Advances in Heat Transfer, Volume 36

ADVANCES IN HEAT T R A N S F E R Volume 36

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Thomas F. Irvine, Jr., 1922-2001, Memorial Volume

A dvances 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

Serial Associate Editors

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 36

ACADEMIC PRESS An imprint of Elsevier Science Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

This book is printed on acid-free paper. (~) Copyright 2002, Elsevier Science (USA). All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. The appearance of code at the bottom of the first page of a chapter in this book indicates the Publisher's consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2002 chapters are as shown on the title pages. If no fee code appears on the chapter title page, the copy fee is the same for current chapters. 0065-2717/02 $35.00 Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press An imprint of Elsevier Science 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http ://www. academicpress.com Academic Press 84 Theobolds Road, London WC1X 8RR, UK http://www.academicpress.com International Standard Book Number: 0-12-020036-8 International Standard Serial Number: 0065-2717 PRINTED IN THE UNITED STATES OF AMERICA 02 03 04 05 06 07 MB 9 8 7 6 5 4 3 2

1

CONTENTS

Contributors . . . . . . . . . . . . . . . . . . . . . . . .

ix

In M e m o r y of Professor T h o m a s F. Irvine, Jr . . . . . . . . . .

xi

Preface

xv

. . . . . . . . . . . . . . . . . . . . . . . . . .

Inverse Design of Thermal Systems with Dominant Radiative Transfer FRANCIS H. R. F R A N ~ A , JOHN R. H O W E L L , OFODIKE A . F~ZEKOYE, AND JUAN CARLOS MORALES

I. Introduction . . . . . . . . . . . . . . . . . . . . . . A. Rationale for an Inverse Approach to Design B. Differences from Conventional Design . C. Outline of This C h a p t e r .

1

. .

2 4 5

II. List of Symbols . . . . . . . . . . . . . . . . . . . . III. Mathematics of Inverse Design . . . . . . . . . . . . . . A. B. C. D.

Regularization Methods. Other A p p r o a c h e s . Literature Review . State of the Art in Inverse Design

6 7

. . .

7 26 27 28

IV. Inverse Design of Linear Systems D o m i n a t e d by Radiative Transfer . . . . . . . . . . . . . . . . . . A. Systems with Surface Radiative Exchange. B. Radiative Systems with Participating Media .

. .

V. Design of Thermal Systems with Highly Nonlinear Characteristics . . . . . . . . . . . . . . . . A. B. C. D.

31 . .

31 36

49

Techniques for Treating Inverse Nonlinear Problems . Inverse Design of Systems with Nongray Medium . Inverse Heat Source Design Combining Radiation and Conduction . Inverse Boundary Design Combining Radiation and Convection .

50 51 63 73

VI. Imposing Additional Constraints on the Regularized Solution . . . . . . . . . . . . . . . . . .

93

A. Energy Generation Shape Constraint B. Imposing a Reduced Number of Uniform Heat Flux Devices C. Final Remarks on Imposing Additional Constraints

. .

93 98 103

vi

CONTENTS

VII. Conclusions

104

. . . . . . . . . . . . . . . . . . . . . .

A. The Case for Inverse Design

.

.

.

104

.

B. A r e a s for F u r t h e r R e s e a r c h .

105

107

References . . . . . . . . . . . . . . . . . . . . . . .

Advances in Temperature Measurement P. R. N. CHILDS I. I n t r o d u c t i o n

111

. . . . . . . . . . . . . . . . . . . . . .

A. T e m p e r a t u r e Scales. B. Overview o f M e t h o d s

.

.

C. Selection C r i t e r i a

II. M e a s u r e m e n t

111 113

.

.

113

118

Methods . . . . . . . . . . . . . . . . . .

A. Invasive M e t h o d s

118

136

B. Semi-Invasive M e t h o d s C. N o n i n v a s i v e M e t h o d s .

III. Conclusions

.

142

165 166 166

. . . . . . . . . . . . . . . . . . . . . .

Nomenclature

. . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . .

Swirl Flow Heat Transfer and Pressure Drop with Twisted-Tape Inserts RAJ M.

I. I n t r o d u c t i o n

MANGLIK AND ARTHUR E. BERGLES

. . . . . . . . . . . . . . . . . . . . . .

A. G e n e r a l B a c k g r o u n d B. Historical O v e r v i e w .

.

184

.

184

190

.

II. S i n g l e - P h a s e F l o w a n d H e a t T r a n s f e r

. . . . . . . . . . .

A. E n h a n c e m e n t Characteristics B. T h e r m a l - H y d r a u l i c Design C o r r e l a t i o n s .

III. Two-Phase Flow and Heat Transfer A. G e n e r a l C o m m e n t s . B. S u b c o o l e d F l o w Boiling C. Bulk Boiling .

.

197 197 213

.

. . . . . . . . . . . .

230

. .

D. C o n d e n s a t i o n H e a t Transfer.

IV. M o d i f i e d T a p e s a n d C o m p o u n d

230 232 243 245

Techniques . . . . . . . . .

247

A. Modified T w i s t e d - T a p e Inserts .

247

B. C o m p o u n d E n h a n c e m e n t T e c h n i q u e s .

249

V. P e r f o r m a n c e E v a l u a t i o n C r i t e r i a . . . . . . . . . . . . . . A. Single-Phase F l o w B. T w o - P h a s e F l o w .

.

. .

VI. C o n c l u d i n g R e m a r k s A. S u m m a r y

.

250 250 251

.

. . . . . . . . . . . . . . . . . . .

B. R e c o m m e n d a t i o n s for F u t u r e W o r k

253

. .

253 .

255

CONTENTS

Nomenclature

vii

. . . . . . . . . . . . . . . . . . . . . .

256

. . . . . . . . . . . . . . . . . . . . . . .

257

Index . . . . . . . . . . . . . . . . . . . . . . . .

267

Subject Index . . . . . . . . . . . . . . . . . . . . . . . .

