. 19, 905–916. 132. Zhang, Z. M., Gentile, T. R., Migdall, A. L., and Datla, R. U. (1997). Transmittance measurements for filters of optical density from one to ten. Appl. Opt. 36, 8889–8895. 133. Zhang, Z. M. (1999). Optical properties of a slightly absorbing film for oblique incidence. Appl. Opt. 38, 205–207.
284
Z. M. ZHANG ET AL.
134. Cravalho, E. G., Tien, C. L., and Caren, R. P. (1967). Effect of small spacing on radiative transfer between two dielectrics. J. Heat Transfer 89, 351–358. 135. Polder, D. and Van Hove, M. (1971). Theory of radiative heat transfer between closely spaced bodies. Phys. Rev. B 4, 3303–3314. 136. Loomis, J. J. and Maris, H. J. (1994). Theory of heat transfer by evanescent electromagnetic waves. Phys. Rev. B 50, 18,517–18,524. 137. Pentry, J. B. (1999). Radiative exchange of heat between nanostructures. J. Phys.: Condensed Matter 11, 6621–6633. 138. Mulet, J.-P., Joulain, K., Carminati, R., and Greffet, J.-J. (2001). Nanoscale radiative heat transfer between a small particle and a plane surface. Appl. Phys. Lett. 78, 2931–2933. 139. Whale, M. D. (2000). Optimization of the spacing effect of thermal radiation for microscale thermophotovoltaic devices. In ‘‘Heat Transfer and Transport Phenomena in Microscale’’, pp. 339–346. Begell House, New York. 140. Whale, M. D. and Cravalho, E. G. (2002). Modeling and performance of microscale thermophotovoltaic energy conversion devices. IEEE Trans. Energy Conv. 17, 130–142. 141. Pan, J. L., Choy, H. K. H., and Fonstad, C. G., Jr. (2000). Very large radiative transfer over small distances from a black body for thermophotovoltaic applications. IEEE Trans. Electron Dev. 47, 241–249. 142. Zhang, Z. M. and Fu, C. J. (2002). Unusual photon tunneling in the presence of a layer with a negative refractive index. Appl. Phys. Lett. 80, 1097–1099. 143. Zhang, Z. M. and Fu, C. J. (2003). Transmission enhancement by using a negativerefractive-index layer. Microscale Thermophys. Eng (in press). 144. Smith, D. R. and Kroll, N. (2000). Negative refractive index in left-handed materials. Phys. Rev. Lett. 85, 2933–2936. 145. Gralak, B., Enoch, S., and Tayeb, G. (2000). Anomalous refractive properties of photonic crystals. J. Opt. Soc. Am. A 17, 1012–1020. 146. Notomi, M. (2000). Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap. Phys. Rev. B 62, 10,696– 10,705. 147. Shelby, R. A., Smith, D. R., and Schultz, S. (2001). Experimental verification of a negative index of refraction. Science 292, 77–79. 148. Valanju, P. M., Walser, R. M., and Valanju, A. P. (2002). Wave refraction in negativeindex media: Always positive and very inhomogeneous. Phys. Rev. Lett. 88, 187,401. 149. Kakaes, K. (2002). Theorists doubt claims for perfect lens. Science 296, 1380–1381. 150. Mehta, C. L. (1963). Coherence-time and effective bandwidth of blackbody radiation. Il Nuovo Cimento 28, 401–408. 151. Wolf, E. (1978). Coherence and radiometry. J. Opt. Soc. Am. 68, 6–17. 152. Carminati, R. and Greffet, J. J. (1999). Near-field effects in spatial coherence of thermal sources. Phys. Rev. Lett. 82, 1660–1663. 153. Mandel, L. and Wolf, E. (1995). ‘‘Optical Coherence and Quantum Optics’’. Cambridge University Press, Cambridge. 154. Zhang, Z. M. (1998). Far-infrared radiation modulators using high-Tc superconductors. J. Heat Transfer 120, 24–29. 155. Kumar, A. R., Zhang, Z. M., Boychev, V. A., and Tanner, D. B. (1999). Far-infrared absorptance of YBa2Cu3O7- films in the normal state. Microscale Thermophys. Eng. 3, 5–15. 156. Kumar, A. R., Zhang, Z. M., Boychev, V. A., Tanner, D. B., Vale, L. R., and Rudman, D. A. (1999). Far-infrared transmittance and reflectance of YBa2Cu3O7- films on Si substrates. J. Heat Transfer 121, 844–851. 157. Zhang, Z. M. and Flik, M. I. (1993). Predicted absorptance of YBa2Cu3O7/YSZ/Si multilayer structures for infrared detectors. IEEE Trans. Appl. Supercond. 3, 1604–1607.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
285
158. Kumar, A. R. and Zhang, Z. M. (1999). Far-infrared spectroscopy of high-temperature superconducting films on silicon substrates. In ‘‘Recent Research Developments in Applied Spectroscopy’’ (S. G. Pandalai, ed.), Vol. 2, pp. 92–112. Research Signpost, India. 159. Gabriel, C. J. and Nedoluha, A. (1971). Transmittance and reflectance of systems of thin and thick layers. Optica Acta 18, 415–423. 160. Potter, R. F. (1981). Optical properties of lamelliform materials. SPIE Proc. 276, 204–213. 161. Harbecke, B. (1986). Coherence and incoherence reflection and transmission of multilayer structures. Appl. Phys. B 39, 165–170. 162. Pfrommer, P., Lomas, K. J., Seale, C., and Kupke, C. (1995). The radiation transfer through coated and tinted glazing. Solar Energy 54, 287–299. 163. Filinski, I. (1972). The effects of sample imperfections on optical spectra. Physica Status Solidi B 49, 577–588. 164. Carniglia, C. K. (1979). Scalar scattering theory for multilayer optical coatings. Opt. Eng. 18, 104–115. 165. Swanepoel, R. (1984). Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films. J. Phys. E: Sci. Instrum. 17, 896–903. 166. Gunde, M. K. and Aleksandrov, B. (1991). Infrared optical constants and roughness factor functions determination: the HTHRTR method. Appl. Opt. 30, 3186–3196. 167. Petrich, R. and Stenzel, O. (1994). Modeling of transmittance, reflectance and scattering of rough polycrystalline CVD diamond layers in application to the determination of optical constants. Opt. Mater. 3, 65–76. 168. Ohlı´ dal, I., Vizd’a, F., and Ohlı´ dal, M. (1995). Optical analysis by means of spectroscopic reflectometry of single and double layers with correlated randomly rough boundaries. Opt. Eng. 34, 1761–1768. 169. Mitsas, C. L. and Siapkas, D. I. (1995). Generalized matrix method for analysis of coherence and incoherent reflectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates. Appl. Opt. 34, 1678–1683. 170. Kushev, D. B. and Zheleva, N. N. (1996). Incoherent transmissivity and reflectivity of an absorbing-plane plate or layer with linear variations in thickness. Appl. Opt. 35, 113–116. 171. Le´rondel, G. (1999). Fresnel coefficients of a rough interface. Appl. Phys. Lett. 74, 2740– 2742. 172. Montecchi, M., Montereali, R. M., and Nichelatti, E. (2001). Reflectance and transmittance of a slightly inhomogeneous thin film bounded by rough, unparallel interfaces. Thin Solid Films 396, 262–273. 173. Beckmann, P. and Spizzichino, A. (1987). ‘‘The Scattering of Electromagnetic Waves from Rough Surfaces’’. Artech House, Morwood, MA. 174. Davies, H. (1954). The reflection of electromagnetic waves from a rough surface. Proc. Inst. Electrical Eng. 101, 209–214. 175. Torrance, K. E. and Sparrow, E. M. (1967). Theory for off-specular reflection from roughened surfaces. J. Opt. Soc. Am. 57, 1105–1114. 176. Nayar, S. K., Ikeuchi, K., and Kanade, T. (1991). Surface reflection: physical and geometrical perspectives. IEEE Trans. Pattern Anal. Machine Intel. 13, 611–634. 177. He, X. D., Torrance, K. E., Sillion, F. X., and Greenberg, D. P. (1991). A comprehensive physical model for light reflection. Comput. Graph. 25, 175–186. 178. Peng, S. and Morris, G. M. (1995). Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings. J. Opt. Soc. Am. A 12, 1087–1096. 179. Naqvi, S. S. H., Krukar, R. H., McNeil, J. R., Franke, J. E., Niemczyk, T. M., Haaland, D. M., Gottscho, R. A., and Kornblit, A. (1994). Etch depth estimation of largeperiod silicon gratings with multivariate calibration of rigorously simulated diffraction profiles. J. Opt. Soc. Am. A 11, 2485–2493.
286
Z. M. ZHANG ET AL.
180. Hering, R. G. and Smith, T. F. (1970). Surface roughness effects on radiant transfer between surfaces. Int. J. Heat Mass Transfer 13, 725–739. 181. Toor, J. S. and Viskanta, R. (1968). A numerical experiment of radiant heat interchange by the Monte Carlo method. Int. J. Heat Mass Transfer 11, 883–897. 182. Zaworski, J. R., Welty, J. R., and Drost, M. K. (1996). Measurement and use of bi-directional reflectance. Int. J. Heat Mass Transfer 39, 1149–1156. 183. Zaworski, J., Welty, J. R., Palmer, B. J., and Drost, M. K. (1996). Comparison of experiment with Monte Carlo simulations on a reflective gap using a detailed surface properties model. J. Heat Transfer 118, 388–393. 184. Zhou, Y. H., Zhang, Z. M., Tsai, B. K., and DeWitt, D. P. (2001). Effects of radiative properties of surfaces on radiometric temperature measurement. In ‘‘Proceedings of the Ninth International Conference on Advanced Thermal Processing of Semiconductors (RTP’2001)’’. Anchorage, Alaska, pp. 179–188. 185. Hapke, B. (1993). ‘‘Theory of Reflectance and Emittance Spectroscopy’’. Cambridge University Press, Cambridge. 186. Greenberg, D. P., Torrance, K. E., Shirley, P., Arvo, J., Ferwerda, J. A., Pattanaik, S., Lafortune, E., Walter, B., Foo, S.-C., and Trumbore, B. (1997). ‘‘A Framework for Realistic Image Synthesis’’. SIGGRAPH 97, ACM Annual Conference Series, pp. 477–494. 187. Stover, J. C., Bernt, M. L., Church, E. L., and Takacs, P. Z. (1994). Measurement and analysis of scatter from silicon wafers. SPIE Proc. 2260, 182–191. 188. Bjuggren, M., Krummenacher, L., and Mattsson, L. (1997). Characterization of engineering surfaces by infrared scattering. Opt. Eng. 36, 874–882. 189. Stover, J. C. (1995). ‘‘Optical Scattering: Measurement and Analysis’’. SPIE-The International Society for Optical Engineering, Bellingham, Washington. 190. Coulombe, S. A., Minhas, B. K., Raymond, C. J., Naqvi, S. S. H., and McNeil, J. R. (1998). Scatterometry measurement of sub-0.1 mm linewidth gratings. J. Vac. Sci. Technol. B 16, 80–86. 191. Zhou, Y. H., Shen, Y. J., Zhang, Z. M., Tsai, B. K., and DeWitt, D. P. (2002). A Monte Carlo model for predicting the effective emissivity of the silicon wafer in rapid thermal processing furnaces. Int. J. Heat Mass Transfer 45, 1945–1949. 192. Adams, B., Hunter, A., Yam, M., and Peuse, B. (2000). ‘‘Determining the Uncertainty of Wafer Temperature Measurements Induced by Variations in the Optical Properties of Common Semiconductor Materials’’. Proceedings of the 197th Meeting of the Electrochemical Society, Vol. PV2000-9, pp. 363–374. 193. Marx, E., Malik, I. J., Strausser, Y. E., Bristow, T., Poduje, N., and Stover, J. C. (1998). Round robin determination of power spectral densities of different Si wafer surfaces. SPIE Proc. 3275, 26–36. 194. Xun, X., Peng, C., Saito, K., and Mansuripur, M. (2001). Scattering measurements on optical disks and their relation to media noise. Appl. Opt. 40, 4728–4737. 195. Whitehouse, D. J. (1997). Surface metrology. Meas. Sci. Technol. 8, 955–972. 196. Greenwood, J. A. and Williamson, J. B. P. (1966). Contact of nominally flat surfaces. Proc. R. Soc. Lond. 295, 300–319. 197. Majumdar, A. (1991). Effect of interfacial roughness on phonon radiative heat conduction. J. Heat Transfer 113, 797–805. 198. Vorburger, T. V., Dagata, J. A., Wilkening, G., and Lizuka, K. (1998). Charaterization of surface topography. In ‘‘Beam Effects, Surface Topography, and Depth Profiling in Surface Analysis’’ (A. W. Czanderna, T. E. Madey, and C. J. Powell, eds.). Plenum Press, New York. 199. Zhou, Y. H. and Zhang, Z. M. (2003). Radiative properties of semitransparent silicon wafers with rough surfaces. J. Heat Transfer (in press).
OPTICAL AND THERMAL RADIATIVE PROPERTIES
287
200. Bawolek, E. J., Mohr, J. B., Hirleman, E. D., and Majumdar, A. (1993). Light scatter from polysilicon and aluminum surfaces and comparison with surface-roughness statistics by atomic force microscopy. Appl. Opt. 32, 3377–3400. 201. Majumdar, A. and Tien, C. L. (1990). Fractal characterization and simulation of rough surfaces. Wear 136, 313–327. 202. Stover, J. C., Ivakhnenko, V. I., and Scheer, C. A. (1998). Comparison of surface PSD’s calculated from both AFM profiles and scatter data. SPIE Proc. 3275, 37–46. 203. Ohlı´ dal, I., Ohlı´ dal, M., Franta, D., and Tykal, M. (1999). Comparison of optical and non-optical methods for measuring surface roughness. SPIE Proc. 3820, 456–467. 204. Koh, J., Lu, Y., Wronski, C. R., Kuang, Y., Collins, R. W., Tsong, T. T., and Strausser, Y. E. (1996). Correlation of real time spectroellipsometry and atomic force microscopy measurements of surface roughness on amorphous semiconductor thin films. Appl. Phys. Lett. 69, 1297–1299. 205. Birkebak, R. C. (1971). Optical and mechanical RMS surface roughness comparison. Appl. Opt. 10, 1970. 206. Zhu, Q., Sin, S., and Zhang, Z. M. (2002). ‘‘Surface Characterization of the Rough Side of Silicon Wafers’’. Proceedings of the 21th Southeastern Conference on Theoretical and Applied Mechanics, pp. 49–55, Orlando, FL. 207. Wiesendanger, R. (1994). ‘‘Scanning Probe Microscopy and Spectroscopy: Methods and Applications’’. Cambridge University Press, Cambridge. 208. Thomas, T. R. (1999). ‘‘Rough Surfaces’’. Imperial College Press, London. 209. Binnig, G. and Rohrer, H. (1982). Scanning tunneling microscopy. Helv. Phys. Acta 55, 726–735. 210. Binnig, G., Quate, C. F., and Gerber, Ch. (1996). Atomic force microscope. Phys. Rev. Lett. 56, 930–933. 211. Wyant, J. C., Koliopoulos, C. L., Bhushan, B., and Basila, D. (1986). Development of a three-dimensional noncontact digital optical profiler. J. Tribol. 108, 1–8. 212. Germer, T. A., Asmail, C. C., and Scheer, B. W. (1997). Polarization of out-of-plane scattering from microrough silicon. Opt. Lett. 22, 1284–1286. 213. Germer, T. A. (2000). Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry. Phys. Rev. Lett. 85, 349–352. 214. Germer, T. A. (1997). Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness. Appl. Opt. 36, 8798–8805. 215. Sung, L., Mulholland, G. W., and Germer, T. A. (1999). Polarized light-scattering measurements of dielectric spheres upon a silicon surface. Opt. Lett. 24, 866–868. 216. Germer, T. A. (2001). Polarized light scattering by microroughness and small defects in dielectric layers. J. Opt. Soc. Am. A 18, 1279–1288. 217. Germer, T. A. and Asmail, C. C. (1999). Polarization of light scattered by microrough surfaces and subsurface defects. J. Opt. Soc. Am. A 16, 1326–1332. 218. Lai, L. and Irene, E. A. (2000). Si/SiO2 interface roughness study using Fowler-Nordheim tunneling current oscillations. J. Appl. Phys. 87, 1159–1164. 219. Toh, S. L., Quan, C., Woo, K. C., Tay, C. J., and Shang, H. M. (2001). Whole field surface roughness measurement by laser speckle correlation technique. Opt. Laser Technol. 33, 427–434. 220. Mu¨ller, T., Kumpe, R., Gerber, H. A., Schmolke, R., Passek, F., and Wagner, P. (2001). Techniques for analysing nanotopography on polished silicon wafers. Microelectron. Eng. 56, 123–127. 221. Barradas, N. P. (2001). Rutherford backscattering analysis of thin films and superlattices with roughness. J. Phys. D 34, 2109–2116.
288
Z. M. ZHANG ET AL.
222. Zant, P. V. (2000). ‘‘Microchip Fabrication’’, 4th edn. McGraw-Hill, New York. 223. Vatel, O., Dumas, P., Chollet, F., Salvan, F., and Andre, E. (1993). Roughness assessment of polysilicon using power spectral density. Jpn. J. Appl. Phys. Part 1 32, 5671–5674. 224. Courboin, D., Grouillet, A., and Andre, E. (1995). Surface roughness of ion implanted (100) silicon studied by atomic force microscopy. Surface Sci. 342, L1111–L1115. 225. Vorburger, T. V., Marx, E., and Lettieri, T. R. (1993). Regimes of surface roughness measurable with light scattering. Appl. Opt. 32, 3401–3408. 226. Marx, E. and Vorburger, T. V. (1990). Direct and inverse problems for light scattered by rough surfaces. Appl. Opt. 29, 3613–3626. 227. Dumas, P., Bouffakhreddine, B., Amra, C., Vatel, O., Andre, E., Galindo, R., and Salvan, F. (1993). Quantitative microroughness analysis down to the nanometer scale. Europhys. Lett. 22, 717–722. 228. Borovikov, V. A. and Kinber, B. Ye. (1994). ‘‘Geometrical Theory of Diffraction’’. The Institute of Electrical Engineers, London. 229. Asmail, C. C. (1991). Bidirectional scattering distribution function (BSDF): a systematized bibliography. J. Res. Natl. Inst. Stand. Technol. 96, 215–223. 230. Philips-Invernizzi, B., Dupont, D., and Caze´, C. (2001). Bibliographical review for reflectance of diffusing media. Opt. Eng. 40, 1082–1092. 231. Ogilvy, J. A. (1987). Wave scattering from rough surfaces. Rep. Prog. Phys. 50, 1553–1608. 232. Tang, K., Yang, Y., and Buckius, R. O. (1999). Theory and experiments on scattering from rough interfaces. Annu. Rev. Heat Transfer 10, 100–140. 233. Nicodemus, F. E. (1970). Reflectance nomenclature and directional reflectance and emissivity. Appl. Opt. 9, 1474–1475. 234. Greffet, J. J. and Nieto-Vesperinas, M. (1998). Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law. J. Opt. Soc. Am. A 15, 2735–2744. 235. Snyder, W. C., Wan, Z., and Li, X. (1998). Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s law. Appl. Opt. 37, 3464–3470. 236. Snyder, W. C. (1998). Reciprocity of the bidirectional reflectance distribution function (BRDF) in measurements and models of structured surfaces. IEEE Trans. Geosci. Remote Sensing 36, 685–691. 237. Kriebel, K. T. (1996). On the limited validity of reciprocity in measured BRDFs. Remote Sensing Environ. 58, 52–62. 238. Dimenna, R. A. and Buckius, R. O. (1994). Electromagnetic theory predictions of the directional scattering from triangular surfaces. J. Heat Transfer 116, 639–645. 239. Cohn, D. W., Tang, K., and Buckius, R. O. (1997). Comparison of theory and experiments for reflection from microcontoured surfaces. Int. J. Heat Mass Transfer 40, 3223–3235. 240. Dimenna, R. A. and Buckius, R. O. (1994). Quantifying specular approximations for angular scattering from perfectly conducting random rough surfaces. J. Thermophys. Heat Transfer 8, 393–399. 241. Yang, Y. and Buckius, R. O. (1995). Surface length scale contributions to the directional and hemispherical emissivity and reflectivity. J. Thermophys. Heat Transfer 9, 653–659. 242. Sa´nchez-Gil, J. A. and Nieto-Vesperinas, M. (1991). Light scattering from random rough dielectric surfaces. J. Opt. Soc. Am. A 8, 1270–1286. 243. Nieto-Vesperinas, M. and Sa´nchez-Gil, J. A. (1992). Light scattering from a random rough interface with total internal reflection. J. Opt. Soc. Am. A 9, 424–436. 244. Nieto-Vesperinas, M., Sa´nchez-Gil, J. A., and Maradudin, A. A. (1991). Multiple light scattering from metal and dielectric rough surfaces. Waves Random Media 1, S157–S163. 245. Tsai, B. K. and DeWitt, D. P. (1994). Macroscopic spread function analysis for subsurface scattering of semitransparent materials. J. Thermophys. Heat Transfer 8, 202–207.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
289
246. Meister, G., Rothkirch, A., Spitzer, H., and Bienlein, J. (2001). Width of the specular peak perpendicular to the principal plane for rough surfaces. Appl. Opt. 40, 6072–6080. 247. Asmail, C., Hsia, J., Parr, A., and Hoeft, J. (1994). Rayleigh scattering limits for low-level bidirectional reflectance distribution function measurements. Appl. Opt. 33, 6084–6091. 248. Soto-Crespo, J. M. and Nieto-Vesperinas, M. (1989). Electromagnetic scattering from very rough random surfaces and deep reflection gratings. J. Opt. Soc. Am. A 6, 367–384. 249. Berman, D. H. and Perkins, J. S. (1985). The Kirchhoff approximation and first-order perturbation theory for rough surface scattering. J. Acoust. Soc. Am. 78, 1045–1051. 250. Thorsos, E. I. (1988). The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum. J. Acoust. Soc. Am. 83, 78–92. 251. Bourlier, C., Berginc, G., and Saillard, J. (2001). Theoretical study of the Kirchhoff integral from a two-dimensional randomly rough surface with shadowing effect: application to the backscattering coefficient for a perfectly-conducting surface. Waves Random Media 11, 91–118. 252. Liszka, E. G. and McCoy, J. J. (1982). Scattering at a rough boundary-extensions of the Kirchhoff approximation. J. Acoust. Soc. Am. 71, 1093–1100. 253. Ishimaru, A. and Chen, J. S. (1991). Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation. Waves Random Media 1, 21–34. 254. Torrance, K. E. and Sparrow, E. M. (1966). Off-specular peaks in the directional distribution of reflected thermal radiation. J. Heat Transfer 88, 223–230. 255. Torrance, K. E. and Sparrow, E. M. (1967). Directional emittance of an electric nonconductor as a function of surface roughness and wavelength. Int. J. Heat Mass Transfer 10, 1709–1716. 256. Tang, K., Dimenna, R. A., and Buckius, R. O. (1997). Regions of validity of the geometric optics approximation for angular scattering from very rough surfaces. Int. J. Heat Mass Transfer 40, 49–59. 257. Tang, K. and Buckius, R. O. (1998). The geometric optics approximation for reflection from two-dimensional random rough surfaces. Int. J. Heat Mass Transfer 41, 2037–2047. 258. Houchens, A. F. and Hering, R. G. (1967). Bidirectional reflectance of rough metal surfaces. Prog. Astronautics Aeronautics 20, 65–89. 259. Stagg, B. J. and Charalampopoulos, T. T. (1991). Surface-roughness effects on the determination of optical properties of materials by the reflection method. Appl. Opt. 30, 4113–4118. 260. Burnell, J. G., Nicholas, J. V., and White, D. R. (1995). Scattering model for rough oxidized metal surfaces applicable to radiation thermometry of reformer furnaces. Opt. Eng. 34, 1749–1755. 261. Shen, Y. J. and Zhang, Z. M. (2002). Design and characterization of a bidirectional refletometer. In ‘‘Proceedings of the 12th International Heat Transfer Conference’’. (J. Taine, ed.), Vol. 1, pp. 615–620, Elsevier, Paris. 262. Shen, Y. J., Zhang, Z. M., Tsai, B. K., and DeWitt, D. P. (2001). Bidirectional reflectance distribution function of rough silicon wafers. Int. J. Thermophys. 22, 1311–1326. 263. Neumann, L., Neumann, A., and Szirmay-Kalos, L. (1999). Compact metallic reflectance models. Comput. Graph. Forum 18, C161–C171. 264. Lalonde, P. and Fournier, A. (1997). Generating reflected directions from BRDF data. Comput. Graph. Forum 16, C293–C300. 265. Cook, R. L. and Torrance, K. E. (1981). A reflectance model for computer graphics. Comput. Graph. 15, 307–316. 266. Ward, G. J. (1992). Measuring and modeling anisotropic reflection. Comput. Graph. 26, 265–272.
290
Z. M. ZHANG ET AL.
267. Phong, B. T. (1975). Illumination for computer generated pictures. Commun. ACM 18, 311–317. 268. Tada, H., Abramson, A. R., Miaoulis, I. N., Wong, P. Y., Nieva, P. and Zavracky, P. (1998). ‘‘Effect of Surface Patterning in Thin Film Structures on the Thermal Radiative Properties During Rapid Thermal Processing’’. Proceedings of the 1998 International Mechanical Engineering Congress and Exposition, Anaheim, CA. 269. Hebb, J. P., Jensen, K. F., and Thomas, J. (1998). The effect of surface roughness on the radiative properties of patterned silicon wafers. IEEE Trans. Semocond. Manuf. 11, 607–614. 270. Warnick, K. F. and Chew, W. C. (2001). Numerical simulation methods for rough surface scattering. Waves Random Media 11, R1–R30. 271. Tsang, L., Kong, J. A., Ding, K.-H., and Ao, C. O. (2001). ‘‘Scattering of Electromagnetic Waves–Numerical Simulations’’. Wiley, New York. 272. Fung, A. K. and Chen, M. F. (1985). Numerical simulation of scattering from simple and composite random surfaces. J. Opt. Soc. Am. A 2, 2274–2284. 273. Chen, M. F. and Fung, A. K. (1988). A numerical study of the regions of validity of the Kirchhoff and small-perturbation rough surface scattering models. Radio Sci. 23, 163–170. 274. Thorsos, E. I. and Jackson, D. R. (1991). Studies of scattering theory using numerical methods. Waves Random Media 1, S165–S190. 275. Macaskill, C. and Kachoyan, B. J. (1993). Iterative approach for the numerical simulation of scattering from one- and two-dimensional rough surfaces. Appl. Opt. 32, 2839–2847. 276. Yablonovitch, E. (1982). Statistical ray optics. J. Opt. Soc. Am. 72, 899–907. 277. Tang, K. and Buckius, R. O. (2001). A statistical model of wave scattering from random rough surfaces. Int. J. Heat Mass Transfer 44, 4059–4073. 278. Tran, P., Celli, V., and Maradudin, A. A. (1994). Electromagnetic scattering from a twodimensional, randomly rough, perfectly conducting surface: iterative methods. J. Opt. Soc. Am. A 11, 1686–1689. 279. Maystre, D., Saillard, M., and Ingers, J. (1991). Scattering by one- or two-dimensional randomly rough surfaces. Waves Random Media 1, S143–S155. 280. Tang, K. and Buckius, R. O. (1998). Bi-directional reflection measurements from twodimensional microcontoured metallic surfaces. Microscale Thermophys. Eng. 2, 245–260. 281. Tran, P. and Maradudin, A. A. (1994). The scattering of electromagnetic waves from a randomly rough 2D metallic surface. Opt. Commun. 110, 269–273. 282. Tran, P. and Maradudin, A. A. (1993). Scattering of a scalar beam from a twodimensional randomly rough hard wall: Dirichlet and Neumann boundary conditions. Appl. Opt. 32, 2848–2851. 283. Tsang, L., Chan, C. H., and Pak, K. (1994). Backscattering enhancement of a twodimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulations. J. Opt. Soc. Am. A 11, 711–715. 284. Pak, K., Tsang, L., Chan, C. H., and Johnson, J. (1995). Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations. J. Opt. Soc. Am. A 12, 2491–2499. 285. Johnson, J. T., Tsang, L., Shin, R. T., Pak, K., Chan, C. H., Ishimaru, A., and Kuga, Y. (1996). Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data. IEEE Trans. Antennas Propagation 44, 748–756. 286. Majumdar, A. and Tien, C. L. (1990). ‘‘Reflection of Radiation by Rough Fractal Surfaces’’. 5th AIAA/ASME Thermophysics and Heat Transfer Conference, ASMEHTD-Vol. 137, pp. 27–35. Seattle, WA.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
291
287. Majumdar, A. and Bhushan, B. (1990). Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J. Tribol. 112, 205–216. 288. Berry, M. V. (1979). Diffractals. J. Phys. A 12, 781–797. 289. Tang, K., Kawka, P. A., and Buckius, R. O. (1999). Geometric optics applied to rough surfaces coated with an absorbing thin film. J. Thermophys. Heat Transfer 13, 169–176. 290. McGurn, A. R. and Maradudin, A. A. (1989). An analogue of enhanced backscattering in the transmission of light through a thin film with a randomly rough surface. Opt. Commun. 72, 279–285. 291. Lu, J. Q., Maradudin, A. A., and Michel, T. (1991). Enhanced backscattering from a rough dielectric film on a reflecting substrate. J. Opt. Soc. Am. B 8, 311–318. 292. Duparre´, A. (1995). Light scattering of thin dielectric films. In ‘‘Handbook of Optical Properties’’ (R. E. Hummel and K. H. Guenther, eds.), pp. 273–303. CRC Press, Boca Raton, FL. 293. Ohlı´ dal, I. and Navra´til, K. (1995). Scattering of light from multilayer systems with rough boundaries. Prog. Opt. 34, 249–331. 294. Maradudin, A. A., Lu, J. Q., Michel, T., Gu, Z.-H., Dainty, J. C., Sant, A. J., Me´ndez, E. R., and Nieto-Vesperinas, M. (1991). Enhanced backscattering and tranmission of light from random surfaces on semi-infinite substrates and thin films. Waves Random Media 3, S129–S141. 295. Pak, K., Tsang, L., and Johnson, J. (1997). Numerical simulations and backscattering enhancement of electromagnetic waves from two-dimensional dielectric random rough surfaces with the sparse-matrix canonical grid method. J. Opt. Soc. Am. A 14, 1515–1529. 296. Ishimaru, A., Chen, J. S., Phu, P., and Yoshitomi, K. (1991). Numerical, analytical, and experimental studies of scattering from very rough surfaces and backscattering enhancement. Waves Random Media 1, S91–S107. 297. Blumberg, D. G., Freilikher, V., Fuks, I., Kaganovskii, Y., Maradudin, A. A., and Rosenbluh, M. (2002). Effects of roughness on the retroreflection from dielectric layers. Waves Random Media 12, 279–292. 298. Freilikher, V., Kanzieper, E., and Maradudin, A. A. (1997). Coherent scattering enhancement in systems bounded by rough surfaces. Phys. Rep. 288, 127–204. 299. Jakeman, E. (1991). Non-Gaussian statistical models for scattering calculations. Waves Random Media 3, S109–S119. 300. Gue´rin, C.-A. (2002). Scattering on rough surfaces with alpha-stable non-Gaussian height distributions. Waves Random Media 12, 293–306. 301. Orden, B. L. (1992). Bidirectional reflectance and specularity of twelve spacecraft thermal control materials. J. Thermophys. Heat Transfer 6, 672–679. 302. Simonson, R. J., Altamirano, M. M., Champetier, R. J., Cohen, M. R., Snyder, J., and Stapp, R. S. (1993). A new design for an ultrahigh vacuum surface analysis chamber with optical scatter measurement capability. Rev. Sci. Instrum. 64, 1287–1291. 303. Martı´ nez-Anto´on, J. C. and Bernabeu, E. (1994). Automatic three-dimensional spectrogoniometer for determination of optical properties and surface parameters. Appl. Opt. 33, 6059–6061. 304. Zipin, R. B. (1966). A preliminary investigation of the bidirectional spectral reflectance of V-grooved surfaces. Appl. Opt. 5, 1954–1957. 305. O’Donnell, K. A. and Mendez, E. R. (1987). Experimental study of scattering from characterized random surfaces. J. Opt. Soc. Am. A 4, 1194–1205. 306. McGuckin, B. T., Haner, D. A., Menzies, R. T., Esproles, C., and Brothers, A. M. (1996). Directional reflectance characterization facility and measurement methodology. Appl. Opt. 35, 4827–4834.
292
Z. M. ZHANG ET AL.
307. Aslan, M., Yamada, J., Mengu¨c, M. P., and Thomasson, J. A. (2002). ‘‘Radiative Properties of Individual Cotton Fibers: Experiments and Predictions’’. 8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, St. Louis, MO, AIAA 2002-3325. 308. Cunnington, G. R. and Lee, S. C. (2002). ‘‘Bidirectional Reflectance Measurements of Spacecraft Thermal Control Materials’’. 8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, St. Louis, MO, AIAA 2002-3326. 309. Proctor, J. E. and Barnes, P. Y. (1996). NIST high accuracy reference reflectometerspectrophotometer. J. Res. Natl. Inst. Stand. Technol. 101, 619–627. 310. Barnes, P. Y., Early, E. A. and Parr, A. C. (1998). ‘‘Spectral Reflectance’’. NIST Special Publication 250-48, US Government Printing Office, Washington, DC. 311. Birkebak, R. C. and Eckert, E. R. G. (1965). Effects of roughness of metal surfaces on angular distribution of monochromatic reflected radiation. J. Heat Transfer 87, 85–94. 312. Anderson, S., Pompea, S. M., Shepard, D. F., and Castonguay, R. (1988). Performance of a fully automated scatterometer for BRDF and BTDF measurements at visible and infrared wavelengths. SPIE Proc. 967, 159–170. 313. Feng, X., Schott, J. R., and Gallagher, T. (1993). Comparison of methods for generation of absolute reflectance-factor values for bidirectional reflectance-distribution function studies. Appl. Opt. 32, 1234–1242. 314. Germer, T. A. and Asmail, C. C. (1999). Goniometric optical scatter instrument for outof-plane ellipsometry measurements. Rev. Sci. Instrum. 70, 3688–3695. 315. Murray-Coleman, J. F. and Smith, A. M. (1990). The automated measurement of BRDFs and their application to luminaire modeling. J. Illum. Eng. Soc. 19, 87–99. 316. White, D. R., Saunders, P., Bonsey, S. J., van de Ven, J., and Edgar, H. (1998). Reflectometer for measuring the bidirectional reflectance of rough surfaces. Appl. Opt. 37, 3450–3454. 317. Sandmeier, S., Muller, C., Hosgood, B., and Andreoli, G. (1998). Sensitivity analysis and quality assessment of laboratory BRDF data. Remote Sensing Environ. 64, 176–191. 318. Torrance, K. E. and Sparrow, E. M. (1965). Biangular reflectance of an electric nonconductor as a function of wavelength and surface roughness. J. Heat Transfer 87, 283–292. 319. Voss, K. J., Chapin, A., Monti, M., and Zhang, H. (2000). Instrument to measure the bidirectional reflectance distribution function of surfaces. Appl. Opt. 39, 6197–6206. 320. Ford, J. N., Tang, K., and Buckius, R. O. (1995). Fourier transform infrared system measurement of the bidirectional reflectivity of diffuse and grooved surfaces. J. Heat Transfer 117, 955–962. 321. Zhu, Q. Z., Shen, Y.-J., and Zhang, Z. M. (2002). ‘‘Bidirectional Reflectance Measurement of Microstructured Silicon Surfaces’’. Proceedings IMECE’2002-Paper No. 32757, New Orleans, LA. 322. Perlmutter, M. and Howell, J. R. (1963). A strongly directional emitting and absorbing surface. J. Heat Transfer 282–283. 323. Demont, P., Huetz-Aubert, M., and N’Guyen, H. T. (1982). Experimental and theoretical studies of the influence of surface conditions on radiative properties of opaque materials. Int. J. Thermophys. 3, 335–364. 324. Hesketh, P. J., Zemel, J. N., and Gebhart, B. (1986). Organ pipe radiant modes of periodic micromachined silicon surfaces. Nature 324, 549–551. 325. Hesketh, P. J., Gebhart, B., and Zemel, J. N. (1988). Measurements of the spectral and directional emission from microgrooved silicn surfaces. J. Heat Transfer 110, 680–686. 326. Hesketh, P. J., Zemel, J. N., and Gebhart, B. (1988). Polarized spectral emittance from periodic micromachined surfaces. I. Doped silicon: the normal direction. Phys. Rev. B 37, 10,795–10,802.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
293
327. Hesketh, P. J., Zemel, J. N., and Gebhart, B. (1988). Polarized spectral emittance from periodic micromachined surfaces. II. Doped silicon: angular variation. Phys. Rev. B 37, 10,803–10,813. 328. Zemel, J. N. (1988). Black body radiation and transport processes from microstructures. Comments Condensed Matter Phys. 14, 1–20. 329. Wang, T. K. and Zemel, J. N. (1991). Polarized spectral emittance from periodic micromachined surfaces. III. Undoped silicon: the normal direction in shallow lamellar gratings. Infrared Phys. 32, 477–488. 330. Wang, T. K. and Zemel, J. N. (1992). Polarized spectral emittance from periodic micromachined surfaces. IV. Undoped silicon: normal direction in deep lamellar gratings. Appl. Opt. 31, 732–736. 331. Wang, T. K. and Zemel, J. N. (1993). Polarized spectral emittance from periodic micromachined surfaces. V. Undoped silicon: angular measurement in shallow lamellar gratings. Appl. Opt. 32, 2021–2027. 332. Greffet, J. J., Carminati, R., Joulain, K., Mulet, J. P., Mainguy, S., and Chen, Y. (2002). Coherent emission of light by thermal sources. Nature 416, 61–64. 333. Shchegrov, A. V., Joulain, K., Carminati, R., and Greffet, J.-J. (2000). Near-field spectral effects due to electromagnetic surface excitations. Phys. Rev. Lett. 85, 1548–1551. 334. Maruyama, S., Kashiwa, T., Yugami, H., and Esashi, M. (2001). Thermal radiation from two-dimensionally confined modes in microcavities. Appl. Phys. Lett. 79, 1393–1395. 335. Sai, H., Yugami, H., Akiyama, Y., Kanamori, Y., and Hane, K. (2001). Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region. J. Opt. Soc. Am. A 18, 1471–1476. 336. Kanamori, Y., Hane, K., Sai, H., and Yugami, H. (2001). 100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask. Appl. Phys. Lett. 78, 142–143. 337. Petrou, A. and McComb, B. D. (1991). Magnetospectroscopy of confined semiconductor systems. In ‘‘Landau Level Spectroscopy’’ (G. Landwehr and E. I. Rash, eds.), pp. 682– 775. Elsevier, Amsterdam. 338. Vasko, F. and Kuznetsov, A. V. (1999). ‘‘Electronic States and Optical Transitions in Semiconductor Heterostructures’’. Springer-Verlag, Berlin. 339. Gaponenko, S. V. (1998). ‘‘Optical Properties of Semiconductor Nanocrystals’’. Cambridge University Press, Cambridge. 340. Woggon, U. (1997). ‘‘Optical Properties of Semiconductor Quantum Dots’’. SpringerVerlag, Berlin. 341. M. L. Sadowski, M. Potemski, and M. Grynberg, eds. (1999). ‘‘Optical Properties of Semiconductor Nanostructures’’. Kluwer Academic, Dordrecht. 342. Shah, J. (1996). ‘‘Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures’’. Springer-Verlag, Berlin. 343. Efros, Al.L. and Rosen, M. (2000). The electronic structure of semiconductor nanocrystals. Ann. Rev. Mat. Sci. 30, 475–521. 344. Braun, M. B., Forchel, A., Zull, H., and Reithmaier, J. P. (1997). Exciton wave packets in In0.135Ga0.865As/GaAs quantum wires. Phys. Rev. B 55, 9290–9293. 345. Kayanuma, Y. (1986). Wannier exciton in microcrystals. Solid State Commun. 59, 405– 408. 346. Einevoll, G. T. (1992). Confinement of excitons in quantum dots. Phys. Rev. B 45, 3410– 3417. 347. Ding, Z., Quinn, B. M., Haram, S. K., Pell, L. E., Korgel, B. A., and Bard, A. J. (2002). Electrochemistry and electrogenerated chemiluminescence from silicon nanocrystal quantum dots. Science 296, 1293–1297.
294
Z. M. ZHANG ET AL.
348. Chen, P., Piermarocchi, C., and Sham, L. J. (2001). Control of exciton dynamics in nanodots for quantum operations. Phys. Rev. Lett. 87, 067401. 349. Stintz, A., Liu, G. T., Li, H., Lester, L. F., and Malloy, K. J. (2000). Low-threshold current density 1.3-mm InAs quantum-dot lasers with the dots-in-a-well (SWELL) structure. IEEE Photon. Technol. Lett. 12, 591–593. 350. Huang, M. H., Mao, S., Feick, H., Yan, H., Wu, Y., Kind, H., Weber, E., Russo, R., and Yang, P. (2001). Room-temperature ultraviolet nanowire nanolasers. Science 292, 1897–1899. 351. Canham, L. T. (1990). Silicon quantum wire array fabricated by electrochemical and chemical dissolution of wafers. Appl. Phys. Lett. 57, 1046–1048. 352. Koshida, N. and Koyama, H. (1992). Visible electroluminescence from porous silicon. Appl. Phys. Lett. 60, 347–349. 353. Kanemitsu, Y. (1995). Light emission from porous silicon and related materials. Phys. Rep. 263, 1–91. 354. Canham, L., ed. (1997). ‘‘Properties of Porous Silicon’’. INSPEC, the Institute of Electrical Engineers, London. 355. Amato, G., Delerue, C., and von Bardeleben, H.-J., eds. (1997). ‘‘Structural and Optical Properties of Porous Silicon Nanostructures’’. Gordon & Breach Sci. Publ, Amsterdam. 356. Delerue, C., Allan, G., and Lannoo, M. (1993). Theoretical aspects of the luminescence of porous silicon. Phys. Rev. B 48, 11,024–11,036. 357. Brus, L. (1994). Luminescence of silicon materials: chains, sheets, nanocrystals, nanowires, microcrystals, and porous silicon. J. Phys. Chem. 98, 3575–3581. 358. John, G. C. and Singh, V. A. (1995). Porous silicon: theoretical studies. Phys. Rep. 263, 93–151. 359. Murayama, K., Miyazaki, S., and Hirose, M. (1995). Excitation and recombination process in porous silicon. Solid State Commun. 93, 841–846. 360. Voos, M., Delalande, C., Ben Dahan, M., Wainstain, J., Titkov, A. N., and Halimaoui, A. (1995). Excitation spectroscopy of the visible photoluminescence of porous silicon. Solid State Commun. 94, 651–654. 361. Lockwood, D. J. and Wang, A. G. (1995). Quantum confinement induced photoluminescence in porous silicon. Solid State Commun. 94, 905–909. 362. Tsuboi, T., Laiho, R., and Pavlov, A. (1995). Temperature dependence of radiative and non-radiative transitions in porous silicon. Thin Solid Films 255, 216–218. 363. Finkbeiner, S. and Weber, J. (1995). Interpretation of the temperature dependence of the strong visible photoluminescence of porous silicon. Thin Solid Films 255, 254–257. 364. Hipwell, M. C. and Tien, C. L. (1999). Short time-scale radiative transfer in light-emitting porous silicon. Int. J. Heat Mass Transfer 42, 249–261. 365. Theiß, W., Henkel, S., and Arntzen, M. (1995). Connecting microscopic and macroscopic properties of porous media: choosing appropriate effective medium concepts. Thin Solid Films 255, 177–180. 366. Astrova, E. V. and Tolmachev, V. A. (2000). Effective refractive index and composition of oxidized porous silicon films. Mater. Sci. Eng. B 69–70, 142–148. 367. Diesinger, H., Bsiesy, A., Herino, R., and Gelloz, B. (2000). Effect of the quantum confinement on the optical absorption of porous silicon, investigated by a new in-situ method. Mater. Sci. Eng. B 69–70, 167–170. 368. Lohner, T., Fried, M., Petrik, P., Polgar, O., Gyulai, J., and Lehnert, W. (2000). Ellipsometric characterization of oxidized porous silicon layer structures. Mater. Sci. Eng. B 69–70, 182–187. 369. Setzu, S., Ferrand, P., and Romestain, R. (2000). Optical properties of multilayered porous silicon. Mater. Sci. Eng. B 69–70, 34–42.
OPTICAL AND THERMAL RADIATIVE PROPERTIES
295
370. Carr, G. L., Perkowitz, S., and Tanner, D. B. (1985). Far-infrared properties of inhomogeneous materials. Infrared Millimeter Waves 13, 171–263. 371. Kore, L. and Bosman, G. (1999). Feasibility of porous silicon as a primary material in solar cells. Solar Energy Mater. Solar Cells 57, 31–48. 372. Bilyalov, R. R., Ludemann, R., Wettling, W., Stalmans, L., Poortmans, J., Nijs, J., Schirone, L., Sotgiu, G., Strehlke, S., and Levy-Clement, C. (2000). Multicrystalline silicon solar cells with porous silicon emitter. Solar Energy Mater. Solar Cells 60, 391–420. 373. Adamian, Z. N., Hakhoyan, A. P., Aroutiounian, V. M., Barseghian, R. S., and Touryan, K. (2000). Investigations of solar cells with porous silicon as antireflection layer. Solar Energy Mater. Solar Cells 64, 347–351. 374. Martin-Palma, R. J., Vazquez, L., Martinez-Duart, J. M., Schnell, M., and Schaefer, S. (2001). Antireflective porous-silicon coatings for multicrystalline solar cells: the effects of chemical etching and rapid thermal processing. Semiconductor Sci. Technol. 16, 657–661. 375. Martı´ n-Palma, R. J., Va´zquez, L., Herrero, P., Martı´ nez-Duart, J. M., Schnell, M., and Schaefer, S. (2001). Morphological, optical and electrical characterization of antireflective porous silicon coatings for solar cells. Opt. Mater. 17, 75–78. 376. Charrier, J., Le Gorju, J., Haji, J., and Guendouz, M. (2000). Optical waveguides fabricated from oxidised porous silicon. J. Porous Mater. 7, 243–246. 377. Bondarenko, V. P., Dorofeev, A. M., and Kazuchits, N. M. (1995). Optical waveguide based on oxidized porous silicon. Microelectron. Eng. 28, 447–450. 378. Lammel, G., Schweizer, S., and Renaud, P. (2001). Microspectrometer based on a tunable optical filter of porous silicon. Sensors Actuators A 92, 52–59. 379. Mu¨ller, F., Birner, A., Go¨sele, U., Lehmann, V., Ottow, S., and Fo¨ll, H. (2000). Structuring of macroporous silicon for application as photonic crystals. J. Porous Mater. 7, 201–204. 380. Yablonovitch, E., Gmitter, T. J., and Leung, K. M. (1991). Photonic band structure: the face-centered-cubic case employing nonspherical atoms. Phys. Rev. Lett. 67, 2295– 2298. 381. Yablonovitch, E., Gmitter, T. J., Meade, R. D., Rappe, A. M., Brommer, K. D., and Joannopoulos, J. D. (1991). Donor and acceptor modes in photonic band structure. Phys. Rev. Lett. 67, 3380–3383. 382. Sigalas, M. M., Ho, K. M., Biswas, R., and Soukoulis, C. M. (1998). Theoretical investigation of defects in photonic crystals in the presence of dielectric losses. Phys. Rev. B 57, 3815–3820. 383. Yablonovitch, E. (1993). Photonic band-gap structures. J. Opt. Soc. Am. B 10, 283–295. 384. Ho, K. M., Chan, C. T., Soukoulis, C. M., Biswas, R., and Sigalas, M. (1994). Photonic band gaps in three dimensions: new layer-by-layer periodic structures. Solid State Commun. 89, 413–416. 385. Ozbay, E., Tuttle, G., McCalmont, J. S., Sigalas, M., Biswas, R., Soukoulis, C. M., and Ho, K. M. (1995). Laser-micromachined millimeter-wave photonic band-gap cavity structures. Appl. Phys. Lett. 67, 1969–1971. 386. Ozbay, E., Michel, E., Tuttle, G., Biswas, R., Sigalas, M., and Ho, K. M. (1994). Micromachined millimeter-wave photonic band-gap crystals. Appl. Phys. Lett. 64, 2059– 2061. 387. Fleming, J. G. and Lin, S. -Y. (1999). Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 mm. Opt. Lett. 24, 49–51. 388. Lin, S. Y., Hietala, V. M., Lyo, S. K., and Zaslavsky, A. (1996). Photonic band gap quantum well and quantum box structures: a high-Q resonant cavity. Appl. Phys. Lett. 68, 3233–3235. 389. Loncar, M., Nedeljkovic, D., Doll, T., Vuckovic, J., Scherer, A., and Pearsall, T. P. (2000). Waveguiding in planar photonic crystals. Appl. Phys. Lett. 77, 1937–1939.
296
Z. M. ZHANG ET AL.
390. Bayindir, M., Cubukcu, E., Bulu, I., Tut, T., Ozbay, E., and Soukoulis, C. M. (2001). Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals. Phys. Rev. B 64, 195113-1-195113/7. 391. Kosaka, H., Kawashima, T., Tomita, A., Notomi, M., Tamamura, T., Sato, T., and Kawakami, S. (1998). Superprism phenomena in photonic crystals. Phys. Rev. B 58, R10096–R10099. 392. Knight, J. C., Birks, T. A., and Russell, P. St. (2001). ‘‘Holey’’ silica fibers. In ‘‘Optics of Nanostructured Materials’’ (V. A. Markel and T. F. Geroge, eds.), pp. 39–71. Wiley, New York. 393. Pendry, J. B. and MacKinnon, A. (1992). Calculation of photon dispersion relations. Phys. Rev. Lett. 69, 2772–2775. 394. Ochiai, T., and Sakoda, K. (2001). Nearly free-photon approximation for twodimensional photonic crystal slabs, Phys. Rev. B 64, 045108-1–04510811. 395. Sigalas, M. M., Ho, K.-M., Biswas, R., and Soukoulis, C. M. (2001). Photonic crystals. In ‘‘Optics of Nanostructured Materials’’ (V. A. Markel and T. F. Geroge, eds.), pp. 1–38. Wiley, New York. 396. Sakoda, K. (1997). Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice. J. Opt. Soc. Am. B 14, 1961–1966. 397. Sakoda, K. (1999). Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals. Opt. Expr. 4, 167–176. 398. Sakoda, K., Sasada, M., Fukushima, T., Yamanaka, A., Kawai, N., and Inoue, K. (1999). Detailed analysis of transmission spectra and Bragg-reflection spectra of a twodimensional photonic crystal with a lattice constant of 1.15 mm. J. Opt. Soc. Am. B 16, 361–365. 399. Sakoda, K. (2001). ‘‘Optical Properties of Photonic Crystals’’. Springer-Verlag, Berlin. 400. Villeneuve, P. R., Fan, S., and Joannopoulos, J. D. (1996). Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency. Phys. Rev. B 54, 7837–7842. 401. Lin, S. Y., Fleming, J. G., Sigalas, M. M., Biswas, R., and Ho, K. M. (1999). Photonic band-gap microcavities in three dimensions. Phys. Rev. B 59, R15579–R15582. 402. Lin, S. Y., Fleming, J. G., Chow, E., Bur, J., Choi, K. K., and Goldberg, A. (2000). Enhancement and suppression of thermal emission by a three-dimensional photonic crystal. Phys. Rev. B 62, R2243–R2246.
ADVANCES IN HEAT TRANSFER VOL. 37
Microchannel Heat Exchanger Design for Evaporator and Condenser Applications
MAN-HOE KIM1, SANG YONG LEE2, SUNIL S. MEHENDALE3, and RALPH L. WEBB4 1
R&D Center, DA Network Business, Samsung Electronics Co., Ltd., 416 Maetan-3Dong, Suwon 442-742, South Korea, Tel.: þ82-31-218-5010; fax: þ82-31-218-5195. E-mail: [email protected], 2 Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Science Town, Daejeon 305-701, South Korea, Tel.: þ82-42-869-3026; fax: þ82-42-869-8207. E-mail: [email protected], 3 Delphi Harrison Thermal Systems, 200 Upper Mountain Road, Lockport, NY 14094, USA, Tel.: þ1-716-438-4843; fax: þ1-716-439-3168. E-mail: [email protected], 4 Department of Mechanical Engineering, Pennsylvania State University, 206 Reber Building, University Park, PA 16802, USA, Tel.: þ1-814-865-0283; fax: þ1-814-863-4848. E-mail: [email protected]
I. Introduction Compact and microchannel heat exchangers are widely used in modern air-conditioning, heat pump, and refrigeration systems for a variety of residential, industrial, automotive and process industry applications. One of the principal ideas behind using compact and micro-scale heat exchangers is their potential for enhanced heat transfer coefficients. The multilouvered fin surfaces are commonly used on the air-side of aluminum brazed heat exchangers associated with extruded flat microchannel tubes as shown in Fig. 1. Using such multiple-channel heat exchangers enables the transfer of large amounts of heat by increasing the heat transfer surface area while enjoying higher heat transfer coefficients. In this article we will present the state of the art in extruded microchannel heat exchanger design in mobile and residential air-conditioning and heat pump applications. The focus will be on evaporator and condenser designs, where a two-phase tube-side flow and air-side crossflow arrangement are most commonly used. The article will provide a critical review of existing work, and put forward new data and novel modeling methods relevant to the applications of interest. Emerging Advances in Heat Transfer Volume 37 ISSN 0065-2717
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FIG. 1. Two different types of heat exchangers.
applications of extruded microchannel heat exchangers and current difficulties arising from a lack of understanding flows in small passages and headers are also addressed.