279

References Author

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CONTRIBUTORS

Numbers in parentheses indicate the pages on which the authors'contributions begin.

ARTHUR E. BERGLES (183), Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, USA P. R. N. CHILDS (111), Thermo-Fluid Mechanics Research Centre, University of Sussex, Brighton BN1 9QT, UK OFODIKE A. EZEKOYE (1), Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA FRANCIS H. R. FRANgA (1), Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA JOHN R. HOWELL(1), Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA RAJ M. MANGLIK (183), Thermal Fluids and Thermal Processing Laboratory, Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221, USA JUAN CARLOS MORALES (1), Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA

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IN MEMORY OF PROFESSOR THOMAS F. IRVINE, JR. JUNE 25, 1922 TO JUNE 2, 2001

Professor Thomas F. Irvine, Jr., passed away suddenly at his home in St. James, New York, on June 2, 2001, at the age of 78. He would have observed his 79th birthday on June 25, 2001. Volume 36 of Advances in Heat Transfer is dedicated in his memory to honor his contributions to the fields of heat transfer and fluid mechanics as a permanent memorial to one of the founding editors of the series. Professor Irvine was born on June 25, 1922, in New Jersey and raised in Pittsburgh, Pennsylvania. He attended college at the Pennsylvania State University where he received his bachelor of science degree in electrical

xii

IN MEMORY OF PROFESSOR THOMAS F. IRVINE, JR.

engineering in 1946. From there, he went to the University of Minnesota where he studied mechanical engineering under Professor E. R. G. Eckert. It was at the University of Minnesota that he received both his master's degree and his doctoral degree in mechanical engineering. After completing his Ph.D. dissertation in 1956, he joined the faculty of the Mechanical Engineering Department of the University of Minnesota, first as an assistant professor from 1956 to 1958 and then as an associate professor from 1958 to 1959. In 1959, he joined the faculty of the Mechanical Engineering Department of the North Carolina State University as Professor of Mechanical Engineering, where he remained until 1961. In 1961, he accepted an appointment as Professor of Engineering and the first Dean of the College of Engineering of the newly formed State University of New York at Stony Brook. It was under his leadership that the new College of Engineering got its start, and construction of the facilities and staffing of the academic departments were accomplished. He held the position of Dean of Engineering from 1961 to 1972, at which time he stepped down as Dean to return to full-time teaching and research, which he continued until his death this year. Professor Irvine's scientific contributions have been published in more than 200 refereed papers in the international literature. His published works range from novel techniques for the measurement of thermophysical properties, experimental studies in boiling heat transfer and two-phase flow and heat transfer, to fluid mechanical and heat transfer characteristics of rheological fluids. In addition to his interest in fundamental research, he also pursued the practical applications of such phenomena. Among these pursuits was the development of a concept for continuous-flow electrophoresis, development of a precision falling-needle viscometer, and the development of methods to measure diffusivities of translucent fluids. In addition to his many contributions in the scientific literature, Professor Irvine edited many books, including Heat Transfer Reviews, 1953-1969 [with E. R. G. Eckert, Pergamon Press (1971)], Steam and Air Tables in SI Units [with J. P. Hartnett, Hemisphere (1976)], Heat Transfer Reviews, 1970-1975, [with E. R. G. Eckert, Pergamon Press (1977)], Steam and Gas Tables with Computer Equations [with P. E. Liley, Academic Press (1984)], and Heat Transfer Reviews, 1976-1986 [with E. R. G. Eckert, R. J. Goldstein, and J. P. Hartnett, Wiley Interscience (1990)]. Professor Irvine was a pioneer in creating and maintaining a variety of publications for the dissemination of scientific information. He was the first editor of the Journal of Heat Transfer of the ASME, serving in that capacity from 1960 to 1963. He was one of the founding editors and coeditor of the Academic Press monograph series Advances in Heat Transfer since its inception in 1964 until the present (36 volumes). He was the coeditor of the Pergamon Press textbook series Pergamon Unified Engineering Series from

o o ~

IN MEMORY OF PROFESSOR THOMAS F. IRVINE, JR.

Xlll

1967 to 1974 (18 volumes). He was the coeditor of the graduate textbook series by Hemisphere/McGraw-Hill entitled Series in Fluids and Thermal Engineering from 1972 to 1982 (24 volumes). In addition, he was the cofounder and coeditor of Heat Transfer Research [with J. P. Hartnett, Wiley Interscience, and Begell House (1972 to present)], Heat Transfer Asian Research [with J. P. Hartnett, Wiley Interscience (1974 to present)], and Previews of Heat and Mass Transfer [with J. P. Hartnett, Rumford (1976 to present)], all of which still serve to gather scientific contributions from around the world and publish them in a coordinated and accessible format. He also served on the editorial advisory boards of the International Journal of Heat and Mass Transfer and International Communications in Heat and Mass Transfer. Professor Irvine was very active in a number of domestic and international scientific organizations for many years until his death. He actively participated in many important functions of the Assembly for International Heat Transfer Conferences, UNESCO of the United Nations, and the International Center for Heat and Mass Transfer. He was a member of the founding committee for the International Center for Heat and Mass Transfer and was an ASME representative to the Scientific Council of ICHMT. He was a member of the ASME for 42 years and was elected fellow of that society. He served on the K-7 Thermophysical Properties Committee and the admissions committee of the ASME, and he was also a fellow of the American Association for the Advancement of Science. In 1992, he was awarded the prestigious Heat Transfer Memorial Award by the ASME. His many international friends and colleagues provide evidence of his tireless efforts to promote open international scientific exchange between many nations, particularly during politically disadvantageous times, most prominently with the People's Republic of China, the Republic of China, Republic of Korea, Japan, and the states of the former Soviet Union. He enjoyed visiting professorship and visiting research scientist positions abroad at the Technical University of Munich, University of Belgrade, AERE Harwell, and the University of Florence. He lectured around the world in these countries, as well as in Turkey, Germany, Yugoslavia, Czechoslovakia, Romania, Poland, and India. On November 17, 2001, many of his family, colleagues, and former students gathered at a memorial symposium to honor his memory at the State University of New York at Stony Brook. We will miss him but, as time goes by, we will be bolstered by his memory, for rarely has such kindness, generosity, understanding, and scientific excellence been embodied in one person as it was in him. G. A. Greene J. P. Hartnett Y. I. Cho