II. Single- and Two-phase Flows in Microchannels A. INTRODUCTION Condensers and evaporators are essential components of air-conditioning and refrigerating equipment. The hydraulic diameter of the refrigerant channels employed in most of the compact heat exchangers are typically within the range of 0.5–2 mm because higher heat transfer performance can be achieved with such small flow passages. Besides, not only the smaller quantity of the working fluid (refrigerant) is required but also less of the raw materials are consumed in constructing thermal systems with the smaller heat exchangers (Cornwell and Kew [1]). Mehendale et al. [2] classified the size range of the refrigerant channels into four categories; micro size (Dh ¼ 1–100 mm), meso size (Dh ¼ 0.1–1 mm), compact heat exchanger size (Dh ¼ 1–6 mm) and conventional heat exchanger size (Dh>6 mm). However, it is quite common to use the term ‘‘microchannel’’ for the meso size and even to somewhat larger size (up to compact heat exchanger size); thus the size range of the channel covered in this article (0.1–5 mm) will be simply termed as the ‘‘microchannel’’ in this article. The flow and heat transfer studies for single-phase flows through the channels with their size (hydraulic diameter) mostly smaller than 1.0 mm are well summarized in the work of Mehendale et al. [2]. However, there are relatively fewer studies reported on two-phase flow. Some of the works on
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two-phase frictional pressure drop and boiling heat transfer in the extremely narrow channels (35–110 mm) have been made by Moriyama et al. [3,4]. Several other works on the two-phase flow within the channels in the size range of 0.13–2.5 mm have been summarized also by Mehendale et al. [2]. In single-phase channel flows, once the fluid properties are given, the hydrodynamic and heat transfer phenomena can be solely characterized by the Reynolds number, which is the ratio between the inertia and the viscous forces. Often, with an extremely small flow passage, the Knudsen number may become important. Knudsen number is defined as: Kn ¼ =D
ð1Þ
where, and D denote the mean free path of the molecules and the channel dimension (diameter), respectively. However, in most cases, is very small ( 6 108 m for air at 1.0 atm condition and even smaller for the liquid flows), and accordingly the value of Kn becomes very small. In other words, for the flow passage smaller than say 1.0 mm, the concept of the fluid continuum may start to break down, which is not the size range of our interest. According to the review paper by Mehendale et al. [2], Pfahler et al. [5] had suggested a rarefied gas effect to explain the reduction of the friction factor. That is, the rarefied gas effect first becomes significant for 0.001
ð2Þ
WeL ¼ jL2 DL =
ð3Þ
2 WeG ¼ jG DG =
ð4Þ
liquid Weber number
and gas Weber number
may be needed. In microchannels, the values of Eo are small (Eo<1); in other words, the surface tension force becomes predominant over the gravitational force. This implies that the flow direction (orientation) plays only a minor role in microchannel flow (Damianides and Westwater [6],
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Triplett et al. [7] and the flow regimes are influenced by the surface wettability (Barajas and Panton [8]). Nevertheless, either of WeL or WeG is in the range of 1–100 and the effect of the inertia force becomes important as well as the surface tension force. Similar case appears with the two-phase flow in a micro-gravity condition (Zhao and Rezkallah [9], Rezkallah [10], Bousman et al. [11], Lowe and Rezkallah [12]) For the air–water flow under the atmospheric pressure condition with Eo being an order of unity, the characteristic length (diameter) of the channel is approximately 3 mm. This length scale is close to 5 mm suggested by Fukano and Kariyasaki [13] as the upper bound of the microchannel size determined from the flow pattern experiments. In the present article, although some discussions are on the single-phase microchannel flow, emphasis is placed on the two-phase flow characteristics for circular and rectangular channels with their hydraulic diameters smaller than 5 mm, mostly ranging from 0.1 to 5 mm.
B. SINGLE-PHASE FLOWS According to the review work by Peng and Wang [14] for microchannels ranging from 0.1 to several hundred microns, the fully-developed turbulent convection is known to start at the Reynolds number about 1000–1500. Besides, they reported that the heat transfer behavior at the transition and the laminar regions was affected not only by the fluid velocity (and temperature) but also by the size of the microchannel. Another work by the same authors (Peng et al. [15]) with rectangular multiple microchannels having hydraulic diameters of 0.133–0.343 mm, transition to turbulence occurs at the Reynolds number as low as 200, which is much lower than with the conventional tubes. However, Rao and Webb [16] claimed that the apparent discrepancies of friction could be attributed to the flow maldistribution in multiple microchannels. According to them, flow maldistribution is expected in multiple microchannels because of manufacturing tolerances and poor manifold design. They concluded that the flow characteristics in single microchannels agreed well with those in macro-scale channels. 1. Single-phase Pressure Drop a. Laminar Flows Circular Tubes. Olsson and Sunden [17] measured the frictional pressure drop of air for fully developed flows in tubes ranging from 2 to 20 mm in
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diameters. The maximum error in measuring the friction factor was reported to be 8% for laminar flows. For a laminar flow within a circular tube, the friction factor satisfies f ReD ¼ 16
ð5Þ
and the data points stay within 5%. They also noted that the transition to the turbulent flow starts at Re ¼ 2300 for the tubes larger than 3 mm. However, the onset of transition is shifted towards somewhat higher Reynolds numbers as the tube size becomes smaller, which is contradictory to the observation of Peng and Wang [14]. They explained that this is probably due to the entrance effect in their experiments. Nevertheless, the size effect is minor in this case, and Eq. (5), developed for single-phase flows in conventional size tubes (Dh>5 mm), is considered acceptable for laminar flows in circular microchannels. Non-circular Channels. For a rectangular channel with the aspect ratio of 9 and the hydraulic diameter ranging from 1.5 to 6 mm, Olsson and Sunden [17] compared their measured results for air flow with the equation f ReDh ¼ 20:9
ð6Þ
reported by Shah and London [18], and confirmed that all their data points were within 5%. The constant shown in the right hand side of Eq. (6) depends on the aspect ratio of the rectangular channel. The transition to turbulence begins at the Reynolds number of 2100. According to Hartnett and Kostic [19], for various aspect ratios of the rectangular channels, the friction factors are well represented by the polynomial function as: f ReDh ¼ 24ð1 1:3553AR þ 1:9467AR2 1:7012AR3 þ 0:9564AR4 0:2537AR5 Þ
ð7Þ
Here, AR stands for the aspect ratio of the channel, and the Reynolds number is based on the channel hydraulic diameter. Equation (7) also agrees well with the measured results of Mishima et al. [20] ((1.07, 2.45, 5) mm 40 mm), Wambsganss et al. [21] (3.18 19.08 mm2) and Lee and Lee [22] ((0.4, 1, 2, 4) mm 20 mm). This implies that Eq. (7) also represents the friction factors satisfactorily for microchannels.
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For other cross sectional shapes, different friction factors should be used and the values of f ReDh are found in most of the convection heat transfer texts such as Kakac and Yener [23]. b. Turbulent Flows Circular Tubes. Fully turbulent flow is established at the Reynolds number between 3000 and 4000. From the experimental results for the air flow through the circular tubes ranging 2–20 mm I.D., Olsson and Sunden [17] reported that, except for the 2 mm case, the well-known Blasius correlation represents the experimental data within the experimental error. The Blasius correlation is expressed as follows: 0:25 f ¼ 0:079ReD
ð8Þ
With the 2 mm-tube, fully turbulent flow was reached at slightly higher Reynolds number compared with the larger tubes. Moreover, the friction factor appeared to be lower than predicted by Blasius correlation by about 8%. Tran et al. [24] measured single-phase friction factors as the preliminary tests to validate their instrumentation and data reduction method for boiling heat transfer experiments with a round tube (D ¼ 2.46 mm) using R-12. Measured data for isothermal, turbulent flows agreed with Blasius correlation within 15% over the Reynolds number range of 4000–12 000. Later, Tran et al. [25] performed the similar experiments for round tubes (D ¼ 2.46 and 2.92 mm) using a different fluid, R-134a. In this case, the friction factor was close to or slightly lower than the Blasius prediction (Eq. (8)) at the Reynolds number greater than 3000. Lin et al. [26] investigated the local frictional pressure drop through capillary tubes (1.5 m long with 0.66 and 1.17 mm I.D., respectively) using R-12 as the working fluid. The ranges of the temperature, pressure, degree of subcooling and the mass velocity at the tube inlet are 290–326 K, 6.3– 13.2 bar, 0–17 K and (1.44–5.09) 103 kg/m2 s, respectively. The measured results are about 20% higher than the values predicted with Blasius correlation for smooth tubes. They explained this to be due to the large values of the relative roughness ("/D) with capillary tubes, which affects the shear stress at the tube wall remarkably. To take account of the relative roughness factor, the following Churchill’s correlation (Churchill [27]) for the friction factor was suggested to use. f ¼8
"
8 Re
1=2
1 þ ðA þ BÞ3=2
#1=12
ð9Þ
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MICROCHANNEL HEAT EXCHANGER
Here, 8 3916 2 > > > > > = < 7> 6 1 7 6 A ¼ 2:457 ln6 0:9 ; 7 " 5> > 4 7 > > > > ; : þ0:27 Re D
B¼
37 530 16 Re
ð10Þ
In this study, the Reynolds number range was 4.64 103–3.76 104, and the roughness values were 2.0 106 (for the 0.66 mm-tube) and 3.5 106 m (for the 1.17 mm-tube), respectively. The Churchill correlation predicts the experimental data with the standard relative errors of 6.1 and 4.8% for 0.66 and 1.17 mm I.D., respectively. As a preliminary work to the two-phase experiments, Zhang and Webb [28] checked the single-phase pressure drop for R-134a, R-22 and R-404A flowing in a multi-circular-port extruded aluminum tube with hydraulic diameter of 2.13 mm and copper tubes of 6.25 and 3.25 mm. The friction factor was predicted within 10% using the Blasius friction correlation. In general, it can be stated that the simple and conventional Blasius correlation is applicable to the microchannels provided that the surface roughness is negligibly small relative to the tube diameter. Non-circular Channels. For the friction factor with single rectangular channels, Olsson and Sunden [17] showed that the transition to turbulence begins at the Reynolds number 2100, and the measurements coincide with the Blasius line (Eq. (8)) at the Reynolds number 4000 based on the hydraulic diameter. However, the friction factor becomes higher than predicted at the larger value of the Reynolds number. Similar experiments have been performed by Mishima et al. [20], Wambsganss et al. [21], Tran et al. [24,25,29] and Lee and Lee [22] using various two-phase mixtures such as air– water, R-12 and R-134a. Wambsganss et al. [21] reported that the transition to turbulence starts at 2700 that is somewhat higher than critical Reynolds number for large pipes. The measured data of Tran et al. [24,29] with R-12 coincide with the Blasius correlation within 15%. Recent experiments by Tran et al. [25] show that the measured results are close to or slightly smaller than the values predicted by the Blasius correlation. Lee and Lee [22] also confirmed that the Blasius correlation is applicable to the microchannels. For R-134a flow in multi-port flat extruded aluminum tubes with their hydraulic diameters ranging from 0.44 and 1.56 mm, Webb and Ermis [30] reported that the friction factors are typically 10–15% below the values predicted by Eq. (8). However, as stated in their paper, the friction factor is very sensitive to the uncertainty of the cross-sectional flow area information provided by the manufacturers and this might cause errors in deducing the friction factors.
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As a whole, for the case of the single-phase flow in the microchannels (mostly in the range between 0.4 and 5 mm), the existing correlations for the conventional size tubes (larger than 5 mm) could be used without serious discrepancies. For the smaller channels (say, below 0.1 mm), there are some of the works (as cited by Mehendale et al. [2]) reporting that the single-phase pressure drop deviates substantially from that with the conventional size. However, Rao and Webb [16] claimed that, even for the smaller channels (10–400 mm in hydraulic diameters), the fluid flow behavior follows the same hydraulic resistance laws as macro-scale flows and the variation is less than 20% in both the laminar and turbulent regimes. They reasoned that the deviation mostly observed with the multipass channels is attribute to the anomalous flow behavior due to the flow mal-distribution. 2. Single-phase Heat Transfer a. Laminar Flows Circular Tubes. For fully developed laminar flows in circular tubes of conventional heat exchangers (larger than 5 mm in diameter), the heat transfer coefficient appears constant. In other words, the Nusselt numbers for the constant-heat-flux and constant-wall-temperature conditions are expressed, respectively, as: NuD ¼ 4:36
ðfor constant heat fluxÞ
ð11Þ
NuD ¼ 3:66
ðfor constant wall temperatureÞ
ð12Þ
For arbitrary wall heating condition, Nusselt number stays between 3.66 and 4.36 and can be obtained by superposing the constant-wall-temperature solutions. No peculiar phenomenon has been reported on the laminar heat transfer for microchannels, and Eqs. (11) and (12) may be used without any serious error in this case. Non-circular Channels. For rectangular channels, the Nusselt number for the fully developed flow under the constant heat flux condition is expressed as (Hartnett and Kostic [19]): NuDh ¼ 8:235ð1 2:0421AR þ 3:0853AR2 2:4765AR3 þ 1:0578AR4 0:1861AR5 Þ
ð13Þ
For other cross sectional shapes, the value of the Nusselt numbers are listed in most of the convection heat transfer texts such as in Kakac and Yener [23]. Similar to the case of the friction factors, one may use Eqs. (11)–(13) for microchannels.
305
MICROCHANNEL HEAT EXCHANGER
b. Turbulent Flows Circular Tubes. Wambsganss et al. [31] have reported that their measured values approached the Petukhov–Popov correlation (Petukov [32]) within 5% accuracy with the increase of the Reynolds number. NuD ¼
ð f =8ÞReD Pr 1:07 þ 12:7ð f =8Þ1=2 ðPr2=3 1Þ 4
6
ð0:5 < Pr < 2000; 10 < ReD < 5 10 Þ
ð14Þ
In their work, the tube diameter was 2.92 mm and R-113 was used as the test fluid. According to the recent work by Tran et al. [24], the following Dittus– Boelter equation represents the measured data within 2/þ6% for a R-12 flow in a 2.46 mm-tube. 0:4 NuD ¼ 0:023Re0:8 D Pr
ð0:7 < Pr < 160; ReD 10 000; L=D > 10Þ ð15Þ
Yan and Lin [33,34] obtained the heat transfer coefficient for a R-134a flow through a circular tube of 2 mm in diameter, and confirmed that, for the mass flux larger than 200 kg/m2 s (which corresponds to the turbulent flow in this case), both the Dittus–Boelter equation (Eq. (15)) and the following Gnielinski correlation well represent the measured data: NuD ¼
ð f =2ÞðReD 1000ÞPr pffiffiffiffiffiffiffiffi ½1 þ ðD=LÞ2=3 ðPr=Prw Þ0:11 1 þ 12:7 f =2ðPr2=3 1Þ 6
ð0:5 < Pr < 2000; 3000 < ReD < 5 10 Þ
ð16Þ
Non-circular Channels. Tran et al. [24,29] have performed experiments with a rectangular channel (4.06 1.70 mm2, Dh ¼ 2.4 mm) using R-12, and reported that the measured values are within 2/ þ 6% from the Dittus– Boelter equation, similar to the circular tube case. For multipass tubes with internal microfins, it is worthwhile to look in to the work by Webb and Ermis [30]. They used flat extruded aluminum tubes with their hydraulic diameters ranging from 0.44 to 1.564 mm, and R-134a was used as a working fluid. They reported that their all-liquid heat transfer data fall 10–15% below the Petkhov–Popov correlation [32] that has been developed for conventional plain circular tubes. This, again, attributed to the uncertainty of the tube size information provided by the manufacturer. For the tubes with the smaller scales (i.e., from several-ten to -hundred microns), heat transfer correlations are well summarized by Mehendale
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M.-H. KIM ET AL.
et al. [2]. When the tube size goes down to the micron-scale, the relative surface roughness at the tube wall should be taken into account seriously. In sum, there seems no other dependence of the heat transfer phenomena on the channel size than with the conventional size. Thus, it can be concluded that, for both circular and non-circular microchannels, the turbulent heat transfer correlations for the macro-size channels can be needed without any significant error.
C. TWO-PHASE FLOWS In two-phase flows, the pressure drop and the heat transfer performance are strongly dependent on the flow patterns. The flow patterns, in turn, are determined by the fluid properties (density and viscosity of gas and liquid, surface tension, and surface wettability), channel geometry (hydraulic diameter, angle of orientation), velocities of the gas and liquid, and also by heating or cooling rate of the tube surfaces. In addition, another important parameter to be considered in two-phase flow problems is the void fraction. However, the phenomenon of two-phase flows in microchannels, whose characteristic dimensions are smaller than a typical bubble size (diameter), is intriguing and must be addressed separately for this application (Wambsganss et al. [35]). Lowry and Kawaji [36] also concluded that the correlations of Taitel et al. [37] on flow transition for large pipes are not valid for narrow channels. As mentioned in Section II.A, the Eotvos number (Eo), defined in Eq. (2), is smaller than the unity with the microchannel twophase flow. The Eotvos number is the reciprocal of the square root of the Confinement number, NCONF, which is also widely adopted in the heat transfer correlations for microchannel two-phase flows, defined as follows:
NCONF
¼ gðL G Þ
0:5 , Dh
ð17Þ
Here, L, G, Dh, and g denote the liquid and gas densities, hydraulic diameter, surface tension and the gravitational constant, respectively. In other words, the following criterion is appropriate to microchannels. Eo ¼ N2 CONF < 1
ð18Þ
In the present section, the flow pattern transition criteria pertinent to the small-scale channels (microchannels) are introduced and then discussions on the pressure drop and heat transfer correlations will be followed.
MICROCHANNEL HEAT EXCHANGER
307
1. Flow Patterns The work of Suo and Griffith [38] may be the first one reported on the two-phase flow in small horizontal channels (in the order of 1-mm I.D. and velocities up to 3 m/s), where capillary slug flow was examined in detail. In this regime, the mixture density and the liquid film around a bubble have been correlated. Also, the conditions under which long horizontal bubbles can exist were established by correlations those authors had developed. Later, several works were reported on the flow pattern identification as summarized in the review article by Ghiaasiaan and Abdel-Khalik [39]. Figures 2 and 3 show the typical flow patterns in small vertical tubes and the horizontal narrow rectangular channels, respectively (Mishima and Hibiki [40], Wambsganss et al. [35]). The types of the flow patterns observed in the small channels are basically similar to those observed in the large pipes. That is, in large vertical pipes, bubbly, slug, churn-turbulent and annular flows are observed (Fig. 2), while in horizontal flows, bubbly, plug, slug, annular, stratified and wavy flows appear. The last two patterns—stratified and wavy flows—do not appear in vertical flows. However, in the case of the small vertical channels, several unique flow patterns are also observed as in (b), (c), (e) and (g) of Fig. 2 (Mishima and Hibiki [40]). In bubbly flow with tiny bubbles (Fig. 2(b)), the bubbles rise with a spiral motion. However, when the bubble size becomes comparable to the tube diameter (Fig. 2(c)), a group of bubbles rise in a row without coalescence. For slug flow (Fig. 2(e)), the ratios of bubble lengths to the tube diameter are generally much larger
FIG. 2. Flow regimes in vertical tubes (Mishima and Hibiki [40]). (a), (d), (f), (h) and (i) : Flow regimes frequently observed in large diameter tubes. (b), (c), (e) and (g): Flow regimes specially appear in capillary tubes.
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M.-H. KIM ET AL.
FIG. 3. Flow patterns in horizontal rectangular channels (Wambsganss et al. [35]).
than those observed in the large tubes. In the churn flow regime, the tiny bubbles in the liquid slugs move rapidly upwards. For horizontal flows, the stratified and wavy flow patterns were not observed in small-scale circular tubes or rectangular channels with their aspect ratios smaller than the unity and a very small vertical dimension. For horizontal rectangular channels, the dimension of the vertical side itself is an important parameter as well as the aspect ratio, as reported by Wambsganss et al. [35] and Troniewski and Ulbrich [41]. Flow patterns in small-diameter horizontal tubes also depend on the contact angle. Barajas and Panton [8] used four kinds of materials with different contact angles to examine the flow pattern in small tubes (1.6 mm I.D.). Basically the flow patterns are very similar to each other except for occurrence of a rivulet flow. A rivulet is a stream of liquid on the tube surface, and twists its way downwards much like a river. The rivulet flow is a new flow regime that replaces wavy flow when the contact angle becomes large. Also, in the large contact-angle range (>90 ), the transition
MICROCHANNEL HEAT EXCHANGER
309
boundaries, except for the plug–slug transition, are sensitive to the change of the contact angle. Taitel and Dukler [42] adopted the Kelvin–Helmholtz instability concept as the transition mechanism for a stratified flow. However, through a series of experiments with tubes of 4–12.5 mm in diameter, Barnea et al. [43] argued that the surface tension predominates over the gravitational force in smaller tubes, and the Kelvin–Helmholtz type instability is not responsible for the stratified–intermittent transition. The model of Barnea et al. [43] takes into account the surface tension force at low gas and liquid flow rates by comparing the gravitational force with the surface tension force. The following equation represents the condition of the onset of intermittent or slug/plug flow:
hG
p 4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L g½1 ðp=4Þ
ð19Þ
Here, hG stands for the gas phase height. Wilmarth and Ishii [44] performed experiments on adiabatic concurrent vertical and horizontal two-phase flows of air and water through rectangular channels with their gaps of 1 and 2 mm. With the vertical flows, all the flow regimes appear in the conventional-size round tubes were observed. For the bubbly-to-slug and slug-to-churn transitions, new models for the distribution parameter C0 are needed for the better prediction. However, for the churn–annular transition, the annular drift-velocity correlation with the zero superficial liquid velocity showed good agreement with the experimental data. For the flows in horizontal channels with their longer side placed vertically (AR>1), the stratified wavy regime did not appear. In this case, the existing model for the transition from the stratified smooth flow to the plug flow was found to be unsatisfactory and a better model should be developed. However, in their work, no report has been made on the horizontal flows in rectangular channels with their longer side placed horizontally (AR<1). Wambsganss et al. [35] reported that the conventional flow pattern maps are not generally applicable to the small rectangular channels on quantitative basis. The flow pattern transition and flow regime maps for microchannels are well summarized in the review work by Ghiaasiaan and Abdel-Khalik [39]. In the present article, we will limit our discussion on the transition criteria between bubbly, slug, churn-turbulent, annular flows since they are the flow patterns mostly observed in small-scale (micro) channels.
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M.-H. KIM ET AL.
a. Bubbly-to-slug Transition For this transition criterion, the experimental results of Hibiki and Mishima [45] are introduced. Their results are in accordance with those of Xu et al. [46]. The observation by Wilmarth and Ishii [44] generally agrees with the results by Ali and Kawaji [47] and Mishima et al. [20], but there are discrepancies in some parts. Hibiki and Mishima [45] and Tripplett et al. [7] reasoned the discrepancies to the different observation methods and definitions of the flow regimes by different researchers. Moreover, the transition between the flow regimes occurs gradually, and is affected also by upstream conditions such as bubble diameters and distribution along the tube cross section at the inlet. Therefore, the transition boundaries should be understood as a band with a certain width proportional to the uncertainty in determining the transition criteria. This explanation is applicable also to other transition boundaries. Let us consider the drift-flux model, UG ¼
jG ¼ C0 j þ UGj
ð20Þ
where j ¼ jL þ jG :
ð21Þ
In Eq. (20), C0 denotes the distribution parameter expressed by Ishii [48] as: rffiffiffiffiffiffi G ð22Þ C0 ¼ 1:35 0:35 L for rectangular channels and C0 ¼ 1:2 0:2
rffiffiffiffiffiffi G L
ð23Þ
for circular tubes. Mishima and Hibiki [40] proposed C0 ¼ 1:2 þ 0:51e0:691D
ð24Þ
for circular tubes ranging from 1 to 4.9 mm, where the effect of the tube size is taken into account. In Eq. (24), the unit of D is mm. For the rectangular channels with 1.0 and 2.4 mm gaps, the void fraction is well correlated with Eq. (22) (Mishima et al. [20]). The similar tendency was reported also in the former paper by the same authors (Mishima et al. [20]) for rectangular ducts with a narrow gap; i.e., the distribution parameter increases when the gap
MICROCHANNEL HEAT EXCHANGER
311
becomes very narrow. These tendencies may be attributed to the centralized void profile and the laminarization of the flow in small or narrow channels. By inserting Eq. (21) into Eq. (20), a relationship between jG and jL at the bubbly-to-slug transition can be obtained as follows: jL ¼
1 1 jG UGj max C0 C0 1
ð25Þ
The concept of amax, the maximum void fraction for the bubbly flow without forming slug bubbles by coalescence, was introduced in the above equation, and the drift-velocity UGj for bubbly flows in a rectangular channel is expressed as UGj
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðs þ wÞ ¼ 0:35 p
ð26Þ
by Xu et al. [46] for rectangular channels or
UGj ¼
pffiffiffi g 1=4 ð1 Þ1:75 2 2L
ð27Þ
by Ishii [48] for circular tubes. However, Hibiki and Mishima [45] pointed out that the drift-velocity becomes negligible in a narrow gap flow. The same conclusion was made with a round tube with its inner diameter smaller than 5 mm (Mishima and Hibiki [40]). Mishima et al. [20] and Wilmarth and Ishii [49] also mentioned previously that the drift-velocity was almost negligible for the gap sizes smaller than 2.4 and 2 mm corresponding to the hydraulic diameters of 4.5 and 3.5 mm, respectively. The final information to be obtained in Eq. (25) is amax. The values of amax are reported to be 0.18 (Griffth and Wallis [50]) or 0.25 (Taitel et al. [37]) for circular tubes, and 0.3 for both circular tubes (Mishima and Ishii [51]) and rectangular channels (Wilmarth and Ishii [44]), respectively. Recently, Hibiki and Mishima [45] showed that the void fraction at the transition (amax) varies from 0.2 to 0.3 depending on the channel gap as follows: max ¼ 0:2 s max ¼ þ 0:15 20Dh max ¼ 0:3
for s < Dh for Dh s 3Dh for s > 3Dh
ð28Þ
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M.-H. KIM ET AL.
In summary, the transition criterion between the bubbly and the slug flows in small channels can be expressed by Eq. (25) along with Eqs. (22), (24), (26)–(28). b. Slug-to-churn Transition On this transition criterion, a number of conflicts are found between the existing reports. The criterion of Hibiki and Mishima [45] coincides with that of Mishima et al. [20], but does not agree with those of Wilmarth and Ishii [44] and Xu et al. [46]. The criteria for bubbly-to-slug and slug-to-annular transitions by Wilmarth and Ishii [44] agree well with those of Ali and Kawaji [47], but not for the slug-to-churn transition. No churn-turbulent flow was observed with the 1 mm-gap flow by Mishima et al. [20]. On the other hand, the slugto-annular criterion for 2 mm-gap flow by Lowry and Kawaji [36] agrees with the slug-to-churn criterion of Wilmarth and Ishii [44]. The experimental data of Triplett et al. [7] are in overall agreement with similar experimental data of Damianides and Westwater [6] and Fukano and Kariyasaki [13], when inconsistencies associated with the flow pattern identification are removed. The criteria to differentiate the churn-turbulent flow from the slug flow regime for vertical flows and the plug flow from the slug flow regime in horizontal flows are rather subjective and depend on the researchers. Thus, in this respect, Fukano and Kariyasaki [13] categorized the churn-turbulent and slug flow regimes in vertical flows and the plug and slug flows in horizontal flows simply into intermittent flows. Then the two-phase flow can be classified into three basic regimes; bubbly, intermittent and annular flows. This classification, except for the rectangular channels with large aspect ratios, enables us to identify the flow regime consistently regardless of the flow orientation. In this respect, as mentioned in Section I, the micro-gravity experiments can be simulated with the microchannel experiments, such as by Galbiati and Andreini [52]. Other reports on the micro-gravity two-phase flow are available; Lowe and Rezkallah [12], Zhao and Rezkallah [9], Rezkallah [10] and Bousman et al. [11]. Especially Zhao and Rezkallah [9] named the bubbly/slug, frothy slug–annular and the annular flows as the surface tension, intermittent and the inertia regions, respectively, and constructed a flow pattern map with the gas and liquid Weber numbers as its axes. c. Slug/churn-to-annular Transition With the two-phase forced convection in the annular flow regime, the heat transfer rate is increased with the increases in quality and mass flux. On the other hand, in the confined bubble regime (i.e., in the slug or churn flow
MICROCHANNEL HEAT EXCHANGER
313
regime), the nucleate boiling predominates and the heat transfer rate is increased with the heat flux (Cornwell and Kew [1]). Thus, in designing compact heat exchangers, prediction of the slug/churn-to-annular transition is very important. Several related works for rectangular microchannels have been reported so far; Lowry and Kawaji [36], Wilmarth and Ishii [44], Hibiki and Mishima [45], Xu et al. [46], and Rezkallah [10]. The details are described below. Lowry and Kawaji [36] assumed that the transition to the annular flow regime starts when the pressure drop by the wall shear and the surface tension force is unbalanced, and proposed the following equations as the transition criterion: ð1 Þ3 1 m jL ¼ ð4 pÞ L 2 2 ¼
jG ðjG þ 2jL Þ
ð29Þ ð30Þ
In the above transition equation, the liquid velocity at the interface is assumed the same with the core-gas velocity, and the interfacial shear stress was neglected. Later, Hibiki and Mishima [45] obtained the transition criterion from the force balance equations for the annular flow configuration based on the results of Ishii [48], Mishima and Ishii [51] and Wilmarth and Ishii [44]. Considering a liquid film with a core-air flow, the force balances for the liquid and gas phases are written as (Hibiki and Mishima [45]): – Liquid phase:
– Gas phase:
pffiffiffi dp 2ðs þ wÞ 2ðs þ wÞ i þ ¼ L g wL wsð1 Þ dz wsð1 Þ
ð31Þ
pffiffiffi dp 2ðs þ wÞ ¼ G g þ i dz ws
ð32Þ
By eliminating the pressure gradient terms from Eqs. (31) and (32), along with the appropriate expressions for the interfacial shear stress ( i) and the wall shear stress ( wL) in the liquid film, respectively, fi fi jG jL 2 ð33Þ i ¼ G Ur2 ¼ G 2 2 1 wL ¼
fLF fLF jL 2 L UL2 ¼ L 2 2 1
ð34Þ
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M.-H. KIM ET AL.
the following relationship is finally obtained: pffiffiffi fi G ðs þ wÞ jG jL 2 fL L ðs þ wÞjL2 ¼1 gws 1 gwsð1 Þ2
ð35Þ
Here, Hibiki and Mishima [45] considered the following criteria as the transition to annular flow: (a) flow reversal in the liquid film surrounding a large bubble; (b) destruction of liquid slugs or large waves by entrainment or deformation. With criterion (a), the film velocity becomes zero and hence jL ¼ 0
ð36Þ
In addition, for the interfacial friction factor, the following correlations were used: fi ¼ 0:005½1 þ 75ð1 Þ
ð37Þ
Here, void fraction a can be obtained from the drift-flux model (Eq. (20)) with the drift-velocity UGj
pffiffiffi g 1=4 ¼ 2 2L
ð38Þ
and the distribution parameter C0 expressed as Eq. (22). With criterion (b), the entrainment condition initiating the flow transition is given by (Ishii [48]) jG
g 1=4 0:2 N mL 2G
ð39Þ
where the viscosity number NL is defined as mL NL ¼ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii1=2 L =ðgÞ
ð40Þ
Xu et al. [46] and Galbiati and Andreini [52] followed the same approach but with different correlations for void fraction and interfacial friction factor. Xu et al. [46] adopted a constant value of 0.75 for the void fraction in
MICROCHANNEL HEAT EXCHANGER
315
Eq. (35), and for the interfacial friction factor, the following correlation proposed by Govan et al. [53] was used: " 1=3 # L n1 fi ¼ CG ReG 1 þ 6ð1 Þ ð41Þ G Here, CG and n1 are given in the paper by Xu et al. [46]. Galbiati and Andreini [52] also adopted Eq. (35) as the transition criterion to annular flow for a horizontal microchannel using the drift-flux model. In their work, the drift-velocity was set to zero and the following distribution parameter, obtained experimentally, was used: C0 ¼ 1=ð0:9522 9:9946A þ 117:92A2 Þ
ð42Þ
A ¼ L j=
ð43Þ
For the interfacial friction factor, fi ¼ fG 1 þ 150ð1 0:5 Þ
ð44Þ
was used. There are some other studies reported on the transition criterion to the annular flow regime, such as by Zhao and Rezkallah [9] and Rezkallah [10]. There, the flow regime map with the gas and liquid Weber numbers taken as the axes. However, the results are not given in mathematical correlation forms that are easy to use. 2. Two-phase Pressure Drop Two-phase pressure drop consists of frictional, accelerational and gravitational terms. Among them, the accelerational and gravitational pressure drops can be estimated easily once the void fraction is obtained properly. Thus, in the present section, only the frictional pressure drop is going to be discussed in detail. Typical works on the two-phase frictional pressure drop are listed in Table I. In two-phase channel flows, the concept of the two-phase frictional multiplier has been widely used. The two-phase frictional multipliers based on the liquid flow rates are defined as follows:
2L ¼
ðdp=dz F ÞTP ðdp=dz F ÞL
ð45Þ
2Lo ¼
ðdp=dz F ÞTP ðdp=dz F ÞLo
ð46Þ
or
316
TABLE I THE STATE-OF-THE-ART Authors
Fujita et al. [62] Triplett et al. [63]
TWO-PHASE FLOW FRICTIONAL PRESSURE DROP
Geometryy–orientation* and size (mm)
R–V, (0.5, 1, 2) 80 R–H, 19.05 3.18 R–H, (1.465, 0.778) 80 C–H, 1, 2.4, 4.9 C–V, 1, 2, 3, 4 R–V, (1.2, 2.4, 5) 40 C–H/V, 0.74–3.07
Lee and Lee [22] Zhang and Webb [28]
R–H, (0.2–2) 10 C–H, 1.1, 1.45 S–H, 1.09, 1.49 R–H, (0.4–4) 20 C, 2.13 (multi-port), 3.25, 6.25
Wang et al. [64]
C, 3, 5, 7, 9
Boiling flow Lazarek and Black [66] Lin et al. [26] Yan and Lin [33] Kureta et al. [34]
C–V, 3.1 C, 0.66, 1.17 C–H, 2 C–H/V, 2-6
Fluid
IN
MICROCHANNELS Suggested correlations
Air/water Air/water Air/water Air/water Air/water
Þ
L ¼ fnð jG Modified Chisholm correlation Separated flow model – Modified Chisholm correlation
Air/glycerine solution N2/ethanol solution Air/water
Beattie correlation Chisholm correlation for large jL Homogeneous flow model
Air/water R-113 R-134a, R-22 R-404A R-22, R-407C R-410A
Modified Chisnolm correlation Modified Friedel correlation
R-113 R-12 R-134a Water
C ¼ 30
Lo ¼ fn(ReLo, ReTP, "/D) 0:1 fTP ¼ 0:11Reeq Martinelli–Nelson model for wall friction loss, annular flow model for accelerational loss
Modified homogeneous flow model and Chisholm correlation
M.-H. KIM ET AL.
Adiabatic two-phase flow Lowry and Kawaji [36] Wambsganss et al. [21] Ali et al. [57] Fukano and Kariyasaki [13] Mishima et al. [20], Mishima and Hibiki [40] Bao et al. [61]
ON
Tran et al. [25]
R-134a, R-12
R–H, 4.06 1.7
R-113
Yang and Webb [71]
R, 2.64 RF, 1.56
R-12
0:12 (rectangular f =fL ¼ 0:435Reeq channel, micro-fin tubes)
Yan and Lin [34] Webb and Ermis [30]
C–H, 2 RF, 0.44–1.56 R, 1.33
R-134a R-134a
1:074 fTP ¼ 498:3Reeq ðdp=dz F Þ ¼ aGm xn (a, m, n: determined from experiments)
Condensing flow
y R, rectangular channel; C, circular tube; S, semi-triangular channel; F, finned channel. * H, horizontal flow; V, vertical flow.
Modified B-coefficient method
MICROCHANNEL HEAT EXCHANGER
C–H, 2.46, 2.92
317
318
M.-H. KIM ET AL.
Here, the numerator of Eq. (45) stands for the two-phase frictional pressure drop while the denominator the single-phase pressure drop for the liquidonly flow. The denominator of Eq. (46) implies the pressure drop gradient for the whole flow considered as in the liquid phase. Another way to express the two-phase frictional pressure drop is to use the homogeneous flow model by introducing the two-phase friction factor in terms of the two-phase Reynolds number. The homogeneous model is often adopted since it is simple to use. Therefore, in this section, two main approaches of the pressure drop research are going to be introduced for both the adiabatic and the phase-changing (evaporation and condensation) flows. a. Adiabatic Flows The two-phase frictional multiplier 2L defined in Eq. (45) is often expressed as a function of the Martinelli parameter defined as
ðdp=dz F ÞL X¼ ðdp=dz F ÞG
1=2
ð47Þ
where the denominator denotes the single-phase pressure drop for the gasonly flow. The classical Chisholm correlation [54] based on the graphical presentation of Lockhart and Martinelli [55] is given as follows:
2L ¼ 1 þ
C 1 þ X X2
ð48Þ
The constant values have been proposed for parameter C for each flow regime (of the liquid and gas phases) in circular tubes (Chisholm [54]) as follows: Liquid phase Turbulent Laminar Turbulent Laminar
Gas Phase C Turbulent 20 Turbulent 12 Laminar 10 Laminar 5
ð49Þ
Lowry and Kawaji [36] have examined the flow patterns of the cocurrent upward air–water flow in 80 mm-wide rectangular passages with the gap size ranging from 0.5 to 2 mm, and measured the pressure drop along the passage. The experiments were conducted under the atmospheric pressure condition and the superficial velocity ranges of air and water were 0.1–20 and 0.05–10 m/s, respectively. They concluded that the Lockhart–Martinelli correlation is an adequate predictor of the two-phase frictional multiplier for the pressure drop, but fails to predict the mass velocity effect. Instead,
MICROCHANNEL HEAT EXCHANGER
319
they found that the two-phase frictional multiplier is largely dependent on the superficial gas velocity but less sensitive to the liquid velocity and the gap width. They showed that the two-phase frictional multiplier is a strong function of the dimensionless gas velocity defined as: jG ¼ jG ½G =gDh ðL G Þ 0:5
ð50Þ
However, the result was not provided in a correlation form, which is easy to use. Wambsganss et al. [21] measured frictional pressure gradient of air–water flow in a rectangular channel with a cross section of 19.05 3.18 mm but in different horizontal orientations (i.e., with the aspect ratios of horizontal between horizontal plates (AR ¼ 1/6) and horizontal between vertical plates (AR ¼ 6)). Mass velocity and the quality ranges were 50–2000 kg/m2 s and 2.5 105–1.0, respectively, and the experiments were carried out under the atmospheric pressure condition. Also they pointed out that the Friedel correlation [56], to be introduced later, over-predicts the data at low values of mass velocity while under-predicts at high values. The over-prediction is large for G<200 kg/m2 s, and the error is generally unacceptable. They proposed the following correlation for parameter C of Eq. (48) as a function of the Martinelli parameter (X) and the all-liquid Reynolds number (ReLo) for limited ranges (mass quality greater than 0.05 and mass velocity less than 350 kg/m2 s, which correspond to X<1, ReLo<2200, respectively). C ¼ aX b
ð51Þ
a ¼ 2:44 þ 0:00939ReLo
ð52Þ
b ¼ 0:938 þ 0:000432ReLo
ð53Þ
For total mass velocity greater than 400 kg/m2 s (ReLo>2200), C ¼ 21 was recommended for the Chisholm correlation (Eq. (48)). Ali et al. [57] measured the frictional pressure drop of air–water flow under the atmospheric pressure condition for rectangular channels (240 mm long, 80 mm wide) with different orientations. Those are vertical upward, horizontal between horizontal plates (H–H), and horizontal between vertical plates (H–V). Two different gap sizes (0.778 and 1.465 mm) were tested. The superficial velocities of the air and water range from 0.15 to 16 and 0.19 to 6.0 m/s, respectively. They proposed to use the separated flow model as fTP ¼ CRem L
ð54Þ
320
M.-H. KIM ET AL.
with ReL ¼
L jL Dh L ð1 Þ
ð55Þ
For void fraction and the values of C and m, measured values were directly used but not provided explicitly in their paper. Fukano and Kariyasaki [13] have performed pressure drop measurements for air–water two-phase flow through circular tubes of 1, 2.4 and 4.9 mm I.D. Flow directions were vertical upward, horizontal and vertical downward. Their superficial velocity ranges cover 0.03–2.0 and 0.1–20 m/s for water and air, respectively. Deviations from the Chisholm correlation [54] are quite large in the range of 1<X<10 for capillary tubes, but the errors decrease with the increase of the tube diameter. Large errors were found in the intermittent (slug) region. They considered the pressure loss consists of the wall friction loss and the expansion loss at the tails of the long slug bubbles (i.e., sudden expansion of liquid flow from a liquid film annulus to a liquid slug), and the latter effect was considered important. However, to calculate the expansion loss, information on the liquid film thickness just upstream of the tail and the length fraction of the liquid slug occupying the tube should be known a priori, which are not easy to get practically. Mishima et al. [20] and Mishima and Hibiki [40] reported that parameter C in Eq. (48) should include the effect of tube size to represent the frictional pressure drop for small tubes less than 5 mm in hydraulic diameter. Mishima and Hibiki [40] have proposed a correlation for parameter C of Eq. (48) as a function of the tube size (hydraulic diameter) as follows: C ¼ 21f1 expð0:319Dh Þg
ð56Þ
The database for Eq. (56) was from the experimental results of Sugawara et al. [58], Sadatomi et al. [59], Ungar and Cornwell [60], Moriyama et al. [4], Mishima et al. [20], and Mishima and Hibiki [40]. The database includes both vertical and horizontal results with round tubes (1.05–9.1 mm in diameters) and rectangular channels ((0.007–17) (20–50) mm2) under the atmospheric pressure condition, and the gas and liquid superficial velocities cover 0.02–50 and 0.002–2.3 m/s, respectively. Bao et al. [61] measured the frictional pressure drop for vertical and horizontal circular tubes of 0.74–3.07 mm using air and water/glycerin mixtures with different concentrations. The superficial velocity ranges of the liquid and air were 0.007–2.4 (0.05
MICROCHANNEL HEAT EXCHANGER
321
values obtained from Chisholm B-coefficient method, Lockhart–Martinelli (Chisholm) correlation (Eqs. (48) and (49)), Friedel, Muller-Steinhagen and Heck, and Beattie correlation. These correlations generally perform satisfactorily for ReL>1000 and it is concluded that fine passage dimensions do not themselves cause any problems to these correlations under the turbulent conditions. Among them, they proposed to use the Beattie correlation for ReTP>1000 as follows: " # 1 9:35 pffiffiffi pffiffiffi ¼ 3:48 4 log10 ð57Þ ReTP f f TP ¼ L ð1 bÞð1 þ 2:5bÞ þ G b b¼ TP ¼
xTP G
G L xL þ ð1 xÞG
ð58Þ
ð59Þ ð60Þ
For the range of ReTP<1000 f ¼
16 ReTP
ð61Þ
Fujita et al. [62] performed a series of the experiments with nitrogen/ ethanol solutions flowing through rectangular channels (gap size ranges from 0.2 to 2.0 mm with the fixed channel width of 10 mm). The superficial velocity ranges of the gas and the liquid were 0.1–14.0 and 0.04–0.8 m/s, respectively. They concluded that the Chisholm correlation could not predict the two-phase pressure drop when the liquid superficial velocity is low and the flow pattern is in an intermittent regime. In their work, surface tension was varied from 0.022 to 0.073 N/m by changing the ethanol concentration. The values of L for channels with small clearances are generally lower than those for channels with larger clearances. For relatively high liquid velocity cases, L is well correlated by the Chisholm correlation (Eq. (48)) irrespective of the ethanol concentrations. On the other hand, with the lower liquid velocity, L depends not only on the X values but also on the ethanol concentrations. Triplett et al. [63] measured the frictional pressure drop of air–water flow under the atmospheric pressure condition through circular channels of 1.1 and 1.45 mm I.D. and semi-triangular (triangular with one corner smoothed) channels with hydraulic diameters of 1.09 and 1.49 mm. The
322
M.-H. KIM ET AL.
superficial velocity ranges of the air and water were 0.02–80 and 0.02–8 m/s, respectively. They compared the measured results with the Friedel correlation [56]
2Lo ¼ E þ
3:24FH Fr0:045 We0:035 2
E ¼ ð1 xÞ þ x
2
L G
ð62Þ
fGo fLo
ð63Þ
F ¼ x0:78 ð1 xÞ0:224 0:91
0:19
ð64Þ 0:7
H¼
Fr ¼
G2 2TP gD
ð66Þ
G2 D TP
ð67Þ
We ¼
L G
mG mL
1
mG mL
ð65Þ
and with the following homogeneous pressure drop model.
dp 2G2 ¼ fTP Dh TP dz
ð68Þ
Here, TP represents the two-phase homogeneous density defined as TP ¼
x 1x þ G L
1
ð69Þ
and fTP denotes the two-phase friction factor which has the Blasius equation form as: 1=4 fTP ¼ 0:079ReTP
ð70Þ
where ReTP ¼
GDh TP
ð71Þ
MICROCHANNEL HEAT EXCHANGER
323
and TP ¼
x 1 x 1 þ G L
ð72Þ
They concluded that the homogeneous pressure drop model well predicts the experimental data for the bubbly and slug flow patterns and at high ReL where the homogeneous-flow assumption is well applicable. Relatively significant deviations are mostly associated with the slug–annular and annular flow patterns, and the slug flow at very low ReL. Similarly, the Friedel correlation (Eq. (62)) well predicts the frictional pressure drop in bubbly and slug flows at high ReL. However, a large deviation is observed at low ReL, and the Friedel correlation turned out to be less accurate than the homogeneous pressure drop model. Very recently, Lee and Lee [22] proposed new correlations, based on the dimensionless parameter introduced by Suo and Griffith [38], for the air– water two-phase pressure drop through horizontal rectangular channels under the atmospheric pressure condition. The channels have small gaps (0.4–4 mm) while the channel width being fixed to 20 mm, which are equivalent to the hydraulic diameters ranging from 0.8 to 6.7 mm, and 305 data points were obtained. The superficial velocity ranges of water and air were 0.03–2.39 and 0.05–18.7 m/s, respectively. Here, the two-phase frictional multiplier was expressed using the Chisholm type correlation, but with modification of parameter C where the effects of the mass flux and gap size were considered as follows: C ¼ fnðl;
; ReLo Þ ¼ Alq
r
ResLo
ð73Þ
Here, l ¼ 2L =L Dh ¼ L j=
ð74Þ ð75Þ
Constant A and exponents q, r and s for different flow regimes (laminar/ turbulent) of gas and liquid phases were determined from the experimental data and listed in their paper. The correlations with the modified C successfully cover wide ranges of the Martinelli parameter (X ¼ 0.303–79.4) and the all-liquid Reynolds number (ReLo ¼ 175–17 700) based on the hydraulic diameter within the deviation range of 10%. Their correlation also well correlates the measured data of Wambsganss et al. [21] and Mishima and Hibiki [40].
324
M.-H. KIM ET AL.
Zhang and Webb [28] measured adiabatic two-phase pressure drop for R-134a, R-22 and R-404A flows in a multi-port extruded aluminum tube with hydraulic diameter of 2.13 mm and copper tubes of 6.25 and 3.25 mm. Their mass velocity, vapor quality and saturation temperature ranges were 200–1000 kg/m2 s, 0.2–0.89 and 20–65 C, respectively. Using their 119 data points, a new correlation was developed as follows by modifying the Friedel correlation (Eq. (62)), that has a mean deviation of 11.5%.
2Lo
2
¼ ð1 xÞ þ 2:87x
2
p pc
1
0:8
0:25
þ1:68x ð1 xÞ
p pc
1:64
ð76Þ
Here, it should be noted that, the exponents on the We and Fr numbers in the Friedel correlation are very small and the We and Fr terms were not used in their correlation. Wang et al. [64] reported the two-phase frictional characteristics based on their own data and other studies. The refrigerants covered were R-12, R-22, R-134a, R-404A, R-407C and R-410A, and the size range of the tubes were 2.13–10 mm in diameters. The range of the mass velocity was 50–1029 kg/ m2 s. Based on a total of 885 data points, the largest database to the authors’ knowledge, modifications to the homogeneous model and the Chisholm correlation were made with their mean deviations around 17–18%. That is, for the homogeneous model,
dp dp ¼
dz dz h
¼ 1:0 þ
ð77Þ
0:67We0:25 expðBo0:25 Þ
ð78Þ
and for the Chisholm type correlation (Eq. (48)),
C ¼ 1:08
mL mG
0:05 0:5FrLo
for F < 1
ð79Þ
and
C¼
0:68487 þ 4:0156Bo0:2812 We0:01ðL =G Þ ðX 0:02Þ0:27
0:73
for F > 1
ð80Þ
MICROCHANNEL HEAT EXCHANGER
325
" 0:67 0:1 #2 D G FrG þ 8 1 FrL F ¼ 0:074 b L
ð81Þ
Here,
FrL ¼
L jL2 ðL G ÞgD
ð82Þ
FrG ¼
2 G jG ðL G ÞgD
ð83Þ
0:5
ð84Þ
b¼ gðL G Þ b. Flows with Phase Change
Boiling (Evaporating) Flows. In the work of Koizumi and Yokohama [65], to calculate the length of the capillary tube in the refrigeration system, flow visualization and pressure drop experiments have been performed. R-22 and the capillary tubes of 1.0 and 1.5 mm I.D. were used in their experiments. The all-liquid Reynolds number ranged from 43 000 to 64 000. They found that the starting point of vaporization (onset of the nucleate boiling) became delayed, and confirmed the flow pattern within the tube to be homogeneous. Thus, with the simple homogeneous model, the pressure distribution along the tube could be predicted successfully if the delay of vaporization was estimated correctly. Lazarek and Black [66] examined the boiling heat transfer and the pressure drop phenomena of a R-113 flow in a vertical round tube with 3.1 mm I.D. They reported that the conventional Chisholm correlation (Eq. (48)) must be modified with C ¼ 30 for the turbulent (liquid)–turbulent (gas) regime with the Martinelli parameter simplified as: Xtt ¼
1x x
0:9 0:5 0:1 G mL L mG
ð85Þ
A model to predict the local frictional pressure drop of two-phase flows during vaporization of R-12 in capillary tubes (0.66 and 1.17 mm I.D., 150 cm long) was proposed by Lin et al. [26] with the same condition for the single phase result as shown in Eq. (9). A total of 238 data points were
326
M.-H. KIM ET AL.
recorded. They derived the following correlation as the two-phase frictional multiplier:
2Lo
8 " 7 0:9 > > > þ0:27 > > 7 0:9 > > þ0:27 :ln ReTP
#916 > > > > = L # 1 1þx G > " > > > ; D
" D
ð86Þ
Here, ReTP was already defined in Eq. (71). Also, as for the two-phase viscosity mTP, difference between the prediction and the measurement is large with the McAdams or Cicchitti models, and mTP was newly defined as mTP ¼
mL mG mG þ xn ðmL mG Þ
ð87Þ
where n is the experimental constants and taken as 1.4 for the range of the mass quality, 0<x<0.25. An experiment for the frictional pressure drop of an evaporating R-134a flow through a horizontal small circular tube with 2.0 mm I.D. has been performed by Yan and Lin [33]. The ranges of mass velocity, heat flux and the saturation temperature were 50–200 kg/m2 s, 5–20 kW/m2 and 5–31 C, respectively. From the experiment, they concluded that the pressure drop is larger for a higher vapor quality at a given mass velocity. Also, an increase in the wall heat flux results in a mild increase of the pressure drop except at a high vapor quality (larger than 0.7). When the vapor quality becomes large, the saturation temperature effect on the pressure drop can be significant. For a higher saturation temperature, the pressure drop appears smaller. Based on the measured data, an empirical correlation was proposed as: 0:1 fTP ¼ 0:11Reeq
ð88Þ
with Reeq
" 0:5 # GD L ¼ ð1 xm Þ þ xm G L
ð89Þ
where xm is the average quality. Close examination of their work shows that Eq. (88) over-predicts their own data by about 200%, though the trend seems correct; thus use of Eq. (88) is not recommended as is. Kureta et al. [67] measured the pressure drop for flow boiling of water in small-diameter tubes (2.0–6.0 mm) under the atmospheric pressure
MICROCHANNEL HEAT EXCHANGER
327
condition. The experiment covers very high heat and mass flux ranges; up to 33 MW/m2 and 102–104 kg/m2 s, respectively. They concluded that the Martinelli–Nelson method well represented the measured data with the acceleration loss calculated based on the annular flow model together with the Smith correlation for the void fraction. Tran et al. [25] performed a series of boiling experiments on the frictional pressure drop of R-134a, R-12 and R-113 flows using 2.46 and 2.92 mm circular tubes and a rectangular channel with 4.06 1.7 mm2. They confirmed that the existing correlations failed to predict the experimental data and proposed a new correlation by modifying the B-coefficient method of Chisholm [68] using Confinement number defined as Eq. (17):
2Lo ¼ 1 þ ðC Here,
2
2
1Þ NCONF x0:875 ð1 xÞ0:875 þ x1:75
ð90Þ
is the dimensionless physical property defined as:
2
¼
ðdp=dz F ÞGo ðdp=dz F ÞLo
ð91Þ
where subscript Go is for entire fluid flowing as a gas phase only, and subscript Lo is for entire fluid as a liquid phase. In Eq. (90), the confinement number was used instead of the B-coefficient since the Confinement number includes the effects of surface tension and hydraulic diameter, and thus accounts for the maximum size of the bubbles confined in a small channel. For the scaling factor C, the optimization process was performed based on 610 data points for three fluids noted above, and determined to be 4.3. Equation (90) is applicable to smooth tubes with the refrigerants tested in their study, and pressures from 138 to 864 kPa, mass velocities from 33 to 832 kg/m2 s, heat fluxes from 2.2 to 90.8 kW/m2, and qualities from 0 to 0.95. Although their correlation is for the frictional pressure drop, it was confirmed by comparing the data of the same author (Tran et al. [69]) that the accelerational pressure drop was not subtracted properly from the total pressure drop when developing Eq. (90). In their test range, the accelerational pressure drop can be as large as 20%, and direct use of their correlation may cause a substantial error. For a boiling flow of R-113 in rectangular channels with their dimension the same with the adiabatic experiments (Lee and Lee [22]), Lee and Lee [70] confirmed that Eq. (73) also predicts the frictional pressure drop reasonably.
328
M.-H. KIM ET AL.
Condensing Flows. As a part of the condensation heat transfer experiment, two-phase flow pressure drop has been measured for R-12 flowing in both rectangular and micro-fin tubes with hydraulic diameters of 2.64 and 1.56 mm by Yang and Webb [71]. The equivalent mass velocity concept proposed by Akers et al. [72] correlates their data for both the plain and micro-fin tubes within 20%. The correlation was given by f 0:12 ¼ 0:435Reeq fL
ð92Þ
where Reeq has been defined in Eq. (89). The above correlation accounts for the fluid properties and should be applicable to other fluids. The ranges of mass velocity and quality were 400–1400 kg/m2 s and 0.1–0.9, respectively, and the saturation temperature was kept to 65 C. The geometry effect is included in fL. They also noted that the pressure drop is dominated by the vapor shear in both the plain and micro-fin tubes. Yan and Lin [34] measured the frictional pressure drop for a condensing flow using the same tube (circular tube of 2 mm) and fluid (R-134a) with their evaporation experiments (Yan and Lin [33]). The mass velocity, saturation temperature and the heat flux ranges were 100–200 kg/m2 s, 25– 50 C and 10–20 kW/m2, respectively. They proposed, based on their data, a friction factor correlation as fTP ¼ 498:3Re1:074 eq
ð93Þ
with the equivalent Reynolds number already defined in Eq. (89). Very recently, Webb and Ermis [30] reported experimental results on the pressure gradient for a R-134a flow in multi-port flat extruded aluminum tubes. Their hydraulic diameter ranges from 0.44 to 1.56 mm which are believed the smallest hydraulic diameters reported in the literature for the condensation pressure drop. The ranges of the mass velocity and the vapor quality were 300–1000 kg/m2 s and 16–87%, respectively. The saturation temperature was kept to 65 C and the heat flux was maintained at 8 kW/m2. Pressure gradient increases with decreasing of the hydraulic diameter for all tubes, and expressed as
dp F dz
¼ aGm xn
ð94Þ
with the appropriate values of a, m and n tabulated in their paper for each channel.