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PREFACE

For over 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 special 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 36 volumes published to date is an indication of the success of our authors in fulfilling this purpose. In recent years, the editors have undertaken to publish topical volumes dedicated to specific fields of endeavor. Several 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. The editorial board expresses their appreciation to the contributing authors of Volume 36 who have maintained the high standards associated with Advances in Heat Transfer. Lastly, the editors acknowledge the efforts of the staff at Academic Press who have maintained the attractive presentation of the published volumes over the years.

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ADVANCES IN HEAT TRANSFER, VOL. 36

Inverse Design of Thermal Systems with Dominant Radiative Transfer

FRANCIS H. R. FRAN(~A, J O H N R. HOWELL, OFODIKE A. EZEKOYE, and JUAN CARLOS MORALES Department of Mechanical Engineering The University of Texas at Austin Austin, Texas 78712, USA

Abstract

The design of thermal systems often involves the specification of two conditions, typically both temperature and heat flux distributions on some surfaces or within media to perform particular tasks. Examples are in annealing ovens, dryers, chambers for rapid thermal processing of semiconductor wafers, utility and chemical furnaces, infrared ovens, and many others. Conventional design techniques require specification of one and only one boundary condition on each surface of a system, requiring trial-and-error solutions to achieve a design that satisfies the second specified condition. Here, the methods of inverse analysis are applied to the ill-conditioned equations that result when two conditions are specified for a particular surface. It is demonstrated that the use of inverse methods can result in multiple designs that each provide the required conditions; that designs are generated that might not be found through the conventional approach; and that the use of inverse design can lead the designer to efficient and novel designs for thermal systems. 9 2002, Elsevier Science (USA).

I. Introduction

Contemporary analysis and design of engineering systems are generally based on a mathematical model of the system. Because of this, the approach to ISBN: 0-12-020036-8

ADVANCES IN HEAT TEANSFER, VOL. 36 Copyright 2002, Elsevier Science (USA). All rights reserved. 0065-2717/02 $35.00

2

FRANCIS H. R. FRAN(~A ET AL.

design is greatly influenced by (1) the background we have achieved in mathematics and (2) the methods of solution that have evolved for the differential and integral equations that describe the behavior of engineering systems. We are generally taught throughout our exposure to mathematical modeling and analysis of engineering systems that 9 One and only one boundary condition or initial condition must be prescribed for each order of derivative in each dimension of a differential equation describing a system. 9 The number of equations available must be equal to the number of unknowns being sought. However, these constraints are not necessary to the mathematical solution of equations unless we use only conventional methods. Imposing these constraints causes us to approach engineering design from a direction that is often less than optimum. We will attempt to show that eliminating these constraints allows more accurate design of thermal systems and also allows the designer more freedom in design, the possibility of increased creativity, and the discovery of unique designs that could not be found through conventional analysis. We will also demonstrate that this newfound capability in design comes at a price.

A.

RATIONALE FOR AN INVERSE APPROACH TO DESIGN

To bolster the claims made in the opening section, consider the following design problem: A utility boiler is to be constructed. To simplify piping and waterwall design, it is desired to have uniform temperature and heat flux on all of the boiler waterwalls. Where should the burners be placed in the furnace to best achieve this result?

Using the conventional approach to designing such a system, the contemporary designer would probably specify a design set that includes the desired temperature of the constant-temperature surfaces and all of the surface and medium physical and transport properties. The designer would then use experience to generate an initial estimate of the burner characteristics and locations. Commercial codes or specific codes generated for solving this particular problem then allow prediction of the heat flux distribution on the furnace walls. In this conventional or f o r w a r d approach, the geometry, properties, standard boundary conditions (one on each surface), and governing equations are all specified. All major CFD/heat transfer codes require input of this type and require specification of one boundary condition for each variable on each boundary element and one condition in each volume element.

INVERSE DESIGN OF THERMAL SYSTEMS

3

This automatically results in having an equal number of equations and unknowns. If the predicted boundary heat flux distribution from this approach is not satisfactory, then the designer must try new burner locations and rerun the program. This iterative process would continue until a satisfactory solution is found or until money and patience are exhausted. The best solution is then chosen from among the design sets that have been specified and solved. Some reflection will indicate that a better approach might be to specify the distributions of both the desired boundary temperature and the heat flux and then solve directly for the necessary heat source distribution in the furnace. The governing equations are of course the same as for conventional design. This is the inverse design approach. Why do we not approach design in this way, since clearly it is better to specify the desired outcome and solve directly for the required design that will achieve it? First, we have been taught that we must not specify two conditions on a boundary. In addition, we find that it is quite possible, depending on the prescribed volume and surface discretization, that setting up inverse equations in this way will result in many more unknowns (e.g., the heat sources and temperatures in every volume element) than equations. In other situations, the number of equations may exceed the number of unknowns. These inequalities are another thing we have been warned against. Indeed, if we do set up the equations and boundary conditions for an inverse solution and then apply standard matrix solution techniques to the resulting set of equations, we will find that solution will quite often be extremely difficult or impossible. The equation set is found to be ill-conditioned because the matrix of coefficients of the equation set is near-singular or singular. In such a case, conventional matrix inversion techniques and iterative solvers will fail. The advantages of finding an inverse design solution for such a problem are great, if it can somehow be achieved. Inverse design avoids expensive conventional iterative solutions; provides an optimal solution rather than the best solution from among the forward solutions for a limited number of design sets generated by the conventional approach; and may generate solutions that are not intuitively obvious and thus would not be found from the trial-and-error approach. The possible set of viable design solutions can be expanded greatly because every set of imposed conditions may result in one or more potentially usable design solutions, whereas, using the conventional approach, every change in conditions requires a full iterative solution to attain a single viable solution. We now turn to methods for carrying through inverse design. These require overcoming or avoiding the mathematical problems indicated earlier. This will require different approaches than are conventionally used in design; however, the advantages of the inverse approach to design of thermal systems are so great that it is worth a careful exploration of the available methods.