MICROCHANNEL HEAT EXCHANGER
329
c. Remarks As introduced in the previous section, there are a number of works reported on the two-phase frictional pressure drop either with or without phase change. In principle, a good adiabatic pressure drop correlation should be applicable to phase-changing flows if the gravitational or accelerational pressure drops are properly counted, and it is true in many cases. For this, of course, appropriate void fraction correlations are needed. However, often, good void fraction correlations are not available because of the phase non-equilibrium at the high heat flux conditions. Thus, in the present paper, in predicting the two-phase pressure drop, the adiabatic, evaporating and the condensing cases were differentiated. In view of this, for accurate prediction, use of the correlations outside the ranges of the variables should be discouraged. Nevertheless, the correlations based on the larger databases and parametric ranges are considered more reliable. For adiabatic two-phase flows in circular tubes with their diameters larger than about 2 mm (up to 10 mm), the work of Wang et al. [64] is recommended, while for the smaller circular tubes (down to 0.7 mm in diameter) the recommendation of Bao et al. [61] may be accepted. For adiabatic flows in rectangular channels with their hydraulic diameters ranging from 0.8 to 6.7 mm, the result by Lee and Lee [22] is recommended to use. The result of Fujita et al. [62] may be applicable to the smaller size, down to 0.4 mm in hydraulic diameter. For boiling flows in circular tubes, the correlations of Lin et al. [26] may be used for small size (0.66–1.17 mm). On the other hand, the result of Kureta et al. [67] is applicable to boiling flows in large tubes (2–6 mm), but was basically for the high heat-flux and high mass-velocity cases. For boiling flows in rectangular channels ranging from 0.8 to 6.7 mm in hydraulic diameter, the adiabatic result of Lee and Lee [22] can be used. For condensing flows, only three correlations (Yang and Webb [71], Webb and Ermis [30], Yan and Lin [34]) were available to the authors, that cover different size ranges or shapes and all those may be recommended for use. 3. Phase Change Heat Transfer In this section, the heat transfer correlations for the boiling and condensing flows are introduced, and the related works are listed in Tables II and III, respectively. a. Boiling (Evaporating) Flows In boiling channels, heat is transferred either by the nucleate boiling mode or the two-phase forced convection mode. The nucleate boiling mode appears to be dominant at high heat flux, low quality conditions. Therefore, the corresponding flow pattern is either bubbly or slug flow. In this case, the
330
TABLE II THE STATE-OF-THE-ART
ON THE
Authors
Geometryy–orientation* and size (mm)
Cornwell and Kew [1], Kew and Cornwell [73]
R–V, 1.2 0.9, 3.25 1.1
BOILING HEAT TRANSFER Fluid
R-113
IN
MICROCHANNELS
Heat transfer mechanismz N
Suggested correlations
Correlation forms were provided Nu ¼ C1 Bo0:7 ReL0:8 Pr0:4 L 0:4 0:5 Nu ¼ C2 Bo0:3 NCONF Re0:8 L PrL
T
Nu ¼ C3 FReL0:8 Pr0:4 L
C–V, 3.1
R-113
N
Wambsganss et al. [31]
C–H, 2.92
R-113
N
hTP ¼ C1 qC2 (C1, C2: functions of quality, geometry and fluid properties)
Tran [75]
C–H, 2.46, 2.92
R-134a
N
Nu ¼ 770ðReLo BoNCONF Þ0:62 ðG =L Þ0:297
N
–
R–H, 4.06 1.7
R-12 R-113
Kuznetsov and Shamirzaev [76]
A–H, gap 0.9
R-318C
T
–
Oh et al. [77]
C–H, 0.75, 1, 2
R-134a
T
hTP =hL ¼ ½240=Xtt ½1=ReTP 0:6
Yan and Lin [33] Kureta et al. [67]
C–H, 2 C–H/V, 2–6
R-134a Water
T –
Modified Kandlikar [78] correlation Nu ¼ axþb a ¼ (9.7 106)D1.93 0:44 b ¼ 1342DRein (Rein: Re at the inlet)
y R, rectangular channel; C, circular tube; A, annulus. * H, horizontal flow; V, vertical flow. z N, nucleate boiling effect; T, two-phase forced convection.
M.-H. KIM ET AL.
0:714 Nu ¼ 30Re0:857 Lo Bo
Lazarek and Black [66]
TABLE III
Authors
Geometryy–orientation* and size (mm)
Yan and Lin [34] Yang and Webb [71,81]
C–H, 2 RF, 1.56 R, 2.64 RF, 1.41, 1.56 R, 1.33, 2.64
Yang and Webb [82]
Webb and Ermis [30] y
RF, 0.44–1.56 R, 1.33
ON THE
CONDENSATION HEAT TRANSFER
Fluid
Heat transfer mechanism
IN
MICROCHANNELS Suggested correlations
R-134a R-12
– Shear-controlled flow
½hTP D=k Pr0:33 Bo0:3 Re ¼ 6:48Re1:04 eq 1=3 hD=kL ¼ 0:0265Re0:8 (Akers et al. [72] correlation) eq Pr
R-134a R-12
Shear-controlled and surface tension regimes
h ¼ hu ½Au =A þ hf ½Af =A (model validation)
R-134a
–
h ¼ bGpxq (b, p, q: experimentally determined parameters)
R, rectangular channel; C, circular tube; F, finned channel. * H, horizontal flow.
MICROCHANNEL HEAT EXCHANGER
THE STATE-OF-THE-ART
331
332
M.-H. KIM ET AL.
rate of heat transfer increases with the increase of the heat flux due to flow agitation by the bubble motion (formation, growth and detachment from the heated wall). On the other hand, the two-phase forced convection mode becomes dominant at the annular flow regime that occurs at high quality, high mass velocity conditions. With this mode, heat is transferred by conduction and convection through the liquid film and the evaporation occurs continuously at the vapor/film interface. Cornwell and Kew [1] and Kew and Cornwell [73] have examined the relationships between flow patterns and heat transfer coefficients in small rectangular vertical channels with their cross section being 1.2 0.9 (75 channels) and 3.25 1.1 mm2 (36 channels). R-113 was used as the test fluid. They identified three different flow patterns, namely, isolated bubble (IB), confined bubble (CB) and annular-slug flow (ASF) through visualization experiments. Isolated bubbles are seen at very low quality region and the bubble sizes are small compared with the channel hydraulic diameter. In this range, the heat transfer coefficient depends on the channel size and the heat flux while independent of the mass quality. In CB region, however, the bubble sizes are equivalent to the channel size and the bubble motions are restricted by the channel wall. They asserted that the Confinement number in Eq. (17) must be used as an important dimensionless parameter in CB region. In this region, the heat transfer coefficient is affected by the mass velocity and the mass quality as well as by the heat flux and the channel size. Also they showed that annular–slug flow pattern exists at the higher quality region, and the heat transfer rate increased as the mass quality and mass velocity increased. They proposed basic forms of the heat transfer coefficient correlations for each region as: Isolated Bubble ðIBÞ Region :
Nu ¼ C1 Bo0:7 ReL0:8 PrL0:4
Confined Bubble ðCBÞ Region :
0:5 Nu ¼ C2 Bo0:3 NCONF ReL0:8 PrL0:4 ð96Þ
Annular-Slug Flow ðASFÞ Region :
Nu ¼ C3 FReL0:8 PrL0:4
ð95Þ
ð97Þ
where Nu ¼
hTP Dh ; kL
Bo ¼
q00 ; Gifg
ReL ¼
Gð1 xÞDh mL
and
PrL ¼
m L cp ð98Þ kL
and constants C1, C2 and C3 are to be determined from experiments. Basically, the nucleate boiling predominates in the IB and CB regions while the two-phase forced convection is becoming dominant in the ASF region.
MICROCHANNEL HEAT EXCHANGER
333
For IB region, Lazarek and Black [66] examined the boiling heat transfer and pressure drop characteristics of a R-113 flow in a vertical tube with 3.1 mm I.D. The heat flux, mass velocity, Reynolds number and the pressure ranges were 14–380 kW/m2, 125–750 kg/m2 s, 860–5500 and (1.3– 4.1) 105 N/m2, respectively. The Boiling number (Bo) ranged from 2.3 104 to 76 104. The quality had no influence upon the heat transfer coefficients and a correlation having the similar form of Eq. (95) for a particular fluid with x ! 0 was provided as: 0:714 Nu ¼ 30Re0:857 Lo Bo
ð99Þ
They also reported that, as Bo increases, the quality at which the suppression of nucleate boiling occurs also increases. For CB-Region, Tran et al. [24,29] measured boiling heat transfer coefficients using R-12 in a horizontal rectangular channel with 4.06 1.7 mm2 and a horizontal circular tube with 2.46 mm in diameter, respectively. Wambsganss et al. [31] also examined the boiling heat transfer coefficients for a R-113 flow within a horizontal circular tube of 2.92 mm in diameter. Tran et al. [74] pointed out that the boiling heat transfer results of Tran et al. [24,29] and Wambsganss et al. [31] correspond to the nucleate-boilingdominant region where the quality ranges up to 0.6–0.8. Tran [75] proposed a correlation using their own data with R-134a and the data of R-12 (Tran et al. [24,29]) and R-113 (Wambsganss et al. [31]) similar to Eq. (96) as: 0:62
Nu ¼ 770ðReLo BoNCONF Þ
G L
0:297
ð100Þ
The correlation is based on 640 data sets and applicable for pressure from 138 to 864 kPa, heat flux from 2.2 to 129 kW/m2, mass velocity from 33 to 832 kg/m2 s and quality from 0.2 to 0.95. The correlation has the mean deviation of 7.53%. For ASF-Region, Kuznetsov and Shamirzaev [76] carried out experiments on the flow boiling of R-318C in a horizontal transparent annulus with the gap size of 0.9 mm. The mass velocity and the heat flux ranges were 200–900 kg/m2 s and 2–110 kW/m2, respectively. They showed that Eq. (100) represented experimental data well, if the quality was less than 0.3. However, errors between the measured data and the predicted values by Eq. (100) became larger as the film thickness decreased. They also reported that the corresponding flow pattern was annular for the quality ranges greater than 0.3. This implies that the Confinement number is not an appropriate parameter to represent the heat transfer characteristics in the annular–slug flow region.
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Oh et al. [77] performed boiling heat transfer experiments for R-134a flows in horizontal tubes (0.75, 1 and 2 mm in diameters) for the mass velocity range of 240–720 kg/m2 s. The quality and the heat flux ranges were 0.1–1.0 and 10–20 kW/m2 s, respectively. The heat transfer mode in the quality ranges larger than 0.1 was the two-phase forced convection, and the following empirical correlation was proposed. hTP 240 1 0:6 ¼ Xtt ReTP hL
ð101Þ
The above equation corresponds to Eq. (97) since parameter F is a function of Xtt and ReTP. Yan and Lin [33] performed experiments on the evaporating heat transfer of a R-134a flow through a horizontal circular tube of 2 mm I.D. with the mass velocity ranged between 50 and 200 kg/m2 s. The heat flux and the saturation temperature varied from 5 to 20 kW/m2 and from 5 to 31 C, respectively. The evaporation heat transfer coefficient for the small tube in their work is about 30–80% higher for most situations than the large pipe case (larger than 8 mm). With the small tube, the evaporation heat transfer coefficient is higher at the higher heat flux except for the high vapor quality region and at the higher saturation temperature. Also, at a low wall heat flux, the heat transfer rate significantly increases with a small rise in the mass velocity. However, the increasing rate becomes gradual as the heat flux increases. Based on their own data, they modified the correlation for the heat transfer coefficient by Kandlikar [78] as follows:
hTP ¼ C1 CoC2 þ C3 BoC4 FrLo ð1 xm Þ0:8 hL
ð102Þ
Here, the convection number, Co, is defined as Co ¼
1 xm xm
0:8 0:5 G L
ð103Þ
and hL ¼ 4:364
D kL
ð104Þ
Also C
C
Ci ¼ Ci;1 ReLoi;2 TR i;3
ði ¼ 1; 2; 3; 4Þ
ð105Þ
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with TR ¼
T Tc
ð106Þ
and Ci,1, Ci,2 and Ci,3 are determined from the measured data and listed in their paper. Wambsganss et al. [31] have obtained local heat transfer coefficients for a R-113 evaporating flow in a small-diameter tube (2.92 mm I.D.). The experiments were conducted with the heat flux range of 8.8–90.75 kW/m2, mass flux range of 50–300 kg/m2 s, and mass quality range of 0–0.9. They showed that the local heat transfer coefficient for evaporating flow in smalldiameter, horizontal tube is a strong function of heat flux, and only weakly dependent on mass velocity and mass quality. High boiling number and the slug-flow pattern result in domination by the nucleation mechanism. Ten (10) different heat transfer correlations have been evaluated. The correlations of Lazarek and Black [66] shown in Eq. (99) predicted the measured data very well. Also a simple form suggested by Stephan and Abdelsalam [79] for nucleate boiling shown below predicted the data well: hTP ¼ C4 q0:745
ð107Þ
Here the value of C4 is 1.2 for R-113 in the pressure range of 140–180 kPa. For a R-12 flow in a horizontal rectangular channel (4.06 1.70 mm2) with the ranges of mass velocity, heat flux, mass quality and the saturation temperature being 54–396 kg/m2 s, 4.1–33.7 kW/m2, 0.15–0.76 and 758– 945 kPa, respectively, the value of C4 in Eq. (107) is reported to be 2.5 (Tran et al. [29]). Generally, it can be written that hTP ¼ C1 qC2
ð108Þ
where C1 and C2 are the functions of quality and geometry as well as fluid properties. Equations (99) and (107) are within a mean deviation of 13%. Kureta et al. [67] have performed an experimental work on the boiling heat transfer of water flows in tubes of 2.0–6.0 mm for the mass velocity of 102–104 kg/m2 s and the heat flux up to 33 MW/m2. The onset of nucleate boiling could be represented as q00 ¼ 3:0 102 ðT Tsat Þ2:6
ð109Þ
and the Nusselt number was expressed as Nu ¼ ax þ b
ð110Þ
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a ¼ ð9:7 106 ÞD1:93
ð111Þ
0:44 b ¼ 1342DRein
ð112Þ
where Rein denotes the Reynolds number at the inlet of the heated section. For a boiling flow of R-113 in rectangular channels (gap sizes ranging from 0.4 to 2 mm with the channel width being fixed to 20 mm), Lee and Lee [70] introduced enhancement factor F between the boiling heat transfer coefficient (hTP) and the single-phase heat transfer coefficient (hL) as: hTP ¼ FhL
ð113Þ
For the ranges of liquid film Reynolds number (ReLF) smaller than 200, the enhancement factor was obtained as follows: F ¼ fnðAR; L Þ ¼ 10:3AR0:398 L0:598
ð114Þ
The above equation is applicable to the laminar (liquid)–turbulent (gas) flow regime which is the most probable case in the evaporating flow within 20% deviation. The mass velocity and the inlet quality ranges were 50– 200 kg/m2 s and 0.15–0.75, respectively, and the heat flux was fixed to 15 kW/m2. The importance of Eq. (114) lies on the fact that the two-phase effect in the heat transfer correlation is directly related to the two-phase effect in the pressure drop ( L) for the laminar (liquid)–turbulent (gas) regime. For the higher values of ReLF (>200) which correspond to the higher mass velocity cases, the Kandlikar’s flow boiling correlation [78] covered the experimental data within 10.7% mean deviation. b. Condensing Flows There have been a number of experimental studies on the condensation of refrigerants in micro-fin tubes with conventional size (say, larger than 5 mm in hydraulic diameter), and the state-of-the-art is well summarized in the review paper by Newell and Shah [80]. On the other hand, there are relatively fewer works reported on the microchannel condensation. The followings are a few of the works available at the present stage. Similar to the boiling flow case (Yan and Lin [33]), Yan and Lin [34] have performed a condensation heat transfer experiment for a R-134a flow through a horizontal tube of 2.0 mm I.D. The condensation heat transfer coefficient appears to be higher than about 10% compared to the large-pipe case (D ¼ 8 mm). Also, the heat transfer coefficient is higher at the low heat
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flux and saturation temperature and at the high mass flux. Based on the measured data, a heat transfer correlation was proposed as: hTP D 0:33 0:3 1:04 PrL Bo Re ¼ 6:48Reeq kL
ð115Þ
Here, Bo and Reeq are already shown in Eqs. (98) and (89), respectively. Operating conditions for Eq. (115) are the same with those for Eq. (93). Through a series of experiments on a R-134a flow in multi-port flat extruded aluminum tubes with their hydraulic diameters ranging from 0.44 to 1.56 mm, Webb and Ermis [30] reported that the condensation heat transfer coefficient increases with decreasing hydraulic diameter for all the tested tubes. They also constructed a correlation for the condensation heat transfer coefficient as: h ¼ bGp xq
ð116Þ
where b, p and q are determined from the measured data. The mass velocity and the quality ranges were 300–1000 kg/m2 s and 16–87%, respectively. The heat flux was 8 kW/m2 and the saturation temperature was 65 C. According to them, as discussed in the work of Yang and Webb [71,81] for the condensation of a R-12 flow, the correlation by Akers et al. [72] hD 0:8 1=3 ¼ 0:0265Reeq PrL kL
ð117Þ
is valid only for the ‘‘shear controlled’’ flow and does not account for the ‘‘surface tension drainage’’ effect. If one operates in a regime with the surface tension drainage, providing an additional enhancement, the surface tension component model of Yang and Webb [82] should be used to account for such an additional effect. The predictive model of Yang and Webb [82] has been validated with the same authors’ data for R-12 and R-134a flows; the model predicts 95% of the condensation data within the range of 16%. D. CONCLUDING REMARKS In the present section, pressure drop and heat transfer characteristics for single- and two-phase flows in microchannels have been reviewed. For single-phase flows through the microchannels considered here (Dh ¼ 0.1– 5 mm, mostly), both the frictional pressure drop and heat transfer correlations developed for large channels are considered valid without serious discrepancies. However, for two-phase microchannel flows, the
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conventional correlations (for large tubes) are unacceptable. In other words, the flow pattern strongly depends on the channel size and configuration as well as on other parameters, and accordingly, the momentum and heat transfer mechanisms change. For the two-phase frictional pressure drop, correlations on two-phase friction factor or the two-phase frictional multiplier have been introduced taking account of the effects of the tube size and configuration, and recommendations were made based on the database size and the ranges of their applicability. Similarly, for the boiling or condensation heat transfer in microchannels, the heat transfer coefficient correlations pertinent to each boiling regime or heat transfer mechanism were introduced. In principle, pressure drop of phase changing flows should be predicted from the adiabatic two-phase frictional correlations if the accelerational and gravitational pressure drops are estimated properly. However, accurate estimation of the accelerational and gravitational pressure drops is available only with the appropriate information on void fraction that depends strongly on the flow pattern. The heat transfer mechanism also depends on the flow pattern that is again related to the void fraction. Since no general void correlation is available in the present stage, and it is really true at the high heat flux condition where the non-equilibrium phenomenon becomes important, the parametric ranges of the pressure drop and heat transfer correlations should be checked carefully before using them.
III. Two-phase Flow Mal-distribution in Microchannel Headers and Heat Exchangers A. INTRODUCTION When multiple-channel heat exchangers are used as evaporators and condensers, there is a possibility of severe mal-distribution of the two-phase flow among the passages of the heat exchanger. Even in the simple case of adiabatic two-phase flow, one typically knows only the bulk vapor quality in the inlet header—not that at the inlet of each channel. Hence, a detailed investigation and knowledge of two-phase flow distribution in heat exchangers is important in view of the following facts: 1. Depending on the extent of mal-distribution, some areas of the heat exchanger may be starved of liquid, causing a spatial variation in core surface temperature (and hence, air temperature). This concern is important, especially in applications connected with human comfort, e.g., in automobile air-conditioning, in which variations in the temperature of air blown into the occupied space through the
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evaporator can cause the occupants to experience discomfort due to insufficient cooling of air ducted to either the driver or the passenger side. 2. In an air-conditioning system, two-phase refrigerant enters the evaporator header, which then routes the refrigerant into successive passes. If the refrigerant distribution among the tubes of any pass is not even, the vapor quality at the exit of different tubes will not be the same, even if the heat load on the evaporator is uniform. This leads to inefficient utilization of the surface area of the heat exchanger for heat transfer. Moreover, there is a possibility that the compressor may be flooded by liquid refrigerant, and this occurrence is highly undesirable from the standpoint of compressor reliability. 3. Without detailed knowledge of the flow distribution, comparisons of heat transfer and pressure drop data (which typically rely on measurements in the headers) to extent correlations are problematic. The difficulty is that when applying the correlations, it is usually assumed that the quality at the inlet of each channel equals the quality in the inlet header. Obviously, such an assumption is dubious. In order to gain insight into the problem of two-phase flow maldistribution in evaporators and condensers, and to study the influence of the several geometric factors and operating conditions on the distribution of the two phases through the channels, we present a critical review of several relevant articles and patents. After reviewing the articles, a summary of important findings and shortcomings in the state-of-the-art will be presented, followed by recommendations for further research.
B. REVIEW
OF
RELEVANT LITERATURE
Cabuk and Modi [83] developed a numerical algorithm to determine the optimum shape of the inner wall of an inlet header that turns the flow by 90 before the fluid flows through the heat exchanger. The ‘‘optimum’’ wall shape is defined as one that minimizes the non-uniformity of flow at the header exit. The flow was assumed to be two-dimensional, inviscid, and uniform across the entrance of the inlet header. This assumption is tantamount to assuming potential flow throughout the inlet header. In the algorithm they developed, the Laplace equation 2 ¼ 0 was solved iteratively until the tangential velocities at the header inlet and exit were less than 1% of the specified normal velocity components. The header wall shapes obtained upon convergence were considered optimal. The inner wall showed a bulge at the corner for aspect ratios up to about 2.0 (see Fig. 4) thus allowing for the possibility of separation just downstream of the bulge. The
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FIG. 4. Optimal header wall shapes for different aspect ratios (Cabuk and Modi [83]).
authors confirmed the presence of separation by applying a laminar boundary layer analysis to the flow and calculating the critical shape factors below which separation would occur. For aspect ratios around 6.0, no such bulge was predicted for the optimal wall shape, and no separation would therefore occur at higher aspect ratios. Checking for separation is important because, as the authors point out, the validity of their potential flow solution as an approximation to high Reynolds number flows is guaranteed only in the absence of separation. Again, the secondary flows induced by threedimensional effects may cause severe non-uniformities in the flow at the header exit. Cabuk and Modi [83] believe that optimal shapes for aspect ratios greater than 2.0 will provide a reasonably uniform velocity profile at the header exit for moderate to high Reynolds number flows without any flow separation. In an effort to understand the mechanism behind the non-uniform distribution of two-phase flow in an automobile evaporator, Asoh et al. [84] experimentally studied the two-phase flow separation characteristics under test conditions simulating those in an actual evaporator. Two test setups were employed: one was used to observe the flow patterns in the main and branch tubes, and the actual liquid and vapor flow rates were actually measured in the second setup. An inlet pipe with 13.6 mm inside diameter (13.9 mm in the second apparatus) fed two-phase refrigerant successively to three branch pipes, each with an inside diameter of 4.5 mm (7.9 mm in the second apparatus). R-113 was the test fluid in both types of tests. The experimental parameters varied were the vapor quality and the mass flow rate at the inlet of the main pipe and the heat load on the branch tubes. In the flow observation study, a stroboscope was used to take photographs of the flow patterns at the rate of eight frames per second. For all test conditions, the flow pattern in the main pipe was either slug flow or froth flow. However, the liquid slug length and the flow rates of both phases were not the same in the three branches. On the average, the liquid slug was the longest in the first (upstream) branch, becoming progressively shorter in the
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second and third (downstream) branch. The flow rate of the vapor and liquid phases flowing into the branch tubes was controlled more by the liquid flow rate in the main pipe than by the vapor flow rate. One of the interesting observations made was that the distribution ratio (defined as the ratio of the mass flow rate of any phase in a branch to the mass flow rate of the same phase in the main branch) of both phases was a maximum at a branch heat load of approximately 130 W and the branch pressure drop simultaneously reached a minimum at this condition. The researchers also attempted to model the pressure drop in the branch pipes assuming both slip flow and homogeneous flow. The slip flow model predicted the measured pressure drop much more accurately than the homogeneous model, thereby emphasizing the importance of accurately accounting for the spatial distribution of vapor and liquid phases in the branch tube. The principles of header design for single-phase flow have been reasonably well known. However, when used with liquid–vapor mixtures, single-phase headers tend to perform very poorly. Samson et al. [85] built and tested models of various header concepts in an effort to develop a header that equally distributes the liquid flow in an air–water flow. The three header concepts they designed and tested were an in-line header, a ‘‘spreader’’ header, and a symmetrical header, shown schematically in Fig. 5. The liquid distribution of the inline header proved to be very unequal, being very sensitive to both total flow rate and quality, and somewhat less sensitive to
FIG. 5. Several different types of header design (Samson et al. [85]).
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M.-H. KIM ET AL.
FIG. 5. (Continued )
variations in the exit pressure. The authors thus rejected the inline header as a final design, in spite of its being a structurally convenient design. In the spreader header, the flow is spread over a relatively large area prior to reaching the heat exchanger tubes. This reduces the flow velocity at the inlet to each tube and the authors expected the reduced velocity to result in a more even distribution of liquid than in the in-line header. The liquid distribution of the spreader header was acceptable at lower qualities; however, at qualities around 50%, this header design channeled almost all of the liquid phase to just one exit tube. For this reason, the spreader header was not chosen for further development. The symmetrical header showed a much better liquid distribution than the first two designs. However, the drawback of this design is that it would be difficult to build compactly because the exit tubes radiate in all directions. Hence, the symmetrical header concept was modified to create a more manageable design, the ‘‘halfsymmetrical’’ header shown in Fig. 5. The authors demonstrated that this
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new design could distribute the liquid in the two-phase flow almost equally at qualities around 25% and less acceptably at qualities greater than 50%. The laminated plate-fin type evaporator is the design used most in modern automobile air-conditioning systems. The laminated evaporator has been designed in two varieties: one version has tanks on both ends of the evaporator to serve as a channel for the transport of refrigerant between different heat exchanger passes; the other type has a central dividing barrier to form a U-shaped channel with a single tank on only one end of the plate. Rong et al. [86] carried out two kinds of adiabatic air–water flow experiments in such a ‘‘U-channel’’ evaporator. In the first set of experiments, the authors observed the liquid distribution in a channel formed by two U-channel plates for a variety of inlet mass flow rates and qualities, with inlet at the top and bottom. The entire channel wall topography was duplicated in a full-size lucite cast to enable flow visualization. In the second set of experiments, a number of U-channel tubes (each tube formed by two plates) were connected in parallel to a common inlet tube, while the outlet of each channel was separately connected to an air–water separator. The evaporator plates investigated had two different types of internal surface protrusion geometries, round dimples and crossribbed bumps, as shown in Fig. 6. In the cross-rib design, rows of the protrusions in the form of ribs are formed at an angle of 22 to the flow direction. When two such plates are stacked face to face, the ribs overlap in a cross pattern with brazed contact points at the middle and both ends of each rib. In the single channel flow visualization tests, neither plate design showed satisfactory liquid distribution (and hence, surface wetting). Surface wetting was particularly poor in the down flow leg and the U-bend area, where many large dry spots were observed for several flow conditions. The round dimple and cross rib designs showed no substantial difference in surface wetting. When the two-phase flow entered at the top of the channel, the U-channel acted as a liquid collector and better surface wetting could be obtained.
FIG. 6. Evaporator plate designs (Rong et al. [86]).
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M.-H. KIM ET AL.
In the multiple tube U-channel assembly, liquid distribution was highly non-uniform under most inlet flow conditions. Top feed invariably resulted in a high liquid flow rate ratio in the first and/or second channels and the downstream channels were almost dry. In contrast with the top feed, the bottom feed resulted in a hydrostatic head in the up-flow leg of the channels, and the water flow lacked enough momentum to pass through such a leg. The static head tended to make the flow rate distribution in bottom feed better, though not perfect, under many conditions, compared to the top feed case. The only exception was at some high inlet quality conditions, under which the first or last channel was dry. The authors rightly infer that the poor wetting of the plate surface in single channels and poor flow distribution in a number of parallel channels would severely limit the heat transfer capability of their U-channel designs. However, it should be noted that their flow visualization studies, carried out with air–water mixtures under adiabatic conditions in lucite may not be fully representative of conditions in an actual aluminum evaporator employing boiling R-134a with different surface characteristics and fluid properties such as surface tension, that affect surface wetting and heat transfer. Similar to their study of two-phase flow distribution in U-channel evaporator plates, Rong et al. [87] also investigated the distribution of single- and two-phase (air–water) flow in actual compact evaporator channels that had round dimples, and through which the flow passed in only one direction. The flow inlet tube was placed horizontally, and the flow in the vertical channels was either upward or downward. The air flow distribution in 3–6 parallel channels was quite uniform over a wide range of test conditions. However, water flow exhibited significant non-uniformity in downward flow due to gravitational effects, with more than 95% of the total water flow passing through the first channel at the lowest flow rates. As the total water flow rate was increased, the flow ratio through the first channel was reduced to about 50% because of the increased momentum carrying water to the downstream channels. In upward flow, the water distribution was comparatively more uniform at low flow rates. As the flow rate increased, more water flowed through the last two channels due to increased kinetic energy of the water, and the distribution became non-uniform. The two-phase flow distribution was also quite uneven over the range of conditions tested. For downward flow, the first channel always received most of the water flow, while the air tended to pass through the last several channels for 3–7 parallel channels. In upward, high quality flows, the water and air distributions were similar to those in downward flow. However, for low air flow rates, a more uniform water distribution was obtained, although the air distribution was not uniform, attaining a maximum at about the middle channel.
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Under most of the test conditions, the flow in the header tube was annular, the liquid film being much thicker at the bottom than at the top. For the rest of the conditions, the flow pattern was slug flow. In order to explain the differences in two-phase flow distribution obtained under various flow conditions, the authors correlated the flow patterns in the header tube with the distribution of air and water in the parallel channels. The authors also attempted to improve the flow distribution through the parallel channels by adding an appropriately oriented partial flow blockage at the joint of the header tube with the channel plate. Using this technique, they were able to achieve substantial improvement in the water distribution for vertically downward flow. Recently, there have been high demands for leveling the distribution of the temperature of air blown out from the evaporator of an automotive airconditioner. To suppress the biased distribution of the temperature of air leaving the evaporator due to mal-distribution of two-phase refrigerant among the evaporator plates, Torigoe and Shimoya [88] conceived of a novel evaporator design, as described in U.S. patent 5,701,760. As shown in Fig. 7(a), the evaporator plate is divided longitudinally into two elongated recesses by a partition rib. Thus, two cores are effectively created: a downstream core and an upstream core, as far as the flow of air is concerned. Figure 7(b) shows the refrigerant flow through two passes of the upstream core and two passes of the downstream core. As is clear from Fig. 7(b), the refrigerant flows in the same direction (upward or downward) in passes of the upstream and downstream cores that overlap each other in the air flow direction. The advantage offered by the flow arrangement shown in Fig. 7(b) can be understood from Fig. 7(c and d). In Fig. 7(c), the liquid and vapor flow distribution in the right hand passes of the upstream and downstream cores of Fig. 7(b) is shown. In the upstream core, because of the influence of gravity, the last few tubes get the most liquid, while the tubes to the left get relatively lower liquid flow. Therefore, a ‘‘hot spot’’ or a local superheated region develops in the left part of the upstream core. From similar considerations, a hot spot develops in the right part of the downstream core. Since the hot and cool regions of the upstream and downstream cores thus overlap each other, the air undergoing cooling in the right pass comes out at a relatively uniform temperature. The same phenomenon occurs in the two left-hand passes, and is shown in Fig. 7(d). It should be noted that since the refrigerant flows in series through the passes of the upstream and downstream cores, the arrangement described in this patent is not very flexible and the heat transfer capability of this design will suffer at higher heat loads due to high refrigerant pressure drop. Watanabe et al. [89] attempted to elucidate the effect of the total mass flow rate, the vapor quality at the header inlet, the heat load on each pass,
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FIG. 7. Evaporator plate configuration and flow arrangement (Torigoe and Shimoya [88]).
and the number of passes on the distribution of two-phase flow in a multipass serpentine automobile evaporator. Their experimental system was designed to simulate a serpentine automobile evaporator and consisted of a horizontal pipe with four vertical upward branch pipes. Each pipe simulated a serpentine pass. To ensure geometric similarity between the model and the actual evaporator, the ratio of the inner diameter of the branch pipes to the main pipe was set at 0.3. The tests were conducted with R-11 at three different flow rates and with inlet vapor qualities ranging from 0 to 40%. The results of the tests can be summarized as follows:Increasing both the mixture flow rate and the quality at the header inlet increased the mass flow ratio in the downstream tube. Under adiabatic conditions, the flow distribution was greatly dominated by the inlet quality in the header. When the
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tubes were heated, the pressure drop along each tube, and not the inlet quality, influenced the flow distribution. In fact the flow distribution in all tubes became very uniform under heat load. The authors measured the difference between the pressure at the bottom of the main tube and that at the outlet of each tube, and found that it could be predicted using previously suggested correlations for the pressure drop along the tube and at each T-junction. They therefore recommend that single T-junction data may be fruitfully employed to predict the flow distribution and pressure drop in multi-pass tubes. The authors correlated their data for the vapor mass flux at the inlet of a branch and in the main tube by the following equation: GG;Pin;j ¼ 6:07GG;M;j
ð118Þ
Similarly, they also provided the following correlation for the fraction of liquid taken off at a junction as a function of the Reynolds number of the vapor in the main tube immediately before the junction: m_ L;Pin;j =mL;M;j ¼ 2:74 105 ReG;M;j 0:0124
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ReG;M;j þ 1:37
ð119Þ
Reimann and Seeger [90] measured the pressure differences between the pipe inlet and the run, and between the pipe inlet and the branch for air– water and steam–water flow through a T-junction. The pipes of the Tjunction had equal inside diameters of 50 mm; the inlet and run pipes were horizontal, while the flow in the branch could be horizontal, vertically upward, or vertically downward. The authors also proposed a model to predict the measured pressure drop in the run and the branch. The model was based on writing the pressure difference between the inlet and the run (or between the inlet and the branch) as the sum of 1. the reversible pressure increase between the inlet and the throat of the vena contracta in the branch pipe, and 2. the pressure difference between the vena contracta and a position further downstream in the branch (modeled according to a sudden expansion). Their model predicted the measured pressure drop data better than some previous models for all branch orientations and both fluid mixtures, in general. The results for a vertically upward branch, however, were not satisfactory. The authors suggest that further detailed measurements of the global or local void fraction are necessary for successfully modeling the twophase pressure drop across a T-junction with a vertically upward branch.
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In the above study by Reimann and Seeger [90], flow separation and pressure drop were measured for only one T-junction. Even though such studies form a good starting point, the knowledge thus obtained may not be directly applicable to the flow in multiple T-junctions, which are found in practice in modern heat exchangers. This fact motivated Kariyasaki et al. [91] to experimentally investigate the flow separation characteristics of an air–water slug flow into three successive horizontal capillary tubes. The twophase mixture passed through an inlet tube with 13.9 mm inside diameter into the branch section and the outlet tube. The branch section consisted of three successive horizontal tubes, each with 4.3 mm inside diameter. All tubes were made of transparent acrylic resin to enable observation of the flow. The flow rates and qualities were selected so that the flow pattern in the inlet tube was slug flow to correspond to actual flow conditions in a serpentine heat exchanger. The results of their study may be summarized as follows: 1. The flow pattern in the branches was either slug or annular flow. When the efflux from the outlet tube was small, the flow patterns in the branches were sometimes different from one another. When gas bubbles in the main tube were sufficiently long, the flow patterns in the branches alternated between slug and annular flow. 2. With an increase in the two-phase efflux from the outlet tube, the distribution of both water and air became more uniform in the three tubes. 3. When the two-phase efflux from the outlet tube was small, the water distribution increased while the air distribution decreased in the direction of flow in the inlet tube. 4. The researchers proposed a model to predict the pressure drop in the horizontal branches. One implication of their model was that the frictional pressure drop in the water slugs as they flowed through the branches was important in determining the distribution of the twophase mixture among the branches. The frictional pressure drop in the water slugs was a function of both the total length of water slugs in the branch and the mean velocity of water flowing in the branch, which in turn were found to depend on the residence time of the water slug at the entrance of the branch.
C. CONCLUDING REMARKS As is amply clear from a study of the literature on two-phase maldistribution, there is insufficient research on refrigerant flow distribution from the main channel (i.e., the tank, header or distributor) of an
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evaporator to each tube. In a direct expansion refrigerator, it is certainly desirable that the refrigerant be distributed according to the heat load on each branch or pass in order to achieve a uniform vapor quality at the outlet manifold and to realize efficient utilization of the heat transfer surface area. However, no method exists to serve as a design tool to predict the two-phase flow distribution for given flow conditions and configurations of the header and the tubes. This limited knowledge is partially due to the complexities of two-phase flow behavior and the many interdependent variables that influence such a flow. Among the geometric factors that affect two-phase flow distribution are: the cross-sectional area of the branch as a fraction of the tank/header/distributor cross-sectional area, the tube spacing, and orientation of the branches. The operating factors that influence the flow distribution are: the total mass flow rate, the vapor quality in the tank, and the heating load on each branch. Another little researched factor that affects the temperature distribution of air leaving the evaporator is the distribution of condensate on the air-side of an evaporator. One more fact that emerges from a study of the literature on two-phase flow distribution is that very little information is available in the form of reliable models and/or correlations to predict the distribution of the liquid and vapor phases in each pass or tube for a given evaporator geometry and operating conditions. Finally, it should be noted that most of the studies that provide limited correlations for the distribution of two-phase flow do so either for a single T-junction, or for a cylindrical inlet tube with cylindrical branch pipes. Further research and good quality data on the distribution of two-phase flow for the complex header geometries found in evaporators is necessary.
IV. Air-side Performance The folded multilouver fin surfaces are commonly used on the air-side of microchannel heat exchangers associated with extruded flat microchannel tubes to enhance the thermal hydraulic performance. The ratio of air- to refrigerant-side surface area for this geometry is commonly quite small compared to the finned tube heat exchangers with round tubes [92]. If the mass velocity is the same, the smaller Dh the better the heat transfer coefficient. Thus, the air- and refrigerant-side heat transfer coefficients for the microchannel heat exchangers are usually similar to or higher than those for the conventional finned round tube heat exchangers. This suggests that the air-side thermal resistance of the microchannel heat exchanger has a more significant effect on the heat exchanger performance compared to the conventional fin-tube heat exchanger. The air-side thermal hydraulic
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performance of microchannel heat exchangers depends on Reynolds number and louver geometry such as fin and louver pitches, louver angle, and flow depth. Webb [93] presented a careful review for some progress in air-side heat transfer augmentation technology in heat exchanger applications. Jacobi and Shah [94] reported a comprehensive and systematic evaluation of the air-side heat transfer enhancement mechanisms in heat exchangers. In this section, the flow structure in the louvered array is presented to provide the better understanding of the flow and its influence on thermal hydraulic performance of folded louvered fins. The air-side heat transfer and pressure drop data and correlations under dry, wet, and frosting conditions are described. The effect of inclination of heat exchangers on the air-side performance under dry and wet conditions is also addressed. Numerical analysis of multilouvered fin surfaces has been reviewed recently by Heikal et al. [95] and Shah et al. [96] and will not be addressed in detail. A. FLOW STRUCTURE
IN THE
LOUVER FIN ARRAY
Before considering the heat transfer and pressure drop characteristics in louvered fin surfaces, the flow structure is presented since the air-side thermal hydraulic performance is strongly dependent on the flow behavior. Thus to improve the air-side performance for the louver fin heat exchangers, it is important to understand the enhanced mechanism of the louvered fin geometry and the effect of the geometrical parameters on the fluid flow. Several important flow factors play important roles on the air-side performance of louvered fin heat exchangers: boundary layer interruption, vortex shedding and the flow efficiency, associated with the geometrical configurations. The louvered fin surface consists of several louver strips that include entrance and exit deflection louvers, intermediate louvers, and redirection louvers as shown in Figs. 8 and 9. A new boundary layer begins to be developed at the leading edge of each louver strip and is then abruptly broken up at the trailing edge resulting in the high heat transfer coefficient in the initial re-growth region. In addition to promoting the boundary layer restarting, above some critical Reynolds number, the louvers can cause vortex shedding. This vortex shedding can affect the boundary layers on the adjacent louvers and it may be used as a mechanism to enhance the local heat transfer. Another important aspect of louvered fin performance is the flow efficiency as shown in Fig. 10. The flow efficiency describes flow pattern as the degree which the fluid follows the louvers, and depends on louver geometry and the Reynolds number. Flow efficiency increases with louver angle and decreases with fin-to-louver pitch ratio. The flow follows the path of least pressure drop. The flow efficiency is one when the flow is parallel to and through the louvers, whereas it is equal to zero when the flow is 100%
MICROCHANNEL HEAT EXCHANGER
FIG. 8. Folded louver-fin heat exchangers (Chang and Wang [147]).
FIG. 9. Schematic of cross section of multilouved fin geometry (Webb [93]).
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FIG. 10. Definition of flow efficiency.
duct directed flow. Apparently, it is important that the flow efficiency need to be increased to get high air-side performance. A considerable number of experimental and numerical studies on the flow configurations of louvered fin geometries have been reported in the open literature. Beauvais [97] appears to be the first to conduct flow visualization tests on the louvered fin array. He used a smoke flow visualization technique for 10:1 scale models and presented a smoke flow pattern through louvered fins. The flow pattern clearly shows that the main flow is nearly parallel to the louvers (louver directed flow). Prior to his work, it was speculated that the fluid flowed mostly between the fins forming the duct (duct directed flow) and the louvers simply acted as a surface roughness which enhanced the performance characteristics of the fins by promoting turbulence. Smith [98,99] developed the semi-empirical model of the pressure drop and heat transfer for the louver fin core. Wong and Smith [100] measured the pressure drop and heat transfer data using a 5 : 1 scale model of a typical louvered fin array and verified that large scale models could be used effectively to evaluate the performance of full scale louvered fin heat exchanger. They also measured local air flow and temperature distributions with hot wire and thermocouple probes. A detailed description of local flow pattern in the louver fin array was similar to that as observed by Beauvais [97]. Davenport [101,102] conducted flow visualization tests similar to those of Beauvais [97] and demonstrated that the flow structure within the louver array was a function of Reynolds number. At low Reynolds number, the developing boundary layers on adjacent louvers became thick enough to effectively block off the air flow passage between the louvers, resulting in nearly axial flow through the fin array. As Reynolds number increased, however, the flow became gradually parallel to the louvers. It was shown
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that the discrepancy between the mean flow and louver angles was 4–7 and it depended on louver geometry and the Reynolds number. He found that near the triangle base, the flow was unaffected by the louvers and flowed straight down the duct. Davenport also measured the flow velocity and turbulence intensity in the louver array using hot wire anemometry. It was observed that the flow at the triangular base of the louver fin was nearly twice that over the louvers, with a higher degree of turbulence outside the wakes of the louvers. The degree of louver bypass flow was proportional to the louver length-to-fin height ratio. Axial velocity measurement revealed developing flow for the first few louvers of each bank, with a larger developing region downstream of the redirection louver. Tura [103] performed flow visualization tests using 10 : 1 scale rectangular and 4 : 1 triangular louvered fin models. He found the louver angle to have a substantial influence on performance, but the length of redirection louver to be insignificant. Below a certain Reynolds number, a flattening of the j curve vs. ReLp was observed. He speculated this falloff of j factor was attributed to a flow transition from forced to natural convection at the lower Reynolds number. At high Reynolds numbers, he observed vortex shedding from the entrance louver and after the redirection louver. The vortices were interrupted by the downstream louvers. Note that the end walls probably affected his results substantially since he used scaled models with only five columns (counted in the spanwise direction) of fins as discussed later. Kajino and Hiramatsu [104] conducted flow visualization tests and numerical analysis of the louvered fin array. They used dye injection and hydrogen bubble techniques for 10 : 1 scale models. It was showed that boundary layers existed on both the upper and lower surfaces of the louver, and a laminar wake existed downstream from each louver. Flow separation was observed on the backside of inlet louvers. A significant fraction of the flow bypasses the louvers at large fin-to-louver pitch ratios because the hydraulic resistance of the duct directed flow region is substantially lower than that of the louver directed flow region. When the fin-to-louver pitch ratio is reduced, the hydraulic resistance of the duct directed flow region increases and results in a higher mean flow angle as more of the flow passes through the louvers. Achaichia and Cowell [105] measured air-side performance using 15 samples and found the same flow pattern proposed by Davenport [101]. Figure 11 shows heat transfer data for three louver fin geometries with plate fins and 11 mm tube pitch. Two types of flow structure exist in the louver fin array: louver directed flow and duct directed flow. At the high Reynolds number end, the Stanton number (St) data are almost parallel (but lower than) to the Pohlhausen equation for laminar boundary flow over a flat plate (louver directed flow). The curves tend to approach the equation for
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FIG. 11. Heat transfer characteristics for different louver fin geometries as function of Reynolds number (Achaichia and Cowell [105]).
fully developed laminar duct flow (duct directed flow) at the lowest Reynolds number. The ‘‘duct flow’’ line is for fully developed laminar flow in a rectangular channel with the aspect ratio of 0.30. The aspect ratios for the three samples are 0.23–0.37. This flow behavior is consistent with Davenport’s flow observations. The flattening of the Stanton number curve at low Reynolds number is observed. Davenport [101] and Achaichia and Cowell [105] conjectured this phenomenon was attributed with the transition of flow pattern from louver directed flow to duct directed flow. However, Tura [103] conjectured that this performance degradation was attributed to a transition from forced to free convection at the low Reynolds numbers. Achaichia and Cowell [106] also conducted numerical analysis of fully developed periodic laminar flow in the louver fin array. Figure 12 shows that how the mean flow angle (Fa) depends on the Reynolds number and the louver geometric parameters. The mean louver angle at high Reynolds numbers approaches the louver angle (La) within a few degrees, and falls off as the Reynolds number decreases. Their analysis supported Davenport’s concept concerning the louver blockage due to the boundary layer development at low Reynolds number flow. They developed a simple polynomial equation to describe the mean flow angle (Fa) as a function of Reynolds number (ReLp), the fin-to-louver pitch ratio (Fp/Lp) and the louver angle (La, degree) as F ¼ 0:936
243 Fp 1:76 þ 0:995L ReLp Lp
ð120Þ
Achaichia and Cowell [105] incorporated the ratio of mean flow angle to louver angle to their heat transfer correlation. If eddies are shed from the louvers, because the flow is not parallel to the louvers, their analysis will
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FIG. 12. Mean flow angle as function of lover angle and Reynolds number. (Achaichia and Cowell [105])
probably not accurately predict the effects of these eddies on the thermal hydraulic performance. Howard [107] performed flow visualization tests on a 10 : 1 scale model of two-dimensional louvered fin array using a dye injection technique. He observed that flow instabilities (i.e., vortex shedding) initiated at a Reynolds number of approximately ReLp ¼ 900 and the instabilities progressed upstream (from the exit end of the array) as Reynolds number increased. He characterized the flow structure as being either ‘‘efficient’’ or ‘‘inefficient’’. This definition is based on the assumption that an efficient flow structure should yield a higher heat transfer coefficient than one having the inefficient flow pattern. Howard’s definitions are consistent with the definitions of duct directed flow and louver directed flow. He also observed a transient region between inefficient and efficient flow, just as a transition region exists from laminar to turbulent flow. It was shown that the transition occurred at Lp/ Fp ¼ 0.7–0.8 for a 20 louver angle. The flow structure was considered to be efficient if Lp/Fp>0.8 and inefficient if Lp/Fp<0.7. Antoniou et al. [108] measured local velocity and turbulence levels in a louvered fin array using a 16:1 scale model (La ¼ 20 and Fp/Lp ¼ 1.7). They found the mean flow angle gradually changed in the first few louvers to reach a constant fully developed value less than the louver angle. The flow alignment with the louver was lower after the redirection louver. Above a certain critical Reynolds number, they noticed vortices to shed off the trailing edge of the louvers resulted in increasing the local turbulence levels. The effect of these vortices propagated in a band in the mean flow direction.
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The turbulence level and the band width increased as Reynolds number increased. This flow configuration was probably influenced by the end walls, however, since they used the scale model with only seven columns of fins. Webb [109] and Webb and Trauger [110] performed flow visualization tests for the geometrical parameters such as louver pitch, louver angle and fin pitch using large scale models in a water channel. They defined the term ‘‘flow efficiency’’ (Fe) that describes the degree of alignment for a particular geometry and flow pattern as the ratio of actual transverse distance to ideal transverse distance (Fig. 10). Fe ¼
N actual traverse distance ¼ D ideal traverse distance
ð121Þ
Achaichia and Cowell [105,106] used the ratio of mean flow angle to louver angle (Fa/La) rather than Webb’s flow efficiency definition. These terms are related by the simple expression as Fe ¼
N tanðF Þ F ¼ ffi D tanðL Þ L
ð122Þ
This approximation (Fe ¼ Fa/La) is valid within 3% for 0.2
Re Lp Fe ¼ Fe
1:35 0:61 Lp L Fp 90
for ReLp > Re Lp
ð123Þ
ð124Þ
where Fe ¼ 0:95
Lp Fp
0:23
;
Re Lp ¼ 828
0:34 L 90
ð125Þ
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Cowell et al. [112] showed that heat the transfer for multilouvered fin geometry improves as the flow pattern becomes louver directed flow and the rate of falloff of the mean flow angle is a function of Fp/Lp. They presented the following equation for maximum flow angle when the Reynolds number is infinity: ðF Þmax ¼ 0:936 1:76
Fp þ 0:995L Lp
ð126Þ
They conjectured that the falloff in Stanton number would begin to be detectable when the critical flow angle ratio, Fa/(Fa)max ¼ 0.95. They proposed the following critical Reynolds number at which this falloff begins: Re Lp ¼ 4860=ð0:936 1:76ðFp =Lp Þ þ 0:995L Þ
ð127Þ
Note that this equation was based on Achaichia and Cowell’s model [106] which did not agree so well with Webb and Trauger’s test data [110]. Bellows [113] performed ink-in-water flow visualization tests over a Reynolds number (ReLp) range of 50–600 using six stereolithography 10.5 : 1 scale models. He suggested that the guidelines for scale model size and number of fins to keep duct walls from affecting flow in the center of the louver array: Nc >
3dt Fp
ð128Þ
where dt is the transverse distance that flow travels through the model. He compared flow efficiency for a twelve- to a six-column model and found them to differ by 100%. Note that most of past experimental studies on the louvered fin geometry have been performed using test sections which incorporated only three to ten columns of fin in the transverse direction [108,114–118]. When too few columns of fins are used, instead of louver directed flow, the flow separates and passes over the louvers causing a large increase in pressure drop that is not representative of a full-scale heat exchanger [119]. His flow efficiency measurement results showed that an absolute percent deviation of 57% compared to Achaichia and Cowell’s numerical simulation [106]. He speculated that this large discrepancy was attributed to the fact that the numerical simulation model neglected the developing flow. He developed a correlation for flow efficiency as a function of ReLp, Fp/Lp, and La.
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300 Fp þ 1:34L 10 5 ReLp Lp Fe ¼ L
ð129Þ
The above equation is similar to that defined by Achaichia and Cowell [106]. However, this correlation was developed using only a limited data set, and the flow velocity used in the Reynolds number was not defined. Bellow observed that the flow behavior in the louvered fin array was generally laminar in the Reynolds number rage considered (ReLp ¼ 50–600) with vortex shedding occurring within the louver array at ReLp>400. Namai et al. [120] performed flow visualization using three dimensional 5 : 1 scale models with louvered and non-louvered fins. They used dye injection and hydrogen bubble techniques and investigated the effects of louvers, fin geometry such as louver angle (La ¼ 10, 20, 30 ), louver pitch (Lp ¼ 3.25, 5.0, 6.5 mm), louver length, and fin fold configuration. It was shown that the louvers interrupted the boundary layer development and enhanced heat transfer, while in the case of a non-louvered fin the boundary layer developed from the leading edge and no interruption was observed at ReDh ¼ 300. They reported that the mean flow angle increased with increasing louver angle and decreasing louver pitch. At the high Reynolds number (ReDh ¼ 1500), a small oscillation occurred in the wakes of the upstream region and this oscillation propagated the downstream louvers and was amplified to a mixed dynamic flow in the downstream region, suggesting that these strong mixing effects enhanced heat transfer. In the case of triangular fin geometry, the strong three-dimensional characteristics existed in the louver fin array due to the inclined fins and louvers and the secondary flow near the apex by the boundary layer growth. They also reported that non-louvered region of fins deteriorated the heat transfer performance due to the growth of the boundary layer. Springer and Thole [121] conducted flow visualization tests using a 20 : 1 scale model of louver geometry with La ¼ 27 , Fp/Lp ¼ 0.76, Lp ¼ 27.9 mm, Fth ¼ 2.28 mm and Nl ¼ 17. They conducted the CFD simulations to determine the number of columns necessary to obtain periodic flow conditions. The CFD results showed that the end walls have a significant effect on the flowfield for the five column model, making it impossible to achieve a periodic flow in the section, whereas the flowfield for the 19 column model indicates louver directed flow like periodic flow conditions. Thus, they used 19 column model for the tests. However, they did not provide guidelines to estimate the number of columns of fins necessary for a given geometry to obtain periodic flow conditions. The measured flow features confirmed that the flow was louver directed for this particular geometry at ReLp1 ¼ 230, 450 and 1016. They reported that at ReLp1 ¼ 230 the
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boundary layer on the upstream side of the louver was thicker than on the downstream side of the louver, indicating that the heat transfer coefficients of the downstream side were higher than the upstream side of the louver. However, this difference of the boundary layer thickness between the upstream and downstream sides of the louver decreased as Reynolds number increased. It was shown the wake region downstream of the louver was observed at all three Reynolds numbers. At the high Reynolds number (ReLp1 ¼ 1016) the wake region of the upstream louver propagated to the downstream louver. On the other hand, at the low Reynolds number (ReLp1 ¼ 230) the downstream louver was not affected by the wake region occurred by the upstream louver. They measured velocity fluctuations for a range of Reynolds numbers (ReLp1 ¼ 1000–1900) behind the fifth streamwise louver to quantify the flow unsteadiness in the wake region of a fully developed louver. The time-resolved velocity measurements indicated that for ReLp1 1000 there was a distinct frequency that occurred where the Strouhal number was constant at St ¼ 0.17 in the range of ReLp1 ¼ 1000– 1900. Springer and Thole [122] extended their previous research [121] and measured detailed flowfield in the entry region of six different louver geometries (20 : 1 scale models) with La ¼ 27 and 39 , Fp/Lp ¼ 0.76–1.52, Lp ¼ 27.9 mm and Nl ¼ 17. They reported the similar findings as previous studies. They presented that the effect of the mean flow angle is more dependent on Fp/Lp than La. This trend is in disagreement with Bellows [113] who reported that louver angle (La) plays a larger role than Fp/Lp in flow efficiency. This discrepancy may be attributed to the fact that their results are based on a limited data set. From the time resolved velocity measurements for La ¼ 27 , they found an earlier onset of unsteadiness in the louver array with increasing Fp/Lp, suggesting the fast propagation of instabilities for the larger Fp/Lp can be occurred. For Fp/Lp ¼ 0.76, a distinct frequency component (St ¼ 0.17) was seen at ReLp1 ¼ 1000, whereas for Fp/ Lp ¼ 0.91 and 1.52 the frequency components of St ¼ 0.19 and 0.22 began to be apparent at ReLp1 ¼ 800 and 400, respectively. However, they did not relate their flowfield measurements to heat transfer and pressure drop data. Beamer et al. [123] reported a study on flow visualization tests and twodimensional CFD analysis to investigate flow phenomena and heat transfer enhancement mechanisms in multilouved fins. They used two 10.5 : 1 scale models: the first one was a conventional louver array with 12 fins, and the second one was identical to the first one except that walls were added to divide the model into sections of 3, 6, and 3 fins to investigate the wall effects on the flow phenomena. The models were tested in two water tunnels using both hydrogen bubble and dye injection techniques to visualize streamlines. They reported that the flow efficiency decreased near the wall due to the wall
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FIG. 13. A comparison of flow efficiency of multilouvered fins (Beamer et al. [123]).
effect that the wall inhibited flow perpendicular to the longitudinal axis of the fin. Figure 13 shows their flow efficiency results compared with those by Achaichia and Cowell [106] and Sahnoun and Webb [111]. Achaichia and Cowell’s correlation [106] overestimates the flow efficiency because the mean flow angle equation is developed based on the fully developed periodic flow and the developing flow region at the beginning of each louver bank is neglected in their numerical study. The CFD flow efficiency results [123] and flow visualization measurements show good agreement. Both results display a well discernable knee below that the flow efficiency drops off rapidly. The data of Sahnoun and Webb [111] show a similar trend, except that the Reynolds number (ReLp) at which the knee occurs and flow efficiency above the knee are considerably high. This rapid drop off of the flow efficiency at low Reynolds numbers may be attributed to the flow transition from the louver directed flow to duct directed flow due to the increased boundary layer thickness along the louvers. This boundary growth also impedes heat transfer. However, the effect of flow efficiency on pressure drop is negligible. The overall pressure drop is a result of friction drag and form drag. For high Reynolds numbers, the flow is predominantly aligned with the louvers and pressure drop is primarily due to friction loss effects. As Reynolds number decreases flow pattern becomes duct directed flow and the portion of friction loss is decreased and form drag increases due to the roughening effect of the louvers on the duct flow. The increase in form drag offsets most of the decrease in friction loss thereby minimizing the effect of flow efficiency on overall pressure drop. The CFD study on one fin with a periodic boundary
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condition showed that flow approached and entered the fin array parallel to the longitudinal axis, flow then developed and aligned with the leading bank of louvers, reversed at the redirection louver, developed and aligned with the trailing banks of louvers, exited the fin parallel to the longitudinal axis of the fin. They reported that evidence of flow separation at the leading edge, and vorticity in the wake behind the trailing edge of the louvers was observed in the CFD vorticity plots, which were not shown in their paper. However, they did not provide details of louver geometry and their methodology. Dejong and Jacobi [119] presented flow visualization, pressure drop, and mass (heat) transfer data for five louvered fin geometries with La ¼ 18–28 , Fp/Lp ¼ 1.09–1.7, Lp ¼ 11.9 mm, and Fh ¼ 70.6 mm, focused on the understanding of physical processes rather than the development of correlations. They described the flow behavior in louvered fin array under spanwise periodic conditions (far from heat exchanger walls), with special emphasis on the role of vortex shedding as follows. At very low Reynolds numbers flow passes through the ducts between columns of fins (duct directed flow), and as Reynolds number increases the flow efficiency increases to some maximum value (louver directed flow). Flow efficiency decreases with Fp/Lp and increases with louver angle. The flow follows the path of least resistance. Two competing effects are at work. First, duct directed flow has a non-zero degree angle of attack to the louvers and thus has higher form drag than louver directed flow where the angle of attack approaches zero degree. Second, friction drag is lower for developing flow through a narrow duct, such as the small passageway between louvers. As the louver angle increases, the louver gap increases in size and the duct diameter decreases. As Fp/Lp decreases, the duct diameter decreases as well. Above a critical Reynolds number, the flow becomes unsteady. First, fin wakes become unsteady, and as the Reynolds number is increased further, small-scale spanwise vortices begin to be shed from the leading edges of fins. The vortices are smaller and result in less mixing than in the similar offset-strip fin geometry. While vortices cause a distinct increase in heat transfer in offset-strip fin geometry [124,125], the increase in heat transfer in the louvered fin geometry is less than 5% (within the experimental uncertainty). There is no noticeable effect on pressure drop. In the louvered fin geometry, a fin is not positioned in the wake region of the fin directly upstream, and thus downstream of the redirection louver the Reynolds number at which the fins begin to shed vortices is not strongly dependent on location in the array. Heat transfer coefficients in the array are also approximately constant throughout the array except for a decrease on the redirection louver and in the recovery zone downstream of the redirection louver under some cases. Heat transfer is not noticeably affected by vortex shedding within the Reynolds number range of interest.
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Dejong and Jacobi [119] also reported the effects of array walls on flow, pressure drop, and heat transfer behavior and the extent to which these effects penetrate an array. It was shown that, compared to flow far from the walls where spanwise periodic conditions exist, flow near the walls is characterized by lower flow efficiency, deviations in the flow velocity, large separation and recirculation zones between louvers, and unsteady flow at Reynolds numbers where the flow farther from the walls is steady. The flow angle of incidence to the louvers is much higher because of lower flow efficiency near the walls. The flow separates at the leading edge due to the larger adverse pressure gradient, and a large recirculation zone forms on the downstream side of the fin. A small second recirculation zone forms at the trailing end of the fin because flow passing along the downstream end of the fin interacts with the first recirculation zone, resulting in a region of high shear, which is resolved by a secondary small recirculation zone. The presence of the first recirculation zone causes a flow expansion along the upstream side of the next louver downstream. Above a certain Reynolds number, the adverse pressure gradient associated with this expansion results in a third recirculation zone. These separation and recirculation bubbles have a destabilizing effect on the flow, and thus the fins begin to shed vortices at a lower Reynolds number than fins in the center of the array. These vortices are of a larger scale than vortices shed from the fins farther from the walls. These wall effects are confined several columns next to the walls. Heat transfer in this region is lower than average for the array at low Reynolds numbers but somewhat higher than average at higher Reynolds number. At all Reynolds numbers, the walls cause a large increase in pressure drop. They presented the minimum number of columns such that flow in the center of the array can be representative of flow in a heat exchanger with many more columns. Nc ¼ Nwall þ
Lp Nl tanðL Þ Fp
ð130Þ
where Nwall ¼ 6L < 30 Nwall ¼ 8L 30 Nwall ¼ 9L 30
and or and
Fp =Lp > 0:9 Fp =Lp 0:9 Fp =Lp 0:9
Here Nl is the number of louvers from the inlet to the redirection louver. This equation is sufficient for most cases. However, for unusual cases with a much larger Nl than normal combined with a large louver angle or very
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small fin-to-louver pitch ratio, an additional minimum boundary must be placed on Nwall. Nwall
Lp Nl tanðL Þ Fp
ð131Þ
This formula suggested that eight to ten columns are necessary, but they recommended that at least 12 columns be used for typical geometries. Note that for typical heat exchangers, the walls have a very small effect on heat transfer and pressure drop. When modeling heat exchangers, wall effects can be ignored unless the total number of columns of fins is less than ten times the number of columns affected by the walls. Tafti et al. [126,127] conducted numerical calculations for both fully developed and developing flow and heat transfer in multilouvered fin arrays in the range from low Reynolds number laminar flow to an unsteady transitional regime. From fully developed flow calculations for Fp/Lp ¼ 1.11 and La ¼ 25 , they confirmed the flow structure observed by many previous researchers: it is duct directed flow at low ReLp, whereas it is louver directed flow at high ReLp. It was found that the flow was steady up to ReLp ¼ 782. At very low Reynolds numbers (ReLp ¼ 0.7 and 43) no flow separation was observed on the louver surface, while at ReLp ¼ 162 the flow was separated at the trailing edge resulting in a steady circulating wake. As ReLp increased further beyond ReLp ¼ 782, the flow became unsteady with vortex shedding from the leading edge of the fin. In addition at low Reynolds number, the thermal wake of upstream louver affected significantly the thermal boundary layer of downstream louver. In the unsteady flow regime, the vortices behave as large-scale mixers, bring the free stream air to their downstream side and eject the near-wall air on their upstream side, resulting in enhancement of the average heat transfer. For a single louver geometry with Fp/Lp ¼ 1.0 and La ¼ 30 , they performed high fidelity numerical simulations to map the onset of instabilities, their propagation and characteristic frequencies. They presented the detailed transition mechanism from a steady laminar flow to laminar unsteady and subsequently to chaotic. Figure 14 shows the schematic of flow structure and propagation of instabilities as a function of Reynolds number. Note that this figure only shows the initial appearance of features for the sake of clarity. At the lowest Reynolds number (ReLp1 ¼ 10), the flow is laminar and no separation is observed in the array. At ReLp1 ¼ 100, small recirculation zones are found to exist in the wake regions of all the louvers, and the recirculation bubbles become stronger and larger as the Reynolds number increases. At ReLp1 ¼ 200, two steady separation bubbles appear on the redirection louver and the exit louver. At ReLp1 ¼ 300, three recirculation zones are found to
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FIG. 14. Development and propagation of instabilities in the multilouvered fins (only the initial appearance of instabilities is shown) (Tafti et al. [127]).
exist on the following louvers (1 and 5) of the entrance and redirection louvers, and the exit louver. The first instability with the Von Karman vortex street appears in the wake region of the exit louver at ReLp1 ¼ 400 with a characteristic frequency (St ¼ 0.084). This instability is caused by the interaction of the leading edge shear layer separated at the top on the exit louver with the trailing edge shear layer separated at the bottom. As the Reynolds number increases further, the initial wake instability propagates upstream louvers from the wake region of the exit louver, whereas the flow field does not indicate any other instability mechanism developing in the louver array. At ReLp1 ¼ 900, additional free shear layer (Kelvin–Helmholtz type) instabilities with a characteristic frequency of St ¼ 0.17 have developed on the last three louvers near the exit of the array. Similar observations were reported by Howard [107], who found that the vortex shedding was initiated at the exit end of the louver array and the instability propagated upstream as Reynolds number increased. Time resolved velocity measurements by Springer and Thole [121] in multilouvered fin geometries also showed that vortex shedding existed in the array at ReLp1 ¼ 1000, but was not evident ReLp1 900. However, their measured fundamental frequency is St ¼ 0.17, which agrees exactly with Tafti et al. [127]. Zhang et al. [128] also reported that the characteristic frequencies were 0.14–0.17 in their numerical studies of infinite arrays of inline and staggered parallel fins. When ReLp1 ¼ 1300, vortex shedding is established upstream of the redirection louver, whereas
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the flow tends towards a chaotic state in the downstream half of the array. They also reported that the transition to unsteady flow and subsequent vortex shedding could significantly enhance heat transfer. Note that DeJong and Jacobi [119] reported that heat transfer was not noticeably affected by vortex shedding within the Reynolds number range of interest. Tafti and Zhang [129] extended their previous work [127] to multiple louver geometries and investigated the effects of geometrical parameters on flow transition in multilouved fins. It was shown that the flow tends to become unstable earlier as the louver angle and thickness are increased. They presented that the interior louver bank instabilities (onset, propagation, and characteristic frequencies) are independent of the exit wake instability. On the other hand, the exit wake instability is influenced by the exit and interior louver geometries. Louver angle and louver thickness affect substantially the initiation of the exit wake and interior bank instabilities, whereas the effect of fin pitch on the onset of instability is relatively small. They defined a Reynolds number (Res) to estimate the onset of instabilities in the interior louvers: Res ¼ ReLp1
2n Fp cos F Fp cos F Fth
ð132Þ
Here n is an integer value of the number of louvers corresponding length from the end of the entrance louver to the center of the redirection louver. It was shown that this simple model could predict well the initiation of instability in the interior louvers comparing with the measurement by DeJong and Jacobi [119]. They reported that a staggered plate array is more prone to instabilities than a louvered array, based on the estimation results using their model for the stagger plate fin geometries studied by DeJong and Jacobi [124] and Mochizuki and Yagi [130]. As the fin pitch, louver angle, and fin thickness are increased, the exit wake instability propagates more rapidly into the upstream louver bank. The effect of the fin spacing on the characteristic frequency in the interior louvers is larger than the louver pitch or the fin thickness, while the exit wake frequencies scale with the projected length of the exit louver in the flow direction. Zhang and Tafti [131] conducted numerical studies on the effects of thermal wakes on heat transfer in multilouved fins. They classified thermal wake interference in the louver fin array into two types of interference: interand intra-fin interference. The interference occurred between adjacent columns of louver fins is defined as inter-fin type and the interference occurred on subsequent louvers of the same column of fin is defined as intrafin type. The inter- and intra-fin interference can be occurred in both louver directed and duct directed flows. However, the inter-fin interference is
366
M.-H. KIM ET AL.
dominant when the flow is louver directed, whereas the latter is dominant when the flow is duct directed. Both types of interference affect significantly the average heat flux, while the effect of thermal wake interference on the heat transfer coefficient depends on the location of thermal wakes near the louver. It was shown that that the thermal wake effect can be expressed as functions of the flow efficiency and the fin-to-louver pitch ratio. The thermal wake effect on the heat transfer coefficient is negligible at the high flow efficiency, while its effect increases with decreasing the flow efficiency and fin-to-louver pitch ratio.
B. DRY CONDITIONS Many different air-side heat transfer data and correlations in brazed aluminum flat tube heat exchangers with folded or plate louvered-fin surfaces have been reported in the open literature. Table IV summarizes the j and f correlations including the range of geometrical parameters. Unless stated otherwise all data reported are for the Colburn j factor and Fanning friction f factor defined as [132] j ¼ St Pr2=3 o ¼
ho Pr2=3 m Vc cpo o
ð133Þ
Ac m 21 Po 1 1 2 2 f ¼ ðKc þ 1 Þ 2 1 þ ð1 Ke Þ Ao 1 ðm Vc Þ2 2 2 ð134Þ 1. Heat Transfer Data Davenport [101,102] developed a heat transfer correlation using 32 single-row louver fin geometries with flow depth of 40 mm and fin-to-louver pitch ratio of Fp/Lp ¼ 1.03–2.13. The ratio of louver length to fin height ranges 0.62–0.93. The copper louver fin (Fth ¼ 0.75 mm) with flat crests soldered to 0.2 mm thick brass tubes to define triangular air passages as shown in Fig. 8 (Type A). By incorporating the louver pitch rather than hydraulic diameter, the number of variables in the j correlation was reduced to four. The j correlation includes dimensional length scale terms that are expressed in millimeters, and contains the louver height (Lh) rather than the louver angle (La). The correlation does not contain the flow depth because all the samples have the same flow depth of Fd ¼ 40 mm. This indicates that his correlation is more suitable for the heat exchangers with large flow depth and does not reflect the effect of flow depth. The validity of the correlation
TABLE IV SUMMARY
OF
j
AND
f CORRELATIONS
Reference
Correlations
Geometry
Dry
Davenport [102]
For 300 < ReDh < 4000: 1:1 Ll Lh0:33 Fh0:26 j ¼ 0:249ReL0:42 p Fh For 70 < ReDh < 900: 0:89 Ll Fh0:23 L0:2 f ¼ 5:47ReL0:72 Lh0:37 p p Fh For 1000 < ReDh < 4000: 0:33 1:1 Lh Ll H 0:46 f ¼ 0:494ReL0:39 p Fh Fh where Lh ¼ ðLp =2Þ sinðL Þ
Ns ¼ 32 geometries Fp =Lp ¼ 1:03–2:13 Fp ¼ 2:01–3:35 mm Lp ¼ 1:5–3:0 mm L ¼ 8:34–30:4 Lh ¼ 0:186–0:46 mm Ll ¼ 5:0–11:7 mm Fd ¼ 40 mm Fh ¼ 7:8; 12:7 mm Dh ¼ 1:8–2:99 mm Fth ¼ 0:075 mm
Dry
Achaichia and Cowell [105]
For 75 < ReDh < 3000: 0:09 0:04 F Tp Fp Pr2=3 Re0:59 j ¼ 1:544 Lp L Lp Lp 243 Fp þ 0:995L where F ¼ 0:936 1:76 Lp ReLp For 150 < ReDh < 3000:
Ns ¼ 15 geometries Fp =Lp ¼ 1:18–4:11 Fp ¼ 1:65–3:33 mm Lp ¼ 0:81–1:4 mm L ¼ 20–30 Lh ¼ 0:139–0:334 mm Ll ¼ 5:0–11:0 mm
f ¼ 0:895fA1:07 Fp0:22 Lp0:25 Tp0:26 ð2Lh Þ0:33
MICROCHANNEL HEAT EXCHANGER
Surface condition
0:318 logðReLp 2:25Þ
where fA ¼ 596ReLp
367
(Continued)
368
TABLE IV (CONTINUED) Surface condition
Reference
Geometry
For ReDh < 150:
Fd ¼ 20:8; 41:6 mm Fh ¼ 6:0–12:0 mm Td ¼ 16:0 mm Dh ¼ 2:69–5:02 mm Fth ¼ 0:05 mm
f ¼
0:25 0:83 Fp0:05 L1:24 10:4ReL1:17 p Tp ð2Lh Þ p
where Lh ¼ ðLp =2Þ sinðL Þ Dry
Sunden and Svantesson [138]
For 65 < ReLp < 750: 0:0206 0:285 0:0671 0:243 Fp Fh Lh Fd j ¼ 3:67Re0:591 Lp Lp Lp Lp Lp 0:022 1:085 0:067 0:31 Fp Fh Lh Fd f ¼ 9:2ReL0:54 p Lp Lp Lp Lp where Lh ¼ ðLp =2Þ sinðL Þ
Dry
Chang et al. [141]
For 100 < ReLp < 700: j¼
"0:438 0:291Re0:589 Lp
f ¼ 0:805Re0:514 Lp
Fp Lp
0:72
Fh Lp
1:22
Ll Lp
1:97
Ns ¼ 6 geometries Fp =Lp ¼ 1:1–3:8 Fp ¼ 1:5–2:0 mm Lp ¼ 0:5–1:4 mm Fh ¼ 8:0–12:5 mm Lh ¼ 0:16–0:39 mm Ll ¼ 5:0–10:3 mm Fd ¼ 37:0–57:4 mm Fth ¼ 0:04–0:06 mm Ns ¼ 18 geometries Fp =Lp ¼ 1:06–1:67 Fp ¼ 1:8–2:2 mm Lp ¼ 1:318–1:693 mm L ¼ 28 Fh ¼ 16:0; 19:0 mm Ll ¼ 12:15–17:18 mm
M.-H. KIM ET AL.
Correlations
Fd ¼ 22–44 mm Fth ¼ 0:16 mm Dh ¼ 3:007–3:720 mm " ¼ 7:88–11:172 Dry
Dillen and Webb [144]
Ar ¼
1 Arr
Dhe ¼
if Arr ¼
Tp Do Ll >1 Fp Fth
4ðTp Do Ll ÞðFp Fth Þ 2ðTp Do Ll þ Fp Fth Þ
Ae ¼ 2Fd ðTp Do Ll þ Fp Fth Þ 0:5 0:5 S1 S2 þNl þ b¼2 Lp Lp Dry
Dillen and Webb [144]
For 400 < ReDh < 1000 : 0:31 0:4 Ae ADl Ll Lp L Fp þ CDl þ 4:5 bReL0:6 p Ao Ao Ao 90 Fh
f ¼ fe
For 1000 < ReDh < 4000 :
Model verification using data from Davenport [102] Ns ¼ 32 geometries Fp =Lp ¼ 1:03–2:13 Fp ¼ 2:01–3:35 mm Lp ¼ 1:5–3:0 mm L ¼ 8:34–30:4 Fh ¼ 7:8; 12:7 mm Lh ¼ 0:186–0:46 mm Ll ¼ 5:0–11:7 mm Fd ¼ 40 mm Fth ¼ 0:075 mm Dh ¼ 1:8–2:99 mm Do ¼ 1:5 mm
MICROCHANNEL HEAT EXCHANGER
For 400 < ReDh < 4000 : 0:201 0:139 ! Pr2=3 L Fp he Ae þ 0:0874Zf kLl bReL0:555 j¼ p 90 Fh m Vc Zo Ao he Dhe ¼ 10:81 þ 12:63Ar 1:61A2r 18:86A0:5 where r k Tp Do Ll Ar ¼ Arr if Arr ¼ 1 Fp Fth
Model verification using data from Davenport [102] Ns ¼ 32 geometries (Continued)
369
370
TABLE IV (CONTINUED) Surface condition
Reference
Geometry
Correlations
f ¼ fe
0:41 0:53 Ae ADl Ll Lp L Fp þ CDl þ 1:14 bReL0:43 p Ao Ao Ao 90 Fh
fe Dhe Vc 0:164 ¼ 32:72 þ 18:73Ar 37:04Ar0:5 Ar ¼ Nl Ll Fth
where ADl
for L < 8
for 8 L 12 cosðL Þ ¼ for L > 12 0:222 þ 0:283=sinðL Þ
CDl ¼ 0:8 CDl
Dry
Webb et al. [145]
For 400 < ReDh < 4000 : j¼
Pr2=3 m Vc Zo Ao
0:195 0:0522 ! Fp 0:581 L he Ae þ 0:744Zf kLl bReLp 90 Fh
he Dhe ¼ 10:81 þ 12:63Ar 1:61A2r 18:86Ar0:5 k Tp Do Ll Ar ¼ Arr if Arr ¼ 1 Fp Fth
where
Ar ¼
1 Arr
if Arr ¼
Tp Do Ll >1 Fp Fth
Model verification using data from Davenport [102] and Chang and Wang [146] Ns ¼ 57 geometries Fp =Lp ¼ 0:97–2:13 Fp ¼ 1:80–3:35 mm Lp ¼ 1:318–3:0 mm L ¼ 8:34–30:4 Fh ¼ 7:8–19:0 mm
M.-H. KIM ET AL.
CDl ¼ 2p sinðL Þ
Fp =Lp ¼ 1:03–2:13 Fp ¼ 2:01–3:35 mm Fp ¼ 2:01–3:35 mm Lp ¼ 1:5–3:0 mm L ¼ 8:34–30:4 Fh ¼ 7:8; 12:7 mm Lh ¼ 0:186–0:46 mm Ll ¼ 5:0–11:7 mm Fd ¼ 40 mm Fth ¼ 0:075 mm Dh ¼ 1:8–2:99 mm Do ¼ 1:5 mm
Dhe ¼
4ðTp Do Ll ÞðFp Fth Þ 2ðTp Do Ll þ Fp Fth Þ
Dry
Webb et al. [145]
For 400 < ReDh < 1000 : 0:233 0:628 Ae ADl Ll Lp L Fp þ CDl þ 6:242 bReL0:759 f ¼ fe p Ao Ao Ao 90 Fh For 1000 < ReDh < 4000 : f ¼ fe
fe Dhe Vc 0:164 ¼ 32:72 þ 18:73Ar 37:04A0:5 r Ar ¼ Nl Ll Fth ¼ 2p sinðL Þ for L < 8 ¼ 0:8 for 8 L 12 cosðL Þ for L > 12 ¼ 0:222 þ 0:283=sinðL Þ
where ADl CDl CDl CDl
0:521 0:0772 Ae ADl Ll Lp L Fp þ CDl þ 0:876 bReL0:555 p Ao Ao Ao 90 Fh
Model verification using data from Davenport [102] and Chang and Wang [146] Ns ¼ 57 geometries Fp =Lp ¼ 0:97–2:13 Fp ¼ 1:80–3:35 mm Lp ¼ 1:318–3:0 mm L ¼ 8:34–30:4 Fh ¼ 7:8–19:0 mm Lh ¼ 0:186–0:663 mm Ll ¼ 5:0–17:18 mm Fd ¼ 22–44 mm
MICROCHANNEL HEAT EXCHANGER
Ae ¼ 2Fd ðTp Do Ll þ Fp Fth Þ 0:5 0:5 S1 S2 þNl þ b¼2 Lp Lp
Lh ¼ 0:186–0:663 mm Ll ¼ 5:0–17:18 mm Fd ¼ 22–44 mm Fth ¼ 0:075; 0:16 mm Dh ¼ 1:8–3:74 mm Do ¼ 1:5; 5:0 mm
(Continued)
371
372
TABLE IV (CONTINUED) Surface condition
Reference
Correlations
Geometry
Fth ¼ 0:075; 0:16 mm Dh ¼ 1:8–3:74 mm Do ¼ 1:5; 5:0 mm Dry
Chang and Wang [146]
For 100 < ReLp < 1000 :
Dry
Chang and Wang [147]
For 100 < ReLp < 3000 : 0:26 0:51 0:26 0:82 L Fl Td Ll j ¼ 1:18Re0:505 Lp 90 Lp Lp Lp 0:25 0:097 Tp Fth Lp Lp For 100 < ReLp < 3000 ðfolded fin geometryÞ :
Ns ¼ 91 geometries Fp =Lp ¼ 0:45–4:11 Fp ¼ 0:51–3:33 mm Lp ¼ 0:5–3:0 mm L ¼ 8:43–35 Fh ¼ 6:0–20:0 mm Ll ¼ 2:13–18:5 mm Fd ðTd Þ ¼ 15:6–57:4 mm
M.-H. KIM ET AL.
"0:192 "l0:0956 j ¼ 0:436ReL0:559 p f ¼ 0:862ReL0:488 "0:706 "l1:04 p
Ns ¼ 27 geometries Fp =Lp ¼ 0:97–1:67 Fp ¼ 1:8–2:2 mm Lp ¼ 1:318–1:860 mm L ¼ 28 Fh ¼ 16:0; 19:0 mm Ll ¼ 12:15–17:18 mm Fd ¼ 22–44 mm Fth ¼ 0:16 mm Dh ¼ 3:007–3:740 mm " ¼ 7:778–11:704 "l ¼ 0:432–0:628
j ¼ ReL0:49 p Chang et al. [149]
Ll Lp
0:68 0:28 0:05 Tp Fth Lp Lp
f ¼ f1 f2 f3
For ReLp < 150 : 3:04 Fp ln 1:0 þ Lp !!1:435 0:48
3:01 Fth Dh 3:01 f2 ¼ ln þ0:9 ln 0:5ReLp Fp Lp ð0:805ðFp =Fh ÞÞ
f1 ¼ 14:39ReLp
f3 ¼
0:308 0:308 0:1167ðTp =Do Þ 0:35 Fp Fd L e Ll Ll
For 150 < ReLp < 5000 : f1 ¼
ð0:6049ð1:064=L0:2 ÞÞ 4:97ReLp
!!0:527 0:5 Fth þ0:9 ln Fp
Tp ¼ 7:51–25:0 mm Fth ¼ 0:0254–0:16 mm Dh ¼ 0:824–5:02 mm
Ns ¼ 91 geometries (data collected from eight different sources) Fp =Lp ¼ 0:45–4:11 Fp ¼ 0:51–3:33 mm Lp ¼ 0:5–3:0 mm L ¼ 8:43 35 Fh ¼ 6:0–20:0 mm Ll ¼ 2:13–18:5 mm Fd ðTd Þ ¼ 15:6–57:4 mm Tp ¼ 7:51–25:0 mm Fth ¼ 0:0254–0:16 mm Dh ¼ 0:824–5:02 mm
MICROCHANNEL HEAT EXCHANGER
Dry
0:27 0:14 0:29 0:23 L Fp Fl Td 90 Lp Lp Lp
(Continued)
373
374
TABLE IV (CONTINUED) Surface condition
Kim and Bullard [148]
Geometry
Correlations
f2 ¼
2:966 Fp 0:7931ðTp =ðTp Do ÞÞ Dh ln 0:3ReLp Lp Ll
f3 ¼
Tp Dm
0:0446
ln 1:2 þ
1:4 !3:553 Lp L0:477 Fp
For 100 < ReLp < 600 : j ¼ ReL0:487 p
0:257 0:13 0:29 0:235 0:68 L Fp Fh Fd Ll 90 Lp Lp Lp Lp
0:279 0:05 Tp Fth Lp Lp
f ¼ ReL0:781 p
M.-H. KIM ET AL.
Dry
Reference
0:444 1:682 1:22 0:818 1:97 L Fp Fh Fd Ll 90 Lp Lp Lp Lp
Ns ¼ 45 geometries Fp =Lp ¼ 0:59–0:82 Fp ¼ 1:0–1:4 mm Lp ¼ 1:7 mm L ¼ 15 29 Fh ¼ 8:15 mm Ll ¼ 6:4 mm Fd ¼ 16:0–24:0 mm Td ¼ 16:0–25:4 mm Tp ¼ 10:15; 11:15 mm Fth ¼ 0:1 mm
Wet
Kim and Bullard [189]
For 80 < ReLp < 300 : 0:25 0:171 0:29 0:248 L Fp Fh Fd j ¼ ReL0:512 p 90 Lp Lp Lp 0:68 0:275 0:05 Ll Tp Fth Lp Lp Lp 0:395 2:635 1:22 0:823 1:97 L Fp Fh Fd Ll 90 Lp Lp Lp Lp
Fd ¼ 16:0–24:0 mm Td ¼ 16:0–25:4 mm Tp ¼ 10:15; 11:15 mm Fth ¼ 0:1 mm
MICROCHANNEL HEAT EXCHANGER
f ¼ ReL0:798 p
Ns ¼ 30 geometries Fp =Lp ¼ 0:59–0:82 Fp ¼ 1:0–1:4 mm Lp ¼ 1:7 mm L ¼ 15–29 Fh ¼ 8:15 mm Ll ¼ 6:4 mm
375
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M.-H. KIM ET AL.
ranges over Reynolds numbers (ReDh ¼ 300–4000) based on the hydraulic diameter (Dh), not the louver pitch (Lp) as used in the correlations. This lower limit of Reynolds number (ReDh ¼ 300) for the correlation was attributed to the flow pattern transition and/or experimental error at low Reynolds number as described earlier. As Reynolds number decreases, the boundary layer on the louvers becomes thicker to block the flow passage between louvers and the flow behavior changes from the louver directed flow to the duct directed flow. This flow behavior at low Reynolds numbers results in inconsistent heat transfer characteristics at high Reynolds number. Another possibility will be experimental error because at low Reynolds number measurement uncertainty increases due to the smaller temperature difference [133] and effect of natural convection [103]. Davenport reported about 95% of the heat transfer data ( j factors) had been correlated within 6%. Achaichia and Cowell [105] measured the air-side performance of a wide range of louver fin surfaces with inline tube layout. All the samples consist of one or two rows of flat brass tubes and copper plate louver fins with thickness of 0.05 mm as shown in Fig. 8 (Type B). They also found the significant drop in the heat transfer coefficients at low Reynolds number as observed by Davenport [102], and suggested that this was due to the flow pattern changing the alignment from the louver to the fin direction and in turn resulted in the heat transfer characteristics undergoing a transition from flat plate behavior to that typical for duct flow. They developed a simple polynomial equation to describe the mean flow angle (Fa) as a function of the louver angle (La), the fin-to-louver pitch ratio (Fp/Lp) and Reynolds number (ReLp) as F ¼ 0:936
243 Fp þ 0:995L 1:76 Lp ReLp
ð135Þ
Note that this correlation does not include the number of louvers. This relation was included in the correlation as shown in Table IV. In addition, they found the effects of tube-to-louver pitch ratio and fin-to-louver pitch ratio on the heat transfer correlation is very small and obtained the following simple expression with neglecting the geometrical parameters j ¼ 1:18ReL0:58 p
F Pr2=3 L
ð136Þ
Apparently, this correlation equation implied that the flow pattern in the louver array affected significantly the air-side heat transfer performance. As the mean flow angle or Fa/La decreases with decreasing Reynolds number,
MICROCHANNEL HEAT EXCHANGER
377
the j factor decreases. They reported this simple equation predicted all the heat transfer data for ReLp>75 to within 10%. However, they measured the heat transfer data using the heat exchanger samples with extremely small non-louvered length (0.5 mm), indicating that their correlation may overpredict the heat transfer data for the heat exchanger geometries having moderate non-louver length. Aoki et al. [134] measured heat transfer characteristics of different louver fin arrays such as louver angles, louver and fin pitches, and reported that the heat transfer coefficients at low face velocity decreased with increasing fin pitch. This indicates the flow pattern may change with increasing fin pitch at low Reynolds number flow. They also found that heat transfer coefficient increased with louver angle, reached a maximum at angle of 28–30 , and decreased again. Suga and Aoki [135] proposed the following optimum geometrical relation between the fin-to-louver pitch ratio and the louver angle in the range of ReLp1 ¼ 64–450 (based on face velocity), La ¼ 20–30 and Fp/Lp 1.125. Fp ¼ 1:5 tanðL Þ Lp
ð137Þ
Note that this relation was developed based on their two-dimensional numerical study under the assumption of steady state laminar boundary layer flow. The boundary layer interruptions create the wake region, and self-sustained flow unsteadiness. Thus the models based on the boundary layer approximation are not adequate and do not accurately predict the heat transfer data. Sunden and Svantesson [136,137] measured air-side performance for six louver fin surfaces and all the louver fin surfaces were found to be more efficient than the corresponding smooth surface. They presented the j factor correlation as a dimensionless form [138]. Heikal et al. [95] argued Sunden and Svantesson’s correlation should be treated with caution because each heat exchanger sample contained five geometric parameters and were based on a limited number (six) of samples. Especially, the dependence of fin-tolouver pitch ratio on the j factor is significantly smaller than proposed by most other studies as shown in Table IV. Webb and Jung [139] measured air-side performance of the brazed aluminum heat exchangers using six louver fin geometries which were three standard and three splitter louver fin geometries. They reported the brazed aluminum design gave 90% higher heat transfer coefficient for only 25% high pressure drop compared to a plain plate fin design with fin density (472 fins/m) and 9.5 mm diameter tube. They also presented the j factors
378
M.-H. KIM ET AL.
against the Reynolds number (ReLp), but did not provide correlation equation. Rugh et al. [140] measured heat transfer coefficients over the range of Reynolds number (ReDh ¼ 150–300) using a high-density louvered fin and flat-tube heat exchanger (2000 fins/m) with a splitter plate (Fig. 8, Type D). They reported a louvered fin heat exchanger produced a 25% increase in heat transfer and a 110% increase in pressure drop relative to a plain fin, but note that they tested only one heat exchanger. Chang et al. [141] developed the j factor correlation by Colburn analogy, based on test data for 18 samples of folded louvered fin heat exchangers. Their correlation contains the finning factor (" ¼ Ao/Ato) defined by McQuiston [142], which represents the geometrical parameters. The finning factor dependence on the j factor is significant ( j / "0.438), but its effect on the j factor is totally different compared to the plane finned round tube heat exchanger data from McQuiston ( j / "0.15) [142] and Kayansayan ( j / "0.362) [143]. This discrepancy probably is attributed to the different flow characteristics between two heat exchangers. The fin spacing is a major parameter in representing the finning factor that increases with decreasing the fin spacing. For the louver fin geometry, the smaller fin pitch may increase the flow efficiency and so does the j factor. On the other hand, for the round-tube heat exchangers the smaller fin spacing results in degradation of heat transfer performance due to the channel effects [143]. They reported that their simple correlation was found to predict 92% of the heat transfer data to be within 10%. However, the louver angle was not included in the correlation because all the heat exchanger samples considered had the same louver angle of 28 . Sahnoun and Webb [111] developed an analytical model to predict the heat transfer coefficient for louvered fin geometry. They calculated the heat transfer coefficients for each louver fin element such as the internal louvers, inlet and outlet louvers, and redirection louvers using the Polhausen solution for laminar flow over a flat plate. In case of the internal louvers, the velocity over the louver is calculated associated with the newly developed flow efficiency (Eqs. (123)–(125)). The heat transfer coefficient in the nonlouvered end regions is estimated using a fully developed laminar flow solution. The model for the j factor vs. Reynolds number (ReDh) was validated using Davenport’s data [102] and it was shown that the mean deviations were 8 and 20% at high and low Reynolds numbers, respectively. Dillen and Webb [144] developed a rationally based semi-empirical heat transfer correlation, based on the analytical model of Sahnoun and Webb [111] and the model verified using Davenport’s data [100]. They claimed that 100 and 82% of the data were predicted within 20 and 10%,
MICROCHANNEL HEAT EXCHANGER
379
respectively. Note that the thermal conductivity is missing in the last term of their heat transfer correlation (Eq. (33) of Dillen and Webb’s paper [144]). Webb et al. [145] extended previous work [144] by including Chang and Wang’s data [146] and developed the heat transfer correlation, which 95% of the data predicted within 20%. Chang and Wang [146] reported that Sahnoun and Webb’s analytical model and Dillen and Webb’s rationally based model [144] could predict their heat transfer data for their 27 samples within the standard deviations of 7.4 and 6.5%, respectively. Chang and Wang [146] presented that rationally based semi-empirical model could be used to predict the heat transfer data quite reasonably, while Heikal et al. [95] argued that their models [111,144] are quite complex to use and correlated the data with accuracy similar to that of Davenport’s simple correlation [102]. Chang and Wang [146] developed the j factor correlation using 27-folded louver fin heat exchanger geometries which were Chang et al.’s 18 samples [141] plus nine samples. The area ratio ("l) and finning factor (") as geometrical parameters were included in the correlation, but the louver angle was not contained again because all the test samples had the same louver angle of 28 . Their correlation indicated that 85% of the data could predict within 10%. Chang and Wang [147] collected the heat transfer data for 91 heat exchanger samples (76-folded louver fin and 15 plate louver fin surfaces) from nine different sources and developed the j factor correlation by a trial and error process. They claimed that the correlation could predict 87.8% of all the data to within 15% and a mean deviation of 8.21%. Note that they neglected the fin-to-louver pitch ratio (Fp/Lp) in the correlation. They also developed the j correlation using the folded fin louvered fin data with excluding Achaichia and Cowell’s data [105] for the 15 plate louvered fin geometries. It was shown that that 89.3% of the folded fin data were correlated within 15% and a mean deviation of 7.55%. They also proposed a simple equation for a quick engineering evaluation of heat transfer coefficient as j ¼ 0:42ReL0:496 p
ð138Þ
They claimed this simple equation could predict 88.2 and 70.7% of all the data collected within 25 and 15%, respectively. Jeon and Lee [118] measured the local heat transfer characteristics using 27 multilouvered fin heat exchanger samples of 15:1 scale models. They found that at low Reynolds number (ReLp) the local average heat transfer coefficient of the downstream louver banks are lower than those of the upstream louver banks due to the thermal wake effects on the downstream
380
M.-H. KIM ET AL.
louver banks. They developed the following heat transfer correlation for a wide range of variables (ReLp ¼ 100–1800, Fp/Lp ¼ 1.1–3.3, La ¼ 20–40 ) j ¼ 0:991
ko pL 0:112 Fp 0:456 2=3 ReL0:596 Pro p cpo mo 180 Lp
ð139Þ
They reported that 93 and 66% of the heat transfer data were predicted within 20 and 10%, respectively, but they did not compare this correlation with any other results. Note that their scale models used only 5– 9 columns of fins and so the flow through the models may not be representative of full-scale heat exchangers. Kim and Bullard [148] measured air-side performance using 45 heat exchangers with different louver angles (15–29 ), fin pitches (1.0, 1.2, 1.4 mm) and flow depths (16, 20, 24 mm). All the heat exchangers have the fin-to-louver pitch ratios of Fp/Lp<1. Their j factor correlation was developed for the air-side Reynolds numbers of ReLp ¼ 100–600 within a rms errors of 14.5%, base on Chang and Wang’s correlation [147]. They compared several different heat transfer correlations using two different heat exchanger configurations (Table V) that represented the condenser and evaporator for mobile a/c, respectively (Fig. 15). Two sample heat exchangers are similar except louver angle and flow depth; flow depth of type I is 16 mm while type II is 41.76 mm. Sunden and Svantesson’s [138], Chang and Wang’s [147], Kim and Bullard’s [148] correlations reflect the effect of different flow depths, while the other correlations have the similar j factors for both heat exchangers since the flow depth is not included in their correlations. Davenport’s correlation [102] was developed using heat exchanger samples with the flow depth of 40 mm. The j factors from his correlation are smaller than the others for the heat exchanger type-I, while they are similar to Kim and Bullard [148] and Chang and Wang’s [147] correlation data for the heat exchanger type-II, indicating that Davenport’s correlation [102] is more suitable for the heat exchangers with larger flow depth and does not reflect the effect of flow depth. The j factors from Table V HEAT EXCHANGER SAMPLES
FOR
COMPARISON
OF
CORRELATIONS
Heat exchanger La ( ) Fp (mm) Lp (mm) Ll (mm) Fh (mm) Tp (mm) Td (mm) Fth (mm) type I II
23 30
1.4 1.6
1.7 1.55
6.4 7.16
8.15 8.8
10.15 10.24
16 41.76
0.1 0.1
MICROCHANNEL HEAT EXCHANGER
FIG. 15. A comparison of j factor for multilouvered fins.
381
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Achaichia and Cowell’s correlation [105] are larger than those from the others, suggesting that the heat exchangers with smaller non-louvered end regions have relatively higher heat transfer coefficients. Jeon and Lee’s j factor correlation [118] indicated relatively larger j values compared to the others, and this probably was attributed to their scaled heat exchanger models that had only 5–9 columns of fins as described earlier. Sunden and Svantesson’s j factor correlation [138] has large variation with the flow depth, compared to other correlations, but it is hard to explain why the j factors show such behavior because they developed the correlation using only six different heat exchanger geometries. 2. Pressure Drop Data Davenport [102] measured the pressure drop characteristics from isothermal test conditions, and the resulted f factor equation was described as Ac m 21 Po ðKc þ Ke Þ ð140Þ f ¼ Ao 1 ðm Vc Þ2 They found Reynolds number based on louver pitch was effective for the correlation, suggesting the relevance of laminar boundary layers. By incorporating this Reynolds number (ReLp), they developed the f correlations by including four and five variables at high and low Reynolds (ReDh) numbers, respectively (Table IV). Note that the correlations are dimensional form and all lengths are in millimeters. They reported 95% of the data had been correlated within 10%, while Chang et al. [149] presented Davenport’s f correlation gave the mean deviation of 17.5% for their data bank for 91 samples collected from nine different sources. Achaichia and Cowell [105] could not find the friction factor in the standard form at high Reynolds number (150
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Davenport [102] wherever it required. The Achaichia and Cowell’s f correlations are dimensional form and all dimensions are in millimeters. Chang et al. [149] reported this f correlation overpredicted significantly their data for 91 samples and the mean deviation was 102.5%, because the heat exchanger samples had two tube rows. Kim and Bullard [148] also presented the correlation overpredicted the data because they used the heat exchanger samples with extremely small non-louvered end regions (0.5 mm). Sunden and Svantesson [138] defined the f factor as core friction, with neglecting flow acceleration, entrance and exit effects. f ¼
Ac 2Po Ao m Vc2
ð141Þ
They developed the f factor correlation in dimensionless form using six different louver fin configurations. Their correlation includes the flow depth, but the heat exchanger samples considered have the relatively large flow depth in the range of 37–57.4 mm, suggesting their correlation will be more suitable for the heat exchangers having large flow depth. Sahnoun and Webb [111] developed an analytical model for the friction factor for folded louvered fin geometry using the method similar to as used in the heat transfer case. It accounts for all the dimensions in the louver array such as the inlet/outlet, internal and redirection louvers, and the nonlouvered end regions, and the number of louvers. Their model included profile drag term which was determined using the equations proposed by Blevins [150] for an inclined plate. The effect of friction and momentum of the developing flow associated with the flow fraction that bypasses the louvers is considered in the model, while the entrance and exit losses are neglected. It was shown that mean error ratios were 9 and 12% at high and low Reynolds numbers (ReDh). Dillen and Webb [144] developed a rationally based semi-empirical friction correlation using Davenport’s data [102] based on the analytical model of Sahnoun and Webb [111]. The resulted correlation predicted 81% of the Davenport’s friction data within 10%. Webb et al. [145] extended their previous study [144], and developed an improved semi-empirical model using the data by Davenport [102] and Chang and Wang [146]. They claimed that 91 and 76% of the data were predicted within 20 and 15%, respectively. Chang et al. [141] tried to derive a correlation using the finning factor (") as a geometrical parameter, but the reasonable accuracy could not be achieved. Thus they developed the f correlation in dimensionless form using a multiple regression method for their 18 samples and claimed that 95% of the data were correlated within 15%. However, Chang and Wang [146]
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reported this f correlation could predict 85% of the data for their 27 samples within 20%. Chang and Wang [146] developed the simple f correlation using the finning factor (") and area ratio ("l) as geometrical variables. It was shown that the correlation could predict 85% of the data for their 27 samples within 10%. Chang et al. [149] developed the f factor correlations by a trial and error process, based on the data bank that Chang and Wang [147] collected when developing their general heat transfer correlation. Chang and Wang [147] reported earlier that the variation of the friction factors vs. the Reynolds number was extremely large and it was not easy to accurately correlate the friction factor. The proposed correlation gave a mean deviation of 9.21% and it was shown that 83% of the data could be correlated within 15%. Note that their friction factor correlation includes large parameters with simple form. Kim and Bullard [148] developed the f correlation using 225 data points for 45 different louver fin geometries with Fp/Lp<1. It was shown that their correlation could predict the data within a rms error of 7%. Fig. 16 depicts comparison of several different f correlations using two different aluminum heat exchangers. The definition of f factors in Fig. 16 is the same as Eq. (124), except Davenport [102] and Sunden and Svantesson [138]. Davenport [102] obtained f factors from the isothermal conditions (Eq. (140)) and this may yield very small error since the density change will be very small due to temperature variation across the heat exchanger. Sunden and Svantesson [138] defined f factor as core friction (Eq. (141)), but this effect will not be significant. The f factors from Achaichia and Cowell’s correlation [105] are larger than those from the others, again since the nonlouvered end portions of the fin is very small. The Sunden and Svantesson’s [138] and Kim and Bullard’s [148] correlations reflect the effect of flow depth on the pressure drop, while other correlations do not reflect. This suggests that the flow depth is one of the important parameters, which influences the friction factor significantly. Kim and Bullard [151] reported their correlation predicted well the pressure drop data of the large flow depth heat exchanger for a mobile CO2 air-conditioner. Figure 16 compares the several different f factor correlations. The friction correlations of Kim and Bullard [148] and Chang et al. [149] are quite similar for heat exchanger type-I. However, for heat exchanger type-II with large flow depth, the Kim and Bullard’s f factors [148] are significantly larger than those obtained from Chang et al.’s correlation [149]. This may be due to the lack of the flow depth parameter in their correlations [149], suggesting that the flow depth is one of the important parameters for pressure drop correlation. The recent friction correlation of Chang et al. [149] includes flow depth parameter only for low Reynolds number flow (ReLp<150).
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FIG. 16. A comparison of f factor for multilouvered fins.