4

B.

FRANCIS H. R. FRAN(~A E T AL.

DIFFERENCES FROM CONVENTIONAL DESIGN

1. Existence

It is quite possible to prescribe an inverse design problem that has no physically acceptable solution. This is not the case for inverse analysis of experimental data, where measurable variables at a boundary imply the existence of real physical conditions elsewhere in the system that provide the measured values (so long as the mathematical model being used is accurate). A mathematical solution to a prescribed inverse design problem may be possible, but it might require negative absolute temperatures, heat sinks rather than sources, or other conditions that cannot be obtained in practice or are very undesirable in a practical design. Sometimes these results come about because the designer has imposed unrealistically tight requirements on the comparison between the inverse solution and the imposed conditions. In that case, a physically real solution may be salvaged by allowing somewhat greater disagreement between the imposed conditions and the predicted results from the inverse solution. (After all, a 5% variation from the imposed conditions may be quite acceptable in a design because there may easily be that much uncertainty in properties or other factors used in the system model.) However, there is no guarantee that a solution exists to a particular inverse design problem just because the designer has specified a desired outcome. 2. Uniqueness

The question of uniqueness is often brought up with regard to inverse solutions of experiment-based problems. Is it not possible to find many solutions that satisfy the imposed conditions? The answer is often yes, and this bedevils the researcher who is using inverse methods to determine physical properties or boundary conditions at a remote location. How can the researcher know whether the found solution reflects the real physical value that is being sought by the inverse solution? The presence of multiple solutions is another reason that inverse design differs, at least philosophically, from inverse solutions applied to experiment. The designer using inverse methods is happy to have multiple solutions so long as they satisfy the prescribed design set within some prescribed error bounds and are physically attainable. Multiple solutions allow the designer a choice, and the solution that is the least expensive and easiest to implement can be chosen from among them. It will be demonstrated that in many cases, slight relaxation on the acceptable accuracy of the design to provide the imposed conditions will give solutions that are much more acceptable to

INVERSE DESIGN OF THERMAL SYSTEMS

5

the designer from practical considerations (smoothness, for example). These solutions may have a much different shape and/or magnitude than the solutions that provide only slightly better agreement with the desired input design conditions.

3. Physics of Design Problems We will demonstrate that the physics of the energy exchange in a design problem should be well understood before an inverse design is attempted. In some cases, a design problem can be posed that is ill-conceived by the designer. This may force the inverse procedure to appear to provide unacceptable behavior, such as poor accuracy or lack of convergence. For example, the designer may pose a problem in which the conditions on the design surface are prescribed (given temperature and heat flux distributions, for example). If the designer also specifies the system geometry in such a way that the heaters to be designed have a small influence on the design surface, then the inverse solution may appear to give poor results. A wide range of solutions for the heater characteristics may provide almost the same conditions on the design surface because the influence of the heaters is small. This result is not a failure in inverse design; rather, it indicates that the imposed system characteristics were poorly thought out. The characteristics of an inverse solution provide guidance to the designer in such a case, but as with all design problems, some careful preliminary analysis can eliminate some less than useful effort. Thus, the designer still needs to bring experience and understanding to bear whether forward or inverse design is to be used. The equation set describing the physics of the problem can often be formulated in many ways. The portions of the equation set that are illconditioned require the most care in solution and can often be identified by careful thought on the part of the designer. The equations can then be formulated in such a way that the inverse portion of the solution is required for only a subset of the entire equation set. After inverse solution of the subset, the remaining equations may be solved by more conventional schemes. An example of such an approach is described in Section V,D. C. OUTLINE OF THIS CHAPTER

The organization of this chapter is somewhat nontraditional. The methods of solution for inverse problems are introduced first to solve a simple but insightful inverse design. Following this is a review of the literature, which traditionally would precede other sections. Because the literature review refers extensively to the methods used in the papers, it is felt that explanation of these methods had best be done first. Following these sections, results of

6

FRANCIS H. R. FRANt~A ET AL.

inverse thermal design are presented for radiating enclosures without and with a participating medium, followed by an inverse thermal design of systems involving multimode heat transfer. The proposed examples, although developed to treat specific cases of inverse design, illustrate general characteristics and difficulties related to such problems, as well as possess a detailed explanation of the applied methodology. In this way, they can form a useful guide for different applications. II. List of Symbols a A A b b

Ce cp

eb

components of matrix A; radiative linear absorption coefficient, m -1 area, m 2 matrix of coefficients components of vector b vector of known quantities coefficients of the weighted sum of gray gases model specific heat, J/kg.K dimensionless emissive power, (Eb-Eoc)