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C. WET CONDITIONS When the heat exchanger surface temperature is below the dew point of incoming moist air, the surface is subjected to condensation of moisture contained in air. A dropwise, a filmwise or a combined (stripwise) condensation will be formed on the surface depending on the wettability of the surfaces. A dropwise condensation forms on a surface having poor wettability, whereas filmwise condensation occurs on a surface having good wettability. Once water condensation occurs on the heat exchanger surface, it accumulates until it is drained by gravitational, capillary, or air shear force. Condensate water retained on the surface and droplets carried by the air stream have an important effect on the thermal hydraulic performance of heat exchangers as well as comfort level in residential and mobile airconditioning applications. Water retained on the heat exchanger surface generally increases the pressure drop across the heat exchanger by restricting the flow of moist air. The amount of pressure drop increase due to condensation water depends on the heat exchanger geometry and surface condition as well as operating conditions. Water retention influences significantly on the sensible heat transfer coefficient, but the direction and magnitude of this effect also appears to be dependent on the heat exchanger geometry and air-side Reynolds number. Condensation also causes waterspray-discharge, corrosion of aluminum fins and the emission of undesired odors with conditioned air. Retained condensate provides a medium for biological activity that might cause the odors. Thus, it is very important to understand the mechanism of condensate retention and drainage and its effects on heat exchanger performance. 1. Heat Exchanger Performance Under Wet Conditions a. Finned Tube Heat Exchangers Lewis [152] presented the straight-line law using the heat and mass transfer analogy (Lewis relation) to determine the air outlet condition of a wetted heat exchanger. It appears to be the first rational step toward a real understanding of the dehumidifying and cooling heat exchangers. This law was developed based on the assumption that the surface temperature was constant throughout the heat exchanger. The straight-line law indicates that the point representing air state over a wet surface must lie on the straight line on the psychrometric chart joining the point for the initial air-condition and the point on the saturation curve corresponding to the constant surface temperature. Chilton and Colburn [153] suggested the modified Reynolds (Chillton–Colburn) analogy for flow over tubes and tube banks and plane surfaces over a wide range of Prandtl and Schmidt numbers. Knaus [154] and Pownall [155] confirmed the validity of straight-line law. Note that this
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law is valid only for a cooling surface whose temperature is constant. However, in actual case the surface temperature varies at each point along the air passage. Goodman [156] extended the application of this straight-line law to wetted heat exchangers in which the surface temperature varies and presented a method for determining the change of air state through a wet surface. However, his method did not account for the effect of dehumidification on the fin efficiency. Ware and Hacha [157] and Bryan [158,159] presented experimental data for heat and mass transfer on a bare tube and extended surface cooling coils and they proposed data analysis methods based on Lewis relation. Myers [160] proposed log mean enthalpy difference methods for the cooling coils. Bettanini [161] performed experimental study on simultaneous heat and mass transfer between moisture air and a simple vertical plane surface. He found that both dropwise and filmwise condensation enhanced sensible heat transfer under dehumidifying conditions, and a heat transfer enhancement for dropwise condensation was greater than that for filmwise condensation. Bettanini reported that the sensible heat transfer enhancements were attributed to the combined effects of mass transfer and surface roughness. Studies on combined heat and mass transfer for fin-and-round tube heat exchangers were performed by several investigators [162–184]. Guillory and McQusiton [162] reported that sensible heat transfer and pressure drop for a parallel-plate heat exchanger under dehumidifying conditions were increased due to an increase in surface roughness associated with condensate retention. A similar result for a very simple two parallel plate heat exchanger was reported by Tree and Helmer [163]. An increase in sensible heat transfer and pressure drop was found under wet conditions in the fully turbulent flow regimes. McQuiston [164–166] measured sensible heat transfer and pressure drop data for parallel plate heat exchangers under wet conditions and found that the surface condition and type have an important effect on the heat exchanger performance. Measured heat transfer rates for dropwise condensation were typically higher than for filmwise condensation. Several investigators (Threlkeld [167], McQuiston [168], Elmahdy and Biggs [169], Wu and Bong [170], Hong and Webb [171], and Liang et al. [172]) provided the fin efficiency for the fully wetted heat exchanger surfaces. Wu and Bong [170] also presented the overall fin efficiency for the partially wet surface and reported that only when the fin is partially wet the overall fin efficiency depends significantly on the relative humidity. Hill and Jeter [174] developed a linear sub-grid model for the air-conditioner’s cooling and dehumidifying coil, which is an evaporator. They showed that the singlepass, cross flow arrangement of the model was adequate to model the counter cross flow heat exchangers. Mirth et al. [175] investigated performance analysis of the cooling coil based on ARI Standard [176], and
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Hu et al. [177] analyzed the effect of the shape of the condensation water on the fin surface on thermal performance characteristics. Wang et al. [178] reported the heat transfer coefficient for wet surface was smaller than that of dry surface for low Reynolds numbers based on tube diameter, whereas friction factors under wet conditions were 60–120% higher than for dry conditions and insensitive to the inlet relative humidity. Chuah et al. [179] investigated dehumidifying performance of chilled water coils with variation of water flow rate. Kim and Jacobi [180] investigated condensation accumulation effects on air-side heat transfer and pressure drop characteristics for plain- and slit-fin-and-tube heat exchangers. Korte and Jacobi [181] measured heat transfer, pressure drop, and dynamic retention data using two plain-fin-and-tube heat exchangers, which were identical except the fin spacing (one had 3.18 mm, and the other had 6.35 mm). They presented the effects of fin spacing, Reynolds number, and surface condition on condensation retention. Recently, some investigations for plane and louver fin heat exchangers under wet conditions were conducted [182–184]. However, little data exist on the air-side performance of louvered fin aluminum microchannel heat exchangers with dehumidification. b. Brazed Aluminum Heat Exchangers Webb and Jung [185] studied the application of brazed aluminum heat exchangers to the residential air-conditioner and showed that the heat transfer rate of the brazed aluminum heat exchanger was 50% higher than that of a conventional heat exchanger. They reported that condensate could be removed well from the heat exchanger surface, so it could be used as an evaporator for the residential air-conditioning system. Chiou et al. [186] measured thermal hydraulic performance of two serpentine automotive evaporators. They reported that the sensible heat transfer coefficient and pressure drop for wet surfaces were larger than those for dry surfaces, because the condensate acted as a surface roughness and this roughness caused turbulent mixing of the flow. McLaughlin and Webb [187,188] studied the heat transfer and pressure drop characteristics of louvered fin automotive evaporators with the larger flow depth (Fd 50 mm) and conducted condensate visualization tests. McLaughlin and Webb [186] reported the effect of condensate on thermal hydraulic performance depended significantly on the louver configurations. Smaller fin pitches caused fin bridging, causing the heat transfer to decrease slightly as the friction increased, while small louver pitches resulted in louver blockage, causing both heat transfer and friction to decrease. Kim and Bullard [189] measured air-side heat transfer and pressure drop data using 30 different brazed aluminum heat exchangers with different geometrical parameters under dehumidifying conditions. The heat
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exchangers tested are the same as those for their study of dry conditions [148]. The test was conducted for air-side Reynolds number in the range of 80–300 and tube-side water flow rate of 320 kg/h. The dry- and wet-bulb temperatures of the inlet air were 27 and 19 C, respectively and the inlet water temperature was 6 C. For small louver angle (La<27 ), they found that sensible heat transfer coefficients for all fin pitches considered in their study are smaller than those for dry conditions [148], indicating the amount of condensate retained on the surface increased and the condensate may significantly impair heat transfer. Pressure drop for wet conditions is consistently higher than that for dry conditions [148] and the incremental rate of pressure drop with decreasing fin pitch decreases with face velocity, suggesting that higher face velocity improves condensate drainage as air shear force increases. They developed j and f correlations for wet conditions with rms errors of 16.9 and 13.6% (Table IV), respectively, and compared them with several dry surface correlations as shown in Figs. 15 and 16. 2. Effects of Inclination on the Air-side Performance Inclined heat exchangers have been used in air-conditioning and heat pump applications for the cost effective compact systems, although these configurations may deteriorate the system performance. There are some publications on the effect of an inclination angle on the heat transfer and the pressure drop of the heat exchangers. However, most of the published data have considered the bare-tube banks, high-fin tube banks and conventional finned tube heat exchangers. Only three publications in the open literature [190–192] are available for the inclination effects on the thermal hydraulic performance of multilouvered-fin-and-flat-tube heat exchangers. Osada et al. [190] studied the effect of inclination on the heat transfer and pressure drop characteristics of folded multilouvered fins under dry and wet conditions. They used a singlerow test piece with larger flow depth (Fd ¼ 58 mm), louver angle (La ¼ 35 ), Fp/Lp ¼ 1.5/1.2>1, and three redirection louvers. A single row of folded multilouvered fin was brazed at both sides with the aluminum plates, which had the same grooves as the actual evaporator tube. The Peltier cooling modules were used for regulating the temperature of the tube walls. The core size (tube (plate) length fin height) of the test piece was 120 10 mm2. However, 20 mm sections of the top and bottom were covered by thin acrylic resin plates. Thus the humid air entered in the normal direction to the 80 mm section between covered sections. The range of the inclination angles from the vertical position was 30 60 (leeward inclination: <0 , windward inclination: >0 ). Osada et al. [190] reported that both the leeward and windward inclinations improved fin performance and
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inclination toward windward direction had better performance. The heat transfer coefficients for wet conditions were smaller than those for dry conditions except >30 , whereas the pressure drops were larger than those for dry conditions except 7 45 . When ¼ 15 , louver blockages were observed at upstream region between the leading edge and first redirection louver of the fin. They speculated that the air shear force (leeward direction) acting on the condensate water in the folded fin bends was balanced by the gravitational force (windward direction) at this particular inclined angle ( ¼ 15 ). These louver blockages upstream the fin caused the flow pattern to be duct-directed flow and thus the pressure drops decreased. They suggested that the optimum installation angle of this particular evaporator was >30 . However, they did not present whether a downstream duct was present or not. The presence of the downstream duct can affect significantly the fin performance as will be discussed later. Kim et al. [191] investigated the effect of inclination angle (0, 30, 45, and 60 ) on the heat transfer and pressure drop characteristics of a brazed aluminum heat exchanger under dry and wet conditions. The heat exchanger has folded multilouvered fins with La ¼ 27 , Fp ¼ 1.4 mm, Lp ¼ 1.7 mm, and Fd ¼ 20 mm. The louver length, tube pitch and fin height are 6.4, 10.15 and 8.15 mm, respectively, and the tested core size is 403 255 mm2. The experiments were conducted for the Reynolds numbers of 100–300. The inlet air temperature for the dry conditions was maintained at 21 C, whereas for the wet conditions, the inlet dry- and wet-bulb temperatures were 27 and 19 C, respectively. The pressure drops were measured with (Ptotal) and without (Pduct) the heat exchanger installed to obtain the net pressure drops (Pnet ¼ PtotalPduct) across the heat exchanger. For ¼ 45 , an upstream duct is also attached to investigate the effect of upstream turning on the performance of the heat exchanger. The sensible heat transfer coefficients and pressure drops for dehumidifying conditions were affected significantly by the air shear force at high Reynolds numbers, especially in case of larger inclination angles, and the gravitational force played an important role in condensate drainage. The sensible heat transfer coefficients for <0 (leeward inclination) were 3–19% higher than those for >0 (windward inclination) for ReLp ¼ 100–300, reflecting the greater difference in condensate drainage patterns at higher Reynolds numbers. However, the heat transfer data for both dry and wet conditions were neither influenced significantly by the inclination angle (60<<60 ), nor by the presence or absence of an upstream duct, while the pressure drops increased consistently with the inclination angle. When the upstream duct was present, the pressure drops were 7–16% higher than those for the case of absence of the upstream duct, due to the additional turn required for the air to flow through the folded fins of the heat exchanger. The total pressure drops for
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FIG. 17. Flow configuration of inlet and exit air flow (Kim et al. [191]).
the forward (windward direction) inclination were larger than those of the backward (leeward direction) inclination, reflecting the asymmetry associated with the louver geometry as shown in Fig. 17. When the upstream duct was not present, Kim et al. [191] proposed friction factor correlation for both total and net (heat exchanger only) pressure drops of the inclined heat exchanger based on the vertical position’s correlations [148,189] within rms errors 10.4 and 8.8% for dry conditions and 11.8 and 8.2% for wet conditions, respectively. However, note that the correlation was developed using the experimental data obtained from only one heat exchanger. Kim et al. [192] reported that the effect of inlet air humidity condition on the air-side heat transfer and pressure drop characteristics for an inclined brazed aluminum heat exchanger. They measured thermal hydraulic performance for the air-side Reynolds numbers of 80–400 using a folded multilouvered fin heat exchanger with La ¼ 27 , Fp ¼ 2.1 mm, Lp ¼ 1.4 mm and Fd ¼ 27.9 mm. The inclination angles from the vertical position were 0, 14, 45, and 67 (leeward direction). The heat transfer data were obtained for wet conditions only and the pressure drop data were measured for both dry (adiabatic) and wet conditions. The pressure drops for dry conditions were measured with isothermal condition, whereas for wet conditions the pressure drops were measured with variation of inlet humidity. The inlet air temperature and relative humidity range were 12 C and 60–90%, respectively. The sensible heat transfer coefficients decreased with the
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increase of air inlet humidity, since higher inlet humidity caused more condensate accumulation on the coil surface, which in turn acted as another thermal resistance for low Reynolds number flow case. However, the inlet humidity effect on the heat transfer coefficient was not significant for the small inclination angles ( 45 ). These results were similar to those of Wang et al. [178,184] who reported the effect of inlet humidity on the heat transfer was negligible for the conventional finned round tube heat exchangers. Its effect increased with inclination angle, and for ¼ 67 the heat transfer coefficients decreased significantly with the increase of air inlet humidity. The air-side pressure drops for both dry and wet conditions increased systematically with face velocity and inclination angle. The pressure drops for wet conditions were 3–14% larger than those for dry condition at the same face velocity. However, for ¼ 67 a significant pressure drop increase was observed, and this result was similar to that of Kim et al. [191] who reported that the pressure drop increased significantly when 60 . The inlet humidity had no significant influence on pressure drops, a result similar to that for the conventional finned round tube heat exchangers with fully wet surface [178]. However, Kim and Bullard [151] reported that the data obtained earlier in a nearly identical wind tunnel with a microchannel heat exchanger having a smaller fin-to-louver pitch ratio (Fp/Lp ¼ 1.4/1.7<1) and larger flow depth (Fd ¼ 41.8 mm) showed a significant effect of air inlet humidity on the air-side pressure drops at ¼ 0 . This difference could be attributed to the heat exchanger geometry as pointed out by Wang et al. [184] who found the effect of inlet humidity on the friction depended on the heat exchanger geometry, especially the longitudinal tube pitch. The heat exchanger tested in the study [192] has a larger fin-to-louver pitch ratio (Fp/Lp ¼ 2.1/1.4>1) and smaller flow depth (Fd ¼ 27.9 mm), so the effect of condensate amount on the surface may be smaller compared to the heat exchanger with smaller fin pitch and greater flow depth, suggesting the inlet humidity effect on the pressure drops depends on heat exchanger configuration, such as Fp/Lp and Fd. For a fixed inlet humidity, the heat transfer coefficients reach a maximum when ¼ 14 and then decrease with the increase of inclination angle. This initial increase may be due to the effect of gravitational force promoting condensate drainage. The cause of the subsequent decrease is less clear, as will be discussed below. Osada et al. [190] reported that both the leeward and windward inclinations increased the heat transfer coefficients and inclination toward windward direction had better performance. Recently, Kim et al. [191] reported also a modest inclination toward the leeward promoted drainage and so the heat transfer coefficients increased, but the windward inclination decreased the heat exchanger performance because the condensate had to drain by flowing against the airflow direction. This may be due
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FIG. 18. Louver array at the inlet and outlet of the heat exchanger.
to the different geometry of the heat exchangers used in their studies. Osada et al. [190] used a single row test piece with larger flow depth (Fd ¼ 58 mm), louver angle (La ¼ 35 ), Fp/Lp ¼ 1.5/1.2>1, and three redirection louvers, while Kim et al. [191] used a full-scale heat exchanger (Fd ¼ 20 mm, La ¼ 27 , and Fp/Lp ¼ 1.4/1.7<1) with one redirection louver. Another possibility can be due to the louver directions at the inlet and outlet of the heat exchangers. Osada et al. [190] installed the heat exchangers so that the inlet and outlet louvers turned the flow upward and downward, respectively, while Kim et al. [191,192] installed their heat exchanger in the opposition direction as shown in Fig. 18. This indicates the louver directions at the inlet and outlet of the heat exchanger affect significantly the thermal hydraulic performance for both dry and wet conditions when the heat exchanger is inclined from the vertical position.
3. Surface Treatment of Heat Exchanger Fins a. Surface Coating The wettability of the surface affects significantly on the nature of condensate retention and drainage, and therefore overall heat exchanger performance under wet conditions. The static contact angle or dynamic
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contact angles (advancing and receding contact angles) are usually used for indicating surface wettability, and wettability can be quantified by measuring contact angle. Smaller contact angles indicate more wettable surfaces. The hydrophilic coating can effectively reduce the contact angle and promote the condensate drainage. Thus, hydrophilic coatings in wet heat exchangers have been used to improve wettability of the fin surfaces and to protect the fins from corrosion. Several different types of surface coatings were developed to promote wettability, to provide anti-corrosion properties, and to eliminate undesired odors [193–201]. Hong and Webb [201] classified the surface treatment types as organic, inorganic, and organic–inorganic coatings. The conventional hydrophilic coatings usually exhibit good wettability in the initial stage, but the repeated wet/dry cycles cause hydrophilicity to lose gradually with time. Min et al. [202] and Min and Webb [203] studied the durability of hydrophilicity of the coated surfaces using the wet/dry cycling methods. Koga and Takeshige [193] developed a new type of process for a surface coating that added antimicrobial functions to the hydrophilic surface, and this process could eliminate undesired odors. Min and Webb [204] conducted condensation visualization tests using four typical fin surface materials, including aluminum, copper, two commercial coatings on aluminum, and established an experimental method for observing condensate formation and drainage on vertical fin surfaces. As the receding contact angle increased, the condensate retention first increased, reached a maximum at about 40 , and then began to decrease. They found that oil contamination increased the receding contact angles of all surfaces tested, indicating the fabricating process oil made wettability of the surface become poor. b. Effect of Surface Coating on Heat Exchanger Performance Mimaki [205] reported that the pressure drop of the wet coils with hydrophilic-coated fins was reduced up to 40–50%, whereas the cooling capacity of the system with surface treated coil was improved by 2–3% compared to one using untreated fins. Note that he did not provide the fin geometry tested, nor did he describe details of operating conditions and data reduction method. Wang and Chang [206] studied heat and mass transfer characteristics for 11 different louver geometries. They reported that the hydrophilic coating had little effect on the thermal hydraulic performance for dry coil as expected, whereas the sensible heat transfer coefficient for wet conditions was not affected significantly by the surface coating, and the friction losses were 15–40% lower than those without coating. Ha et al. [207,208] investigated the effects of wet/dry cycles on the performance of the wet heat
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exchanger with several different hydrophilic surfaces, and proposed pressure drop model for wet surfaces. Hong and Webb [209] investigated the effects of surface coatings on three different fin geometries (wavy, lanced, and louver). The effects of coatings on the wet and dry heat transfer coefficient were negligible, while the wet-todry pressure drop ratio was 1.2 at the face air velocity of 2.5 m/s. The sensible heat transfer coefficient under wet conditions was 10–30% less than that for dry conditions, depending on the particular surface geometry. Min et al. [202] investigated the long-term hydraulic performance of wet heat exchangers with and without hydrophilic coatings. They found that both advancing and receding contact angles of the uncoated heat exchanger reduced significantly as the wet/dry cycling increased due to formation of a transparent aluminum oxide scale on the coil surface. Thus, the wet-to-dry pressure drop ratio was decreased substantially as the number of cycles was increased. They also presented that the wet coil pressure drop was a linear function of the receding contact angle, while it could not be correlated with the advancing contact angle. This indicates that the receding contact angle is a key parameter of the condensate formation and drainage on the heat exchanger surface. c. Condensate Carryover Phenomena For vertical fins at low air velocity, condensate retained on the surface drains down the fins and exits at the base of the heat exchanger. If the air velocity is sufficiently high, then some of the condensate will be separated from the coil and become entrained as droplets in the downstream flow. This is called condensate carryover, and it is not a desired operating condition. The onset of condensate carryover will be closely related on the fin geometry and wettability. Understanding of condensate carryover in wet heat exchangers is important, but very little work has been reported. Brown et al. [210] developed an analytical model to predict the trajectory of a spherical particle/droplet placed in a uniform fluid flow. The model was described in general terms and expressed the horizontal carryover distance as a function of free stream velocity, gravitational force, fluid density, drag coefficient, vertical travel distance, droplet radius and droplet density. The model was compared to an experiment in which polypropolyene spheres of the diameter of 3.0 mm were dropped from a height of 133 mm in a uniform air velocity. The results from the analytical model agreed well with the experimental data. Note use of the model requires information of the droplet diameter and the droplet release height. Mathur [211] also presented model to predict the condensate carryover distance at a given face velocity, coil height and for a range of water droplet diameter.
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Hong [212] developed a condensate carryover model based on the assumption that condensate droplets were entrained from the liquid film on the fin surface, with a focus on establishing the carryover velocity for initiation. Min and Hong [213] investigated the condensate carryover phenomena in wet heat exchangers using two wavy fin-and-tube coils with 492 fins/m and 317.5 mm fin height. They reported that the receding contact angle was a key factor controlling the condensate carryover characteristics, and the observed carryover caused by entrainment of bridged condensate. The condensate carryover for a coil having a 10 receding contact angle was substantially lower than a coil having a 70 receding contact angle at the same face velocity. Numerous condensate droplets and bridges were seen on the surface of the 70 receding contact angle coil, but few were observed for the 10 receding contact angle coil. d. Modeling Condensate Retention It is very important to develop a physical model, which can predict the condensate retention and drainage on the fin surfaces of dehumidifying cooling heat exchangers, but very limited work is available in the open literature. Some work has been reported for modeling condensate film drainage during the condensation of pure vapor without vapor flow force [214–216]. Rudy and Webb [214] developed a general analytical model to predict the portion of surface that was covered by condensate retention on a horizontal integral low-fin tube. They reported that retention was intensified for a heat exchanger with small fin spacing since they did not account for the surface tension effects. Rudy and Webb [215] and Webb et al. [216] developed the combined models taking into account the effects of surface tension induced drainage. Jacobi and Goldschmidt [217] developed a simple model of condensate drainage retained as bridges formed between adjacent fins. The model was qualitatively successful in explaining the crossover in sensible heat transfer coefficient for wet conditions they measured. Korte and Jacobi [181] developed a general condensate retention model and compared to experimental data. D. FROSTING CONDITIONS When the heat exchanger surface temperature is below the freezing point of water and the dew point of the inlet moist air, the heat exchanger surface is treated as a frosting surface. A very thin frost layer can enhance the heat transfer due to increased surface area and roughness, however the frost continues to accumulate on the surface and gradually block the airflow passages. The blockage of the airflow passages reduces airflow through the
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heat exchanger and degrades the system capacity and its efficiency. Eventually, defrost cycle is required to melt the accumulate frost and to mitigate the performance degradation due to the frost in refrigeration and heat pump applications. However, system performance and comfort level are also impaired during defrost cycle, and it is required to extend the operating cycle between the two defrosting cycles. Therefore, the effect of the frost on the heat exchanger and system performance should be known for intelligent design and operation of the system. Water retention and drainage issues during defrost cycle are also important because the refreezing of melted water can result in structural changes in the frost layer that may increase both the frost density and conductivity. There are many studies on the frost properties and frost growth mechanism in simple geometries [218–267], and the effects of frost growth on the conventional finned tube heat exchanger [268–286] and system performance [287–294]. Frost properties such as crystalline structure, density and thermal conductivity affect inherently the performance of heat exchanger. Frost formation is affected by surface condition (hydrophilic or hydrophobic surface), but its effect on the frost thickness also depends on frost growth stage. Hayashi et al. [232] originally classified frost growth stages into three categories (crystal growth, frost layer growth, and frost layer full growth period), but they did not consider condensation period. Padki et al. [248] conducted an extensive literature survey and their brief review attempted to classify some of the important issues on frost formation. O’Neal and Tree [245] and Kondepudi and O’Neal [277] reviewed the literature on frost formation in simple geometries, and on the effects of frost formation and growth on finned tube heat exchanger performance, respectively. However, only one study dealing with the effects of frost on the microchannel heat exchangers has been published in the open literature to date [295]. Itoh et al. [295] studied the effect of air-side configurations on the thermal hydraulic performance of microchannel heat exchangers. Fig. 19 and Table VI present the schematic and specifications of heat exchangers considered in their studies, respectively. They investigated pressured drop, flow rate, and heat transfer rate for three different heat exchangers as a function of the duration of operation during frosting, and compared them with those for the baseline heat exchanger. They performed the experiment using R-22 as a refrigerant, and inlet air (dry-/wet-bulb) and refrigerant temperatures were 2/1 and 8 C, respectively. After around 20 min from the operation, the heat transfer rate for the prototype-1 decreased below 50% of the baseline, which was a fin and tube heat exchanger having offset slit fins only in the rear low. This was mainly due to small flow rate due to large pressure drop, indicating the part of air passage was blocked in by
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FIG. 19. Heat exchanger configurations. (Itoh et al. [295]).
frost. To resolve the problems involved, they considered the prototype heat exchanger-2 and 3 with separate plate and fins protruded from the front of tubes as shown in Fig. 19. Although flow rate increases due to smaller pressure drop, heat transfer rates are lower than the baseline, indicating fin efficiencies are small due to the large fin height and Fd>Td. As shown in Fig. 20, heat transfer rate is constant or increases due to increased surface area and roughness at the beginning of the frosting process. However, after around 20 min, heat transfer rates for all the heat exchanger configurations considered are below 75% of the baseline. They reported that the air-side configuration should be thoroughly modified in order to apply the
TABLE VI
Items
Fin type
Fin height, Fh (mm) Fin pitch, Fp (mm) Louver pitch, Lp (mm) Louver angle, La ( ) Tube depth, Td (mm) Tube diameter, Do (mm) Flow depth, Fd (mm)
OF
HEAT EXCHANGERS Microchannel heat exchangers
Fin and tube heat exchanger (baseline)
Prototype-1
Prototype-2
Prototype-3
Plane fin in the front low and offset slit fin in the rear low
Folded louver fin
Folded louver fin with splitter plate
Folded louver fin with splitter plate
25 1.6 – – – 7.0 31.25
8.1 1.4 1.4 30 20 2.0 20
16.2 1.8 1.2 40 16 3.0 20
16.2 1.5 1.0 20, 30, 40 16 3.0 24
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FIG. 20. Heat exchanger performance for different design options (Itoh et al. [296]).
microchannel heat exchangers to the evaporator for the heat pump applications. E. CONCLUDING REMARKS The literature survey indicates that a considerable amount of studies have been conducted on the air-side performance characteristics of louvered-finmicrochannel-tube heat exchangers under dry conditions. The flow structure in the louvered array is essential to provide the better understanding of the flow and its influence on thermal hydraulic performance of folded louvered fins. Although the overall flow patterns and the performance of this geometry seem to be well understood, detailed information for developing flow and heat transfer in multi-louvered fin arrays in the range of low Reynolds laminar flow to an unsteady transitional flow regime is not
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available. Recently numerical modeling results on the complex unsteady flow configuration of the louvered fin arrays have been reported [126–129] and further study is required to provide valuable information on thermal hydraulic characteristics of this heat exchanger surface. On the other hand, very few data exist on the study of folded louver-finmicrochannel-tube heat exchangers under wet and frosting conditions. The effect of condensate retention on sensible heat transfer under wet conditions appears to be dependent on the particular heat exchanger configuration and operating condition as well as the data reduction methods associated with the definition of fin efficiency for wet surface. Although many investigators have reported the effects of condensate retention on fin-and-tube heat exchanger performance, few studies have focused on surface wetability effects or made observations regarding the quantity and nature of retained condensate for microchannel heat exchangers. The effect of inclination of microchannel heat exchangers on the air-side performance under dry and wet conditions is also important, but few reports are available. It is desirable to explore the use of microchannel heat exchangers in air-conditioning and heat pump applications, where normal operating conditions give rise to condensate retention and frost formation on the air-side heat transfer surface.
V. Heat Exchanger Applications Significant progress has been made in the past 15 years in use of heat exchangers having small channel size. The automotive industry has made significant progress in using air-cooled evaporators and condensers having small hydraulic diameter passages for refrigerant flow. The automotive heat exchanger technology involves ‘‘brazed aluminum’’ construction. Such brazed aluminum heat exchangers offer high performance and low size and weight. This technology is under serious consideration for residential airconditioning. However, such technology must compete with the low cost finand-tube technology using mechanically expanded round tubes. Webb and Lee [296] provide a performance and cost comparison of brazed aluminum and expanded round tube technology for equal performance constraints. This analysis showed that the brazed aluminum is a little more costly. Hence, penetration of brazed aluminum technology into the HVAC industry will likely occur first where reduced size and weight are valued. Examples of this need are window units and transport refrigeration. Microchannel heat exchangers are finding interest in cooling of electronic equipment. Typically, the Central Processor Unit (CPU) in desktop computers has been cooled by a small aluminum heat sink attached to the
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CPU. This ‘‘active heat sink’’ has a small fan attached to it for air flow. Because of the increasing power of the CPU, and size and noise constraints, practical cooling limits of the ‘‘active heat sink’’ are being approached as discussed by Saini and Webb [297]. Serious work is underway to evaluate possibilities for cooling the CPU using liquid or two-phase cooling in multiple, parallel microchannels.
A. BRAZED ALUMINUM CONDENSERS The automotive air-cooled condenser (Fig. 21(b)), uses a flat extruded multi-port tube of very small hydraulic diameter, such as illustrated in Fig. 22. These condensers have replaced round tube condensers illustrated in
FIG. 21. Fin-and-tube and brazed aluminum condensers.
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FIG. 22. Photographs of tubes tested by Webb and Ermis [30].
Fig. 21(a). A flat tube presents less projected frontal area to the air stream, and hence will have reduced air-side pressure drop for equal frontal area. Further, use of ‘‘parallel-flow’’ refrigerant circuiting (Fig. 21(b)) improves refrigerant-side performance, relative to serpentine circuiting (Fig. 21(a)) that is typically used in round tube condensers. Parallel flow circuiting means that the total refrigerant flow is divided among a number of tubes each joined to a header pipe (manifold). The condensers typically use more than one pass between the two headers. Brazed aluminum technology is also used for automotive evaporators, radiators, charge-air coolers, and condensers for residential air-conditioners. Figure 21(b) illustrates a parallel flow condenser, and shows the method of dividing the flow at the manifold headers to provide multi-passes on the refrigerant side. As shown in Fig. 21(b), the number of tubes in parallel is reduced with each subsequent pass. This maintains high vapor velocity (and high condensing coefficients) in each pass. For fixed mass velocity, and vapor quality, the condensing coefficient increases as the tube diameter is decreased. Excessive refrigerant pressure drop is avoided by using multiple tubes in parallel to provide a reasonable cross-section flow area. Because of the small minor diameter of the flat tube, the air-side pressure drop is significantly smaller than will occur for condensers made with round
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tubes (Fig. 21(a)). As the tube minor diameter is reduced, the tube-side hydraulic diameter decreases. Commercial automotive condensers are used today with tube-side hydraulic diameters as small as 0.44 mm. Kim and Bullard [92] measured thermal hydraulic performance of two brazed aluminum heat exchangers using a refrigerant R-22 and system performance of a residential window room air-conditioner with the brazed aluminum condensers. Heat transfer rates per unit core volume of brazed aluminum heat exchangers were 14–331% higher than those of the conventional finned round-tube heat exchangers. When a brazed aluminum heat exchanger was installed to the window air-conditioning system as a condenser, the similar system performance was obtained, while refrigerant charge, condenser core volume and weight could be reduced by 35, 55 and 35%, respectively. Smaller refrigerant charge and condenser weight imply reduced cycling losses due to the refrigeration migration and the thermal mass of the heat exchanger, and therefore better seasonal performance.
B. TUBE-SIDE DESIGN
OF THE
AUTOMOTIVE CONDENSER
The parallel flow condenser uses multiple tubes in each pass. Webb and Ermis [30] report condensing data in Fig. 22, small hydraulic diameter tube geometries and discuss design issues related to the effect of tube hydraulic diameter. The number of tubes in each pass (Nt,p) is selected by the designer, as is the number of passes. The mass velocity in each pass is calculated as G ¼ m_ =ðNt;p Aci Þ where m_ is the mass flow rate and Aci is the flow area of one tube. Small tube minor diameter is preferred, because this will provide more air-side surface area for the same tube pitch, and lower pressure drop, because the small minor diameter provides less profile drag on the tube. Further, the required refrigerant charge will be reduced. To obtain the same circuit pressure drop with a small Dh tube as for a larger Dh, one may reduce the number of passes and/or reduce the mass velocity. To obtain the same condenser pressure drop for the same number of passes, the reduced Dh tubes will operate at a moderately lower mass velocity than a higher Dh tube. In the limit, a tube having smaller minor diameter will be preferred over a tube having larger minor diameter, even if the small Dh tube has no higher hAi/L than the larger tube. This is because of the benefits derived on the air-side. The condenser design involves selecting the required number of passes for the specific tube, and further selecting the optimum number of tubes/pass in each pass. One may decrease the number of tubes in successive passes of the condenser, because the lower average vapor quality will cause the dPf/dz to decrease.
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C. BRAZED COPPER AIR-COOLED HEAT-EXCHANGERS Recent technology advances have occurred in the use of copper to make air cooled heat exchangers that can compete with brazed aluminum heat exchangers for certain applications. This brazing process is a recent development and is described by Sundberg et al. [298]. The copper industry has developed the ‘‘Cupro-Braze’’ brazing process, which uses a Cu–Ni–Sn– P braze alloy that melts at approximately 600 C, which is the same temperature used for aluminum brazing. Either vacuum or inert gas furnaces may be used. No flux is required for brazing, and the braze alloy may be applied either in paste form or flame sprayed. This process provides very strong, corrosion resistant joints. Such heat exchangers use a welded copper (or brass) tube made of 0.15 mm wall copper strip. A corrugated insert may be placed in the tube and brazed to the flat tube wall to provide structural strength and small hydraulic diameter. Webb [299] describes the technology and performance of brazed copper–brass automotive radiators. Because copper has 100% higher thermal conductivity than aluminum, copper fins may be thinner than aluminum fins. Thus, a copper/brass heat exchanger will typically have lower air pressure drop than an aluminum heat exchanger for the same frontal area and air flow rate. Such copper/brass technology will maintain structural strength at higher operating temperatures than is possible with brazed aluminum. An ideal application for copper/brass technology is charge-air coolers. Another possible application for copper/brass technology is in electronic equipment cooling using water as the working fluid in a thermo-syphon heat rejection system. This is described by Webb and Yamauchi [300]. Automotive charge-air coolers is a new application for the CuproBrazeTM heat exchanger. This is because charge-air coolers operate at a sufficiently high operating temperature that brazed aluminum does not have sufficient structural strength.
D. ELECTRONIC EQUIPMENT COOLING Microchannel heat exchangers are finding interest in cooling of electronic equipment. One concept involves transferring the heat from the CPU to a two- or single-phase liquid that flows through multiple, parallel microchannels that are etched in a silicon wafer. The heat absorbed in the microchannel heat exchanger is dissipated in a separate water or air-cooled heat exchanger. Koo et al. [301] have proposed use of two-phase cooling using multi-microchannels machined in a silicon substrate. Their concept is illustrated in Fig. 23. They propose to use an electro-kinetic pump to force subcooled water through the microchannels, where vaporization occurs.
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FIG. 23. Schematic of a microchannel heat sink from Koo et al. [301].
Koo et al. [301] have provided predicted results (supported by singlechannel tests) for a 25 mm square heat sink having either 100 mm wide, 150 mm channel depth with 125 channels in parallel. Koo et al. [301] performed single-channel boiling tests in a 50 70 mm2 channel crosssection, 20 mm long and compared their predictions with the single-channel data. The data and predictions agreed reasonably well. Jiang et al. [302] constructed and tested a multi-microchannel heat sink similar to the concept of Koo et al. [301]. They used 40 parallel channels and 100 mm square etched in a 20 29 0.5 mm thick silicon wafer with water as the working fluid. Flow mal-distribution is a major concern in any application involving multiple, parallel channels. This will be discussed later. The previously noted ‘‘thermo-syphon’’ device tested by Webb and Yamauchi [300] consists of a boiler, the connecting pipe, and an air-cooled condenser. The working fluid is boiled in a small chamber mounted on the CPU, and the vapor is condensed in the tubes of the remotely located aircooled condenser and gravity drained to the boiler. They built and tested two versions of the thermo-syphon device. One is made of aluminum and uses R-134a working fluid. The other is copper and uses water working fluid. The copper unit was designed to reject 100 W at the balance point on the 80 mm diameter fan curve. The design has 54 mm finned width 70 mm tube length and uses a tube cross-section is 16 1.95 with 0.15 mm wall thickness. A 0.6 mm thick Cu sintered surface is applied to the bottom surface of the boiler. E. WORKING FLUIDS Brazed aluminum heat exchangers have been quite successful for current automotive air-conditioning applications. However, present environmental concerns may place severe limits on permissible working fluids. Future desirable working fluids appear to be limited to air, water, propane, and CO2. Kim et al. [303] gave a critical review of fundamental process and system design issues for CO2 vapor compression systems. They presented
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the characteristics of CO2 as a refrigerant and the potential applications of the transcritical cycle technology in various refrigeration, air-conditioning and heat pump systems. The automotive industry is presently considering replacement of R-134a with CO2 for automotive air-conditioning. The operating pressures range from subcritical (3–4 MPa) to 10–15 MPa and thermophysical properties vary dramatically near the critical region, where specific heat and Prandtl number approaches infinity. This operating pressure of a CO2 airconditioner is approximately 10 times higher than is required for R-134a. Fortunately, it is possible to design a brazed aluminum heat exchanger for this high operating pressure (10–15 MPa). Pettersen et al. [304] reported microchannel heat exchangers (gas cooler and evaporator) could be used for CO2 air-conditioning systems, but several issues remained to be investigated, including frosting formation of outdoor heat exchangers and condensate drainage at low face velocity, for residential systems. Figures 24 and 25 show gas cooler and evaporator configuration designed by Pettersen et al. [304]. Water may be the preferred working fluid for electronic cooling, such as the thermo-syphon concept described by Webb and Yamauchi [300]. Brazed aluminum cannot be used with pure water in a closed system, because aluminum is susceptible to pinhole corrosion. For this case, copper construction is preferred. F. FLOW DISTRIBUTION CONCERNS Much of the future heat exchanger technology using microchannels for the tube-side liquid (or two-phase) fluids involve use of multiple tubes connected in parallel. The fluid is supplied to a manifold, and then distributed to the multiple parallel tubes. It is difficult to get uniform flow distribution into the multiple branch channels. This is particularly true, if a two-phase fluid is used. One should be particularly careful in selecting a manifold design for distributing liquid in multiple parallel channels. Rao and Webb [16] and Webb and Rao [305] provide CFD analysis that show that the data for multiple-channels may be significantly affected by poor flow distribution in the headers. Several important studies have been done concerning twophase flow distribution. Among these are Azzapardi [306], and Reimann and Khan [307]. The inertia of the liquid will tend to promote maldistribution of two-phase flows. Another concern is the orientation of the headers whether they are oriented in the vertical or horizontal directions. It is possible that vertical headers will give better flow distribution than horizontal headers. However, vertical headers may not be practical for evaporators, because condensate
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FIG. 24. A prototype microchannel CO2 gas cooler for a car air-conditioning system. (Pettersen et al. [304]).
drainage from the fins will be poorer than for horizontally oriented headers. There is very little practical information on two-phase flow distribution. G. MODEL
FOR
MICROCHANNEL HEAT EXCHANGERS
Microchannel heat exchangers are commonly used as both indoor and outdoor heat exchangers for prototype CO2 systems. In the CO2 systems, a gas cooler operates at a supercritical state and differs substantially from
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FIG. 25. Cross-section of the header pipe and microchannel tube of a prototype CO2 evaporator for a car air-conditioning system (Pettersen et al. [304]).
conventional condensers, while the evaporator is operated at conditions similar to those for the conventional evaporators. Therefore, a model of the microchannel evaporator for the CO2 system can be developed based on methods similar to those used for the conventional finned tube heat exchangers. Since microchannel technology is relatively new, however, there are few publications about heat exchanger design issues [151,296,304,308– 310]. Kim and Bullard [151] developed a detailed finite volume model for a multi-slab microchannel evaporator and validated the model for a two-slab prototype evaporator. Several correlations for air- and refrigerant-side heat transfer and friction loss were compared before selecting appropriate correlations for the model. They reported their model predicted the experimental data with reasonable accuracy, and could be used for the performance analysis and designing of a microchannel evaporator. Yin et al. [308] developed a gas cooler model for CO2 air-conditioning systems and compared simulation results with experimental data. They performed simulation work for several design options of gas coolers using the gas cooler model associated with a system design model, and presented multi-slab gas cooler design is the more effective way to improve the system performance. Bullard and Hrnjak [309] and Yin et al. [310] reported COP can be improved 11% and discharge pressure reduced 0.5 MPa by a 2 C more reduction in the refrigerant exit temperature as shown in Fig. 26. This indicates a CO2 transcritical cycle is so sensitive to the refrigerant exit condition and counter flow configuration is important for the gas cooler
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FIG. 26. Optimal discharge pressure for a CO2 air-conditioning system (Bullard and Hrnjak [309]).
FIG. 27. Gas cooler design for a CO2 air-conditioning system (Yin et al. [310]).
design since refrigerant-side has a large temperature glide. Yin et al. [310] proposed a multi-slab gas cooler design (Fig. 27(b)), and reported the new design offered better performance than the commonly used multi-pass design (Fig. 27(a)). For the given heat exchanger volume, they reported that a newly designed cross-counter flow gas cooler can be improved system capacity and COP by 3–4 and 5%, respectively, compared to the old design (Fig. 27(a)). H. CONCLUDING REMARKS In the past, ‘‘microchannel’’ heat exchangers have been made of brazed aluminum. However, new copper fabrication technology is creating opportunities for copper air-cooled heat exchangers having flat tubes. By using a brazed, corrugated insert, high strength and small hydraulic diameter can be achieved. Such heat exchangers may be closely weight
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competitive with brazed aluminum technology. Use of copper also creates the possibility for water as the working fluid, without the need for corrosion inhibiters. Substantial advances have been made in extrusion technology to form small hydraulic diameter aluminum tube extrusions. Semi-analytically based correlations exist for prediction of two-phase heat transfer (condensation and evaporation) and pressure drop. However, equivalent understanding of flow distribution in the headers does not exist and further work is needed. Good predictive ability exists for air-side heat transfer and pressure drop. New applications for brazed microchannel heat exchangers are expected, especially where small size and weight are important. Air-conditioning applications are obvious. Electronic equipment cooling is a new and exciting application for both brazed copper and aluminum air-cooled heat exchangers. Use of very small micro-channels (e.g., 50 mm) will allow liquid cooling with a very small heat input area.
VI. Conclusion The state of the art in extruded microchannel heat exchanger design for evaporator and condenser applications is critically reviewed. The article covers single- and two-phase flows in microchannels, two-phase flow distribution issues in the extruded-microchannel design, air-side flow patterns and performance data under dry, wet and frosting conditions. Emerging applications and current difficulties arising from a lack of information on flows in small passages and manifolds are also reported. For single-phase flows within microchannels, both the frictional pressure drop and heat transfer correlations developed for large channels are considered valid without serious discrepancies. However, for two-phase microchannel flows, the conventional correlations for large tubes are unacceptable. This is because the flow pattern strongly depends on the channel size, and hence, the momentum and heat transfer mechanisms become different. Since no general correlation on the pressure drop and heat transfer is available in the present stage, the parametric ranges of each specific correlation should be checked carefully before using them. Relatively fewer works have been reported on the condensation than on the evaporation. No acceptable methodology has been reported to predict the two-phase flow distribution for arbitrary flow conditions and header-tube configurations. The operating factors that influence the flow distribution are: total mass flow rate, vapor quality at the inlet of the distributor, and the heating load on each branch. Very little information is available in the form of models and/or correlations to predict the distribution of the liquid and
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vapor phases in each pass or tube for a given evaporator geometry and operating conditions. Further research and high-quality data on the distribution of two-phase flow for the complex header geometries found in various evaporators are necessary. Considerable literature and correlations have been published to predict the air-side performance, particularly of louver fins under dry conditions. Some information exists on air-side characteristics for de-humidification. These results are influenced by the wettability of the fins. Detailed information on developing flow and heat transfer in louvered fin arrays for the transitional and low-Reynolds laminar flow regimes is not available. It is desirable to explore the use of microchannel heat exchangers in airconditioning and heat pump applications, where normal operating conditions give rise to condensate retention and frost formation on the air-side heat transfer surface. Little information exists related to frosting on the louver-fin microchannel heat exchangers. Conventionally, microchannel heat exchangers have been made of brazed aluminum. However, new copper fabrication technology is creating opportunities for copper air-cooled heat exchangers having flat tubes. New applications for brazed microchannel heat exchangers are expected especially where small size and weight are important. Air-conditioning applications are obvious. Electronic equipment cooling is a new and exciting application for both brazed copper and aluminum air-cooled heat exchangers. Copper microchannel heat exchangers allow higher temperature operation than is possible with aluminum; one such application is for charge-air coolers.
Acknowledgements This work was partly supported by a grant from the Critical Technology Project of the Ministry of Science and Technology of Korea and in part by the Brain Korea 21 Project. The first author is grateful for supporting of this work to the Air Conditioning and Refrigeration Center (ACRC) at the University of Illinois at Urbana-Champaign.
Nomenclature A Ac AR C
area [m2] minimum free-flow area of air-side [m2] aspect ratio (¼ s/w) [–] parameter in Chisholm correlation [–]
Co cp D Dh dt
distribution parameter [–] specific heat [J/kg K] diameter [m] hydraulic diameter [m] transverse distance that flow travels [m]
MICROCHANNEL HEAT EXCHANGER
F f Fd Fe Fh Fp Fth Fa G g h h ifg j j jG k Kc Ke L L Lh Ll Lp La m_ Nc Nl Ns Nt,p p P q00 s T Td Tp U u Vc Vo w X x z
ratio of boiling heat transfer to single-phase heat transfer [–] friction factor [–] flow depth [m] flow efficiency [–] fin height [m] fin pitch [m] fin thickness [mm] mean flow angle [ ] mass velocity [kg/m2 s] gravitational acceleration [m/s2] height [m] heat transfer coefficient [W/m2 C] latent heat of vaporization [J/kg] Colburn j-factor [–] superficial velocity [m/s] dimensionless superficial gas velocity [–] thermal conductivity [W/m C] abrupt contraction coefficient [–] abrupt expansion coefficient [–] pipe (tube) length [m] heat exchanger length [m] louver height [m] louver length [m] louver pitch [m] louver angle [ ] mass flow rate [kg/s] number of columns [–] umber of louvers [–] number of heat exchanger samples [–] number of tubes in each pass [–] pressure [Pa] pressure drop [Pa] heat flux [W/m2] channel height [m] temperature [ C] tube depth [m] tube pitch [m] velocity [m/s] face velocity [m/s] maximum air velocity [m/s] volumetric flow rate [m3/min] channel width [m] Martinelli parameter [–] quality [–] axial coordinate [m]
GREEK LETTERS a
void fraction [–]
" "
2 Zf Zo
l m
413
dimensionless physical property in Eq. (90) [–] pipe surface roughness [m] finning factor (Ao/Ato) [–] two-phase frictional multiplier [–] fin efficiency [–] surface effectiveness [–] mean free path [m] dimensionless parameter defined as Eq. (74) [–] dynamic viscosity [Ns/m2] kinematic viscosity [m2/s] inclination angle [ ] density [kg/m3] contraction ratio (Ac/Afr) [–] surface tension [kg/s2] shear stress [Pa] dimensionless parameter defined as Eq. (75) [–] stream function [–] correction factor [–]
DIMENSIONLESS NUMBERS Bo Co Eo Fr Kn NCONF NmL Nu Re ReLp
ReLp1 Pr St We
Boiling number Convection number Eotvos number Froude number Knudsen number Confinement number Viscosity number Nusselt number Reynolds number Reynolds number based on louver pitch and maximum air velocity (VcLp/) Reynolds number based on louver pitch and face velocity (uLp/) Prandtl number Stanton number Weber number
SUBSCRIPTS 1 2 A app c CONV cor
inlet outlet average value apparent critical two-phase forced convection correlation
414 D Dh duct eq f fr G Gj Go h h i in j L l LF Lo
M.-H. KIM ET AL.
diameter hydraulic diameter duct equivalent flooded (condensation) frontal gas (vapor) drift value entire flow as gas phase homogeneous height interface inlet junction liquid length liquid film entire flow as liquid phase
Lp M m max min net o NcB p Pin r sat total TP t th tt u
louver pitch main tube mean value maximum minimum net air-side nucleate boiling pitch inlet of a T-junction branch relative value saturation total two-phase tube thickness turbulent (liquid)–turbulent (gas) unflooded (condensation)
References 1. Cornwell, K. and Kew, P. A. (1993). Boiling in small parallel channels. In ‘‘Energy Efficiency in Process Technology’’ (P. A. Plivachi, ed.), pp. 624–638. Elsevier, New York. 2. Mehendale, S. S., Jacobi, A. M., and Shah, R. K. (2000). Fluid flow and heat transfer at micro- and meso-scales with application to heat exchanger design. Appl. Mech. Rev. 53(7), 175–193. 3. Moriyama, K., Inoue, A., and Ohira, H. (1992). The thermohydraulic characteristics of two-phase flow in extremely narrow channels—the frictional pressure drop and heat transfer of boiling two-phase flow, analytical model. Trans. JSME 58(546), 393–400. 4. Moriyama, K., Inoue, A., and Ohira, H. (1992). The thermohydraulic characteristics of two-phase flow in extremely narrow channels—the frictional pressure drop and void fraction of adiabatic two-component two-phase flow. Trans. JSME 58(546), 401–407. 5. Pfahler, J., Harley, J., Bau, H. H., and Zemel, J. (1991). Gas and liquid flow in small channels, micromechanical sensors, actuators and systems. ASME DSC 32, 49–60. 6. Damianides, C. A. and Westwater, J. W. (1988). ‘‘Two-phase Flow Patterns in a Compact Heat Exchanger and in Small Tubes’’. Proc. 2nd UK National Conf. on Heat Transfer, pp. 1257–1268. 7. Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., and Sadowski, D. L. (1999). Gas– liquid two-phase flow in microchannels. Part I: Two-phase flow patterns. Int. J. Multiphase Flow 25, 377–394. 8. Barajas, A. M. and Panton, R. L. (1993). The effect of contact angle on two-phase flow in capillary tubes. Int. J. Multiphase Flow 19, 337–346. 9. Zhao, L. and Rezkallah, K. S. (1993). Gas–liquid flow patterns at microgravity conditions. Int. J. Multiphase Flow 19(5), 751–763. 10. Rezkhallah, K. S. (1996). Weber number based flow-pattern maps for liquid–gas flows at microgravity. Int. J. Multiphase Flow 22(6), 1265–1270. 11. Bousman, W. S., McQuillen, J. B., and Witte, L. C. (1996). Gas–liquid flow patterns in micro-gravity: Effects of tube diameter, velocity and surface tension. Int. J. Multiphase Flow 22(6), 1035–1053.
MICROCHANNEL HEAT EXCHANGER
415
12. Lowe, D. C. and Rezkallah, K. S. (1999). Flow regime identification in microgravity twophase flows using void fraction signals. Int. J. Multiphase Flow 25, 433–457. 13. Fukano, T. and Kariyasaki, A. (1993). Characteristics of gas–liquid two-phase flow in a capillary tube. Nucl. Eng. Design 141, 59–68. 14. Peng, X. F. and Wang, B. X. (1994). ‘‘Liquid Flow and Heat Transfer in Microchannels With/Without Phase Change’’. Heat Transfer 1994, Proc. 10th Int. Heat Transfer Conference, Vol. 5, pp. 159–177. 15. Peng, X. F., Peterson, G. P., and Wang, B. X. (1994). Frictional flow characteristics of water flowing through rectangular microchannels. Exp. Heat Transfer 7, 249–264. 16. Rao, P. and Webb, R. L. (2000). ‘‘Effects of Flow Mal-distribution in Parallel Microchannels’’. Paper NHTC-2000 12102, Proc. 2000 National Heat Transfer Conference, Pittsburgh, PA. 17. Olsson, C. and Sunden, B. (1994). ‘‘Pressure Drop Characteristics of Small-sized Tubes’’. ASME Paper No. 94-WA/HT-1. 18. Shah, R. K. and London, A. L. (1978). Laminar flow forced convection in ducts. In ‘‘Advances in Heat Transfer Suppl. 1., pp. 196–202. Academic Press, New York. 19. Hartnett, J. P. and Kostic, M. (1989). Heat transfer to Newtonian and non-Newtonian fluids in rectangular ducts. Adv. Heat Transfer 19, 247–356. 20. Mishima, K., Hibiki, T., and Nishihara, H. (1993). Some characteristics of gas–liquid flow in narrow rectangular ducts. Int. J. Multiphase Flow 19(1), 115–124. 21. Wambsganss, M. W., Jendrzejczyk, J. A., France, D. M., and Obot, N. T. (1992). Frictional pressure gradients in two-phase flow in a small horizontal rectangular channel. Exp. Thermal Fluid Sci. 5, 40–56. 22. Lee, H. J. and Lee, S. Y. (2001). Pressure drop correlations for two-phase flow within horizontal rectangular channels with small heights. Int. J. Multiphase Flow 27, 783–796. 23. Kakac, S. and Yener, Y. (1995). ‘‘Convective Heat Transfer’’, 2nd edn., p. 139. CRC Press. 24. Tran, T. N., Wambsganss, M. W., and France, D. M. (1996). Small circular- and rectangular-channel boiling with two refrigerants. Int. J. Multiphase Flow 22(3), 485–498. 25. Tran, T. N., Chyu, M.-C., Wambsganss, M. W., and France, D. M. (2000). Two-phase pressure drop of refrigerants during flow boiling in small channels: An experimental investigation and correlation development. Int. J. Multiphase Flow 26, 1739–1754. 26. Lin, S., Kwok, C. C. K., Li, R.-Y., Chen, Z.-H., and Chen, Z.-Y. (1991). Local frictional pressure drop during vaporization of R-12 through capillary tubes. Int. J. Multiphase flow 17, 95–102. 27. Churchill, S. W. (1977). Frictional equation spans all fluid flow regimes. Chem. Eng. 84, 91–92. 28. Zhang, M. and Webb, R. L. (2001). Correlation of two-phase friction for refrigerants in small-diameter tubes. Exp. Thermal Fluid Sci. 25, 131–139. 29. Tran, T. N., Wambsganss, M. W., France, D. M., and Jendrzejczyk, F. A. (1993). Boiling heat transfer in a small, horizontal, rectangular channel. AIChE Symp. Ser. 89(295), 253–261. 30. Webb, R. L. and Ermis, K. (2001). Effect of hydraulic diameter on condensation of R-134a in flat, extruded aluminum tubes. J. Enhanced Heat Transfer 8, 77–90. 31. Wambsganss, M. W., France, D. M., Jendrzejczyk, F. A., and Tran, T. N. (1993). Boiling heat transfer in a horizontal small-diameter tube. Trans. ASME, J. Heat Transfer 115, 963– 972. 32. Petukov, B. S. (1970). Heat transfer and friction in turbulent pipe flow with variable physical properties. In ‘‘Advances in Heat Transfer’’, pp. 503–564. Academic Press, New York. 33. Yan, Y.-Y. and Lin, T.-F. (1998). Evaporation heat transfer and pressure drop to refrigerant R-134a in a small pipe. Int. J. Heat Mass Transfer 41, 4183–4194.
416
M.-H. KIM ET AL.
34. Yan, Y.-Y. and Lin, T.-F. (1999). Condensation heat transfer and pressure drop of refrigerant R-134a in a small pipe. Int. J. Heat Mass Transfer 42, 697–708. 35. Wambsganss, M. W., Jendrzejczyk, J. A., and France, D. M. (1991). Two-phase flow patterns and transitions in a small, horizontal, rectangular channel. Int. J. Multiphase Flow 17(3), 327–342. 36. Lowry, B. and Kawaji, M. (1988). Adiabatic vertical two-phase flow in narrow flow channels. AIChE Symp. Ser. 84(263), 133–139. 37. Taitel, Y., Barnea, D., and Dukler, A. E. (1980). Modeling flow pattern transition for steady upward gas–liquid flow in vertical tubes. AIChE J. 26, 345–354. 38. Suo, M. and Griffith, P. (1964). Two-phase flow in capillary tubes. J. Basic Eng. 86, 576–582. 39. Ghiaasiaan, S. M. and Abdel-Khalik, S. I. (2001). Two-phase flow in microchannels. Adv. Heat Transfer 34, 145–254. 40. Mishima, K. and Hibiki, T. (1996). Some characteristics of air–water two-phase flow in small diameter vertical tubes. Int. J. Multiphase Flow 22(4), 703–712. 41. Troniewski, L. and Ulbrich, R. (1984). Two-phase gas–liquid flow in rectangular channels. Chem. Eng. Sci. 39(4), 751–765. 42. Taitel, Y. and Dukler, A. E. (1976). A model for predicting flow regime transition in horizontal and near horizontal gas–liquid flow. AIChE J. 22, 47–55. 43. Barnea, D., Lulinkski, Y., and Taitel, Y. (1983). Flow in small diameter pipes. Can. J. Chem. Eng. 16, 617–620. 44. Wilmarth, T. and Ishii, M. (1994). Two-phase flow regimes in narrow rectangular vertical and horizontal channels. Int. J. Heat Mass Transfer 37(12), 1749–1758. 45. Hibiki, T. and Mishima, K. (2001). Flow regime transition criteria for upward two-phase flow in vertical narrow rectangular channels. Nucl. Eng. Design 203, 117–131. 46. Xu, J. L., Cheng, P., and Zhao, T. S. (1999). Gas–liquid two-phase flow regimes in rectangular channels with mini/micro gaps. Int. J. Multiphase Flow 25, 411–432. 47. Ali, M. I. and Kawaji, M. (1991). ‘‘The Effect of Flow Channel Orientation on Two-phase Flow in a Narrow Passage Between Flat Plates’’. Proc. ASME/JSME Thermal Engineering Conference, Vol. 2, pp. 183–190. 48. Ishii, M. (1977). ‘‘One Dimensional Drift-flux Model and Constitutive Equations for Relative Motion Between Phases in Various Two-phase Regimes’’. ANL Report ANL-77-47. 49. Wilmarth, T. and Ishii, M. (1997). Interfacial area concentration and void fraction of twophase flow in narrow rectangular vertical channels. Trans. ASME, J. Heat Transfer 119, 916–922. 50. Griffith, P. and Wallis, G. P. (1961). Two-phase slug flow. Trans. ASME, J. Heat Transfer 83, 307–320. 51. Mishima, K. and Ishii, M. (1984). Flow regime transition criteria for upward two-phase flow in vertical tubes. Int. J. Heat Mass Transfer 27(5), 723–737. 52. Galbiati, L. and Andreini, P. (1994). Flow pattern transition for horizontal air–water flow in capillary tubes. A microgravity ‘‘equivalent system’’ simulation. Int. Comm. Heat Mass Transfer 21(4), 461–468. 53. Govan, A. H., Hewitt, G. F., Owen, D. G., and Burnett, G. (1989). Wall shear stress measurements in vertical air–water annular two-phase flow. Int J. Multiphase Flow 15, 307–325. 54. Chisholm, D. (1967). A theoretical basis for the Lockhart–Martinelli correlation for twophase flow. Int. J. Heat Mass Transfer 10, 1767–1778. 55. Lockhart, R. W. and Martinelli, R. C. (1949). Proposed correlation of data for isothermal two-phase flow two-component flow in pipes. Chem. Eng. Prog. 45, 39–48. 56. Friedel, L. (1979). ‘‘Improved Friction Pressure Drop Correlation for Horizontal and Vertical Two-phase Flow’’. European Two-phase Group Meeting, Ispra, Italy, Paper No. 2.