/Eb, ref Eb

radiative black body emissive power, W/m 2 f filter value of Tikhonov regularization, Eq. (25) f~ shape constraint coefficient F radiation configuration factor G6~ volume-to-volume direct exchange area per unit of length normalized by AX GS volume-to-surface direct exchange area per unit length normalized by AX height of two-dimensional enclosure, m H identity matrix I singular value index i dimensionless conductive or convective J flux, Eqs. (64) to (67) k thermal conductivity, W/m K; index of conjugate gradient step K kernel of integral equation; regularization parameter of conjugate gradient method L width of two-dimensional enclosure, m L smoothing operator m number of rows of A N total number of enclosure surface elements NcR conduction/radiation parameter,

ka / 4a T3ef

n p p Pr q Q

number of columns of A number of retained singular values A-orthogonal vector, Eq. (32) Prandtl number, #cp/k dimensionless energy flux, Q/Eb, ref energy flux, W/m 2 Q" volumetric energy generation rate, W / m 3 R residual norm Re Reynolds number, pUmH/# r residual vector SG surface-volume direct exchange area per unit length normalized by X SS surface-surface direct exchange area per unit length normalized by AX sG dimensionless volumetric energy generation rate, ~"/aEb, ref S diagonal matrix of singular values of A t normalized temperature, T/Tref T absolute temperature, K u singular function, Eq. (2); dimensionless velocity, U/Um U fluid velocity, m/s Um mean fluid velocity, m/s U matrix orthogonal to matrix V v singular function, Eq. (2) V matrix orthogonal to matrix U; velocity vector in Eq. (61) u, v column vectors for orthonormal vectors used in SVD w singular value; width of plate, m W matrix of singular values x dimensionless coordinate position, X/L; components of vector x X coordinate distance, m x solution vector x* regularized solution vector y dimensionless coordinate distance, Y/H Y coordinate distance, m

INVERSE DESIGN OF THERMAL SYSTEMS

7

GREEK SYMBOLS et Tikhonov regularization parameter, Eq. (19); parameter in conjugate gradient algorithm, Eq. (27); relaxation parameter, Eq. (78) 13 parameter in conjugate gradient iteration, Eq. (30) surface emissivity MTSVD correction term, Eq. (14) minimized Tikhonov functional 3' order ofTikhonov regularization, Eq. (19); medium element F number of medium elements 11 thermal efficiency of thermal process norm of side constraint, Eq. (18); Tikhonov regularization function Ix singular value of kernel K, Eq. (2); dynamic viscosity, N.s/m 2 p density, kg/m 3 tr Stefan-Boltzmann constant, 5.6704 x 10 -8 W/m2.K4; components of S optical thickness, a H percentage error in inverse solution, Eq. (13)

SUBSCRIPTS correction term; conduction design surface gas heater surface heater source HS index of surface subdivisions or singular values; ith derivative i index of surface subdivisions J conjugate gradient step index k iteration number n rank of coefficient matrix N outgoing 0 based on p retained singular values P radiative R reference value ref T total (radiation plus conduction) on bounding surface W initial estimate 0 1,2,3,4 on surfaces 1,2,3, or 4 oc surroundings C D g H

llI. Mathematics of Inverse Design A. REGULARIZATION METHODS

Inverse problems require special methods for treating the ill-conditioned nature of the equations that describe the physical system being analyzed. Various approaches have been developed and tried. This section describes

FRANCIS H. R. FRAN~A ET AL.

8

some of the most common approaches that work well for inverse thermal design problems. First, it is useful to gain insight into the reasons for the ill-conditioned behavior of the equations under study by analyzing the analytical solution of a radiation inverse design problem that results explicitly in a Fredholm integral equation of the first kind. Consider the radiative enclosure shown in Fig. 1. Suppose that Fig. 1 describes an annealing furnace, where a continuous strip of metal is being passed through the furnace as surface 1. In this case, the mass flow rate of the metal sets the required energy flux to be imposed to provide a given temperature profile on the metal as it passes through the furnace, and the temperature profile is set by the requirements of the annealing process. Therefore, both the net heat flux and the temperature (or, equivalently, the emissive power) on surface 1 are inputs to the thermal design problem, and the designer wishes to find the necessary emissive power distribution and energy input for the heating elements on surface 2 of the furnace that will provide these conditions. For convenience of presentation, surface 2 is assumed to be a black body. When the net energy exchange equations for radiative transfer are written in dimensionless form for a particular element on surface 1, the emissive power eb2(X2) on the upper surface is obtained from an integral equation: K(Xl,X2)eb2(x2)dx2 -- ebl(Xl) - - - - q T I ( X l ) 2=0

(1)

E1

(where the radiative and net heat fluxes are the same in this case, qr = qR). The kernel K(Xl,X2) is related to the radiation geometric configuration factor by dFdxl-dx2 = K(Xl, x2)dx2. The dimensionless variables are defined as X l =-- X 1 / L , x 2 - - X 2 / L , eb = (Eb - E b o c ) / E b , ref, and qr = QT/Eb, ref. The domain of the problem is defined by 0 < Xl, x2 < 1. Because both dimensionless emissive power ebl(Xl) and radiative heat flux qrl(xl) on surface

,d

L

7

black surface ~ _ _ _ _

!

?

x2 ~

Eboc

[ !

I ! !

a

" Xl

/

Ebl, QT1, el

! !

FIG. 1. Geometry for inverse design example.

n

Eb ~

INVERSE DESIGN OF THERMAL SYSTEMS

9

1 have been specified, the right-hand-side of Eq. (1). is known, whereas the emissive power on the upper surface, eb2(X2), is unknown. This equation is a Fredholm integral equation of the first kind, which is notoriously ill-posed. By itself, Eq. (1) cannot be solved for a unique distribution eb2(x2). However, Eq. (1) may be written for various locations xl on surface 1. The resulting set of linear equations, while still ill-conditioned, offers some hope of providing a solution for eb2(X2).

1. A n a l y t i c a l Solution

A better understanding of the inherent difficulty of solving Eq. (1) can be achieved by examining its analytical solution. First, a singular value expansion of the kernel K is performed: (3O

K(xl, x2) - Z

~iui(xl)vi(x2)'

(2)

i=1

where ix i, ui, and vi are the singular values and singular functions of K. The functions ui and vi are orthonormal, which implies that 1 if if

(ui, uj) - (vi, vj) -

0

i-j i-~j '

(3)

where the inner product (ui, uj) is defined as

(Ui, Uj) --

Ji

ui(~)uj(~)d ~.

(4)

As a consequence of these properties, it follows that the emissive power on the upper surface can be directly calculated from O(3

eb2(x2) -- Z i=1

(ui, ebl - qT1/~;1) vi(x2).

(5)

~i

For smooth kernel operators, as is the case for the proposed inverse problem, the singular values ~i decrease faster than i -1/2. Noting that the singular values are in the denominator of the infinite series of Eq.(5), the emissive power on the upper surface will converge only if the inner p r o d u c t (ui, e b l - qT1/8l) decreases faster than i -1/2 from some point in the summation. This is known as the Picard condition. Clearly, the summation of Eq. (5) will not converge for all possible imposed conditions on the bottom surface. In fact, the Picard condition is a strong requirement that is not satisfied for most imposed temperature and heat flux distributions on the bottom surface. When two conditions are

10

FRANCIS H. R. FRAN~A ET AL.

imposed on one of the boundaries, it is much more likely that there will be no exact solution for the problem; the summation in Eq. (5) will simply diverge. A solution can be realized only if additional constraints are imposed to stabilize or regularize the problem, although this inevitably introduces an error into the solution [1]. 2. Numerical Discretization

For many physical systems the linear inverse problem resulting from radiation exchange in the absence of conduction or convection can be formulated in terms of a matrix operator A and a vector of unknowns x as a result of numerical discretization of the domain: A. x -

b.

(6)

Considering the problem of Fig. 1, the continuous domain can be divided into m elements of uniform size, Ax, in each surface. The continuous relation of Eq. (1) is replaced by the algebraic relations: m Z j=l

1 Fi-jeb2,j -- ebl,i -- - - q v l , i ,

(7)

E1

where ebl,~ and qTl,i are the dimensionless emissive power and radiative heat flux in element A1,; located on the bottom surface; eb2,j is the dimensionless emissive power of element A2,j located on the upper surface; and F;_j is the view factor between A~,i and A2,j. The indices i a n d j span all the elements on the bottom and upper surfaces, respectively. Equation (7) can be written for each of the m elements on the bottom surface, forming a linear system on the m unknowns eb2j. The elements of A, x, and b are, respectively, ai,j = Fi-j, xj - eb2,j, and bi = ebl,i - qTl,i/E1. To demonstrate the characteristics of inverse problems, consider an enclosure with aspect ratio H / L = 0.5 and with emissivity on the bottom s u r f a c e o f E1 - - 0.8. The end surfaces are black and at zero absolute temperature. The inverse problem consists of finding the emissive power on the upper surface, eb2(x2), that satisfies both a prescribed emissive power and a parabolic total net heat flux on the lower surface, e b l ( x , ) = 1.0 and q T I ( X l ) - 16x 2 - 1 6 X l - 6. The negative value of qrl indicates that surface 1 is being heated by surface 2. Then, the system formed by Eq. (7) can be solved to lead to the emissive power distribution on the upper surface, as shown in Fig. 2. As seen, the necessary emissive power on the upper surface presents steep oscillations between large absolute numbers having alternating signs. This is clearly unsatisfactory, as the emissive power must be a positive number. Besides, it is desirable to find smooth, well-behaved solutions.

11

INVERSE DESIGN OF THERMAL SYSTEMS

1.0E5

.

.

.

.

i , , ,

,

.

.

.

5.0E4

.....

9! eb2

0.0

-1.0E5

.

i

'!

'ii:

.......................................

~" "

'i.... ....i....i'i.................................... .

0.0

0.2

.

0.4

. . 0.6

.

.

. 0.8

X2 F I G . 2. E m i s s i v e p o w e r o n t h e u p p e r s u r f a c e f r o m d i r e c t i n v e r s i o n o f s y s t e m o f e q u a t i o n s . ebl = 1.0, q T l ( X l )

= 16X 2 -- 16Xl -- 6; H / L

= 0.5,~1 = 0.8, m = n = 30.

Solutions such as the one shown in Fig. 2 are typically obtained when it is attempted to solve the system of equations formed by Eq. (7) by a conventional matrix solver, such as Gaussian elimination or LU decomposition. These solutions occur because the system of equations was obtained from the discretization of the set of Fredholm integral equations of the first kind, Eq. (1). When an ill-posed problem is discretized, the inherent difficulties found in the analytical solution are carried over. Analogous to expanding the kernel K, matrix A can be decomposed into a combination of orthogonal matrices and singular values, the so-called singular value decomposition (SVD), and as happens to the singular values of the kernel K of the continuous problem, the singular values of the matrix of coefficients A decay gradually to zero. Therefore, the SVD plays an essential role in the solution of ill-conditioned systems, for it allows an accurate diagnosis of the level of ill-conditioning of the problem. In addition, many regularization methods are based on SVD. 3. Singular Value Decomposition

A matrix A c R mxn (the number of rows, m, and the number of columns, n, are not necessarily the same) can be decomposed according to A = U . S. V r.

(8)

12

F R A N C I S H. R. FRANt~A E T AL.

Matrices U = ( U l , . . . , U n ) C R mxn and V - (vl, . . . , vn) c R nxn are orthonormal. This means that the column vectors ui and vi (the singular vectors of A) form an orthogonal basis: 1 Ui " Uj "~- u

"u

--

0

if if

i-j i ~ j"

(9)

In Eq. (8), S c R n• is a diagonal matrix formed by the singular values r of matrix A. The singular values are nonnegative numbers, and are ordered such that r _> r _> ... _> r _> 0. The SVD decomposition presents a close relation to the eigenvalue decomposition of the symmetric semidefinite matrices A. A r - U . S 2 9V T and AT.A - V. S 2 9U T. That is, the square of the singular values of A corresponds to the decomposition eigenvalues of A. A r a n d A r. A. As will be discussed later, this property relates the SVD to the conjugate gradient iterative scheme. An immediate application of singular value decomposition is that the solution vector can be directly obtained from a linear combination of the vectors l~i, where the coefficients are given by (uf. b)/txi" X --

u i=1

An inspection of the singular values type of ill-conditioning of matrix A.