MICROCHANNEL HEAT EXCHANGER
417
57. Ali, M. I., Sadatomi, M., and Kawaji, M. (1993). Adiabatic two-phase flow in narrow channels between two flat plates. Can. J. Chem. Eng. 71, 657–666. 58. Sugawara, S., Katsuta, K., Ishihara, I., and Muto, T. (1967). ‘‘Consideration on the Pressure Loss of Two-phase Flow in Small-diameter Tubes’’. Proc. 4th National Heat Transfer Symp. of Japan, pp. 169–172 (in Japanese). 59. Sadatomi, Y., Sato, Y., and Saruwatari, S. (1982). Two-phase flow in vertical non-circular channels. Int. J. Multiphase Flow 8, 641–655. 60. Ungar, E. K. and Cornwell, J. D. (1992). ‘‘Two-phase Pressure Drop of Ammonia in Small Diameter Horizontal Tubes’’. AIAA 92-3891. 61. Bao, Z.-Y., Bosnich, M. G., and Haynes, B. S. (1994). Estimation of void fraction and pressure drop for two-phase flow in fine passages. Trans. Inst. Chem. Eng. 72A, 532–625. 62. Fujita, H., Ohara, T., Hirota, M., and Furuta, H. (1995). ‘‘Gas–Liquid Flows in Flat Channels with Small Channel Clearance’’. Proc. 2nd International Conference on Multiphase Flow, ‘95-Kyoto’, April 3–7, 1995, pp. IA3-37–IA3-44. 63. Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., and LeMoul, A. (1999). Gas–liquid two-phase flow in microchannels. Part II: Void fraction and pressure drop. Int. J. Multiphase Flow 25, 395–410. 64. Wang, C.-C., Chiang, S.-K., Chang, Y.-J., and Chung, T.-W. (2001). Two-phase flow resistance of refrigerants R-22, R-410A and R407C in small diameter tubes. Trans. Inst. Chem. Eng. 79(A), 553–560. 65. Koizumi, H. and Yokohama, K. (1980). Characteristics of refrigerant flow in a capillary tube. ASHRAE Trans. Part 2 86, 19–27. 66. Lazarek, G. M. and Black, H. S. (1982). Evaporative heat transfer, pressure drop and critical heat flux in a small vertical tube with R-113. Int. J. Heat Mass Transfer 25, 945–960. 67. Kureta, M., Kobayashi, T., Mishima, K., and Nishihara, H. (1998). Pressure drop and heat transfer for flow-boiling of water in small-diameter tubes. JSME Int. J. Ser. B 41(4), 871–879. 68. Chisholm, D. (1983). ‘‘Two-phase Flow in Pipelines and Heat Exchangers’’. Longman, New York. 69. Tran, T. N., Chyu, M.-C., Wambsganss, M. W., and France, D. M. (1999). ‘‘Two-phase Pressure Drop of Refrigerants During Flow Boiling in Small Channels: An Experimental Investigation and Correlation Development’’. Proc. Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, pp. 293–302. Begell House, New York. 70. Lee, H. J. and Lee, S. Y. (2001). Heat transfer correlation for boiling flows in small rectangular horizontal channels with low aspect ratios. Int. J. Multiphase Flow 27, 2043–2062. 71. Yang, C. Y. and Webb, R. L. (1996). Frictional pressure drop of R-12 in a small hydraulic diameter extruded aluminum tubes with and without micro-fins. Int. J. Heat Mass Transfer 39(4), 801–809. 72. Akers, W. W., Deans, H. A., and Crosser, O. K. (1959). Condensation heat transfer within horizontal tubes. Chem. Eng. Prog. Symp. Ser. 55(29), 171–176. 73. Kew, P. A. and Cornewell, K. (1995). ‘‘Confined Bubble Flow and Boiling in Narrow Spaces’’. Proc. 10th Int. Heat Transfer Conference, pp. 473–478. 74. Tran, T. N., Wambsganss, M. W., Chyu, M.-C. and France, D. M. (1997). ‘‘A Correlation for Nucleate Flow Boiling in Small Channel’’. Proc. Int. Conf. Compact Heat Exchangers for Process Industries, pp. 291–304. 75. Tran, T. N. (1998). ‘‘Pressure Drop and Heat Transfer Study of Two-phase Flow in Small Channels’’. Ph.D. Dissertation, Texas Tech University. 76. Kuznetsov, V. V. and Shamirzaev, A. S. (1999). ‘‘Two-phase Flow Pattern and Flow Boiling Heat Transfer in Non-circular with a Small Gap’’. Proc. Two-Phase Flow Modelling and Experimentation, pp. 249–253.
418
M.-H. KIM ET AL.
77. Oh, H. K., Katsuta, M., and Shibata, K. (1998). ‘‘Heat Transfer Characteristics of R-134a in a Capillary Tube Heat Exchanger’’. Proc. 11th Int. Heat Transfer Conference, Vol. 6, pp. 473–478. 78. Kandlikar, S. G. (1990). A general correlation for saturated two-phase flow boiling heat transfer inside horizontal and vertical tubes. Trans. ASME, J. Heat Transfer 112, 219–228. 79. Stephan, K. and Abdelsalam, M. (1980). Heat transfer correlations for natural convection boiling. Int. J. Heat Mass Transfer 23, 73–87. 80. Newell, T. A. and Shah, R. K. (2001). An assessment of refrigerant heat transfer, pressure drop, and void fraction effects in microfin tubes. HVAC&R Res. 7(2), 125–153. 81. Yang, C. Y. and Webb, R. L. (1996). Condensation of R-12 in small hydraulic diameter extruded aluminum tubes with and without micro-fins. Int. J. Heat Mass Transfer 39(4), 791–800. 82. Yang, C. Y. and Webb, R. L. (1997). A predictive model for condensation in a small hydraulic diameter tubes having axial micro-fins. Trans. ASME, J. Heat Transfer 119, 776– 782. 83. Cabuk, H. and Modi, V. (1989). ‘‘On the Design of Optimum Flow Headers for Heat Exchangers’’. ASME, Heat Transfer Equipment Fundamentals, Design, Applications, and Operating Problems, HTD Vol. 108, pp. 161–167. 84. Asoh, M., Hirao, Y., Aoki, Y., Watanabe, Y., and Fukano, T. (1991). ‘‘Phase Separation of Refrigerant Two-phase Mixture Flowing Downward into Three Thin Branches from a Horizontal Header Pipe’’. Proc. ASME/JSME Thermal Eng. Conf., Vol. 2, pp. 159–164. 85. Samson, E. B., Stark, J. A., and Grote, M. G. (1987). ‘‘Two-phase Flow Header Tests’’. SAE Technical Paper No. 871440. 86. Rong, X., Kawaji, M., and Burgers, J. G. (1996). ‘‘Gas–Liquid and Flow Rate Distributions in Single End Tank Evaporator Plates’’. SAE Technical Paper No. 960375. 87. Rong, X., Kawaji, M., and Burgers, J. G. (1995). ‘‘Two-phase Header Flow Distribution in a Stacked Plate Heat Exchanger’’. FED Vol. 225, pp. 115–122. 88. Torigoe, E. and Shimoya, M. (1997). ‘‘Refrigerant Evaporator, Improved for Uniform Temperature of Air Blown out Therefrom’’. U.S. Patent 5,701,760. 89. Watanabe, M., Katsuta, M., and Nagata, K. (1995). ‘‘Two-phase Flow Distribution in Multi-pass Tube Modeling Serpentine Type Evaporator’’. Proc. ASME/JSME Thermal Eng. Conf., Vol. 2, pp. 35–42. 90. Reimann, J. and Seeger, W. (1986). Two-phase flow in a T-junction with a horizontal inlet; Part II: Pressure differences. Int. J. Multiphase Flow 12, 587–608. 91. Kariyasaki, A., Nagashima, T., Fukano, T., and Ousaka, A. (1995). ‘‘Flow Separation Characteristics of Horizontal Two-phase Air–Water Flow into Successive Horizontal Capillary Tubes Through T-junction’’. Proc. ASME/JSME Thermal Eng. Conf., Vol. 2, pp. 75–82. 92. Kim, M.-H. and Bullard, C. W. (2002). Performance evaluation of a window room airconditioner with microchannel condensers. ASME J. Energy Resources Technol. 124, 47–55. 93. Webb, R. L. (1994). ‘‘Principles of Enhanced Heat Transfer’’, pp. 87–165. John Wiley & Sons, Inc. 94. Jacobi, A. M. and Shah, R. K. (1998). Air-side flow and heat transfer in compact heat exchangers: A discussion of enhancement mechanisms. Heat Transfer Eng. 19(4), 1–13. 95. Heikal, M. R., Drakulic, R., and Towell, T. A. (2000). Multi-louvered fin surfaces. In ‘‘Recent Advances in Analysis of Heat Transfer for Fin Type Surfaces’’ (B. Sunden and P. J. Heggs, eds.), pp. 211–250. WIT Press, Southampton, UK. 96. Shah, R. K., Heikal, M. R., Thonon, B., and Tochon, P. (2000). Progress in the numerical analysis of compact heat exchanger surfaces. Adv. Heat Transfer 34, 363–443. 97. Beauvais, F. N. (1965). ‘‘An Aerodynamic Look at Automotive Radiators’’. SAE Technical Paper No. 650470.
MICROCHANNEL HEAT EXCHANGER
419
98. Smith, M. C. (1968). ‘‘Gas Pressure Drop of Louvered-fin Heat Exchangers’’. ASME Paper 68-HT-27. 99. Smith, M. C. (1972). ‘‘Performance Analysis and Model Experiments for Louvered Fin Evaporator Core Development’’. SAE Technical Paper No. 720078. 100. Wong, L. T. and Smith, M. C. (1973). ‘‘Airflow Phenomena in the Louvered-fin Heat Exchanger’’. SAE Technical Paper No. 730237. 101. Davenport, C. J. (1980). ‘‘Heat Transfer and Fluid Flow in Louvered Triangular Ducts’’. Ph.D. Thesis, Department of Mechanical Engineering, Lanchester Polytechnic. 102. Davenport, C. J. (1983). Correlation for heat transfer and friction characteristics of louvered fin. AICHE Symp. Ser. 79(25), 19–27. 103. Tura, R. (1986). ‘‘Heat Transfer and Air Flow Phenomena in Multilouvered Ducts’’. Ph.D. Thesis, CNAA, Lanchester Polytechnic. 104. Kajino, M. and Hiramatsu, M. (1987). Research and development of automotive heat exchangers. In ‘‘Heat Transfer in High Technology and Power Engineering’’ (W.-J. Yang and Y. Mori, eds.), pp. 42–432. Hemisphere Publishing Co, Washington, DC. 105. Achaichia, A. and Cowell, T. A. (1988). Heat transfer and pressure drop characteristics of flat and louvered plate fin surface. Exp. Thermal Fluid Sci. 1, 147–157. 106. Achaichia, A. and Cowell, T. A. (1988). ‘‘A Finite Difference Analysis of Fully Developed Periodic Laminar Flow in Inclined Louvre Arrays’’. Second UK National Conference on Heat Transfer, Vol. 1, pp. 883–897. 107. Howard, P. (1987). ‘‘An Analytical Model for Heat Transfer and Friction Characteristics of a Multi-louvered Fin Heat Exchanger’’. MS Thesis, The Pennsylvania State University. 108. Antoniou, A. A., Heikal, M. R., and Cowell, T. A. (1990). Measurements of local velocity and turbulence levels in arrays of louvered plate fins. Heat Transfer 4, 105–110. 109. Webb, R. L. (1990). ‘‘The Flow Structure in the Louvered Fin Heat Exchanger Geometry’’. SAE Technical Paper No. 900722. 110. Webb, R. L. and Trauger, P. (1991). Flow structure in the louvered fin heat exchanger geometry. Exp. Thermal Fluid Sci. 4, 205–217. 111. Sahnoun, A. and Webb, R. L. (1992). Prediction of heat transfer and friction for the louver fin geometry. J. Heat Transfer 114, 893–900. 112. Cowell, T. A., Heikal, M. R., and Achaichia, A. (1995). Flow and heat transfer in compact louvered fin surfaces. Exp. Thermal Fluid Sci. 10, 192–199. 113. Bellows, K. D. (1997). ‘‘Flow Visualization of Louvered-fin Heat Exchangers’’. ACRC TR-124, Air Conditioning and Refrigeration Center (ACRC), University of Illinois at Urbana-Champaign. 114. Lee, Y. N. (1986). Heat transfer and pressure drop characteristics of an array of plates aligned at angles to the flow in a rectangular duct. Int. J. Heat Mass Transfer 29(10), 1553– 1563. 115. Zhang, H. and Lang, X. (1989). The experimental investigation of oblique angles and interrupted plate lengths for louvered fins in compact heat exchangers. Exp. Thermal Fluid Sci. 2, 100–106. 116. Lue, S. S., Huang, H. Z., and Tao, W. Q. (1993). Experimental study on heat transfer and pressure drop characteristics in the developing region for arrays of obliquely positioned plates of nonuniform length. Exp. Thermal Fluid Sci. 7, 30–38. 117. Huang, H. and Tao, W. Q. (1993). An experimental study on heat/mass transfer and pressure drop characteristics for arrays of nonuniform plate length positioned obliquely to the flow direction. ASME J. Heat Transfer 115, 568–575, 338–345. 118. Jeon, C. D. and Lee, J. (2001). Local heat transfer characteristics of louvered plate fin surfaces. ASHRAE Trans. 107(1), 338–345.
420
M.-H. KIM ET AL.
119. DeJong, N. C. and Jacobi, A. M. (1999). ‘‘Flow, Heat Transfer, and Pressure Drop Interactions in Louvered-fin Arrays’’. ACRC TR-146, Air Conditioning and Refrigeration Center (ACRC), University of Illinois at Urbana-Champaign. 120. Namai, K., Muramot, H., and Mochizuki, S. (1998). ‘‘Flow Visualization in the Louvered Fin Heat Exchanger’’. SAE Technical Paper No. 980055. 121. Springer, M. E. and Thole, K. A. (1998). Experimental design for flow-field studies of louvered fins. Exp. Thermal Fluid Sci. 18, 258–269. 122. Springer, M. E. and Thole, K. A. (1999). Entry region of louvered fin heat exchangers. Exp. Thermal Fluid Sci. 19, 223–232. 123. Beamer, H. E., Ghosh, D., Bellows, K. D., Huang, L. G., and Jacobi, A. M. (1998). ‘‘Applied CFD and Experiment for Automotive Compact Heat Exchanger Development’’. SAE Technical Paper No. 980426. 124. DeJong, N. C. and Jacobi, A. J. (1997). An experimental study of flow and heat transfer in parallel-plate arrays: Local, row-by-row and surface average behavior. Int. J. Heat Mass Transfer 40, 1365–1378. 125. DeJong, N. C., Zhang, L. W., Jacobi, A. M., Balachandar, S., and Tafti, D. K. (1998). A complementary experimental and numerical study of the flow and heat transfer in offset strip-fin heat exchangers. ASME J. Heat Transfer 120, 690–698. 126. Tafti, D. K., Zhang, L. W., and Wang, G. (1999). Time-dependent calculation procedure for fully developed and developing flow and heat transfer in louvered fin geometries. Num. Heat Transfer, 35, Part A, 225–249. 127. Tafti, D. K., Wang, G., and Lin, W. (2000). Flow transition in a multilouvered fin array. Int. J. Heat Mass Transfer 43, 901–919. 128. Zhang, L. W., Balachandar, S., Tafti, D. K., and Najjar, F. M. (1997). Heat transfer enhancement mechanisms in inline and staggered parallel-plate fin heat exchangers. Int. J. Heat Mass Transfer 40(10), 2307–2325. 129. Tafti, D. K. and Zhang, X. (2001). Geometry effects on flow transition in multilouvered fins—onset, propagation, and characteristic frequencies. Int. J. Heat Mass Transfer 44(22), 4195–4210. 130. Mochizuki, S. and Yagi, Y. (1982). ‘‘Characteristics of Vortex Shedding in Plate Arrays’’. Flow Visualization II—Proc. of the 2nd International Symposium on Flow Visualization, Hemisphere Publishing Corporation, pp. 99–103. 131. Zhang, X. and Tafti, D. K. (2001). Classification and effects of thermal wakes on heat transfer in multilouvered fins. Int. J. Heat Mass Transfer 44(13), 2461–2473. 132. Kays, W. M. and London, A. L. (1984). ‘‘Compact Heat Exchangers’’, 3rd edn., pp. 1–49. McGraw-Hill Book Company. 133. Shah, R. K. and Webb, R. L. (1983). Compact and enhanced heat exchangers. In ‘‘Heat Exchanger—Theory and Practice’’ (J. Taborek, G. F. Hewitt, and N. Afgan, eds.), pp. 425–468. Hemisphere Publishing Corporation. 134. Aoki, H., Shinagawa, T., and Suga, K. K. (1989). An experimental study of the local heat transfer characteristics in automotive louvered fins. Exp. Thermal Fluid Sci. 2, 293–300. 135. Suga, K. and Aoki, H. (1995). Numerical study on heat transfer and pressure drop in multilouvered fins. J. Enhanced Heat Transfer 2(3), 231–238. 136. Sunden, B. and Svantesson, J. (1991). Heat transfer and pressure drop from louvered surfaces in automotive heat exchangers. Exp. Heat Transfer 4, 11–125. 137. Sunden, B. and Svantesson, J. (1992). Thermal hydraulic performance of some standard multilouvered surfaces. Int. J. Heat Technol. 10(3–4), 16–32. 138. Sunden, B. and Svantesson, J. (1992). ‘‘Correlation of j- and f-factors for Multilouvered Heat Transfer Surfaces’’. Proc. of the 3rd UK National Conference on Heat Transfer, pp. 805–811.
MICROCHANNEL HEAT EXCHANGER
421
139. Webb, R. L. and Jung, S. H. (1992). Air-side performance of enhanced brazed aluminum heat exchangers. ASHRAE Trans. 98(2), 391–410. 140. Rugh, J. P., Pearson, J. T., and Ramadhyani, S. (1992). ‘‘A Study of a Very Compact Heat Exchanger Used for Passenger Compartment Heating in Automobiles’’. ASME Symp. Ser. HTD-Vol. 201, pp. 15–24. 141. Chang, Y., Wang, C., and Chang, W. (1994). Heat transfer and flow characteristics of automotive brazed aluminum heat exchangers. ASHRAE Trans. 100(2), 643–652. 142. McQuiston, F. C. (1978). Correlations of heat, mass and momentum transport coefficients for plate-fin-tube heat transfer surfaces with staggered tubes. ASHRAE Trans. 84(1), 294–309. 143. Kayansayan, N. (1993). Heat transfer characterization of flat plain fins and round tube heat exchangers. Exp. Thermal Fluid Sci. 6, 263–272. 144. Dillen, E. L. and Webb, R. L. (1994). ‘‘Rationally Based Heat Transfer and Friction Correlations for the Louver Fin Geometry’’. SAE Technical Paper No. 940504. 145. Webb, R. L., Chang, Y., and Wang, C. (1995). ‘‘Heat Transfer and Friction Correlations for the Louver Fin Geometry’’. 1995 Vehicle Thermal Management Systems Conference Proc., pp. 533–541. 146. Chang, Y. and Wang, C. (1996). Air-side performance of brazed aluminum heat exchangers. J. Enhanced Heat Transfer 3(1), 15–28. 147. Chang, Y. and Wang, C. (1997). A generalized heat transfer correlation for louvered fin geometry. Int. J. Heat Mass Transfer 40(3), 533–544. 148. Kim, M.-H. and Bullard, C. W. (2002). Air-side thermal hydraulic performance of multilouvered fin aluminum heat exchangers. Int. J. Refrigeration 25(3), 390–400. 149. Chang, Y., Hsu, K., Lin, Y., and Wang, C. (2000). A generalized friction correlation for louver fin geometries. Int. J. Heat Mass Transfer 43, 2237–2243. 150. Blevins, R. D. (1984). In ‘‘Applied Fluid Dynamics Handbook’’, p. 322. Van Nostrand Reinhold Company, New York. 151. Kim, M.-H. and Bullard, C. W. (2001). Development of a microchannel evaporator model for a CO2 air-conditioning system. Energy 26(10), 931–948. 152. Lewis, W. K. (1922). The evaporation of a liquid into a gas. Trans. ASME 44, 325–340. 153. Chilton, T. H. and Colburn, A. P. (1934). Mass transfer (absorption) coefficients— prediction from data on heat transfer and fluid friction. Ind. Eng. Chem. 26(11), 1183– 1187. 154. Knaus, W. L. (1935). Heat transmission in cooling air with extended surfaces. Refrigerating Eng. 29, 23–26, 82–85. 155. Pownall, H. B. (1935). Rational development and rating of extended air cooling surface. Refrigerating Eng. 30, 211–218. 156. Goodman, W. (1936). Dehumidification of air with coil. Refrigerating Eng. 31, 225–275. 157. Ware, C. D. and Hacha, T. H. (1960). ‘‘Heat Transfer from Humid Air to Fin and Tube Extended Surface Cooling Coils’’. ASME Paper 60-HT-17. 158. Bryan, W. L. (1961). Heat and mass transfer in dehumidifying surface coils. ASHRAE Trans. 67, 393–405. 159. Bryan, W. L. (1962). Heat and mass transfer in dehumidifying extended surface coils. ASHRAE Trans. 68, 237–247. 160. Myers, R. J. (1967). ‘‘The Effect of Dehumidification on the Air-side Heat Transfer Coefficient for a Finned Tube Coil’’. M.S. Thesis, University of Minnesota, Minneapolis. 161. Bettanini, E. (1970). ‘‘Simultaneous Heat and Mass Transfer on a Vertical Surface’’. Heat Exchange in Refrigerating Systems, International Institute of Refrigeration, pp. 309–317.
422
M.-H. KIM ET AL.
162. Guillory, J. L. and McQuiston, F. C. (1973). An experimental investigation of air dehumidification in a parallel plate heat exchanger. ASHRAE Trans. 79(2), 146–151. 163. Tree, D. and Helmer, W. (1976). Heat, mass, and momentum transfer in a parallel plate condensing flow in a parallel heat exchanger. ASHRAE Trans. 82(1), 289–299. 164. McQuiston, F. C. (1976). Heat, mass and momentum transfer in a parallel plate dehumidifying exchanger. ASHRAE Trans. 82(1), 82–106. 165. McQuiston, F. C. (1978). Heat, mass and momentum transfer data for five plate-fin-tube heat transfer surface. ASHRAE Trans. 84(1), 266–293. 166. McQuiston, F. C. (1978). Correlation of heat, mass and momentum transport coefficients for plate-fin-tube heat transfer surfaces with staggered tubes. ASHRAE Trans. 84(1), 294–308. 167. Kuehn, T. H., Ramsey, J. W., and Threlkeld, J. L. (1998). In ‘‘Thermal Environmental Engineering’’, 3rd edn., pp. 289–331. Prentice Hall. 168. McQuiston, F. C. (1975). Fin efficiency with combined heat and mass transfer. ASHRAE Trans. 81(1), 350–355. 169. Elmahdy, A. H. and Biggs, R. C. (1983). Efficiency of extended surfaces with simultaneous heat and mass transfer. ASHRAE Trans. 89(1), 135–143. 170. Wu, G. and Bong, T.-Y. (1994). Overall efficiency of a straight fin with combined heat and mass transfer. ASHRAE Trans. 100(1), 367–374. 171. Hong, K. T. and Webb, R. L. (1996). Calculation of fin efficiency for wet and dry fins. Int. J. HVAC & R Res. 2, 27–41. 172. Liang, S. Y., Wong, T. N., and Nathan, G. K. (2000). Comparison of one-dimensional models for wet-surface fin efficiency of a plate-fin-tube heat exchanger. Appl. Thermal Eng. 20, 941–966. 173. Uv, E. H. and Sonju, O. K. (1992). Heat transfer measurements of circular finned tubes with and without partial condensation. Inst. Chem. Eng. Symp. Ser. 1(129), 295–302. 174. Hill, J. M. and Jeter, S. M. (1991). A linear subgrid cooling and dehumidification coil model with emphasis on mass transfer. ASHRAE Trans. 97(2), 118–128. 175. Mirth, D. R., Ramadhyani, S., and Hittle, D. C. (1993). Thermal performance of chilledwater cooling coils operating at low water velocities. ASHRAE Trans. 99, 43–53. 176. ARI Standard 410-91. (1991). ‘‘Forced Circulation Air-cooling and Air-heating Coils’’. ARI, Arlington, VA, USA. 177. Hu, X., Zhang, L., and Jacobi, A. M. (1994). Surface irregularity effects of droplets and retained condensate on local heat transfer to finned tubes in cross-flow. ASHRAE Trans. 100(1), 375–381. 178. Wang, C., Hsieh, Y., and Lin, Y. (1997). Performance of plate finned tube heat exchangers under dehumidifying conditions. J. Heat Transfer 119, 109–117. 179. Chuah, Y. K., Hung, C. C., and Tseng, P. C. (1998). Experiments on the dehumidification performance of a finned tube heat exchangers. HVAC&R Res. 4(2), 167–178. 180. Kim, G. J. and Jacobi, A. M. (2000). ‘‘Condensate Accumulation Effects on the Air-side Thermal Performance of Slit-fin Surfaces’’. ACRC Report CR-26, University of Illinois at Urbana-Champaign. 181. Korte, C. and Jacobi, A. M. (2001). Condensate retention effects on the performance of plain-fin-and-tube heat exchangers: Retention data and modeling. J. Heat Transfer 123, 926–936. 182. Yin, J. and Jacobi, A. M. (2000). ‘‘Condensate Retention Effects on the Air-side Heat Transfer Performance of Plain and Wavy-louvered Heat Exchanger’’. ACRC Report TR158, University of Illinois at Urbana-Champaign. 183. Wang, C., Lin, Y., and Lee, C. (2000). An airside correlation for plain fin-and-tube heat exchangers in wet conditions. Int. J. Heat Mass Transfer 43(10), 1869–1872.
MICROCHANNEL HEAT EXCHANGER
423
184. Wang, C., Lin, Y., and Lee, C. (2000). Heat and momentum transfer for compact louvered fin-and-tube heat exchangers in wet conditions. Int. J. Heat Mass Transfer 43(18), 3443– 3452. 185. Webb, R. L. and Jung, S. H. (1992). Air-side performance of enhanced brazed aluminum heat exchangers. ASHRAE Trans. 98(2), 391–401. 186. Chiou, C., Wang, C., Chang, Y., and Lu, D. (1994). An experimental study of heat transfer and flow friction characteristics of automotive evaporators. ASHRAE Trans. 100(2), 575–581. 187. McLaughlin, W. J. and Webb, R. L. (2000). ‘‘Wet Air Side Performance of Louver Fin Automotive Evaporators’’. SAE Technical Paper Series 2000-01-0574. 188. McLaughlin, W. J. and Webb, R. L. (2000). ‘‘Condensate Drainage and Retention in Louver Fin Automotive Evaporators’’. SAE Technical Paper Series 2000-01-0575. 189. Kim, M.-H. and Bullard, C. W. (2002). Air-side performance of brazed aluminum heat exchangers under dehumidifying conditions. Int. J. Refrigeration 25(5), 641–652. 190. Osada, H., Aoki, H., Ohara, T., and Kuroyanagi, I. (1999). Experimental analysis for enhancing automotive evaporator fin performance. In ‘‘Proc. of the International Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries’’ (R. K. Shah, ed.), pp. 439–445. 191. Kim, M.-H., Youn, B., and Bullard, C. W. (2001). Effect of inclination on the air-side performance of a brazed aluminum heat exchanger under dry and wet conditions. Int. J. Heat Mass Transfer 44(24), 4613–4623. 192. Kim, M.-H., Song, S. M., and Bullard, C. W. (2002). Effect of inlet humidity condition on the air-side performance of an inclined brazed aluminum evaporator. Int. J. Refrigeration 25(5), 610–619. 193. Koga, Y. and Takeshige, Y. (1990). ‘‘Development of Multi-functional Surface Treatment’’. SAE Technical Paper No. 900596. 194. Nikaido, N., Shirai, S., Iihashi, M., Umemoto, S., and Suzuki, K. (1978). ‘‘Process for Treating the Surface of Aluminum or Aluminum Alloy’’. US Patent 4105511. 195. Rickert, M. E. (1980). ‘‘Process for Rendering Surfaces Permanently Water Wettable and Novel Products Thus-produced’’. US Patent 4181773. 196. Miwa, K. (1987). ‘‘Hydrophilic Fins for a Heat Exchanger’’. US Patent 4664182. 197. Sako, R., Ogino, T., Kanazawa, M., Nishihara, A., Okita, H., and Sakamoto, Y. (1988). ‘‘Method for Hydrophilic Treatment of Aluminum Using an Amphoteric Polymer’’. US Patent 4783224. 198. Mizoguchi, M., Inoue, S., Otsuka, T., and Isoyama, E. (1990). ‘‘Aluminum Heat Exchanger Provided with Fins having Hydrophilic Coating’’. US Patent 4957159. 199. Espeut, K. W. and Barry, L. (1991). ‘‘Hydrophilic Fins for a Heat Exchanger’’. US Patent 5012862. 200. Ishii, T., Yamazaki, K., and Takasawa, R. (1999). ‘‘Water-based Hydrophilic Coatings and a Process for Manufacturing Precoated Fin Materials for Heat Exchangers with Use of Said Coatings’’. US Patent 5916635. 201. Hong, K. and Webb, R. L. (2000). Wetting coatings for dehumidifying heat exchangers. HVAC&R Res. 6(3), 229–242. 202. Min, J., Webb, R. L., and Bemisderfer, C. H. (2000). Long-term hydraulic performance of dehumidifying heat exchangers with and without hydrophilic coatings. HVAC&R Res. 6(3), 257–272. 203. Min, J. and Webb, R. L. (2001). Condensate formation and drainage on typical fin materials. Exp. Thermal Fluid Sci. 25, 101–111. 204. Min, J. and Webb, R. L. (2002). Long-term wetting and corrosion characteristics of hot water treated aluminum and copper fin stocks. Int. J. Refrigeration 25(8), 1054–1061.
424
M.-H. KIM ET AL.
205. Mimaki, M. (1987). Effectiveness of finned-tube heat exchanger coated hydrophilic-type film. ASHRAE Trans. 93(1), 62–71. 206. Wang, C. and Chang, C. (1998). Heat and mass transfer for plate fin-and-tube heat exchangers, with and without hydrophilic coating. Int. J. Heat Mass Transfer 41, 3109–3120. 207. Ha, S., Kim, C., Ahn, S., and Dreister, G. A. (1998). ‘‘Condensate Drainage Characteristics of Plate Fin-and-tube Heat Exchanger’’. Proc. of the International Conference on Heat Exchangers for Sustainable Development, pp. 423–430. Technical University of Lisbon, Lisbon, Portugal. 208. Ha, S., Kim, C., Ahn, S., and Dreister, G. A. (1999). ‘‘Hydraulic Performance of Wet Finand-tube Heat Exchanger for Various Wettability Surfaces’’. Proc. of the International Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, pp. 463–470. Begell House, Inc., New York. 209. Hong, K. and Webb, R. L. (1999). Performance of dehumidifying heat exchangers with and without wetting coatings. J. Heat Transfer 121, 1018–1026. 210. Brown, J. S., Terry, J. L., and Hutter, R. J. (1994). ‘‘An Analytical Prediction of Water Droplet Travel When Discharged from the Face of an Evaporator Core’’. SAE Technical Paper No. 940501. 211. Mathur, G. D. (1999). ‘‘Investigation of Water Carryover from Evaporator Coils’’. SAE Technical Paper Series 1999-01-1194. 212. Hong, K. (1996). ‘‘Fundamental Characteristics of Dehumidifying Heat Exchangers With and Without Wetting Coatings’’. Ph.D. Thesis, The Pennsylvania State University. 213. Min, J. and Webb, R. L. (2000). Condensate carryover phenomena in dehumidifying, finned-tube heat exchangers. Exp. Thermal Fluid Sci. 22, 175–182. 214. Rudy, T. M. and Webb, R. L. (1981). ‘‘Condensate Retention on Horizontal Integral-fin Tubing’’. In ASME Advances in Enhanced Heat Transfer, Vol. HTD-18, pp. 35–41. 215. Rudy, T. M. and Webb, R. L. (1985). An analytical model to predict condensate retention on horizontal integral-fin tubes. J. Heat Transfer 107, 361–368. 216. Webb, R. L., Rudy, T. M., and Kedzierski, M. A. (1985). Prediction of the condensation coefficient on horizontal integral-fin tubes. J. Heat Transfer 107, 369–376. 217. Jacobi, A. M. and Goldschmidt, V. W. (1990). Low Reynolds number heat and mass transfer measurements of an overall counterflow, baffled, finned-tube, condensing heat exchanger. Int. J. Heat Mass Transfer 33, 755–765. 218. Beatty, K. O., Finch, E. B., and Schoenborn, E. M. (1951). Heat transfer from humid air to metal under frosting conditions. Refrigerating Eng. 59, 1203–1207. 219. Kamei, S., Mizushina, T., Kifune, S., and Koto, T. (1952). Research on the frost formation in a low temperature cooler condenser. Jpn. Sci. Rev. 2(3), 317–326. 220. Chung, P. M. and Algren, A. B. (1958). Frost formation and heat transfer on a cylinder surface in humid air cross flow—Part I: Experimental study. Heating Piping Air Conditioning 30(9), 171–178. 221. Chung, P. M. and Algren, A. B. (1958). Frost formation and heat transfer on a cylinder surface in humid air cross flow—Part I: Theoretical study and conclusions. Heating Piping Air Conditioning 30(10), 115–122. 222. Chen, M. M. and Rohsenow, W. (1964). Heat, mass, and momentum transfer inside frosted tubes—experiment and theory. J. Heat Transfer 86, 334–340. 223. Yonko, J. D. and Sepsy, C. F. (1967). An investigation of the thermal conductivity of frost while forming on a flat horizontal plate. ASHRAE Trans. 73(2), 1.1.1–1.1.12. 224. Trammel, G. J., Little, D. C., and Killgore, E. M. (1968). A study of frost formed on a flat plate held at sub-freezing temperatures. ASHRAE J. July, 42–47. 225. Brian, P. L. T., Reid, R. C., and Brazinsky, I. (1969). Cryogenic frost properties. Cryogenic Technol. 5, 205–212.
MICROCHANNEL HEAT EXCHANGER
425
226. Biguria, G. and Wenzel, L. A. (1970). Measurement and correlation of water frost thermal conductivity and density. I & EC Fundam. 9(1), 129–138. 227. Brian, P. L. T., Reid, R. C., and Shah, Y. T. (1970). Frost deposition on cold surfaces. I & EC Fundam. 9(3), 375–380. 228. Sanders, C. T. (1971). ‘‘Some Physical Aspects of Frost Formation on Cooled Surfaces’’. XIII Int. Congress of Refrigeration, pp. 731–740. Washington, DC. 229. Yamakawa, N., Takahashi, N., and Ohthani, S. (1972). Forced convection heat and mass transfer under frost conditions. Heat Transfer-Japanese Res. 1(2), 1–10. 230. Hobbs, P. V. (1974). ‘‘Ice Physics’’. Clarendon Press, Oxford. 231. Sanders, C. (1974). ‘‘The Influence of Frost Formation and Defrosting on the Performance Air Coolers’’. Ph.D. Dissertation, Technische Hogescool, Delft, Netherlands. 232. Hayashi, Y., Aoki, A., Adachi, S., and Hori, K. (1977). Study of frost properties correlating with frost formation types. J. Heat Transfer 99, 239–245. 233. Schneider, H. W. (1978). Equation of the growth rate of frost forming on cooled surface. Int. J. Heat Mass Transfer 21, 1019–1024. 234. Gatchilov, T. S., Ivanova, V. S., and Kaltchev, K. I. (1979). ‘‘Some Aspects of Construction and Exploitation of Extended Surface Air Coolers, Operating Under Frosting Conditions’’. Proc. of 15th Int. Congress of Refrigeration, Vol. 2, pp. 991–995. 235. Gatchilov, T. S. and Ivanova, V. S. (1979). Characteristics of extended surface air coolers during operation under frosting conditions. Int. J. Refrigeration 2(4), 233–236. 236. Marinyuk, B. T. (1980). Heat and mass transfer under frosting conditions. Int. J. Refrigeration 3(6), 366–368. 237. Schulte, D. W. and Howell, R. H. (1982). The effect of air turbulence on the rate of frost growth on a horizontal flat plate. ASHRAE Trans. 88(2), 201–217. 238. Cremers, C. J. and Mehra, V. K. (1982). Frost formation on vertical cylinders in free convection. J. Heat Transfer 104, 3–7. 239. Tokura, I., Saito, H., and Kishinami, K. (1983). Study on properties and growth rate of frost layers on cold surfaces. J. Heat Transfer 105, 895–901. 240. Aoki, K., Katayama, K., and Hayashi, Y. (1983). A study on frost formation (the process of frost formation involving the phenomena of water permeation and freezing). Bull. JSME 26(211), 87–93. 241. Coley, M. B. (1983). The cost of frost. ASHRAE J. September, 29–31. 242. Seki, N., Fukusako, S., Matsuo, K., and Uemura, S. (1984). Incipient phenomena of frost formation. Bull. JSME 27(233), 2476–2482. 243. Abdel-Wahed, R. M., Hifani, M. A., and Sherif, S. A. (1984). Heat and mass transfer from a laminar humid air stream to a plate at subfreezing temperature. Int. J. Refrigeration 7(1), 49–55. 244. O’Neal, D. L. and Tree, D. R. (1984). Measurement of frost growth and density in a parallel plate geometry. ASHRAE Trans. 90(2), 278–290. 245. O’Neal, D. L. and Tree, D. R. (1985). A review of frost formation in simple geometries. ASHRAE Trans. 91(2), 267–281. 246. Seki, N., Fukusako, S., Matsuo, K., and Uemura, S. (1985). An analysis of incipient frost formation. Warme- und Stoffubertragung 19, 9–18. 247. Miller, W. A. (1987). Laboratory examination and seasonal analysis of frosting and defrosting for an air-to-air heat pump. ASHRAE Trans. 93(1), 1474–1489. 248. Padki, M. M., Sherif, S. A., and Nelson, R. M. (1989). A simple method for modeling the frost formation phenomenon in different geometries. ASHRAE Trans. 95(2), 1127–1137. 249. Sami, S. M. and Duong, T. (1989). Mass and heat transfer during frost growth. ASHRAE Trans. 95(1), 158–165.
426
M.-H. KIM ET AL.
250. Sherif, S. A., Raju, S. P., Padki, M. M., and Chen, A. B. (1990). A semi-empirical transient method for modeling frost formation on a flat plate. Heat and Mass Transfer in Frost and Ice, Packed Beds, and Environmental Discharges. In ‘‘Proc. of AIAA/ASME Thermophysics and Heat Transfer Conference’’ (R. V. Arimilli et al., eds.), ASME HTD-Vol. 139, pp. 15–23. 251. Ostin, R. and Andersson, S. (1991). Frost growth parameters in a forced air stream. Int. J. Heat Mass Transfer 34, 1009–1017. 252. Mao, Y., Besant, R. W., and Rezkallah, K. S. (1991). A method of measuring frost density using flush-mounted removable disks. ASHRAE Trans. 97(1), 26–30. 253. Mao, Y., Besant, R. W., and Rezkallah, K. S. (1992). Measurement and correlations of frost properties with airflow over a flat plate. ASHRAE Trans. 98(2), 65–78. 254. Mao, Y., Besant, R. W., and Falk, J. (1993). Measurement and correlations of frost properties with laminar airflow at room temperature over a flat plate. ASHRAE Trans. 99(1), 739–745. 255. Tao, Y.-X., Besant, R. W., and Mao, Y. (1993). Characteristics of frost growth on a flat plate during the early growth period. ASHRAE Trans. 99(1), 746–753. 256. Tao, Y.-X. and Besant, R. W. (1993). Prediction of spatial and temporal distributions of frost growth on a flat plate under forced convection. J. Heat Transfer 115, 278–281. 257. Tao, Y.-X., Mao, Y., and Besant, R. W. (1994). Frost growth characteristics on heat exchanger surfaces: Measurement and simulation studies. Fundamentals of Phase Change: Sublimination and Solidification. ASME HTD-Vol. 286, pp. 29–38. 258. Sahin, A. Z. (1994). An experimental study on the initiation and growth of frost formation on a horizontal plate. Exp. Heat Transfer 7, 101–119. 259. Sahin, A. Z. (1995). An analytical study of frost nucleation and growth during the crystal growth period. Heat Mass Transfer 30, 321–330. 260. Le Gall, R., Grillot, J. M., and Jallut, C. (1997). Modeling of frost growth and densification. Int. J. Heat Mass Transfer 40(13), 3177–3187. 261. Mao, Y., Besant, R. W., and Chen, H. (1999). Frost characteristics and heat transfer on a flat plate under freezer operating conditions: Part I, Experimentation and correlations. ASHRAE Trans. 105(2), 231–251. 262. Chen, H., Besant, R. W., and Tao, Y. X. (1999). Frost characteristics and heat transfer on a flat plate under freezer operating conditions: Part II, Numerical modeling and comparison with data. ASHRAE Trans. 105(2), 252–259. 263. Thomas, L., Chen, H., and Besant, R. W. (1999). Measurement of frost characteristics on heat exchanger fins—Part I: Test facility and instrumentation. ASHRAE Trans. 105(2), 283–293. 264. Storey, B. D. and Jacobi, A. M. (1999). The effects of streamwise vortices on the frost growth rate in developing laminar channel flows. Int. J. Heat Mass Transfer 42, 3787– 3802. 265. Okoroafor, E. U. and Newborough, M. (2000). Minimising frost growth on cold surfaces exposed to humid air by means of crosslinked hydrophilic polymeric coatings. Appl. Thermal Eng. 20, 737–758. 266. Dyer, J. M., Storey, B. D., Hoke, J. L., Jacobi, A. M., and Geogiadis, J. G. (2000). An experimental investigation of the effect of hydrophobicity on the rate of frost growth in laminar channel flows. ASHRAE Trans. 106(1), 143–151. 267. Hoke, J. L., Georgiadis, J. G., and Jacobi, A. M. (2000). ‘‘The Interaction Between the Substrate and Frost Layer Through Condensate Distribution’’. ACRC TR-177, Air Conditioning and Refrigeration Center, University of Illinois at Urbana-Champaign. 268. Stoecker, W. F. (1957). How frost formation on coils affects refrigeration systems. Refrigerating Eng. 65(2), 42–46.
MICROCHANNEL HEAT EXCHANGER
427
269. Stoecker, W. F. (1960). Frost formation on refrigeration coils. ASHRAE Trans. 66, 91–103. 270. Gates, R. R., Sepsy, C. F., and Huffman, G. D. (1967). Heat transfer and pressure loss in extended surface heat exchangers operating under frosting conditions—Part I: Literature survey, test apparatus and preliminary results. ASHRAE Trans. 73(2), 1.2.1–1.2.13. 271. Huffman, G. D. and Sepsy, C. F. (1967). Heat transfer and pressure loss in extended surface heat exchangers operating under frosting conditions—Part II: Data analysis and correlation. ASHRAE Trans. 73(2), 1.3.1–1.3.16. 272. Hosoda, T. and Uzuhashi, H. (1967). Effects of frost on the heat transfer coefficient. Hitachi Rev. 16(6), 254–259. 273. Neiderer, D. H. (1976). Frosting and defrosting effects on coil heat transfer. ASHRAE Trans. 82(1), 467–473. 274. Fisk, W., Archer, K. H., Chant, R. E., Hekmat, D., Offermann, F. J., III, and Perderson, B. S. (1985). Onset of freezing in residential air-to-air heat exchangers. ASHRAE Trans. 91(1B), 145–158. 275. Fisk, W., Archer, K. H., Chant, R. E., Hekmat, D., Offermann, F. J., III, and Perderson, B. S. (1985). Performance of residential air-to-air heat exchangers during operation with freezing and periodic defrosts. ASHRAE Trans. 91(1B), 159–172. 276. Barrow, H. (1985). A note on frosting of heat pump evaporator surfaces. Heat Recovery Syst. 5(3), 195–201. 277. Kondepudi, S. N. and O’Neal, D. L. (1987). The effects of frost growth on extended surface heat exchanger performance: A review. ASHRAE Trans. 93(2), 258–274. 278. Smith, G. R. (1989). Theoretical cooling coil calculations at freezer temperatures to avoid unfavorable coil-frost. ASHRAE Trans. 95(2), 1138–1154. 279. Emery, A. F. and Siegel, B. L. (1990). Experimental measurements of the effects of frost formation on heat exchanger performance. Heat and Mass Transfer in Frost and Ice, Packed Beds, and Environmental Discharges. In ‘‘Proc. of AIAA/ASME Thermophysics and Heat Transfer Conference’’, (R. V. Arimilli et al., ed.), ASME HTD-Vol. 139, pp. 1–7. 280. Ostin, R. and Johannesson, G. (1991). A polymeric approach to counteract frosting in airto-air heat exchangers. Heat Recovery Syst. CHP 11(5), 415–421. 281. Ostin, R. (1992). A study of heat exchange under frosting conditions. Heat Recovery Syst. CHP 12(2), 89–103. 282. Kondepudi, S. N. and O’Neal, D. L. (1993). A simplified model of pin fin heat exchangers under frosting conditions. ASHRAE Trans. 99(1), 754–761. 283. Ogawa, K., Tanaka, N., and Takeshita, M. (1993). Performance improvement of plate finand-tube heat exchangers under frosting conditions. ASHRAE Trans. 99(1), 762–771. 284. O’Neal, D. L., Bryant, J. A., and Parker, B. (1997). ‘‘Impact of a Hydrophobic Coating on the Frost Buildup and Defrost Performance of a Heat Pump Evaporator’’. Proc. of the 45th Oji International Seminar, pp. 181–187. 285. Watters, R. J., O’Neal, D. L., and Yang, J. (2001). Effect of fin staging on frost/defrost performance of a two-row heat pump evaporator at standard test conditions. ASHRAE Trans. 107(2), 240–249. 286. Watters, R. J., O’Neal, D. L., and Yang, J. (2001). Effect of fin staging on frost/defrost performance of a two-row heat pump evaporator under heavy frosting conditions. ASHRAE Trans. 107(2), 250–258. 287. Kerschbaumer, H. G. (1971). ‘‘Analysis of the Influence of Frost Formation on Evaporators and of the Defrost Cycles on Performance and Power Consumption of Refrigeration Systems’’. XIII Int. Congress of Refrigeration, Washington, DC, pp. 719–730. 288. Tantakitti, C. and Howell, R. H. (1986). Air-to-air heat pumps operating under frosting conditions on the outdoor coil. ASHRAE Trans. 97(1), 827–842.
428
M.-H. KIM ET AL.
289. Tassou, S. A. and Marquand, C. J. (1987). Effects of evaporator frosting and defrosting on the performance of air-to-water heat pumps. Appl. Energy 28, 19–33. 290. O’Neal, D. L., Peterson, K. T., Anaand, N. K., and Schliesing, J. S. (1989). Refrigeration system dynamics during reverse cycle defrost. ASHRAE Trans. 95(2), 689–698. 291. Cole, R. A. (1989). Refrigeration loads in a freezer due to hot gas defrost and their associated costs. ASHRAE Trans. 95(2), 1149–1154. 292. Aoki, K., Hattori, M., and Edayoshi, A. (1991). Characteristics of heat pump system operation with frost formation. ASME/JSME Thermal Eng. Proc. 4, 391–396. 293. Martinez-Frias, J. and Aceves, S. M. (1999). Effects of evaporator frosting on the performance of an air-to-air heat pump. J. Energy Resources Technol. 121, 60–65. 294. Martinez-Frias, J. and Aceves, S. M. (1999). Modeling of a heat pump with evaporator air dehumidification for reduced frost formation. J. Energy Resources Technol. 121, 189–195. 295. Itoh, M., Kogure, H., Yoshinaga, S., Hoshino, R., Wakabayashi, N., Kudoh, M., and Kusumoto, H. (1996). ‘‘Study on Parallel-flow-type Heat Exchangers for Residential Heat Pump Systems’’. Proc. of the 1996 JAR Annual Conference, pp. 73–76 (in Japanese). 296. Webb, R. L. and Lee, H. (2000). Brazed aluminum heat exchangers for residential airconditioning. J. Enhanced Heat Transfer 8, 1–14. 297. Saini, M. and Webb, R. L. (2002). ‘‘Modeling Forced Convection in Air Cooled Heat Sinks for Electronics Cooling’’. Proc. ITherm 2002, pp. 1–8, San Diego, CA. 298. Sundberg, R., Tapper, Beal, R., and Melnyk, W. (1995). ‘‘Optimized Brazed Copper– Brass Radiator’’. SAE VTMS-2 Conf., pp. 289–294, London. 299. Webb, R. L. (1997). ‘‘Effect of Material Properties on Performance and Brazing of Aluminum and Copper/Brass Radiators’’. 1997 Vehicle Thermal Management Systems Conference Proceedings, pp. 363–368, SAE. 300. Webb, R. L. and Yamauchi, S. (2002). ‘‘Test Results on a Thermo-syphon Concept to Cool Desktop Computers and Servers’’. Semi-Therm Proc. 2002, pp. 151–158, San Jose, CA. 301. Koo, J. M., Jiang, L., Zhang, L., Zhou, P., Banerjee, S. S., Kenny, T. W., Santiago, J. G., and Goodson, K. E. (2000). ‘‘Modeling of Two-phase Microchannel Heat Sinks for VLSI Chips’’. Proc. of the 14th Annual IEEE International MEMS-01 Conference, pp. 422–426, Interlaken, Switzerland. 302. Jiang, L., Koo, J. M., Zeng, S., Mikkelsen, J. C., Zhang, L., Zhou, P., Santiago, J. G., Kenny, T. W., Goodson, K. E., Maveety, J. G., and Tran, Q. A. (2001). ‘‘Two-phase Microchannel Heat Sinks for an Electrokinetic VLSI Chip Cooling System’’. Proc. of the 17th IEEE SEMI-THERM Symposium, March 20–22, 2001, pp. 153–157, San Jose, CA. 303. Kim, M.-H., Pettersen, J., and Bullard, C. W. (2003). Fundamental process and system design issues in CO2 vapor compression systems. In review for Prog. Energy Combustion Sci.. 304. Pettersen, J., Hafner, A., Skaugen, G., and Rekstad, H. (1998). Development of compact heat exchangers for CO2 air-conditioning systems. Int. J. Refrigeration 21(3), 180–193. 305. Webb, R. L. and Rao, P. (2003). ‘‘Effect of manifold design on flow distribution in parallel micro-channels’’. Proc. of InterPack’03 International Electronic Packaging Technical Conference and Exhibition, Maui, Hawaii. 306. Azzopardi, B. J. (1994). Two-phase flows in junctions. In ‘‘Encyclopedia of Fluid Mechanics’’, Vol. 3, Chapter 25. Gulf Publishing Co. 307. Reimann, J. and Khan, M. (1983). ‘‘Flow Through a Small Brick at the Bottom of a Large Pipe With Stratified Flow’’. Proc. of the 2nd Int. Topical Meeting on Nuclear Reactor Thermohydraulics, Santa Barbara, CA. 308. Yin, J., Bullard, C. W., and Hrnjak, P. A. (2001). R744 gas cooler model development and validation. Int. J. Refrigeration 24(7), 692–701.
MICROCHANNEL HEAT EXCHANGER
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309. Bullard, C. W. and Hrnjak, P. A. (2002). ‘‘Advanced Technologies for Auto A/C Components’’. Proc. of the 7th International Energy Agency Conference on Heat Pumping Technologies, Vol. 1, pp. 112–124. Beijing, China. 310. Yin, J., Bullard, C. W., and Hrnjak, P. A. (2000). Design strategies for R744 gas cooler. In ‘‘Proc. of 4th IIR-Gustav Lorenzen Conference on Natural Working Fluids’’ (E. A. Groll and D. M. Robinson, eds.), pp. 315–322. Purdue University, West Lafayette, IN, USA.