(10)

O'i O"i

allows an accurate diagnosis of the

Rank-deficient problems" in this case, there is a well-determined, large gap in the series of the singular values, indicating that some rows or columns of A are linearly dependent on each other. 9 Discrete ill-posed problems" these problems arise from the numerical discretization of a Fredholm integral equation of the first kind. The singular values decay continuously to very small values, and there is no distinct gap between large and small ones.

9

The features of discrete ill-posed problems are consequences of the fact that the SVD of A is closely related to the SVE of the kernel K. In many cases, the singular values of A, tri, make a good approximation to the singular values bl,i of g ~ whereas the singular vectors ui and vi of A can yield information on the singular functions of K. Figure 3 presents singular values of the matrix A of the problem shown in Fig. 1, as obtained from SVD. The number of singular values equals the number of columns of A (n -- 30). As seen in Fig. 3, the singular values O"i decrease steeply and continuously to 10 -8 , as is typical of discrete ill-posed problems. This explains why the calculated emissive power on the upper surface presents such steep changes in sign, as the inherent oscillations of the

13

INVERSE DESIGN OF T H E R M A L SYSTEMS

10o 10"1 10"2

.

.

.

.

.

lO-J

(Yi

!!!!!!!!!!!!!!!!i!ii!!!!!!!!!!!ii!!!!!!!!! .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . .

10 -4

10-5

10-6 10-7

10-8

. . . .

0

!

5

. . . .

I . ,

10

J

~

I

15

~

~

,

,

~ . .

20

.

I

. . . .

25

r

30

. . . .

35

FIG. 3. Singular values from the singular value expansion of matrix A, m = n - 30.

high-order vectors of Eq. (10) are amplified by the corresponding small values of cri. Smooth solutions will be possible only in the special cases where u/r. b decays faster than cri. An example of this occurs when vector b is first calculated from Eq. (6) using a known, smooth vector x, and then employing Eq. (10) to find x from the obtained b. However, even for this case, if the singular values are sufficiently small as to be below the computer precision, 10 -14 ~ 10 -16, then the solution will be completely corrupted by the round-off error. In the present example, however, the larger values of the product u/r. b are on the order of 10 -3, leading to coefficients (u r. b)/cri as large as 105, which explains the peaks in the emissive power shown in Fig. 2. An inverse design problem can be solved satisfactorily only if additional constraints are applied to stabilize the original problem, which in turn introduces an error into the solution. In general, for increased levels of stabilization, there will be a greater error, which implies that a compromise is necessary. This is the basic idea of regularization methods.

4. Truncated Singular Value Decomposition ( TSVD) TSVD is a direct regularization method for the solution of ill-conditioned problems and follows directly from the SVD solution of singular problems.

14

FRANCIS H. R. FRAN(~A E T AL.

If the system of equations is singular, then some of the singular values of the matrix A will be exactly zero. The vectors vi corresponding to the singular values O"i that are zero form the null-space of A, which means that A.v/-0.

(11)

The vectors ui, whose corresponding singular values are not zero, span the range of A. If the vector b is in the range of A (in other words, if b can be obtained by a linear combination of the vectors u;), then an infinite number of solutions x exists for the problem, as any linear combination of the vectors vi in the null-space of A can be added to x to form a new solution. The SVD solution is obtained by just eliminating, from the linear combination of Eq. (10), the terms related to the null singular values:

P u~.b Xp

-/~1 .__

O'i

v/,

(12)

where ~rp+l,..., an are zero. The vector x as obtained by Eq. (12) is the one that has the minimum 2-norm among all the existing solutions. When the vector b is not in the range of A, there is no solution for the problem. In this case, the SVD can still be applied to find the least-square solution x to the least-squares problem min ]A. x - b{. In the solution of rank-deficient and ill-posed problems, singular values may not be exactly zero, but they decay, continuously or not, to very small numbers (Fig. 3 is a typical example). As discussed in the previous section, nonregularized solutions tend to be very sensitive to small perturbations, and quite often the elements of x present steep oscillations between large positive and negative numbers. TSVD regularization consists of replacing A by a matrix Ap that is mathematically rank deficient. The TSVD solution can be considered to be the SVD solution of a rank-deficient problem, Eq. (12), obtained by keeping only the terms related to the largest p singular values. The TSVD solution is the least-square solution of the problem minlAp 9x - b], where Ap is the rank-deficient matrix that is obtained from Eq. (8) when the n-p smallest singular values erp+l, . . . , ~n of matrix S are replaced by zero. Like any regularization method, TSVD attempts to stabilize the solution of an ill-conditioned problem, although at the expense of introducing a residual error r = ] A p - x - b I into the solution. The stabilization is achieved by eliminating the high-order terms of the canonical relation of Eq. (10), which are responsible for the instabilities of the system, and which are sometimes corrupted by round-off errors. The number of singular values p kept in the solution is the regularization parameter of the TSVD method. For smaller p the system of equations is more stable, but the residual r is

I N V E R S E D E S I G N OF T H E R M A L SYSTEMS

15

larger. Therefore, the TSVD solution requires an optimum balance between p and [r I. Figure 4 presents the emissive power on the upper surface for the example problem of Fig. 1 as obtained from TSVD regularization. It was verified that solutions with physical meaning occur only when p is kept below 6. Otherwise, the emissive power on the upper surface will include negative values, which is not physically acceptable. In Fig. 4, solutions are presented for p = 1, 3, and 5. As seen, the emissive power on the upper surface is smoothed by neglecting the linear combinations related to the smaller singular values. Interestingly, for p = 2, 4, and 6, the solutions are nearly identical to the ones for p - 1, 3, and 5, respectively. This is because the products u/r. b for i = 2, 4, and 6 are as small as 10 -13, so adding these terms to the linear combination of Eq. (12) results in no change. The solutions shown in Fig. 4 were obtained from regularization of the original system of equations. Therefore, they are necessarily approximations for the problem and need to be verified. A simple, safe way to check an inverse solution is to use it as the input of a forward problem; that is, the

22 i,~

f-k

-

20 ............