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ADVANCES IN HEAT TRANSFER VOL. 37
AUTHOR INDEX Numerals in parentheses following the page numbers refer to reference numbers cited in the text
A
Allen, D. G., 120(316) Almaer, S., 102(237), 120(237) Alt, K. W., 180(6) Altamirano, M. M., 264(302) Alvarez, M. M., 91(274) Amanullah, A., 101(226; 227), 125(227), 127(226; 227), 128(227) Amat, L., 213(49) Amato, G., 273(355) Amra, C., 258(227) Anaand, N. K., 397(290) Anderson, A. C., 224(92) Anderson, C. F., 226(128), 240(128) Anderson, S., 264(312), 397(251) Andre, C., 102(232), 103(246; 247), 105(276), 139(246; 247; 276), 140(276) Andre, E., 258(223; 224) Andreini, P., 312(52), 314(52), 315(52) Andreoli, G., 264(317) Andrews, G., 86(82) Ang, H. M., 101(207), 118(207) Anghaie, S., 3(1) Anne-Archard, D., 103(244), 121(244; 325), 139(325), 141(325) Antar, B. N, 3(2; 3) Antoniou, A. A., 355(108), 357(108) Ao, C. O., 263(271) Aoki, A., 397(232) Aoki, H., 377(134; 135), 389(190), 392(190), 393(190) Aoki, K., 397(240; 292) Aoki, Y., 340(84) Arai, K., 99(185), 100(193), 101(201; 202), 139(193), 141(193), 147(185) Archer, K. H., 397(274; 275) Arguello, M. A., 99(191), 118(191), 128(191) ARI Standard, 410-91, 387(176) Arik, B. E., 91(274) Armstrong, R. C., 80(34), 81(34), 82(34), 89(34) Arntzen, M., 274(365)
Aarons, B. L., 79(21) Abdel-Khalik, S. I., 84(71), 300(7), 307(39), 309(39), 310(7), 312(7), 316(63), 321(63) Abdel-Wahed, R. M., 397(243) Abdelsalam, M., 335(79) Abe, Y., 3(4), 72(37; 43) Abedrabbo, S., 214(71; 70) Abid, M., 128(332), 130(332), 139(332), 140(332) Abramson, A. R., 263(268) Aceves, S. M., 397(293; 294) Achaichia, A., 353(105), 354(105; 106), 355(105), 356(105; 106), 357(106; 112), 358(106), 360(106), 367(105), 376(105), 379(105), 382(105), 384(105) Adachi, S., 397(232) Adamian, Z. N., 274(373) Adams, B., 247(192) Addoms, J. N., 37(25) Aellen, T., 216(78) Aerts, L., 120(316d) Afacan, A., 104(254) Ahmed, H., 214(69) Ahn, S., 394(207; 208) AI-Sharif, M., 33(20), 34(20) Aiba, S., 84(49) Ait-Kadi, A., 101(214; 222) Ajdari, A., 89(266) Akabane, K., 100(193), 139(193), 141(193) Akers, W. W., 328(72), 331(72), 337(72) Akiyama, Y., 268(335) Albiter, V., 99(191), 118(191), 128(191) Albrecht, S., 226(102) Aldington, R. J., 108(283), 128(283) Aleksandrov, B., 246(166) Algren, A. B., 397(220; 221) Ali, A. M., 91(277) Ali, M. I., 310(47), 312(47), 316(57), 319(57) Allan, G., 273(356)
431
432
AUTHOR INDEX
Aroutiounian, V. M., 274(373) Arumugam, M., 95(134), 138(134) Arvo, J., 247(186) Asano, H., 32(19) Ashcroft, N. W., 187(22), 188(22), 190(22), 198(22), 199(22), 202(22), 203(22) Ashiwake, K., 92(97), 118(97) Aslan, M., 264(307) Asmail, C. C., 256(212; 217), 258(229), 261(247), 264(314) Asoh, M., 340(84) Astrova, E. V., 274(366) Aubin, J., 125(327), 140(327) Aunins, J. G., 84(50) Ayazi Shamlou, P., 97(170; 171), 144(171), 147(171) Azuma, H., 17(11), 18(12), 60(12), 61(12) Azzopardi, B. J., 407(306)
B Baars, T., 180(4) Bachtold, A., 180(2) Badhan, R. S., 84(54), 99(54), 118(54) Baillargeon, J. N., 216(77) Baker, M. R., 103(241), 127(241) Bakker, A., 101(229), 113(303), 123(303; 323), 124(303), 125(326), 139(340; 339), 140(339), 151(303; 366; 367) Balachandar, S., 361(125), 364(128), 401(128) Balakrishna, M., 142(353), 144(353) Balandin, A. A., 180(5) Bandow, S., 180(3) Banerjee, S. S., 405(301), 406(301) Bang, S. Y., 214(65) Bao, Z.-Y., 316(61), 320(61), 329(61) Barajas, A. M., 300(8), 308(8) Bard, A. J., 273(347) Bardon, J. P., 85(77) Barigou, M., 104(252), 125(252) Barker, J. R., 204(35), 206(35) Barnea, D., 306(37), 311(37), 309(43) Barnes, H. A., 82(41a), 120(312) Barnes, P. Y., 264(309; 310) Barradas, N. P., 256(221) Barrow, H., 397(276) Barry, L., 394(199) Barseghian, R. S., 274(373) Bartels, P. V., 123(321)
Basila, D., 253(211), 254(211) Bates, R. L., 110(286) Bau, H. H., 299(5) Bauer, G., 213(46) Bawolek, E. J., 249(200) Bayazitoglu, Y., 226(128), 240(128) Bayer, M., 180(4) Bayindir, M., 275(390) Beal, R., 405(298) Beamer, H. E., 359(123), 360(123) Beard, B. T. Jr., 213(59), 214(59) Beatty, K. O., 397(218) Beauvais, F. N., 352(97) Beck, M., 216(78) Beckmann, P., 246(173), 251(173), 255(173), 258(173), 260(173), 262(173) Beckner, J. L., 93(104), 113(104), 116(104), 117(104) Beishon, D. S., 79(8) Belaubre, N., 105(276), 139(276), 140(276) Bellows, K. D., 357(113), 359(113; 123) Bemisderfer, C. H., 394(202), 395(202) Ben Dahan, M., 273(360) Benedict, L. X., 226(101) Berginc, G., 261(251), 263(251) Berman, D. H., 261(249) Bernabeu, E., 264(303) Bernt, M. L., 247(187), 258(187) Berry, M. V., 263(288) Bertrand, F., 101(209; 216; 208; 210; 217), 133(208), 139(208; 210; 217), 140(208; 217), 141(217) Bertrand, J., 91(271), 98(172; 173; 174), 103(243), 111(172; 174), 125(327; 328), 128(332), 130(332), 139(173; 328; 332; 343), 140(327; 328; 332), 141(343) Besant, R. W., 397(252; 253; 254; 255; 256; 257; 261; 262) Bettanini, E., 387(161) Bhushan, B., 253(211), 254(211), 263(287) Bienlein, J., 260(246) Biggs, R. C., 387(169) Biguria, G., 397(226) Bilyalov, R. R., 274(372) Binnie, T. D., 214(73) Binnig, G., 252(209; 210) Birchenough, A., 72(39) Bird, R. B., 80(34), 81(34), 82(34), 89(34) Birkebak, R. C., 251(205), 264(311) Birks, T. A., 275(392)
433
AUTHOR INDEX
Birner, A., 181(7), 274(379) Birto de la Fuente, E., 101(211) Bishop, P. L., 84(46) Biswas, R., 274(382), 275(384; 385; 386; 395) Bjuggren, M., 247(188) Black, H. S., 316(66), 325(66), 330(66), 333(66), 335(66) Blanch, H. W., 84(58) Blasinski, H., 105(268) Blevins, R. D., 383(150) Bloomer, I., 225(99; 100) Blumberg, D. G., 263(297) Boebel, F. G., 213(50; 51; 52) Boger, D. V., 83(43), 121(319a) Bohm, R., 115(309) Bohme, G., 104(255), 108(255), 141(348) Bohn, R. B., 226(101) Bohren, C. F., 226(120), 234(120), 235(120) Boisson, H. C., 103(244), 121(244; 325), 139(325), 141(325) Bollenrath, F. M., 96(152), 145(152) Bondarenko, V. P., 274(377) Bong, T.-Y., 387(170) Bonsey, S. J., 264(316) Booth, J., 213(59), 214(59) Borca-Tasciuc, T., 214(72) Born, L., 182(12), 226(12), 238(12) Borovikov, V. A., 258(228) Bosman, G., 274(371) Bosnich, M. G., 316(61), 320(61), 329(61) Bott, T. R., 142(351) Boueke, A., 182(10), 269(10) Bouffakhreddine, B., 258(227) Bourlier, C., 261(251), 263(251) Bourne, J. R., 79(12), 93(114; 115), 115(114; 115), 131(114), 132(114; 115), 137(114; 115), 138(114; 115) Bousman, W. S., 300(11), 312(11) Bousquet, J., 101(224), 141(224) Bouvenot, A., 149(380) Bouwmans, I., 101(229) Bowen, R. L., 106(282), 108(282), 123(282) Boychev, V. A., 226(107), 243(155; 156), 245(107), 246(107) Boyd, I. W., 214(73) Braun, M. B., 271(344) Braun, W., 180(4) Brazinsky, I., 397(225) Brenke, A., 111(291)
Brewster, M. Q., 186(20), 226(20) Brian, P. L. T., 397(225; 227) Brienne, J.-P., 102(231) Bristow, T., 247(193), 251(193), 258(193) Brito de la Fuente, E., 84(64), 101(208; 209; 218; 219; 220; 221; 225; 215), 101(210; 212; 213; 216; 217), 111(215), 116(213), 117(213), 133(208), 139(208; 210; 217), 140(208), 140(217), 141(217) Brodkey, R. S., 91(275a) Brommer, K. D., 274(381) Brothers, A. M., 264(306) Brown, J. S., 395(210) Bruggemann, D. A. G., 86(81) Brumfield, L. K., 23(15) Brus, L., 273(357) Brusstar, M. J., 60(30) Bruxelmane, M., 99(183) Bryan, W. L., 387(158; 159) Bryant, J. A., 397(284) Bsiesy, A., 274(367) Bucher, E., 182(10), 269(10) Buckius, R. O., 258(232), 260(238; 239; 241; 240), 261(240; 256; 257), 263(257; 277; 280; 289), 264(320) Buerli, M., 79(12) Bullard, C. W., 349(92), 374(148; 189), 380(148), 383(148), 384(148; 151), 388(189), 389(148; 191; 192), 390(191), 391(148; 189; 191; 192), 392(151; 191; 192), 393(191; 192), 404(92), 406(303), 409(151; 308; 309; 310), 410(309; 310) Bulter, H., 93(114), 115(114), 131(114), 132(114), 137(114), 138(114) Bulu, I., 275(390) Bur, J., 275(402) Burgers, J. G., 343(86), 344(87) Burmester, S. S. H., 102(238) Burnell, J. G., 263(260) Burnett, G., 315(53) Butler, H., 93(115), 115(115), 132(115), 137(115), 138(115)
C Cabuk, H., 339(83), 340(83) Cahill, D. G., 204(30) Calabrese, R. V., 77(2), 91(269), 125(269)
434
AUTHOR INDEX
Calderbank, P. H., 92(88), 113(88), 116(88), 117(88) Canham, L., 273(354) Canham, L. T., 273(351) Cantu-Lozano, D., 120(316c) Capasso, F., 216(76; 77) Cardona, M., 217(80; 81), 218(80; 81) Caren, R. P., 237(134) Carminati, R., 237(138), 238(152), 268(332; 333) Carniglia, C. K., 246(164) Carpenter, K. J., 108(283), 128(283) Carr, G. L., 204(33), 274(370) Carreau, P. J., 80(31), 81(31), 82(31), 83(31), 89(31), 97(154; 155; 156; 157; 158; 159; 160; 161; 162), 101(217), 106(31), 113(160), 115(156; 157), 117(31; 157; 160), 119(160; 310), 120(160), 121(160), 122(160; 310), 131(155), 133(155), 138(155; 158; 159; 310), 139(217), 140(217), 141(217), 143(161), 147(159; 161), 148(159; 161) Castonguay, R., 264(312) Caze´, C., 258(230) Celli, V., 263(278) Champetier, R. J., 264(302) Chan, C. H., 263(283; 284) Chan, C. T., 275(384) Chang, C., 394(206) Chang, W., 368(141), 378(141), 379(141), 383(141) Chang, Y.-J., 316(64), 324(64), 329(64), 351(147), 368(141), 370(145; 146), 371(145; 146), 372(146; 147), 373(149), 378(141), 379(141; 145; 146; 147), 380(147), 382(149), 383(141; 145; 146; 149), 384(146; 147; 149), 388(186) Chankraiphon, S., 86(150), 96(150) Chant, R. E., 397(274; 275) Chao, D., 72(39) Chapin, A., 264(319) Chapman, F. S., 92(100), 132(100) Charalampopoulos, T. T., 262(259) Charpentier, J.-C., 84(48) Charrier, J., 274(376) Chase, B., 180(3) Chavan, V. V., 95(129; 130; 131; 132; 133; 134; 135; 136), 115(129; 131; 135;136), 117(129; 135), 121(136), 138(130; 134)
Cheesman, D. J., 98(176), 118(176), 125(176), 127(176) Chelikowsky, J. R., 187(25), 188(25), 192(25), 194(25), 195(25), 210(25), 211(25) Chen, A. B., 397(250) Chen, G., 204(29), 214(72; 74), 226(113; 123; 124), 227(113), 239(123; 124) Chen, H., 397(261; 262) Chen, I., 33(20), 34(20) Chen, J., 226(111) Chen, J. S., 261(253), 263(296) Chen, M. F., 263(272; 273) Chen, M. M., 397(222) Chen, P., 273(348) Chen, P. L., 102(237), 120(237) Chen, W., 214(70; 71) Chen, Y., 268(332) Chen, Y. R., 97(9) Chen, Z.-H., 302(26), 316(26), 325(26), 329(26) Chen, Z.-Y., 302(26), 316(26), 325(26), 329(26) Cheng, D. C.-H., 133(335) Cheng, J., 97(162; 160), 113(160), 117(160), 119(160; 310), 120(160), 121(160), 122(160; 310), 138(310) Cheng, P., 310(46), 311(46), 312(46), 313(46), 314(46), 315(46) Cheng, S. X., 214(71) Chevalier, J.-L., 99(182; 183; 184), 105(258), 121(258), 143(182), 149(378; 379) Chew, W. C., 263(270), 105(258), 121(258) Chhabra, R. P., 80(31; 32), 81(31; 32; 39), 82(31; 32; 39), 83(31; 32; 42), 84(45; 47), 85(42), 89(31; 32), 97(160; 162), 106(31; 32), 113(160), 117(31; 160), 119(160), 120(160), 121(160), 122(160) Chiang, S.-K., 316(64), 324(64), 329(64) Chilton, T. H., 386(153) Chiou, C., 388(186) Chisholm, D., 27(18), 318(54), 320(54), 327(68) Cho, A. Y., 216(76; 77) Cho, K., 84(73) Cho, Y. I., 84(70), 85(78a), 142(352) Choi, B. I., 220(88), 224(92), 226(112) Choi, K. K., 275(402) Chollet, F., 258(223)
435
AUTHOR INDEX
Choplin, L., 84(64), 101(208; 218; 220; 225), 120(316a), 133(208), 139(208), 140(208) Chow, E., 275(402) Chow, P., 213(52) Chowdhury, R., 105(257) Choy, H. K. H., 237(141) Chuah, Y. K., 387(179), 388(179) Chung, P. M., 397(220; 221) Chung, T.-W., 316(64), 324(64), 329(64) Church, E. L., 247(187), 258(187) Churchill, S. W., 302(27) Chyu, M.-C., 302(25), 303(25), 317(25), 327(25; 69) Cohen, D., 150(357) Cohen, M. L., 187(25), 188(25), 192(25), 194(25), 195(25), 210(25), 211(25) Cohen, M. R., 264(302) Cohn, D. W., 260(239) Colburn, A. P., 386(153) Cole, R. A., 397(291) Coley, M. B., 397(241) Colin, C., 34(23) Colles, M. J., 214(73) Collias, D. J., 97(169), 121(169), 122(169), 123(169) Collins, F. G., 3(3) Collins, R. W., 251(204), 256(204) Cook, R. L., 263(265) Coran, A. Y., 79(14) Corneille, J. L., 97(154) Cornewell, K., 330(73), 332(73) Cornwell, J. D., 320(60) Cornwell, K., 298(1), 313(1), 330(1), 332(1) Corpstein, R. R., 110(286) Cossor, G., 103(241), 127(241) Costell-Perez, M. E., 120(313) Costich, E. W., 109(285), 110(285) Couderc, J. P., 98(172; 173; 174), 111(172; 174), 139(173) Coulombe, S. A., 247(190) Coulson, J. M., 80(33), 81(33), 125(33) Courboin, D., 258(224) Coutts, T. J., 182(8), 269(8) Cowell, T. A., 353(105), 354(105; 106), 355(105; 108), 356(105; 106), 357(106; 108; 112), 358(106), 360(106), 367(105), 376(105), 379(105), 382(105), 384(105) Coyle, C. K., 94(120), 145(120), 147(120)
Cravalho, E. G., 211(43), 224(43), 237(134; 140) Cremers, C. J., 397(238) Crosser, O. K., 328(72), 331(72), 337(72) Cubukcu, E., 275(390) Cunningham, M., 155(373a) Cunnington, G. R., 264(308) Curran, S. J., 104(254)
D Dagata, J. A., 249(198), 250(198), 251(198) Dainty, J. C., 263(294) Dallavalle, J. M., 85(79) Daly, G., 150(368) Damianides, C. A., 299(6), 312(6) Datla, R. U., 213(45), 226(108), 232(130), 235(132) Davenport, C. J., 352(101; 102), 353(101), 354(101), 366(101; 102), 367(102), 369(102), 370(102), 371(102), 376(102), 378(102), 379(102), 380(102), 382(102), 383(102), 384(102) Davies, H., 246(174), 255(174), 261(174) De Maerteleire, E., 96(153) De Roussel, P., 87(265), 139(265) Deak, A., 96(145) Deans, H. A., 328(72), 331(72), 337(72) Deckwer, W.-D., 84(56) DeJong, N. C., 357(119), 361(119; 125; 124), 362(119), 365(119) DeKee, D., 80(31), 81(31; 39), 82(31; 39), 83(31), 84(47), 89(31), 106(31), 117(31) Dekker, C., 180(2) Del Sole, R., 226(102) Delalande, C., 273(360) Delaplace, G., 102(231; 232; 233; 234; 235), 105(276), 116(235), 117(233), 139(234; 276), 140(276) Delerue, C., 273(355; 356) Demont, P., 267(323) den Hertog, A. P., 94(116) Dengler, C. E., 37(25) Derdouri, A., 139(342), 140(342) Derksen, J., 91(275), 139(275), 140(275) DeRoussel, P., 87(264), 89(264) Dertinger, S. K. W., 89(266)
436
AUTHOR INDEX
Desplanches, H., 99(182; 183; 184), 105(258), 121(258), 143(182), 149(378; 379) Devotta, I., 104(249) DeWitt, D. P., 214(61), 227(61), 247(191; 184), 260(245), 263(184; 262), 264(262), 266(262) Dhir, V. K., 72(39) Dickey, D. S., 91(277), 124(324), 150(358; 366), 151(358) Diesinger, H., 274(367) Dietsche, W., 150(359) Dieulot, J.-Y., 102(231) Dilhac, M., 213(49) Dillen, E. L., 369(144), 378(144), 379(144), 383(144) Dimenna, R. A., 260(238; 240), 261(240; 256) Dimmick, G. R., 94(117) Ding, K.-H., 263(271) Ding, Z., 273(347) Doll, T., 275(389) Donnelly, V. M., 213(48), 225(98) Donnet, J.-B., 79(14) Doraiswamy, D., 116(304) Doraiswamy, L. K., 79(22) Dorofeev, A. M., 274(377) Dorris, G. M., 79(10) Dowing, R., 33(20), 34(20) Drakulic, R., 350(95), 377(95), 379(95) Dream, R. F., 142(354), 149(354) Dreister, G. A., 394(207; 208) Dresselhaus, G., 180(3) Dresselhaus, M. S., 180(3) Drew, H. D., 213(45) Drolen, B. L., 263(301) Drost, M. K., 247(183; 182), 263(182; 183), 264(182) Duarte, G., 139(347a) Dubois, C., 101(214; 222; 223) Ducla, J. M., 105(258), 121(258) Dukler, A. E., 15(9), 23(14), 34(23; 24), 35(14; 24), 306(37), 311(37), 309(42) Dumas, P., 258(223; 227) Duong, T., 397(249) Duparre´, A., 263(292) Dupont, D., 258(230) Dutta, A., 86(83) Dutta, B., 204(33) Dyer, J. M., 397(266)
Dyster, K. N., 125(329), 139(329), 140(329)
E Early, E. A., 264(310), 265(310) Eckert, E. R. G., 264(311) Edayoshi, A., 397(292) Edgar, H., 264(316) Edney, H. G. S., 95(137), 144(137) Edwards, M. F., 79(3), 80(3), 82(41), 95(137; 41), 97(170; 171), 100(194), 106(3), 111(3), 116(194), 118(41; 194), 135(3), 142(355), 143(355), 144(137; 171; 355), 147(171), 149(355), 150(3), 156(3) Efferding, L. E., 79(13) Efros, Al.L., 269(343), 271(343), 272(343), 273(343) Einekel, W.-D., 150(360) Einevoll, G. T., 273(346) Eklund, P. C., 180(3) Elmahdy, A. H., 387(169) Elson, T. P., 97(167), 98(175; 176), 118(175 176), 125(167; 175; 176), 127(167; 175; 176), 156(374) Ema, H., 139(347), 141(347) Emery, A. F., 397(279) Enoch, S., 237(145) Ermis, K., 303(30), 305(30), 317(30), 328(30), 329(30), 331(30), 337(30), 403(30), 404(30) Esashi, M., 268(334) Espeut, K. W., 394(199) Espinosa, T., 101(219) Espinosa-Solares, T., 101(212; 215), 111(215) Esproles, C., 264(306) Etchells, A. W., III, 99(187), 108(187), 116(304), 128(187) Etemad, S., 204(33) Everett, H. J., 109(285), 110(285)
F Fabre, J. A., 15(9), 34(23; 24), 35(24) Fair, R. B., 214(72) Faist, J., 216(76; 78) Fajner, D., 150(369) Falk, J., 397(254) Fan, S., 275(400)
437
AUTHOR INDEX
Fang, S., 180(3) Fangary, Y. S., 104(252), 125(252) Fasano, J. B., 123(323), 139(340; 339) Fath, P., 182(10), 269(10) Feehs, R. H., 92(87) Feick, H., 273(350) Fejfar, A., 182(11), 269(11) Feng, L.-F., 103(248) Feng, X., 264(313) Fenic, J. G., 123(322) Ferrand, P., 274(369) Ferry, D. K., 204(35), 206(35) Ferwerda, J. A., 247(186) Filinski, I., 246(163) Fillaudeau, L., 102(232) Finch, E. B., 397(218) Finkbeiner, S., 273(363) Fisk, W., 397(274; 275) Fleming, J. G., 275(387; 401) Fletcher, D. F., 125(327), 140(327) Flik, M. I., 206(37; 38), 211(43), 220(88), 224(43; 92), 226(112), 233(38), 243(157) Flores, F., 84(55), 99(55), 118(55) Fo¨ll, H., 274(379) Fondy, P. L., 110(286) Fonstad, C. G., Jr., 237(141) Foo, S.-C., 247(186) Forchel, A., 180(4), 271(344) Ford, D. E., 95(134; 138; 139; 140) Ford, J. N., 264(320) Ford, W. N., 99(187), 108(187), 128(187) Foroquet-Murh, L., 101(230), 149(230) Forouhi, A. R., 225(99; 100) Forschner, P., 108(311), 119(311) Fournier, A., 263(264) France, D. M., 301(21), 302(24; 25), 303(21; 24; 25; 29), 305(24; 29; 31), 306(35), 307(35), 308(35), 309(35), 316(21), 317(25), 319(21), 323(21), 327(25; 69), 330(31), 333(24; 29; 31; 74), 335(29; 31) Franke, J. E., 247(179) Franta, D., 251(203), 258(203) Frayce, D., 139(342), 140(342) Freilikher, V., 263(297; 298) Frenkel, A., 204(32) Friberg, A. T., 213(55) Fried, M., 274(368) Friedel, L., 319(56), 322(56) Fu, C. J., 237(142; 143)
Fuchs, L., 91(270) Fujieda, S., 97(164) Fujii, T., 32(19) Fujita, H., 316(62), 321(62), 329(62) Fujita, Y., 14(8), 53(29), 61(33) Fujiyama, H., 49(28) Fukano, T., 300(13), 312(13), 316(13), 320(13), 340(84), 348(91) Fuks, I., 263(297) Fukusako, S., 397(242; 246) Fukushima, T., 275(398), 276(398) Fung, A. K., 263(272; 273) Furling, O., 120(316a) Furuta, H., 316(62), 321(62), 329(62)
G Gabriel, C. J., 246(159) Galbiati, L., 312(52), 314(52), 315(52) Galindo, E., 84(51; 52; 53; 54; 55; 62; 63; 65), 99(51; 52; 53; 54; 55; 63; 191), 103(65), 111(52; 53; 62), 118(51; 52; 53; 54; 55; 63; 191), 127(53), 128(191; 52; 53) Galindo, R., 258(227) Gallagher, T., 264(313) Ganibal, C., 213(49) Gao, F., 204(33) Gaponenko, S. V., 269(339), 271(339), 272(339), 273(339) Gardner, J., 91(278) Garone, P. M., 213(54) Garrison, C. M., 106(280), 108(280) Gasparetto, C., 120(316c) Gaston-Bonhomme, Y., 99(184), 149(378; 379) Gatchilov, T. S., 397(234; 235) Gates, L. E., 113(303), 123(303), 124(303; 324), 150(303) Gates, R. R., 397(270) Ge, X. S., 226(111) Gebhart, B., 268(324; 325; 326; 327) Gegenwarth, H. H., 222(89), 224(89) Gelloz, B., 274(367) Genereux, F., 181(7) Gentile, T. R., 235(132) Geogiadis, J. G., 397(266; 267) Gerber, Ch., 252(210) Gerber, H. A., 256(220) Germer, T. A., 256(212; 213; 214; 215) Gervais, F., 224(93)
438
AUTHOR INDEX
Ghiaasiaan, S. M., 300(7), 307(39), 309(39), 310(7), 312(7), 316(63), 321(63) Ghosh, D., 359(123), 360(123) Giesekus, H., 128(331) Gini, E., 216(78) Gladki, H., 150(361) Gluz, M. D., 93(101; 102; 103), 106(103), 143(101; 102; 103) Gmachl, C., 216(77; 79) Gmitter, T. J., 274(380; 381) Godfrey, J. C., 82(41), 95(41), 93(113; 112), 118(41) Godleski, E. S., 92(99), 137(99) Gokce, O. H., 214(71) Goldberg, A., 275(402) Goldschmidt, V. W., 396(217) Goodman, W., 387(156) Goodson, K. E., 204(31), 206(37; 38), 233(38), 226(112), 405(301), 406(301; 302) Gori, F., 84(73) Gosele, U., 181(7), 274(379) Gotoh, S., 92(90; 91), 118(90; 91), 127(330) Gottscho, R. A., 247(179) Govan, A. H., 315(53) Gralak, B., 237(145) Gray, J. B., 80(28), 142(350), 144(350) Greenberg, D. P., 246(177), 247(186), 263(177) Greenwood, J. A., 248(196) Greffet, J.-J., 237(138), 238(152), 260(234), 268(332; 333) Grenville, R. K., 111(287), 116(304), 139(337) Griffith, P., 307(38), 311(50), 323(38) Grillot, J. M., 397(260) Groisman, A., 158(377) Grosse, P., 226(109) Grossman, E. N., 226(127), 239(127) Grote, M. G., 341(85) Grothe, H., 213(52) Grouillet, A., 258(224) Grynberg, 269(341), 271(341) Gu, X.-P., 103(248) Gu, Z.-H., 263(294) Guendouz, M., 274(376) Gue´rin, C.-A., 263(300) Guerin, P., 97(158; 159; 161), 138(158; 159), 143(161), 147(161), 148(159; 161) Guillory, J. L., 387(162) Gunde, M. K., 246(166)
Gu¨ntherodt, G., 217(81), 218(81) Gupta, R. K., 86(82) Gupta, S. N., 105(267) Guy, C., 97(154) Gyulai, J., 274(368) Gyurov, P., 104(253)
H Ha, S., 394(207; 208) Haaland, D. M., 247(179) Hacha, T. H., 387(157) Hadley, P., 180(2) Hafner, A., 407(304), 408(304), 409(304) Hagedorn, D., 93(111), 145(111) Haji, J., 274(376) Hakhoyan, A. P., 274(373) Hale, T. K., 84(50) Halimaoui, A., 273(360) Hall, K. R., 93(113; 112) Hane, K., 268(335; 336) Haner, D. A., 264(306) Hannote, M., 84(55), 99(55), 118(55) Hansen, S., 87(264), 89(264) Hanssen, L. M., 213(45), 226(108), 232(130) Hapke, B., 247(185) Haram, S. K., 273(347) Harbecke, B., 246(161) Harley, J., 299(5) Harnby, N., 79(3), 80(3), 106(3), 111(3), 135(3), 150(3), 156(3) Hartnett, J. P., 84(70; 74; 75), 85(78a), 142(352), 301(19), 304(19) Hasan, M. N., 72(39) Hashimoto, K., 97(164; 165) Hassager, O., 80(34), 81(34), 82(34), 89(34) Hattori, M., 92(98), 118(98), 397(292) Haus, H. A., 226(122), 235(122) Havas, G., 96(145) Hayashi, Y., 397(232; 240) Hayer, R. E., 104(254) Haynes, B. S., 316(61), 320(61), 329(61) He, X. D., 246(177), 263(177) Heavens, O. S., 226(118), 229(118), 235(118) Hebb, J. P., 211(43), 224(43), 263(269) Heikal, M. R., 350(95; 96), 355(108), 357(108; 112), 377(95), 379(95) Heim, A., 105(256), 145(256), 147(256) Hekmat, D., 397(274; 275)
439
AUTHOR INDEX
Hellmann, J. R., 214(66) Helmer, W., 387(163) Henkel, S., 274(365) Hentrich, P., 111(302) Hepner, L., 79(21) Heppner, D. B., 34(22) Herbst, H., 84(56) Hering, R. G., 247(180), 262(258), 263(180) Herino, R., 274(367) Herrero, P., 274(375) Hertel, B., 213(51; 52) Hesketh, P. J., 268(324; 325; 326; 327) Hewitt, G. F., 142(351), 315(53) Heywood, N. I., 80(26), 150(362) Hibiki, T., 301(20), 303(20), 307(40), 310(20; 40; 45), 311(20; 40; 45), 312(20; 45), 313(45), 314(45), 316(20; 40), 320(20; 40), 323(40) Hicks, R. W., 123(322), 124(324) Hidaka, S., 14(8), 53(29) Hietala, V. M., 275(388) Hifani, M. A., 397(243) Hill, J. M., 387(174) Hindson, W. A. J., 108(283), 128(283) Hipwell, M. C., 273(364) Hirabayashi, H., 127(330) Hirai, N., 94(118), 113(118), 116(118) Hiramatsu, M., 353(104) Hirao, Y., 340(84) Hiraoka, S., 93(109; 110), 96(146), 111(300), 139(146), 141(146) Hirata, Y., 103(242), 127(242) Hirleman, E. D., 249(200) Hirose, M., 273(359) Hirose, T., 103(245) Hirota, M., 316(62), 321(62), 329(62) Hirschland, H. E., 94(120), 145(120), 147(120) Hirschmugl, C. J., 204(33) Hittle, D. C., 387(175) Hjorth, S. A., 101(226; 227; 228), 111(228), 125(227), 127(226; 227), 128(227) Ho, F. C., 150(363) Ho, K.-M., 274(382), 275(384; 385; 386; 395; 401) Hobbs, P. V., 397(230) Hocker, H., 96(148), 111(148), 96(147), 119(147) Hockey, R. M., 104(250), 111(250), 113(250), 116(250)
Hoeft, J., 261(247) Hofstetter, D., 216(78) Hoke, J. L., 397(266; 267) Holland, F. A., 92(100), 132(100) Hong, K. T., 387(171), 394(201), 395(209), 396(212) Hoogendoorn, C. J., 94(116) Hori, K., 397(232) Horsley, R. R., 101(207), 118(207) Hosgood, B., 264(317) Hoshino, R., 397(295), 398(295) Hosoda, T., 397(272) Houchens, A. F., 262(258) Houska, M., 100(195), 116(195), 118(195) Howard, P., 355(107), 364(107) Howarth, L. E., 224(91) Howell, J. R., 186(18), 221(18), 226(18), 232(18), 246(18), 260(18), 267(322) Howell, R. H., 397(237; 288) Hrnjak, P., 409(308; 309; 310), 410(309; 310) Hsia, J., 261(247) Hsieh, Y., 387(178), 388(178), 392(178) Hsu, K., 373(149), 382(149), 383(149), 384(149) Hu, C.-H., 103(248) Hu, R. Y. Z., 84(74) Hu, X. F., 226(111), 387(177), 388(177) Huang, H. Z., 357(116; 117) Huang, L. G., 359(123), 360(123) Huang, M. H., 273(350) Huetz-Aubert, M., 267(323) Huffman, D. R., 226(120), 234(120), 235(120) Huffman, G. D., 397(270; 271) Hummel, R. E., 187(26), 194(26) Humphrey, A. E., 139(341), 141(341) Hung, C. C., 387(179), 388(179) Hunter, A., 247(192) Hutchinson, A. L., 216(76; 77) Hutter, R. J., 395(210)
I Ibrahim, S., 111(289) Ide, Y., 86(150; 151), 96(150; 151) Iihashi, M., 394(194) Ikeuchi, K., 246(176), 249(176) Ikuta, Y., 93(109) Ilegems, M., 216(78)
440
AUTHOR INDEX
Imon, J., 98(177) Ingers, J., 263(279) Inoue, A., 299(3; 4) Inoue, K., 18(12), 60(12), 61(12), 63(35), 275(398), 276(398) Inoue, S., 394(198) Irene, E. A., 256(218) Irvine, T. F., Jr., 84(73), 85(78) Ishiguro, S., 84(75) Ishihara, I., 320(58) Ishii, M., 309(44), 310(44; 48), 311(44; 48; 49; 51), 312(44), 313(44; 48), 314(48) Ishii, T., 394(200) Ishimaru, A., 261(253), 263(285; 296) Ismail, A. F., 98(177) Isoyama, E., 394(198) Itaya, M., 92(97), 118(97) Ito, R., 93(107; 109) Itoh, M., 397(295), 398(295) Ivakhnenko, V. I., 251(202), 258(202) Ivanova, V. S., 397(234; 235) Ivchenko, E. L., 226(106) Iwasaki, A., 72(43)
J Jackson, D. R., 263(274) Jackson, J. D., 182(13) Jacobi, A. J., 361(124), 365(124) Jacobi, A. M., 298(2), 299(2), 304(2), 306(2), 350(94), 357(119), 359(123), 360(123), 361(119; 125), 362(119), 365(119), 387(177; 180; 181; 182), 388(177; 180; 181; 182), 396(181; 217), 397(264; 266) Jacoboni, C., 204(34; 35) Jahoda, M., 111(301), 151(301) Jakeman, E., 263(299) Jallut, C., 397(260) Jane, R. J., 133(335) Janssen, L. P. B. M., 86(260), 123(321) Jaworski, Z., 102(239), 125(239; 329), 127(239), 128(239), 139(329), 140(329) Jellison, G. E., Jr., 213(58), 214(58; 67), 225(94) Jendrzejczyk, F. A., 303(29), 305(29; 31), 330(31), 333(29; 31), 335(29;31) Jendrzejczyk, J. A., 301(21), 303(21), 306(35), 307(35), 308(35), 309(35), 316(21), 319(21), 323(21)
Jensen, K. F., 263(269) Jeon, C. D., 357(118), 379(118), 382(118) Jeter, S. M., 387(174) Jhaveri, A. S., 95(133) Jiang, L., 405(301), 406(301; 302) Jin, L., 214(71) Joannopoulos, J. D., 274(381), 275(400) Johannesson, G., 397(280) John, G. C., 273(358) Johnson, J. T., 263(284; 285; 295) Johnson, S. R., 213(59), 214(59) Jomha, A. I., 100(194), 116(194), 118(194) Joshi, J. B., 101(11), 104(251), 125(251), 128(251) Joulain, K., 237(138), 268(332; 333) Ju, Y. S., 204(31) Jung, S. H., 377(139), 388(185)
K Kachoyan, B. J., 263(275) Kadlec, M., 98(181), 118(181) Kaganovskii, Y., 263(297) Kai, W., 99(188), 147(188) Kajanto, I., 213(55) Kajino, M., 353(104) Kakac, S., 302(23), 304(23) Kakaes, K., 237(149) Kale, D. D., 95(141), 122(141) Kaltchev, K. I., 397(234) Kamata, N., 97(163) Kamei, S., 397(219) Kaminoyama, M., 99(186), 100(192; 193), 139(193), 141(193) Kamiwano, M., 99(186), 100(192; 193), 111(296), 139(193), 141(193) Kanade, T., 246(176), 249(176) Kanamori, Y., 268(335; 336) Kanazawa, M., 394(197) Kandlikar, S. G., 330(78), 334(78), 336(78) Kanel, J. S., 100(200) Kanemitsu, Y., 273(353) Kantyka, T. A., 105(259), 145(259) Kanzieper, E., 263(298) Kappel, M., 133(334), 135(334), 136(334) Karcz, J., 111(290; 295), 125(295), 142(356), 151(290) Kariyasaki, A., 300(13), 312(13), 316(13), 320(13), 348(91) Kashani, M. M., 82(41), 95(41), 118(41)
AUTHOR INDEX
Kashiwa, T., 268(334) Katamine, T., 94(119) Kataoka, K., 139(347), 141(347) Katayama, K., 397(240) Kato, Y., 111(300) Katsube, T., 92(94), 118(94) Katsuta, K., 320(58) Katsuta, M., 330(77), 334(77), 345(89) Katto, Y., 44(26), 47(26) Kawai, N., 275(398), 276(398) Kawaji, M., 3(2; 3; 5), 17(11), 18(12), 60(12), 61(12), 306(36), 310(47), 312(36; 47), 313(36), 316(36; 57), 318(36), 319(57), 343(86), 344(87) Kawakami, S., 275(391) Kawasaki, H., 60(31), 64(31) Kawasaki, K., 17(11), 18(12), 60(12), 61(12) Kawase, Y., 103(245) Kawashima, T., 275(391) Kawka, P. A., 263(289) Kayama, T., 92(92; 95; 96; 98), 118(92; 95; 96; 98), 147(96) Kayansayan, N., 378(143) Kayanuma, Y., 272(345) Kays, W. M., 366(132) Kazuchits, N. M., 274(377) Kedzierski, M. A., 396(216) Keenan, W. A., 222(89), 224(89) Keentok, M., 120(314) Kelkar, J. V., 108(284), 121(320) Keller, R. B., 60(30) Kelly, W. J., 139(341), 141(341) Kemblowski, Z., 120(315) Kenny, T. W., 405(301), 406(301; 302) Keron, A., 108(283), 128(283) Kerschbaumer, H. G., 397(287) Keshock, E. G., 33(20), 34(20) Kew, P. A., 298(1), 313(1), 330(1; 73), 332(1; 73) Khakhar, D. V., 79(375), 87(264; 265), 89(264), 139(265), 156(375) Khan, M., 407(307) Khayat, R. E., 139(342), 140(342) Kida, F., 92(92), 118(92) Kifune, S., 397(219) Killgore, E. M., 397(224) Kim, C., 394(207; 208) Kim, G. J., 387(180), 388(180) Kim, H. R., 213(44)
441
Kim, I., 84(73) Kim, K., 48(27) Kim, M.-H., 349(92), 374(148; 189), 380(148), 383(148), 384(148; 151; 189), 389(148; 191; 192), 390(191), 391(148; 189; 191; 192), 392(151; 191; 192), 393(191; 192), 404(92), 406(303), 409(151) Kinber, B. Ye., 258(228) Kind, H., 273(350) King, C. D., 34(22) Kingston, R. H., 185(16) Kinoshita, M., 3(4) Kiri, Y., 93(107) Kirk, K. M., 61(34) Kishinami, K., 397(239) Kittel, C., 187(21), 188(21), 189(21), 190(21), 192(21), 196(21), 198(21), 199(21), 202(21), 203(21), 208(21), 209(21), 210(21), 220(21) Kleinman, D. A., 224(90; 91) Knaus, W. L., 386(154) Knight, J. C., 275(392) Knittl, Z., 226(119), 235(119) Knoch, A., 121(317) Ko, S.-Y., 83(44) Kobayashi, T., 326(67), 329(67), 330(67), 335(67) Kocka, J., 182(11), 269(11) Koga, Y., 394(193) Kognal, J., 111(294) Kogure, H., 397(295), 398(295) Koh, J., 251(204), 256(204) Koh, J. S., 213(44) Koizumi, H., 325(65) Koliopoulos, C. L., 253(211), 254(211) Koloni, T., 99(190) Konaka, Y., 93(109) Kondepudi, S. N., 397(277; 282) Kong, J. A., 182(14), 226(14), 263(271) Konno, H., 101(204; 205) Koo, J. M., 405(301), 406(301; 302) Kore, L., 274(371) Korgel, B. A., 273(347) Kornblit, A., 247(179) Korte, C., 387(181), 388(181), 396(181) Kosaka, H., 275(391) Koshida, N., 273(352) Kostic, M., 301(19), 304(19) Koto, T., 397(219) Kovacs, T., 91(270)
442
AUTHOR INDEX
Koyama, H., 273(352) Kramers, H. A., 219(83) Kraume, M., 112(305) Krebs, R., 108(311), 119(311) Kriebel, K. T., 260(237) Kristiansen, B., 120(315) Kroll, N., 237(144) Kronig, R. de L., 219(84) Krukar, R. H., 247(179) Krummenacher, L., 247(188) Kuang, Y., 251(204), 256(204) Kuboi, R., 98(179), 100(197), 151(179) Kudoh, M., 397(295), 398(295) Kuehn, T. H., 387(167) Kuga, Y., 263(285) Kuhn, R., 182(10), 269(10) Kumar, A. R., 226(107), 243(155; 156), 245(107), 246(107) Kumar, P., 104(251), 125(251), 128(251) Kumar, S., 226(116; 115) Kumpe, R., 256(220) Kuneewicz, C., 105(268) Kunioka, S., 97(164) Kuo, J. M., 213(53) Kupke, C., 246(162) Kureta, M., 326(67), 329(67), 330(67), 335(67) Kuriyama, M., 99(185), 147(185) Kuroyanagi, I., 389(190), 392(190), 393(190) Kushev, D. B., 246(170) Kusumoto, H., 397(295), 398(295) Kuznetsov, A. V., 269(338), 271(338) Kuznetsov, V. V., 330(76), 333(76) Kwok, C. C. K., 302(26), 316(26), 325(26), 329(26) Kwong, A., 150(363)
L Laba, D., 80(36) Labrie, R., 101(210), 139(210) Lacki, H., 102(240) Lacroix, R., 101(208), 133(208), 139(208), 140(208) Lafon, P., 139(343), 141(343) Lafortune, E., 247(186) Lai, K., 120(316b) Lai, L., 256(218) Laiho, R., 273(362)
Laird, A. D., 27(18) Lalonde, P., 263(264) Lammel, G., 274(378) Landau, L. D., 219(85) Lang, X., 357(115) Lange, C., 84(67) Langer, G., 88(265a), 96(147; 148), 111(148), 119(147) Lannoo, M., 273(356) Larson, R. G., 158(376) Lazarek, G. M., 316(66), 325(66), 330(66), 333(66), 335(66) Le Blevec, J.-M., 101(224), 141(224) Le Gall, R., 397(260) Le Gorju, J., 274(376) Lee, C., 387(183; 184), 388(183; 184), 392(184) Lee, D.-L., 85(78) Lee, H., 400(296), 401(296), 409(296) Lee, H. J., 301(22), 303(22), 316(22), 323(22), 327(22; 70), 329(22), 336(70) Lee, H. P., 213(53) Lee, H. S., 72(41) Lee, J. H., 213(44) Lee, J., 357(118), 379(118), 382(118) Lee, S. C., 264(308) Lee, S. Y., 301(22), 303(22), 316(22), 323(22), 327(22; 70), 329(22), 336(70) Lee, W. Y., 85(78a) Lee, Y. H., 84(57) Lee, Y. N., 357(114) Lee, Y. S., 111(300) Lefever-Button, G., 234(131), 235(131) Lehmann, V., 274(379) Lehnert, W., 274(368) LeMoul, A., 316(63), 321(63) Leng, D. E., 139(340) Leonard, S. W., 181(7) Leong-Poi, L., 120(316) Le´rondel, G., 246(171) Lester, L. F., 273(349) Lettieri, T. R., 258(225), 261(225), 263(225) Leuliet, J.-C., 101(220; 225), 102(231; 232; 233; 234; 235), 105(276), 116(235), 117(233), 139(234; 276), 140(276) Leung, K. M., 274(380) Levy-Clement, C., 274(372) Lewis, W. K., 386(152) Li, H., 273(349) Li, H. H., 225(97)
443
AUTHOR INDEX
Li, R.-Y., 302(26), 316(26), 325(26), 329(26) Li, X., 260(235) Li, Y., 213(53) Liang, S. Y., 387(172) Lienhard, J. H., 185(15), 202(15) Lifshitz, E. M., 219(85) Lin, C.-X., 83(44) Lin, S. Y., 275(387; 388; 401) Lin, S., 302(26), 316(26), 325(26), 329(26) Lin, T.-F., 305(33; 34), 316(33), 317(34), 326(33), 328(33; 34), 329(34), 330(33), 331(34), 334(33), 336(33; 34) Lin, W., 363(127), 364(127), 365(127), 401(127) Lin, Y., 373(149), 382(149), 383(149), 384(149), 387(178; 183; 184), 388(178; 183; 184), 392(178; 184) Lindley, J. A., 79(16; 17; 18; 19; 20), 156(16; 17; 18; 19; 20), 157(19) Lines, P. C., 108(283), 128(283) Ling, A. T., 23(13) Liszka, E. G., 261(252) Little, D. C., 397(224) Littles, J. W., 34(22) Liu, G. T., 273(349) Liu, H. C., 216(77) Lizuka, K., 249(198), 250(198), 251(198) Llinas, J. R., 99(182), 143(182) Lockhart, R. W., 27(17), 318(55) Lockwood, D. J., 273(361) Lohner, T., 274(368) Lomas, K. J., 246(162) Loncar, M., 275(389) London, A. L., 301(18), 366(132) Long, R. E., 84(72) Loomis, J. J., 237(136) Lossev, V., 84(66) Louie, S. G., 226(103) Loulou, T., 85(77) Lowe, D. C., 300(12), 312(12) Lowndes, D. H., 213(58), 214(58) Lowry, B., 306(36), 312(36), 313(36), 316(36), 318(36) Lu, D., 388(186) Lu, J. Q., 263(291; 294) Lu, Y., 251(204), 256(204) Ludemann, R., 274(372) Lue, S. S., 357(116) Lui, R. K., 3(5) Lulinkski, Y., 309(43)
Luo, Y., 111(291) Lyo, S. K., 275(388)
M Macaskill, C., 263(275) MacFarlane, G. G., 225(95) Machon, V., 98(178), 111(301), 121(178), 123(178), 151(301) MacKinnon, A., 275(393) Macosko, C. W., 139(338; 345) Maex, K., 214(63) Magelli, F., 150(369) Mahan, J. R., 185(17), 186(17), 226(17), 260(17) Mainguy, S., 268(332) Major, M., 111(290), 151(290) Majumdar, A., 187(24), 188(24), 202(24), 204(24), 248(197), 249(200; 201), 263(201; 286) Malam, S. C., 213(56) Malik, I. J., 247(193), 251(193), 258(193) Malloy, K. J., 273(349) Manas-Zloczower, I., 141(349) Mandel, L., 238(153) Mann, R., 84(66), 104(253), 125(328), 139(328), 140(328) Manna, L., 136(336) Manoliadis, O., 84(46) Mansuripur, M., 247(194) Mao, S., 273(350) Mao, Y., 397(252; 253; 254; 255) Maradudin, A. A., 260(244), 263(278; 281; 282; 290; 291; 294; 297; 298) Marchilden, E., 155(373a) Marinyuk, B. T., 397(236) Maris, H. J., 237(136) Marquand, C. J., 397(289) Martin-Palma, R. J., 274(374; 375) Martinelli, R. C., 27(17), 318(55) Martinez, A., 84(65; 62), 99(191), 103(65), 111(62), 118(191), 128(191) Martı´ nez-Anto´on, J. C., 264(303) Martinez-Duart, J. M., 274(374; 375) Martinez-Frias, J., 397(293; 294) Martone, J. A., 94(122), 146(122) Maruyama, S., 268(334) Marx, E., 247(193), 251(193), 258(193; 226; 225), 261(225), 263(225)
444
AUTHOR INDEX
Mashelkar, R. A., 86(83), 95(140; 136; 141; 142; 143), 108(284), 115(136), 121(136; 320), 122(141), 123(142; 143), 135(136) Masiuk, S., 102(240) Masucci, S. F., 150(364) Mathur, G. D., 395(211) Matsumoto, K., 72(42) Matsuo, K., 397(242; 246) Mattsson, L., 247(188) Maveety, J. G., 406(302) Mavros, P., 91(271), 103(243), 125(327; 328), 139(328), 140(327; 328) Maystre, D., 263(279) McAuley, K., 155(373a) McCalmont, J. S., 275(385) McCaulley, J. A., 213(48), 225(98) McComb, B. D., 269(337), 271(337) McCoy, J. J., 261(252) McDonald, D. G., 226(127), 239(127) McDonough, R. J., 79(4), 80(4) McGuckin, B. T., 264(306) McGurn, A. R., 263(290) McLaughlin, W. J., 388(187; 188) McLean, T. P., 225(95) McNeil, J. R., 247(179; 190) McQuillen, J. B., 15(9), 34(24), 35(24), 300(11), 312(11) McQuiston, F. C., 378(142), 387(162; 164; 165; 166) Meade, R. D., 274(381) Mehendale, S. S., 298(2), 299(2), 304(2), 306(2) Mehra, V. K., 397(238) Mehta, C. L., 237(150) Meier, J., 182(11), 269(11) Meister, G., 260(246) Melchior, H., 216(78) Melnyk, W., 405(298) Mendez, E. R., 263(294), 264(305) Mengu¨c, M. P., 264(307) Menon, M., 180(3) Menzies, R. T., 264(306) Mermin, N. D., 187(22), 188(22), 190(22), 198(22), 199(22), 202(22), 203(22) Mersmann, A. B., 99(189) Merte, H., 60(30), 61(34), 72(36; 41) Metzner, A. B., 92(87; 84; 85; 86), 113(84; 85), 115(84), 125(85; 86), 137(86) Mezaki, R., 81(40a), 111(40a), 142(40a) Mezic, I., 89(266)
Miaoulis, I. N., 226(125), 263(268) Michel, B. J., 94(120), 145(120), 147(120) Michel, E., 275(386) Michel, T., 263(291; 294) Micko, S., 72(40) Midoux, N., 101(230), 149(230) Migdall, A. L., 235(132) Mikkelsen, J. C., 406(302) Miller, R. C., 224(91) Miller, W. A., 397(247) Milthorpe, J. F., 120(314) Mimaki, M., 394(205) Min, J., 394(203; 204; 202), 395(202), 396(213) Minekawa, K., 103(245) Minhas, B. K., 247(190) Mirth, D. R., 387(175) Misawa, M., 3(1) Mishima, K., 301(20), 303(20), 307(40), 310(20; 40; 45), 311(20; 40; 45; 51), 312(20; 45), 313(45; 51), 314(45), 316(20; 40), 320(20; 40), 323(40), 326(67), 329(67), 330(67), 335(67) Mishra, V. P., 104(251), 125(251), 128(251) Mitra, K., 226(116) Mitsas, C. L., 246(169) Mitschka, P., 98(181), 118(181) Mitson, R. J., 100(198), 121(198) Mitsuishi, N., 84(76), 94(119; 118), 113(118), 116(118) Miwa, K., 394(196) Miyairi, Y., 84(76), 94(119) Miyazaki, K., 3(4) Miyazaki, S., 273(359) Mizoguchi, K., 96(146), 139(146), 141(146) Mizoguchi, M., 394(198) Mizushina, T., 93(107; 108; 109; 110), 397(219) Mochizuki, M., 81(40a), 111(40a), 142(40a) Mochizuki, S., 358(120), 365(130) Modest, M. F., 186(19), 214(65; 66), 226(19), 260(19) Modi, V., 339(83), 340(83) Modine, F. A., 214(67), 225(94) Mohr, J. B., 249(200) Mo¨ller, H., 213(50; 51) Montecchi, M., 246(172) Montereali, R. M., 246(172)
AUTHOR INDEX
Monti, M., 264(319) Moo-Young, M. B., 84(58), 92(88), 113(88), 116(88), 117(88) Moore, I. P. T., 103(241; 242), 127(241; 242) Moreira, R. G., 120(313) Mori, Y. H., 72(37) Morita, T. S., 63(35) Moriyama, K., 299(3; 4) Morris, G. M., 247(178), 268(178) Morton, J. R., 123(322) Mulet, J.-P., 237(138), 268(332) Mulholland, G. W., 256(215) Muller, C., 264(317) Mu¨ller, F., 274(379) Mu¨ller, T., 256(220) Munier, M., 150(365) Murakami, Y., 93(107; 108) Muramot, H., 358(120) Murayama, K., 273(359) Murray-Coleman, J. F., 264(315) Murthy, M. S., 142(353), 144(353) Muskett, M. J., 150(373) Muto, T., 320(58) Muzzio, F. J., 91(274) Myers, K. J., 125(326), 150(366; 367; 368) Myers, R. J., 387(160)
N Nagase, Y., 98(177) Nagashima, A., 72(37) Nagashima, T., 348(91) Nagata, K., 345(89) Nagata, S., 79(5), 80(5), 92(89; 90; 91; 92; 93; 94; 95; 96; 97; 98), 97(163), 118(89; 90; 91; 92; 93; 94; 95; 96; 97; 98), 127(330), 130(333), 131(333), 142(5), 147(96) Najjar, F. M., 364(128), 401(128) Nakajima, M., 92(90; 95), 118(90; 95) Nakamura, M., 97(165) Nakamura, T., 17(11), 18(12), 60(12), 61(12) Nakanishi, T., 180(2) Nakazawa, T., 32(19) Namai, K., 358(120) Nanda, A. K., 214(70) Naqvi, S. S. H., 247(179; 190) Nathan, G. K., 387(172) Navra´til, K., 263(293) Nayar, S. K., 246(176), 249(176)
445
Nedeljkovic, D., 275(389) Nedoluha, A., 246(159) Neiderer, D. H., 397(273) Nelson, R. M., 397(248) Netusil, J., 94(128) Neumann, A., 263(263) Neumann, E., 72(39) Neumann, L., 263(263) Newborough, M., 397(265) Newell, T. A., 336(80) Ng, P., 120(316b) N’Guyen, H. T., 267(323) Nguyen, Q. D., 83(43), 120(312) Nichelatti, E., 246(172) Nicholas, J. V., 263(260) Nicodemus, F. E., 259(233) Niemczyk, T. M., 247(179) Nienow, A. W., 79(3), 80(3), 84(52; 53; 54; 59), 97(167), 98(176; 178; 179), 99(52; 53; 54), 100(197; 198), 101(226; 227), 102(236; 239), 103(242), 106(3), 111(3; 52; 53; 289), 118(52; 53; 54; 176), 121(178; 198; 236), 123(178), 125(167; 176; 227; 236; 239; 329), 127(167; 176; 226; 227; 239; 242; 53), 128(52; 53; 227; 236; 239), 135(3), 139(329), 140(329), 150(3; 373), 151(179), 156(3; 374; 59) Nieto-Vesperinas, M., 260(234; 242; 243; 244), 261(248), 263(294) Nieva, P., 263(268) Nijs, J., 274(372) Nikaido, N., 394(194) Niranjan, K., 79(19; 20), 156(19; 20) 157(19) Nishihara, A., 394(197) Nishihara, H., 301(20), 303(20), 310(20), 311(20), 312(20), 316(20), 320(20), 326(67), 329(67), 330(67), 335(67) Nishikawa, K., 14(8), 53(29) Nishikawa, M., 92(90; 91; 92; 93; 94; 95; 96; 97; 98) 97(163; 164; 165), 118(90; 91; 92; 93; 94; 95; 96; 97; 98), 147(96) Nix, R. M., 213(56) Nolhier, N., 213(49) Norris, P. M., 226(114), 227(114) Norwood, K. W., 92(86), 125(86), 137(86) Notomi, M., 237(146), 275(391) Nouri, J. M., 104(250), 111(250), 113(250), 116(250) Novak, V., 94(126; 127; 125)
446
AUTHOR INDEX
Nunez, M. C., 101(221) Nunhez, J., 139(347a) Nussenzveig, H. M., 219(82)
O Obot, N. T., 301(21), 303(21), 316(21), 319(21), 323(21) Ochiai, T., 275(394) O’Donnell, K. A., 264(305) O’Donovan, E., 120(314) Oesterle, U., 216(78) Offermann, F. J., III, 397(274; 275) Ogawa, K., 81(40a), 111(40a), 142(40a), 397(283) Ogilvy, J. A., 258(231) Ogino, T., 394(197) Ogushi, T., 3(5) Oh, H. K., 330(77), 334(77) Ohara, T., 316(62), 321(62), 329(62), 389(190), 392(190), 393(190) Ohira, H., 299(3; 4) Ohlı´ dal, I., 246(168), 251(203), 258(203), 263(293) Ohlı´ dal, M., 246(168), 251(203), 258(203) Ohmura, N., 139(347), 141(347) Ohno, H., 44(26), 47(26) Ohta, H., 14(8), 17(11), 18(12), 49(28), 53(29), 60(12; 31), 61(12; 32; 33), 63(35), 64(31) Ohta, K., 17(11) Ohthani, S., 397(229) Oka, T., 72(37) Okada, S., 17(11), 18(12), 60(12; 31), 61(12), 64(31) Okamoto, R., 84(49) Okita, H., 394(197) Okoroafor, E. U., 397(265) Oldshue, J. Y., 79(6), 80(6), 84(60), 94(120), 106(6), 108(344), 126(344), 139(344), 145(120), 147(120), 150(6), 156(6; 60) Oliver, D. R., 100(198), 121(198) Olsson, C., 300(17), 301(17), 302(17), 303(17) O’Neal, D. L., 397(244; 245; 277; 282; 284; 285; 286; 290) Onida, G., 226(102) Orr, C., Jr., 85(79) Osada, H., 389(190), 392(190), 393(190) O’Shima, E., 94(121)
Ostin, R., 397(251; 280) Otsuka, T., 394(198) Ottino, J. M., 77(1), 79(375), 87(261; 262; 263; 264; 265), 89(1; 261; 262; 263; 264), 139(265; 338; 345), 156(375) Otto, R. E., 92(87; 84), 113(84), 115(84) Ottow, S., 274(379) Ousaka, A., 348(91) Owen, D. G., 315(53) Ozaki, N., 111(300) Ozbay, E., 275(385; 386) Ozcan-Taskin, N. G., 102(236), 121(236), 125(236), 128(236)
P Pace, G. W., 97(167), 125(167), 127(167) Pacek, A. W., 102(239), 125(239), 127(239), 128(239) Padki, M. M., 397(248; 250) Pahl, M. H., 111(291) Paiella, R., 216(77) Pak, K., 263(283; 284; 295) Palik, E. D., 206(39), 207(39), 213(39), 222(39), 224(39), 225(39), 235(39), 267(39) Palmer, B. J., 247(183), 263(183) Pan, J. L., 237(141) Pandey, D. K., 105(267) Pandit, A. B., 101(11) Pankove, J. I., 206(41), 208(41), 209(41), 211(41), 212(41), 215(41), 217(41) Panton, R. L., 300(8), 308(8) Papastefanos, N., 111(292; 293) Paris, J., 97(158; 159; 161), 138(158; 159), 143(161), 147(161), 148(159; 161) Parker, B., 397(284) Parker, D. J., 104(252), 125(252) Parr, A. C., 261(247), 264(310), 265(310) Pasquali, G., 150(369) Passek, F., 256(220) Patel, K. G., 94(124), 146(124) Patel, M. K., 139(346), 140(346) Pattanaik, S., 247(186) Patterson, G. K., 79(7), 80(7), 156(7) Patterson, W. I., 97(155; 156; 157; 158), 115(156; 157), 117(157), 131(155), 133(155), 138(155; 158) Paul, D. D., 84(71) Pavlov, A., 273(362)
447
AUTHOR INDEX
Pavlushenko, I. S., 93(101; 102; 103), 106(103), 143(101; 102, 103) Pearsall, T. P., 213(59), 214(59), 275(389) Pearson, J. T., 378(140) Peerhossaini, H., 85(77) Peixoto, S., 139(347a) Pell, L. E., 273(347) Pell, T., 213(56) Pendry, J. B., 275(393) Peng, C., 247(194) Peng, S., 247(178), 268(178) Peng, X. F., 300(15; 14) Penney, W. R., 123(323) Pentry, J. B., 237(137) Perderson, B. S., 397(274; 275) Perez, J. E., 94(123) Pericleous, K. A., 139(346), 140(346) Perkins, J. S., 261(249) Perkowitz, S., 274(370) Perlmutter, M., 267(322) Peters, D. C., 93(105; 106), 129(105; 106), 137(105; 106) Peterson, G. P., 300(15) Peterson, K. T., 397(290) Petrich, R., 246(167) Petrik, P., 274(368) Petrou, A., 269(337), 271(337) Pettersen, J., 406(303), 407(304), 408(304), 409(304) Petukov, B. S., 305(32) Peuse, B., 247(192) Pfahler, J., 299(5) Pfrommer, P., 246(162) Philips-Invernizzi, B., 258(230) Phillips, J. M., 206(38), 233(38) Phillips, V. R., 79(19; 20), 156(19; 20), 157(19) Phong, B. T., 263(267) Phu, P., 263(296) Piermarocchi, C., 273(348) Pikus, G., 226(106) Piriou, B., 224(93) Plazl, I., 99(190) Poduje, N., 247(193), 251(193), 258(193) Poggemann, R., 97(166), 143(166), 149(166) Polder, D., 237(135) Polgar, O., 274(368) Pollard, J., 105(259), 145(259)
Pompea, S. M., 264(312) Poortmans, J., 274(372) Porter, C. D., 204(33) Porter, J. E., 84(69) Poruba, A., 182(11), 269(11) Postlethwait, M. A., 214(66) Potemski, 269(341), 271(341) Potter, R. F., 246(160) Powell, F. R., 234(131), 235(131) Pownall, H. B., 386(155) Prakash, O., 105(267) Proctor, J. E., 264(309) Prokopec, L., 121(318) Prud’homme, R. K., 84(72), 97(168; 169), 121(168; 169), 122(169), 123(169)
Q Qiu, D. M., 72(39) Qiu, T. Q., 226(114), 227(114) Quan, C., 256(219) Quaraishi, A. Q., 95(142; 143), 123(142; 143) Quarrington, J. E., 225(95) Quate, C. F., 252(210) Quinn, B. M., 273(347)
R Rai, C. L., 104(249) Rajaiah, J., 86(82) Raju, S. P., 397(250) Ramadhyani, S., 378(140), 387(175) Ramos, H. L., 92(87) Ramsey, J. W., 387(167) Ranade, V. R., 96(149), 123(149) Ranz, W. E., 139(345), 140(345) Rao, A. M., 180(3) Rao, M. A., 120(316c) Rao, P., 300(16), 304(16), 407(16; 305) Rao, P. G., 104(249) Rappe, A. M., 274(381) Rauendaal, R. C., 80(25) Raughley, D. A., 100(199) Rauline, D., 101(224), 141(224) Rautenbach, R., 96(152), 145(152) Ravindra, N. M., 214(70; 71) Raymond, C. J., 247(190) Razeghi, M., 188(27) Reaves, C. M., 213(54)
448
AUTHOR INDEX
Redfield, D., 214(62) Reeder, M. F., 150(366; 368) Regenass, W., 79(12) Reggiani, L., 204(34) Rehr, E. O., 115(309) Reichert, J.-H., 84(67) Reid, R. C., 397(225; 227) Reilly, C. D., 79(19; 20), 102(238), 156(19; 20), 157(19) Reimann, J., 347(90), 348(90), 407(307) Reining, L., 226(102) Reithmaier, J. P., 271(344) Reizes, J. A., 101(207), 118(207) Rekstad, H., 407(304), 408(304), 409(304) Relandeau, V., 102(234), 139(234) Remes, Z., 182(11), 269(11) Renaud, P., 274(378) Reusss, M., 84(61) Rezkallah, K. S., 3(6; 7), 34(21), 300(9; 10; 12), 312(9; 10; 12), 313(10), 315(9; 10), 397(252; 253) Richardson, J. F., 80(32; 33), 81(32; 33), 82(32), 83(32), 89(32), 106(32), 125(33) Richter, E., 180(3) Richter, K., 226(124), 239(124) Richter, W., 213(46) Rickert, M. E., 394(195) Ridley, B. K., 208(42), 209(42) Rieger, F., 94(126; 127; 128; 125), 108(125), 111(294) Rite, R. W., 3(6; 7) Ritter, G., 213(51) Roberts, V., 225(95) Robertson, T., 79(8) Rogne, H., 214(69) Rohlfing, M., 226(103) Rohr, C., 182(9) Rohrer, H., 252(209) Rohsenow, W. M., 23(13), 142(352), 397(222) Romaszewski, R. A., 100(199) Romestain, R., 274(369) Rong, X., 343(86), 344(87) Ronse, G., 102(235), 116(235) Rosen, M., 269(343), 271(343), 272(343), 273(343) Rosenbluh, M., 263(297) Rothkirch, A., 260(246) Roy, S., 214(65) Ruan, R., 102(237), 120(237) Ruckenstein, E., 86(82)
Rudman, D. A., 243(156) Rudy, T. M., 396(214; 215) Rugh, J. P., 378(140) Rushton, J. H., 109(285), 110(285) Russell, P. St., 275(392) Russo, R., 273(350) Ryan, D., 150(367; 368) Ryan, D. F., 86(260) Ryoo, K., 213(44) Rzyski, E., 150(372)
S Saban, S. R., 213(59), 214(59) Sabato, T., 60(31), 64(31) Sadatomi, M., 316(57), 319(57) Sadatomi, Y., 320(59) Sadowski, D. L., 300(7), 310(7), 312(7) Sadowski, M. L., 269(341), 271(341) Sahin, A. Z., 397(258; 259) Sahnoun, A., 356(111), 360(111), 378(111), 379(111), 383(111) Sai, H., 268(335; 336) Saillard, J., 261(251), 263(251) Saillard, M., 263(279) Saini, M., 402(297) Saito, E., 99(186), 100(192) Saito, F., 100(193), 139(193), 141(193) Saito, H., 72(42), 397(239) Saito, K., 247(194) Saito, M., 3(4) Saito, S., 99(185), 147(185), 101(201; 202) Sakamoto, Y., 394(197) Sako, R., 394(197) Sakoda, K., 275(394; 396; 397; 398; 399), 276(398; 399) Salamone, J. J., 93(111), 145(111) Salvan, F., 258(223; 227) Sami, S. M., 397(249) Samson, E. B., 341(85) Sanchez, A., 84(62), 111(62) Sa´nchez-Gil, J. A., 260(242; 243) Sandall, O. C., 94(122; 123; 124), 146(122; 124) Sanders, C. T., 397(228; 231) Sandmeier, S., 264(317) Sant, A. J., 263(294) Santiago, J. G., 405(301), 406(301; 302) Saruwatari, S., 320(59) Sasada, M., 275(398), 276(398)
AUTHOR INDEX
Sasaki, M., 101(203) Sato, D. L., 213(53) Sato, T., 214(60), 275(391) Sato, Y., 320(59) Saunders, P., 264(316) Sawinsky, J., 96(145) Schaefer, S., 274(374; 375) Schaffer, M., 155(373a) Scheer, B. W., 256(212) Scheer, C. A., 251(202), 258(202) Schenkel, G., 80(23) Scherer, A., 275(389) Schirone, L., 274(372) Schliesing, J. S., 397(290) Schmolke, R., 256(220) Schneider, C. P., 222(89), 224(89) Schneider, H. W., 397(233) Schneider, T., 108(311), 119(311) Schnell, M., 274(374; 375) Schoenborn, E. M., 397(218) Schofield, C., 133(335) Schott, J. R., 264(313) Schro¨der, 213(52) Schraud, G., 213(52) Schulte, D. W., 397(237) Schultz, S., 237(147) Schumann, P. A., Jr., 222(89), 224(89) Schumpe, A., 84(56) Schweizer, S., 274(378) Seale, C., 246(162) Seeger, W., 347(90), 348(90) Seichter, P., 84(66), 111(294), 115(306; 307) Seki, N., 397(242; 246) Sengelin, M., 103(244), 121(244) Seo, G., 213(44) Sepsy, C. F., 397(223; 270) Serrano-Carreon, L., 84(63), 99(63), 118(63) Service, R. F., 180(1) Sestak, J., 100(195), 116(195), 118(195) Setzu, S., 274(369) Seville, J. P. K., 104(252), 125(252) Shah, A., 182(11), 269(11) Shah, J., 269(342), 271(342) Shah, R. K., 298(2), 299(2), 301(18), 304(2), 306(2), 336(80), 350(94; 96) Shah, Y. T., 397(227) Sham, L. J., 273(348) Shamirzaev, A. S., 330(76), 333(76)
449
Shang, H. M., 256(219) Shaqfeh, E., 97(168), 121(168) Sharangpani, R., 226(104) Sharma, P. C., 180(6) Shaw, J. A., 150(370) Shchegrov, A. V., 268(333) Shelby, R. A., 237(147) Shen, Y. J., 247(191), 263(261; 262), 264(262), 265(261), 266(261; 262) Shen, Y.-J., 266(321), 267(321) Shengyao, Y., 99(188), 147(188) Shenoy, A., 214(71) Shepard, D. F., 264(312) Sherif, S. A., 397(243; 248) Shervin, C. R., 100(199) Shibata, K., 330(77), 334(77) Shimizu, K., 103(245) Shimoya, M., 345(88), 346(88) Shin, R. T., 263(285) Shin, S., 85(80a) Shinagawa, T., 377(134) Shirai, S., 394(194) Shires, G. L., 142(351) Shirley, E. L., 226(101) Shirley, P., 247(186) Short, D. G. R., 99(187), 108(187), 128(187) Siapkas, D. I., 246(169) Siegal, M. P., 206(38), 233(38) Siegel, B. L., 397(279) Siegel, R., 186(18), 221(18), 226(18), 232(18), 246(18), 260(18) Siegrist, T., 220(88) Sigalas, M., 275(384; 385) Sigalas, M. M., 274(382), 275(395; 401) Siginer, D., 81(39), 82(39) Sikka, K. K., 214(66) Sillion, F. X., 246(177), 263(177) Simonson, R. J., 264(302) Sin, S., 251(206), 258(206) Sinevic, V., 100(197) Singh, V. A., 273(358) Sirtori, C., 216(76) Sivco, D. L., 216(76; 77) Skaugen, G., 407(304), 408(304), 409(304) Skelland, A. H. P., 81(38), 82(38), 94(117), 100(200), 111(288), 113(288), 115(288), 116(38) Slattery, J. C., 80(37), 85(37) Sleicher, C. A., Jr., 25(16)
450
AUTHOR INDEX
Smalley, R. E., 180(3) Smith, A. M., 264(315) Smith, D. L. O., 79(19; 20), 156(19; 20), 157(19) Smith, D. R., 237(144; 147) Smith, D. Y., 219(87), 220(87) Smith, G. R., 397(278) Smith, J. C., 92(99), 137(99) Smith, J. M., 93(104; 105; 106), 113(104), 116(104), 117(104), 129(105; 106), 137(105; 106) Smith, M. C., 352(98; 99; 100) Smith, T. F., 247(180), 263(180) Snyder, J., 264(302) Snyder, W. C., 260(235; 236) Solomon, J., 97(167), 98(178), 121(178), 123(178), 125(167), 127(167) Song, S. M., 389(192), 391(192), 392(192), 393(192) Sonju, O. K., 387(173) Sopori, B., 214(71) Sotgiu, G., 274(372) Soto-Crespo, J. M., 261(248) Soukoulis, C. M., 274(382), 275(384; 385; 390; 395) Sparrow, E. M., 246(175), 249(175), 261(175; 254; 255), 262(175), 264(255; 318) Special issue on Mixing 80(30) Speranza, A. C., 214(70) Spitzer, H., 260(246) Spitzer, W. G., 224(90; 91) Spizzichino, A., 246(173), 251(173), 255(173), 258(173), 260(173), 262(173) Springer, J., 182(11), 269(11) Springer, M. E., 358(121), 359(121), 364(121), 359(122) Stagg, B. J., 262(259) Stalmans, L., 274(372) Stamatoudis, M., 111(292; 293; 298; 299), 151(298; 299) Stanford, T. G., 91(279) Stapp, R. S., 264(302) Stark, J. A., 341(85) Steffe, J., 80(35) Steffe, J. F., 120(313; 316b) Steiff, A., 97(166), 143(166), 149(166) Steinberg, V., 158(377) Steinbichler, M., 72(40) Stenger, M., 104(255), 108(255) Stenzel, O., 246(167)
Stephan, K., 335(79) Stewart, I. W., 80(26) Stintz, A., 273(349) Stoecker, W. F., 397(268; 269) Stone, H. A., 89(266) Stoots, C. M., 91(269), 125(269) Storey, B. D., 397(264; 266) Stover, J. C., 247(187; 189; 193), 249(189), 251(193; 202), 255(189), 258(187; 189; 193; 202), 261(189) Straub, J., 17(10), 72(10; 38; 40) Strausser, Y. E., 247(193), 251(193; 204), 256(204), 258(193) Strehlke, S., 274(372) Strek, F., 111(295), 125(295), 142(356) Stroock, A. D., 89(266) Stubican, V. S., 214(65) Sturm, J. C., 213(54), 214(68) Subbaswamy, K. R., 180(3) Suga, K., 377(135) Suga, K. K., 377(134) Sugawara, S., 320(58) Sumi, Y., 111(296) Sundberg, R., 405(298) Sunden, B., 300(17), 301(17), 302(17), 303(17), 368(138), 377(136; 137; 138), 380(138), 382(138), 383(138), 384(138) Sung, L., 256(215) Suo, M., 307(38), 323(38) Suzuki, K., 72(42), 394(194) Svantesson, J., 368(138), 377(136; 137; 138), 380(138), 382(138), 383(138), 384(138), 377(136; 137) Svelto, O., 215(75) Swanepoel, R., 246(165) Szirmay-Kalos, L., 263(263)
T Tada, H., 92(91), 118(91), 127(330), 263(268) Tafti, D. K., 361(125), 363(126; 127), 364(127; 128), 365(127; 129; 131), 401(126; 127; 128; 129) Taha, I., 225(98) Taitel, Y., 306(37), 311(37), 309(42; 43) Takacs, P. Z., 247(187), 258(187) Takada, T., 17(11) Takahashi, K., 101(201; 202; 203; 204; 205; 206), 113(206), 117(206), 139(206)
AUTHOR INDEX
Takahashi, N., 397(229) Takasawa, R., 394(200) Takasu, S., 60(31), 64(31) Takeshige, Y., 394(193) Takeshita, M., 397(283) Takigawa, T., 139(347), 141(347) Takimoto, T., 118(92; 93), 92(92; 93) Tamamura, T., 275(391) Tamaoki, H., 17(11) Tamhankar, S. S., 79(22) Tanaka, K., 72(37) Tanaka, N., 397(283) Tanaka, S., 93(108) Tang, K., 258(232), 260(239), 261(256; 257), 263(257; 277; 280; 289), 264(320) Tanguy, P. A., 84(64), 101(208; 209; 210; 211; 212; 213; 214; 215; 216; 217; 218; 219; 220; 221; 222; 223; 224), 104(254), 111(215), 116(213), 117(213), 120(316a), 121(223), 133(208), 139(208; 210; 217), 140(208; 217), 141(217; 224) Tanner, D. B., 26(107), 43(155; 156), 45(107), 46(107), 204(33) Tanner, R. I., 81(40) Tantakitti, C., 397(288) Tao, W. Q., 357(116; 117) Tao, Y. X., 397(262) Tao, Y.-X., 397(255; 256) Tapper, , 405(298) Tareef, B. M., 85(80) Tassou, S. A., 397(289) Tatterson, G. B., 77(2), 79(15), 80(15; 27), 91(15; 27; 277; 278; 279), 106(15; 27), 111(15; 27), 150(27), 156(15; 27) Tay, C. J., 256(219) Tayeb, G., 237(145) Taylor, J. S., 92(85), 113(85), 125(85) Tecante, A., 84(64), 101(212; 219; 215) Tellier, C. R., 206(36) Tenefrancia, N., 101(11) Terry, J. L., 395(210) Terry, K., 100(198), 121(198) Thakur, R. P. S., 226(104) Theiß, W., 274(365) Theliander, H., 111(297), 151(297) Theofanous, T. G., 23(15) Thess, A., 180(3) Thibault, F., 101(211; 212; 214; 219) Thole, K. A., 358(121), 359(121; 122), 364(121), 359(122)
451
Thomas, J., 263(269) Thomas, L., 397(263) Thomas, T. R., 251(208) Thomasson, J. A., 264(307) Thompson, P., 213(53) Thonon, B., 350(96) Thorsos, E. I., 261(250), 263(274) Threlkeld, J. L., 387(167) Thurmond, C. D., 213(57) Tien, C. L., 185(15), 202(15), 226(113; 114; 123; 124), 227(113; 114), 237(134), 239(123; 124), 249(201), 263(201; 286) Tillack, B., 213(51) Tilton, J. N., 139(337) Timans, P. J., 206(40), 207(40), 213(40), 214(40; 64; 69), 221(40), 225(40; 96), 226(104) Titkov, A. N., 273(360) Tiwari, K. K., 101(11), 105(257) Tochon, P., 350(96) Toh, S. L., 256(219) Tokura, I., 397(239) Tolmachev, V. A., 274(366) Tomita, A., 275(391) Tong, A. H., 222(89), 224(89) Tong, F. M., 214(70) Toor, J. S., 247(181), 263(181) Torigoe, E., 345(88), 346(88) Torrance, K. E., 246(175; 177), 247(186), 249(175), 261(175; 254; 255), 262(175), 263(177; 265), 264(255; 318) Torres, L., 84(55; 62), 99(55), 111(62), 118(55) Torres, P., 182(11), 269(11) Torrez, C., 102(232), 103(246; 247), 105(276), 139(246; 247; 276), 140(276) Tosser, A. J., 206(36) Touloukian, Y. S., 214(61), 227(61) Touryan, K., 274(373) Towell, T. A., 350(95), 377(95), 379(95) Tragardh, C., 91(270) Trammel, G. J., 397(224) Tran, P., 263(278; 281) Tran, Q. A., 406(302) Tran, Q. K., 101(207), 118(207) Tran, T. N., 302(24; 25), 303(24; 25; 29), 305(24; 29; 31), 317(25), 327(25; 69), 330(31; 75), 333(24; 29; 31; 75; 74), 335(29; 31) Trauger, P., 356(110), 357(110)
452
AUTHOR INDEX
Tree, D., 387(163) Tree, D. R., 397(244; 245) Trefethen, L. M., 226(125) Triplett, K. A., 300(7), 310(7), 312(7), 316(63), 321(63) Troniewski, L., 308(41) Trumbore, B., 247(186) Tsai, B. K., 247(191; 184), 260(245), 263(184; 262), 264(262), 266(262) Tsang, L., 263(271; 283; 284; 295) Tseng, P. C., 387(179), 388(179) Tsong, T. T., 251(204), 256(204) Tsuboi, T., 273(362) Tura, R., 353(103), 354(103), 376(103) Tut, T., 275(390) Tuthill, J. D., 92(87) Tuttle, G., 275(385; 386) Tykal, M., 251(203), 258(203)
U Uchida, S., 61(33) Ueda, T., 48(27) Uemura, S., 397(242; 246) Uhl, V. W., 80(28), 142(350), 144(350) Ulbrecht, J. J., 79(7), 80(7), 95(129; 130; 131; 132; 133; 138; 139; 140; 141; 142; 143), 96(149), 108(284), 115(129; 131), 117(129), 121(320), 122(141), 123(142; 143; 149), 138(130), 156(7) Ulbrich, R., 308(41) Umemoto, S., 394(194) Ungar, E. K., 320(60) Uv, E. H., 387(173) Uzuhashi, H., 397(272)
V Valanju, A. P., 237(148) Valanju, P. M., 237(148) Vale, L. R., 243(156) van de Ven, J., 264(316) van den Akker, H. E. A., 91(273), 139(273), 140(273) van den Bakker, H. E. A., 101(229) van der Molen, K., 91(272) van Dierendonck, L. L., 86(260) van Driel, H. M., 181(7)
Van Hove, M., 237(135) van Maanen, H. E. A., 91(272) Vandenabeele, P., 214(63) Vanecek, M., 182(11), 269(11) Vasko, F., 269(338), 271(338) Vatel, O., 258(223; 227) Vazquez, L., 274(374; 375) Velasco, D., 84(65), 99(191), 103(65), 118(191), 128(191) Venneker, B. C. H., 91(273), 139(273), 140(273) Vernon, M., 225(98) Vernon, R., 15(9), 34(24), 35(24) Verspaille, M., 120(316d) Villeneuve, P. R., 275(400) Viney, L. J., 80(26) Viskanta, R., 247(181), 263(181) Vizd’a, F., 246(168) Vlaev, D., 84(66), 104(253) Vlaev, S. D., 84(66), 104(253), 125(328), 139(328), 140(328) Vlcek, J., 98(178), 121(178), 123(178) Vogel, B., 17(10), 72(10) von Bardeleben, H.-J., 273(355) von Essen, J., 150(371) Voos, M., 273(360) Vorburger, T. V., 249(198), 250(198), 251(198), 258(226; 225), 261(225), 263(225) Voss, K. J., 264(319) Vuckovic, J., 275(389)
W Wachsen, O., 84(67) Wagner, P., 256(220) Wainstain, J., 273(360) Wakabayashi, N., 397(295), 398(295) Wallis, G. P., 311(50) Walser, R. M., 237(148) Walter, B., 247(186) Wambsganss, M. W., 301(21), 302(24; 25), 303(21; 24; 25; 29), 305(24; 29; 31), 306(35), 307(35), 308(35), 309(35), 316(21), 317(25), 319(21), 323(21), 327(25; 69), 330(31), 333(24; 29; 31; 74), 335(29; 31) Wan, Z., 260(235) Wang, A. G., 273(361)
AUTHOR INDEX
Wang, A. T. A., 84(74) Wang, B. X., 300(15; 14) Wang, C., 351(147), 368(141), 370(145; 146), 371(145; 146), 372(146; 147), 373(149), 378(141), 379(141; 145; 146; 147), 380(147), 382(149), 383(141; 145; 146; 149), 384(146; 147; 149), 387(178; 183; 184), 388(178; 183; 184; 186), 392(178; 184), 394(206) Wang, C.-C., 316(64), 324(64), 329(64) Wang, D. I. C., 84(50) Wang, D., 180(6) Wang, G., 363(126; 127), 364(127), 365(127), 401(126; 127) Wang, J.-J., 103(248) Wang, K. L., 180(5; 6) Wang, K., 100(196), 103(248) Wang, T. K., 268(329; 330) Ward, G. J., 263(266) Ward, R. W., 125(326) Ware, C. D., 387(157) Warnick, K. F., 263(270) Watanabe, J., 93(109; 110) Watanabe, M., 345(89) Watanabe, Y., 340(84) Watters, R. J., 397(285; 286) Weakliem, H. A., 214(62) Webb, R. L., 300(16), 303(28; 30), 304(16), 305(30), 316(28), 317(30; 71), 323(28), 324(28), 328(30; 71), 329(30; 71), 331(30; 71; 81; 82), 337(30; 71; 81; 82), 350(93), 351(93), 356(109; 110; 111), 357(110), 360(111), 369(144), 370(145), 371(145), 376(133), 377(139), 378(111; 144), 379(111; 144; 145), 383(111; 144; 145), 387(171), 388(185; 187; 188), 394(201; 202; 203; 204), 395(202; 209), 396(213; 214; 215; 216), 400(296), 401(296), 402(297), 403(30), 404(30), 405(299; 300), 406(300), 407(16; 300; 305), 409(296) Weber, E., 273(350) Weber, J. H., 23(13) Weber, J., 273(363) Weidner, M., 213(51) Wein, O., 98(180), 118(180), 125(180), 126(180), 127(180) Weinspach, P.-M., 97(166), 143(166), 149(166) Weisman, J., 79(13)
453
Welty, J. R., 247(182; 183), 263(182), 264(182) Wenzel, L. A., 397(226) Werner, U., 88(265a), 96(148), 111(148) Westbye, C. J., 3(2) Westwater, J. W., 299(6), 312(6) Wettling, W., 274(372) Whale, M. D., 237(139; 140) White, D. R., 263(260), 264(316) White, J. L., 86(150), 96(150), 86(151), 96(151) Whitehouse, D. J., 247(195), 251(195), 256(195) Whitesides, G. M., 89(266) Wichterle, K., 98(180; 181), 118(180; 181), 125(180), 126(180), 127(180) Wiesendanger, R., 251(207), 252(207), 253(207) Wilkening, G., 249(198), 250(198), 251(198) Wilkinson, W. L., 142(355), 143(355), 144(355), 149(355) Wille, M., 88(265a) Willeke, G., 182(10), 269(10) Williams, G. P., 204(33) Williams, K. A., 180(3) Williams, M. C., 104(254) Williamson, J. B. P., 248(196) Wilmarth, T., 309(44), 310(44), 311(44; 49), 312(44), 313(44) Wilson, J. I. B., 214(73) Wisdom, D. J., 98(178), 121(178), 123(178) Witte, L. C., 300(11), 312(11) Woggon, U., 269(340), 271(340), 273(340) Wolf, E., 182(12), 226(12), 238(12; 151; 153) Wong, L. T., 352(100), 378(100) Wong, P. Y., 226(125), 263(268) Wong, T. N., 387(172) Woo, K. C., 256(219) Woodcock, L. V., 100(194), 116(194), 118(194) Woodson, B. A., Jr., 84(50) Wooten, F., 219(86), 220(86) Wronski, C. R., 251(204), 256(204) Wu, G., 387(170) Wu, X. D., 204(33) Wu, Y., 273(350) Wunsch, O., 141(348) Wyant, J. C., 253(211), 254(211)
454
AUTHOR INDEX
X Xanthopoulos, C., 111(298; 299), 151(298; 299) Xu, H., 214(68) Xu, J. L., 310(46), 311(46), 312(46), 313(46), 314(46), 315(46) Xuereb, C., 91(271), 103(243), 125(327), 128(332), 130(332), 139(332), 140(327; 332) Xun, X., 247(194)
Y Yablonovitch, E., 263(276), 274(380; 381; 383) Yagi, H., 95(144) Yagi, Y., 365(130) Yam, M., 247(192) Yamada, H., 32(19) Yamada, I., 96(146), 139(146), 141(146) Yamada, J., 264(307) Yamakawa, N., 397(229) Yamanaka, A., 275(398), 276(398) Yamaoka, M., 32(19) Yamauchi, S., 405(300), 406(300), 407(300) Yamazaki, K., 394(200) Yan, H., 273(350) Yan, Y.-Y., 305(33; 34), 316(33), 317(34), 326(33), 328(33; 34), 329(34), 330(33), 331(34), 334(33), 336(33; 34) Yanagimoto, M., 92(89), 118(89), 130(333), 131(333) Yang, C. Y., 317(71), 328(71), 329(71), 331(71; 81; 82), 337(71; 81; 82) Yang, H.-H., 141(349) Yang, J., 397(285; 286) Yang, P., 273(350) Yang, Y., 258(232), 260(241) Yap, C. Y., 97(155; 156; 157), 115(156; 157), 117(157), 131(155), 133(155), 138(155) Yee, G., 72(39) Yeh, D. Y., 180(6) Yeh, P., 226(121), 229(121), 234(121), 235(121), 237(121), 243(121) Yener, Y., 302(23), 304(23) Yin, J., 387(182), 388(182), 409(308; 310) Yoda, S., 17(11), 18(12), 60(12), 61(12) Yokohama, K., 325(65) Yokota, T., 101(204; 205)
Yokoyama, T., 92(89), 118(89), 130(333), 131(333) Yonko, J. D., 397(223) Yoshida, F., 84(68), 95(144) Yoshida, S., 63(35) Yoshimura, T., 139(347), 141(347) Yoshinaga, S., 397(295), 398(295) Yoshitomi, K., 263(296) Yoshiyama, T., 32(19) Youcefi, A., 103(244), 121(244) Youn, B., 389(191), 390(191), 391(191), 392(191), 393(191) Yu, S., 100(196) Yuan, H. H. S., 91(277) Yugami, H., 268(334; 335; 336) Yuge, K., 94(121)
Z Zahradnik, J., 84(66) Zak, L., 98(181), 118(181) Zalc, J. M., 91(274) Zant, P. V., 257(222) Zaslavsky, A., 275(388) Zaumseil, P., 213(51) Zavracky, P., 263(268) Zaworski, J. R., 247(182; 183), 263(182; 183), 264(182) Zehner, P., 112(305) Zell, M., 17(10), 72(10) Zemel, J. N., 268(324; 325; 326; 327; 328; 329; 330; 331) Zemel, J., 299(5) Zeng, S., 406(302) Zeppenfeld, R., 99(189) Zhang, H., 264(319), 357(115) Zhang, L. W., 361(125), 363(126), 364(128), 401(126; 128) Zhang, L., 387(177), 388(177), 405(301), 406(301; 302) Zhang, M., 303(28), 316(28), 323(28), 324(28) Zhang, Q.-C., 226(110) Zhang, X., 365(129; 131), 401(129) Zhang, Y., 214(71) Zhang, Z. M., 204(32), 206(38), 213(45; 47), 220(88), 224(92), 226(107; 108; 117; 126), 232(129; 130), 233(38), 234(131), 235(131; 132; 133), 237(126; 142; 143), 238(126), 239(126), 243(154; 155; 156; 157; 158), 245(107), 246(107), 247(191;
AUTHOR INDEX
184), 249(199), 251(206), 258(206), 263(184; 261; 262), 264(262), 265(261), 266(261; 321), 266(262) Zhao, L., 34(21), 300(9), 312(9), 315(9) Zhao, T. S., 310(46), 311(46), 312(46), 313(46), 314(46), 315(46) Zheleva, N. N., 246(170) Zhou, G., 101(223), 121(223) Zhou, J. J., 213(53) Zhou, P., 405(301), 406(301; 302) Zhou, Y. H., 247(191; 184), 249(199), 263(184)
455
Zhu, Q. Z., 251(206), 258(206), 266(321), 267(321) Ziman, J. M., 187(23), 188(23), 190(23), 198(23), 199(23), 202(23), 203(23), 204(23) Zipin, R. B., 264(304), 267(304) Zitny, R., 100(195), 116(195), 118(195) Zlokarnik, M., 80(28a), 106(281) Zory, P. S., Jr., 226(105) Zou, C., 102(237), 120(237) Zull, H., 271(344) Zumer, M., 99(190)
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ADVANCES IN HEAT TRANSFER VOL. 37
SUBJECT INDEX Automotive charge-air coolers, 405 Automotive condenser, tube-side design, 404 Average shear rate, 113–120 and rotational speed, 119
A Absorption by lattice vibrations, 212–213 Absorption coefficient, 206, 212 Absorption processes, 206–214 Absorptivity, 230 Acoustic phonons, 205 Adiabatic flows, 318–325 Agitated vessels, non-Newtonian effects in, 86–87 Air-conditioning system, 339 Air-side performance, 349–401, 412 dry conditions, 366–384 heat transfer data, 366–382 pressure drop data, 382–384 effects of inclination, 389–393 frosting conditions, 396–400 wet conditions, 386–401 Air-side pressure drop, 403 Alkaline etching, 181 Alumina-in-paraffin oil suspensions, 86 Annular flow regime, heat transfer simulation in, 24 Annular liquid film, 37 behavior of, 23 gravity effect on, 39 in microgravity, 51 momentum and energy equations, 24–25 velocity profile, 25 volumetric flow rate, 25 Annular-slug flow (ASF) region, 332–333 Anomalous dispersion, 223 Anti-Stokes shift, 217 Arbitrary numbers of thick and thin layers, 246 Aspect ratios, 339–340 Atomic force microscopy (AFM), 180, 257 Autocorrelation function (ACF), 250–251 Autocorrelation length, 251 Autocovariance function (ACV), 249–250 Automobile evaporator, 340 Automotive air-conditioning, 407 Automotive air-cooled condenser, 402 Automotive air-conditioner, 345
B Baffles, selection, 151 Banbury type mixers, 155 internals, 156 Band gap, 192 Band structure, 186–187, 276 see also electronic band structure Beckmann-Spizzichino model, 262 Bidirectional reflectance distribution function (BRDF), 247, 258–261, 263–267 Bidirectional scattering distribution function (BSDF), 258–263 Bidirectional transmittance distribution function (BTDF), 258–260, 263–267 Bingham number (Bi), 107 Bingham plastic fluid, 83, 84, 141 Bingham plastic model, 82 Blasius correlation, 303 Blasius equation, 322 Block theorem for periodic potential, 190–193 Body diagonal, 189 Boger fluids, 121 Boiling (evaporating) flows, 325–327, 329–336 Boiling experiments in narrow gaps, 60–61 Boiling heat transfer in microchannels, 330–331 Boiling number (Bo), 333 Bose-Einstein distribution, 201 Bose-Einstein statistics, 185 Bragg diffraction, 189 Brazed aluminum condensers, 402–404 Brazed aluminum heat exchangers, 388–389 thermal hydraulic performance, 404 Brazed copper air-cooled heat-exchangers, 405 457
458
SUBJECT INDEX
Brazed microchannel heat exchangers, new applications, 411 Brillouin scattering, 218 Brillouin zone, 189, 192, 198, 199 Bubbly-to-slug transition, 310–312 Burnout in low quality region, 48
measurement in terrestrial and parabolic flight experiments, 40 mechanisms of, 58 Crystal structure diamond, 187 zinc blende semiconductors, 187 Cupro-Braze heat exchanger, 405
C D Carrier scattering by boundary, 206 by defects, 206 by phonons, 205 Central processor unit (CPU), 401, 406 CFD analysis, 359, 407 CFD flow efficiency, 360 CFD modeling, 139 CFD simulations, 358 CFD vorticity plots, 361 Characterization techniques, 256 Charge carrier scattering, 204–206 Chisholm B-coefficient method, 327 Chisholm correlation, 318, 319, 323–325 Circular tubes, 300–305, 310, 320 Circulation times, 133–139 CO2 systems, 406–410 Coherent length, 231 Coherent limit, 232–237 Colburn j factor, 366 Complex refractive index, 184, 230 Condensate carryover phenomena, 395–396 Condensate retention modeling, 396–400 Condensation heat transfer in microchannels, 330–331 Condenser applications, 297–429 Condensing flows, 328, 336–337 Conduction band, 195, 215 Confined bubble (CB) region, 332–333 Confinement number, 327 Consistency coefficient, 84 Convection effects, 91 Convection number (Co), 334 Copper air-cooled heat exchangers, 410 Copper fabrication technology, 412 Critical heat flux (CHF), 2, 9, 13 data, 4 experiments under microgravity, 39–41 in high quality region, 41–45 in moderate quality region, 45–48
Deborah (De) number, 83 De-humidification, 412 Density, 85 Density of electron states, 270 Diamond, crystal structure, 187 Diamond lattice, 200 Dielectric constant, 184 Dielectric functions, 219–226 Dielectric material, 183 Direct gap semiconductors, 208 subband structure, 196 Disturbance waves, 52–53, 59 Double helicone impeller, 154 Drift-flux model, 310 Drude model, 212, 220–222 Dryout phenomena under microgravity, 39–60
E Eddy diffusion, 90 Effective (or apparent) viscosity, 80 Effective shear rate, 148 Elastic phonon scattering, 203–204 Elasticity number (El), 121–123, 128 Electric resistance of gold thin film, 4–7 Electromagnetic wave, 184 Electron scattering, 202–206 Electron-hole pair, 210 Electronic band structure extended-zone scheme, 193 for semiconductors, 193–196 reduced zone-scheme, 193 Electronic band theory, 190–202 Electronic equipment cooling, 405–406 Electrons, number density, 197 Energy bands, 192 Energy conservation equations, 217 Energy reflectivity, 230 Eotvos number, 299
459
SUBJECT INDEX
Evaporator applications, 297–429 Evaporator plate configuration and flow arrangement, 346 designs, 343 Exciton, 210 Extended-zone scheme, 192 Extinction coefficient, 184, 220, 232 Extrusion technology, 411
F f correlations, 367–375 f factor, 366–384 Fabry-Perot resonator, 245 Fanning friction f factor, 366 Fin-to-louver pitch ratio, 354, 363 Finned tube heat exchangers, 386–388 Flow-boiling flow patterns for, 3 microgravity heat transfer in, 1–76 circular tubes, 15–23 effect of gravity, 15–23 experimental apparatus, 10–11 experimental conditions and procedure, 13–14 flow pattern change, 15 future investigations, 71–72 outline of aircraft experiments, 12 preliminary experiments on ground in, 14–15 Flow-boiling heat transfer in narrow channels, 60–71 experimental apparatus and procedure, 61–63 experimental results, 61–68 extremely small gap size, 68 large gap size, 63–64 small gap size, 64–68 summary of experimental results, 70–71 Flow distribution concerns, 407–408 Flow efficiency, 350, 356 definition, 352 Flow fields created by impellers, 124–133 Flow patterns, 307–315 created by impellers, 124–133 for flow boiling, 3 in horizontal rectangular channels, 308 Flow regime map, 35 Flow regimes in vertical tubes, 307 Flow visualization experiments, 113
Flow visualization tests, 352, 353, 358, 361 FLUENT, 140 Fluid mechanics, 91–141 Fourier transform spectrometer, 237 Free-carrier absorption, 211–212 Free carriers, Drude model for, 220–222 Fresnel approximation, 260 Fresnel coefficients, 230, 235 Fresnel reflection, 228 Friedel correlation, 322 Fron113, 3 Froude number (Fr), 26, 107, 109, 110, 136, 142
G GaAs absorption spectrum, 207 energy band structure, 194 exciton absorption, 210 Geometric-optics model, 262–267 Gold thin film as temperature thermometer, 7–9 electric resistance of, 4–7 relation between resistance and temperature, 6 temperature–resistance characteristics, 14 Grashof number (Gr), 107, 142 Gravity effect of reduction on heat transfer coefficient, 31 effect of reduction on heat transfer due to two-phase forced convection, 29 effect of reduction on liquid–vapor behavior and heat transfer classified by mass velocity, quality and heat flux, 22 effect of reduction on wall shear stress, 30 effect on annular liquid film, 39 effect on flow boiling heat transfer in circular tubes, 15–23 annular flow regime, 18–21 bubble flow regime, 16–18 effect on heat transfer data, 56–58 effect on heat transfer due to two-phase forced convection for water from pressure drop data, 33–37 effect on heat transfer for water, 37 effect on interfacial friction factor, 26–28 effect on pressure gradient, 36
460
SUBJECT INDEX
effect on relationships between wall shear stress, interfacial stress, and annular liquid film, 30 effect on two-phase forced convective heat transfer, 23 see also microgravity Gravity-dependent heat transfer due to two-phase forced convection in annular flow regime, 23–39 analytical model, 23–26
H Hagen–Ruben equation, 221 Hamiltonian operator, 190 Header design, 341–343 Heat exchangers applications, 401–411 performance for different design options, 400 specifications, 399 two-phase mal-distribution, 338–349 Heat transfer mixing operations, 141–149 simulation in annular flow regime, 24 Heat transfer coefficient at different gravity levels, 17–21 due to two-phase forced convection, wall shear stress and annular liquid film thickness, 31, 37 effect of gravity reduction, 31 Heat transfer data, 51–53 for different gravity levels, 67, 69 Helical ribbon impeller, 116, 117, 120, 122, 131–133, 139 Helical screw agitator, 116, 132, 133 with draft tube, 116 Herschel–Bulkley fluid, 83 High viscosity Newtonian liquids, power input, 112 Holes, number density, 197 Homogeneous pressure drop model, 322 HVAC industry, 401
I Impeller design, 110, 142, 152 Impeller geometries, power curves for, 111 Impeller speed, 78–79, 83, 111, 113 Impeller–tank assembly, 118
Impeller–tank combination, 111, 117 Impeller–tank geometry, 149 Impeller vessel configuration, 113 Impellers anchor, 129–130 class I, 125–128, 137, 138, 143–144 class II, 127, 129–130, 137, 138, 144–147 class III, 130–133, 137–139, 147–149 gate, 129–130 novel designs, 154 ranges of operations, 157 selection, 151–156 values of Ks for, 114 see also propeller impellers; turbine impellers InAs quantum dots, 180 Inclination effects, 389–393 Incoherent limit, 232–237 Indirect gap semiconductor, 195 Inelastic non-Newtonian liquids, power input, 112 InGaAs/GaAs quantum wires, 180 Ink-in-water flow visualization tests, 357 Interband transition, 206, 208–210, 225–226 Interface reflection at, 227–231 refraction at, 227–231 Interfacial friction factor, 26, 315 measured and calculated ratios, 38 INTERMIG turbine type impeller, 119 Internal transmissivity, 233 International Space Station, 2 Intraband transitions, 211 Inverted annular flow, 48 Isolated bubble (IB) region, 332–333
J j correlations, 367–375 j factor, 366–384
K k-space, 189 Kelvin–Helmholtz instability concept, 309 Kirchhoff approximation, 261 Kirchhoff’s law, 231 Knudsen number, 299 Kramers–Kronig (K–K) dispersion relations, 219–220, 225, 231
SUBJECT INDEX
L Laminar flow, 118, 300–302, 304 Laminar mixing, 87–89 Laminar shear mixing in concentric cylinder configuration, 89 Laminated plate-fin type evaporator, 343 Laplace equation, 339 Lattice vibration absorption by, 212–213 specific heat due to, 202 spring-mass representation, 198 Layered structures, radiative properties of, 226–246 Light scattering method, 254–256 Lightnin A 315 axial flow impellers, 128 Linear heat flux, 47 Liquid flow rate due to disturbance wave effect on heat transfer, 32 Liquid–gas interface behavior, 35 Liquid–vapor behavior, 50–51 analysis of, 53–56 area ratio patterns for, 57 at different gravity levels, 17–21 before critical heat flux, 48–58 for different gravity levels, 67, 69 near CHF condition, 46 Lockhart-Martinelli correlation, 318 Longitudinal acoustic (LA) branch, 199–200 Longitudinal optical (LO) branch, 199–200 Lorentz oscillator model for phonons, 222–224 Louver angle, mean flow angle as function of, 355 Louver fin array, flow structure, 350–366 Louver fins, 412 Low viscosity liquids, power input, 109
M Martinelli–Nelson method, 327 Martinelli parameter, 318, 323 Matthiessen’s rule, 204 Maxwell’s equations, 182 Mean flow angle as function of louver angle, 355 Mean surface, 248 Mechanically agitated systems, summary of experimental studies, non-Newtonian liquids in, 92–105
461
Mechanically agitated vessels, 77–178 Microchannel headers, two-phase maldistribution, 338–349 Microchannel heat exchangers applications, 401 design, 297–429 model for, 408–410 Microchannel heat sink, 406 Microchannels boiling heat transfer in, 330–331 condensation heat transfer in, 330–331 single-phase flows in, 298–300 two-phase flow frictional pressure drop in, 316–317 two-phase flows in, 298–300 Microelectromechanicalsystems(MEMS),181 Microgravity critical heat flux (CHF) experiments under, 39–41 dryout phenomena under, 39–60 typical g-jitter, 12 Microgravity boiling, research history, 2 Microgravity heat transfer in flow boiling, 1–76 effect of gravity, 15–23 experimental apparatus, 10–11 experimental conditions and procedure, 13–14 flow pattern change, 15 future investigations, 71–72 in circular tubes, 15–23 outline of aircraft experiments, 12 preliminary experiments on ground, 14–15 Microgravity pressure drop data, 33–37 Microscale radiative heat transfer (MHRT) regime, 226–227 Microstructured surfaces, radiative properties of, 267–269 Microtechnology, 179–296 Mixing equipment, 78 and its selection, 150–156 scale up, 91–108 typical tank (with a jacket-impeller as, 106 Mixing operations applications, 78 costs of, 77 examples of different types, 79 heat transfer, 141–150 in viscous liquids, 89 mechanisms, 87–91
462
SUBJECT INDEX
pervasive nature of, 78 power input in, 108–124 reduction in variance of concentration of tracer with time, 136–139 relationship between scale and intensity, 134 use of term, 77 see also overmixing; under-mixing Mixing-time measurement curve, 135 Mixing times, 133–139 Molecular beam epitaxy (MBE), 180 Molecular diffusion, 85, 90, 91 Multilayer structures, radiative properties of, 240–246 Multiple tube U-channel assembly, 344
N Nanomachining, 181 Nanotechnology, 179–296 Newtonian fluids, 81 Non-circular channels, 301–305 Non-Newtonian effects in agitated vessels, 86–87 Non-Newtonian fluids, engineering analysis, 81 Non-Newtonian liquids, 77–178 in mechanically agitated systems, summary of experimental studies, 92–105 Non-radiative recombination, 214, 216 Numerical analysis, 353, 354 Numerical modeling, 139–141 Nusselt number (Nu), 107, 142, 147, 335
O Ohm’s law, 183 One-electron model, 190 Optical interferometric microscopy (OIM), 253–254, 257 Optical phonons, 205 Optical properties of semiconductors, 179–296 fundamentals of, 182–226 temperature effect, 213 Overmixing, 78, 80
P Parabolic flight experiments liquid–vapor separater, 11 test loop, 10
Parabolic flight trajectory, 12 Parallel flow brazed aluminum automotive air-cooled condenser, 402 condenser, 403 refrigerant circuiting, 403 Partial coherence, 232 theory, 237–240 Periodic potential, Block theorem for, 190–193 Phase-change flows, 325–328 Phase-change heat transfer, 329–337 Phase shift interferometry (PSI), 253 Phonon–phonon scattering, 202–203 Phonon scattering, 202–204 Phonons, 197–202 carrier scattering by, 205 Lorentz oscillator model for, 222–224 types, 205 Photolithography, 181 Photomultiplier tubes (PMTs), 218 Photon–phonon scattering, 216–218 Photonic crystals, 180–181, 274–275 Photons, 182, 185 confinement of, 180 momentum, 185 Physical–optics models, 261–262 PIB/PB/Kerosene solution, 121 Planck’s distribution, 231 Planck’s law, 186 Plane of incidence, 227 POLYFLOW, 140 Porous silicon (PS), 273–274 Power consumption, 113 for agitation of low viscosity Newtonian liquids, 112 in viscoplastic dilatant or shearthickening fluids, 118 with turbine impellers, 121 Power curves for impeller geometries, 111 for inelastic shearthinning fluids, 120 for propeller impeller, 110 for time-independent fluids, 115 Power input mixing operations, 108–124 viscoelasticity, 120–124 Power number (Po), 109, 111, 113, 118 Power number (Po)-Reynolds number (Re) relationship, 110
463
SUBJECT INDEX
Power spectral density (PSD) function, 248–251 Poynting vector, 184 Prandtl number (Pr), 107, 142 Pressure drop measurements, 3 Propeller impellers, 126, 128 power curve for, 110 Propellers, 125, 127 Pseudoplastic fluids, 117 Pulp and paper industry, 78 Pumping number, 124
Q Quantum confinement, 269–274 Quantum wires, 180
R Radiation penetration depth, 185 Radiation tunneling, 236–240 Radiative properties layered structures, 226–246 microstructured surfaces, 267–269 multilayer structures, 240–246 rough surfaces, 246–269 single layer, 231–240 thin film on substrate, 240–243 Radiative recombinations, 214 Radiative transitions, 215–216 Raman scattering, 218 Rapid thermal processing (RTP), 182, 247, 263 Rayleigh–Rice approximation, 260 Rayleigh–Rice vector perturbation theory, 255 Rectangular channels, 310 Reduced gravity, 2 Reduced zone-scheme, 193 Reflection at interface, 227–231 Refraction at interface, 227–231 Refractive index, 220, 222, 223, 225, 232, 233 Reynolds number (Re), 106, 108–111, 113, 118–122, 124–126, 128, 136, 138, 158, 299–303, 323, 328, 336, 340, 352, 354– 359, 362–365, 376, 377, 382, 390–392 Reynolds stresses, 90 Rheological models, 82, 83 Rheological properties, 81–84 Rigorous coupled-wave analysis (RCWA), 268
Root-mean-square (rms) roughness, 248 Roughness parameters and functions, 248–251 Rushton turbine, 121, 127
S Saturated LN2, 3 SCABA 6SRGT radial flow impeller, 128 Scale of intensity, 134 Scale of segregation, 134 Scale-up factor, 108 Scanning electron microscopy (SEM), 180, 181, 256 Scanning probe microscopy (SPM), 252–253 Scanning tunneling microscope (STM), 252–253 Scanning white light interferometry (SWLI), 253–254 Scattering of electrons and phonons, 202–206 Scattering rate, 221 Schmidt number (Sc), 107 Schro¨dinger equation, 190, 191 Semiconductor industry, 256–258 Semiconductor materials indirect and direct interband transitions, 188 normal transmittance, 233 Semiconductors electronic band structures for, 193–196 number density of carriers, 196–197 optical properties, 179–296 fundamentals of, 182–226 temperature effect, 213–214 thermal radiative properties of, 179–296 Serpentine circuited round tube condenser, 402 Shear stress for turbulent flow of liquid film, 25 Shearthickening, 82, 85, 112, 118, 120 Shearthinning, 82, 85, 91, 112, 113, 126–128, 130, 149 Sherwood number (Sh), 107 Sigma-blade mixer, 155 Silicon absorption spectrum, 207 energy band structure, 194 Single layer, radiative properties of, 231–240 Single-phase flows in microchannels, 298–300 within microchannels, 411
464
SUBJECT INDEX
Single-phase heat transfer, 304–306 Single-phase pressure drop, 300–304 Slug/churn transition, 312 Slug/churn-to-annular transition, 312–315 Snell’s law, 228 Specific heat, 85 due to lattice vibration, 202 Spectral averaging, 235–236 Spectral-directional absorptivity, 231 Spectral-directional emissivity, 231 Spectral regions, 186 Spectral tri-function automated reference reflectometer (STARR), 264 Splicing, 89 Split-off holes, 196 Spring-mass representation of lattice vibration, 198 Stokes shift, 217 Strong confinement, 272–273 Stylus method, 251–252 Surface characterization methods, 251–256 Surface coating, 393–394 effect on heat exchanger performance, 394–395 Surface roughness characterization, 247–258 Surface treatment of heat exchanger fins, 393–296 Suspensions, thermal conductivities of, 85
T Tank-impeller configuration, 141 Tank-impeller geometry, 137 Tanks, selection, 150–151 Temperature oscillation before critical heat flux, 48–58 Tetrahedral coordination, 187 Thermal conductivity, 85 of suspensions, 85 Thermal entrance region, effect on heat transfer, 33 Thermal radiative properties of semiconductors, 179–296 Thermo-physical properties, 84–86 Thermo-syphon device, 406 Thin gold film see gold thin film Thinning of fluid element due to elongation flow, 88 Thixotropic behavior, 112
Thixotropic substances, 82 Three-axis automated scatterometer (TAAS), 265–266 Total integrated scatter (TIS) method, 255 Total transmittance, 235 Transfer matrix method, 243–246 Transition between band and impurity level, 210–211 Transition of wall temperature and heat transfer coefficient along a heated length, 33 Transmission coefficients, 228 Transmittance spectra, 276 Transparent heated tube, 40 delay time of wall temperature when heated transfer coefficient oscillates, 9 development, 4–9 heat transfer characteristics, 14 resistance measurement, 5 step response, 9 structure, 4 temperature response, 8 wall temperature transition after step power input, 8 Transverse electric (TE) wave, 227, 243 Transverse magnetic (TM) wave, 229, 243 Turbine impellers, 125–126, 128 designs, 153 power consumption with, 121 Turbines, 127 Turbulent flows, 302–306 Turbulent mixing, 90–91 Twin-celled secondary flow pattern, 131 Two-phase flow distribution, 411 Two-phase flow frictional pressure drop in microchannels, 316–317 Two-phase flow mal-distribution in microchannel headers and heat exchangers, 338–349 Two-phase flows, 306–337 in microchannels, 298–300 Two-phase forced convection, heat transfer coefficient due to, 37 Two-phase homogeneous density, 322 Two-phase pressure drop, 315–329
U U-channel evaporator plates, 344 Under-mixing, 80
465
SUBJECT INDEX
V Valence band, 195, 215 Vessels, selection, 150–151 Viscoelastic fluids, 83 Viscoelastic liquids, 82 Viscoelastic systems, 117 Viscoelasticity and power input, 120–124 Viscoplastic characteristics, 85, 112 Viscoplastic liquids, 82
W Wavefront, 183 Wavefunctions, 192 Wavelength ranges and characteristic dimensions, 181
Weak confinement, 271–272 Weber number (We), 107, 109, 136, 142 gas, 299 liquid, 299 Weissenberg number (Wi), 83, 107, 128, 136 Working fluids, 406–407
Y YBCO films, 245–246 Yokogawa Electric LR-8100 recorders, 11
Z Zinc blende semiconductors, crystal structure, 187