-

18

o/

,e

-~

.......

!

I

J'7 .

i. .................

i

[ .....

\

........

-,,,,

.

.

.

\

.

'

~i q

eb2 16

. ...................................................

14

p-1

e--p --~-

12

t 0.0

n

n

n

'

I

I

0.2

I

I

I

0.4

I

I

0.6

I

3 ... p-

I

5

I

0.8

I

I

1.0

x2 F I G . 4. E m i s s i v e p o w e r o n t h e u p p e r s u r f a c e f r o m T S V D

5. ebl

= 1.0,

qTl(Xl)

= 16X 2 -- 16Xl -- 6;

H/L

r e g u l a r i z a t i o n : p = 1,3, a n d

= 0.5,81 = 0.8, m = n = 30.

16

FRANCIS H. R. FRAN(~A E T AL.

obtained emissive power distributions of Fig. 4 are inserted into Eq. (7) to find the heat flux on the bottom surface, keeping the original imposed emissive power e b l - - 1 as a boundary condition. Then, this predicted heat flux distribution is compared to the imposed heat flux, q r l ( x l ) = 1 6 x 2 - 1 6 x l - 6 , by qT1, i, imposed

~

qT1, i, inverse

(13)

q T1, i, imposed

The e r r o r ~i can be calculated for each element on surface 1 and then compared to the precision required in the design. It was found that the maximum values of ~i for the solutions corresponding to p - 1, 3, and 5 were 4.39, 0.872, and 0.715%, respectively. If, for example, a maximum error of 1.0% is required in the design, then the TSVD scheme provides two solutions, for p = 3 and 5. An alternative is to choose the inverse solution for the upper surface emissive power from Fig. 4 and the heat flux specified for the lower surface qrl(Xl) as boundary conditions on the forward problem. The emissive power is then predicted for the lower surface and is compared with the specified value. Either this check or the check proposed earlier is equally valid because the forward solution is well conditioned; however, because heat flux boundary conditions are somewhat more difficult to formulate in forward problems compared with temperature boundary conditions, the latter are usually used.

5. Modified Truncated Singular Value Decomposition ( M T S V D )

This method, proposed by Hansen [1], is a more general method than TSVD in the sense that other constraints can be imposed on the vector solution x rather than only imposing a minimum 2-norm. The MTSVD regularization algorithm is a two-step procedure. First, an initial approximation is obtained by setting the regularization parameterp to compute the TSVD solution xp, as given by Eq. (12). Next, a correction term to xp is computed from the terms that were discarded from the linear combination of Eq. (12), i.e., from the numerical null-space of matrix A. In this step, additional constraints or smoothing characteristics are introduced. The constraints are thus applied only to the terms that cause instabilities of the solution. The correction term is calculated after solving the following least-squares problem for q~p: min[L . Vp . ~p - L . xp[,

(14)

INVERSE DESIGN OF THERMAL SYSTEMS

17

where the matrix L represents the smoothing operator that is being applied on the solution. For example, a second derivative finite difference operator L2 is given by 0 i L2 -

0 -2 .

1 . 1

. -2

.

(15)

1 0

The matrix Vp is formed by the remaining n - p singular vectors, i.e., Vp = [Vp+l, . . . , vn].

(16)

The MTSVD or L-order regularized solution is finally obtained from xr~,p - - X p -

Vp

. q~p.

(17)

As with the TSVD scheme, the regularization parameter of the MTSVD method is p. The choice o f p is a compromise between the residual r and the level of smoothness imposed on the solution vector x. As for other regularization methods, a number of different MTSVD solutions satisfy the problem within some prescribed precision, which leaves an option to the designer to select the most suitable one. Figure 5 presents the emissive power on the upper surface using the MTSVD regularization. As with the TSVD solution, it was not possible to obtain physically acceptable solutions when p is set to a value larger than 6. Figure 5 shows only the solutions for p = 1, 3, and 5, as the solutions for p - - 2, 4, and 6 are the same. Comparing Figs. 4 and 5, it can be seen that the solution for p = 5 using TSVD and MTSVD is very similar, but for p = 1 and 3, the MTSVD solutions are smoother than the TSVD solutions, in the sense of the emissive power on the upper surface being more uniform. For p - - 1, the MTSVD-derived emissive power is uniform. This can be explained by the fact that the second-order derivative of the components of vector x was minimized through the operator L2 in the MTSVD regularization. The accuracy of the M T S V D solutions can be verified by means of the error defined in Eq. (13). For all three cases, the error was below 1.0 %: for p = 1, 3, and 5, the maximum value of ~i was 0.751, 0.911, and 0.594 %, respectively. Note that the maximum error for p = 3 exceeds that for either p - 1 or p = 5. This is because the MTSVD algorithm minimizes the residual, which is related to the average rms error. Generally, the maximum local error will also be reduced, but that is not necessarily the case as is observed here.

18

FRANCIS

. . . . . . .

22

/ ........

20 :-

18

:

/ .... i ......

ET AL.

H. R. F R A N t ~ A

!

.

.

.

.

.

.

!

........

/

~.

,

~ ................

,;,~..

g - - - : - " u ~ : e , ~ o , - e q ~ , ~ T v 4 1 ' ' "

I : ~" : .... /- . . . . . . . . . . ; ................. .k. ........ I \

,

:: \

\ . . . .,~...... . . . . i . . . . . . . .. .. .. .. .. .. .. . . . . .

,

-

x

-' \

............................. ~" %

]

eb2 16

................

: ...................................

..................................

;

i

9 :,

14

................................ i

i

:

:

:

. . . . . . 120.0

;......................

j,

012

p=l

o--p

= 3 ......

~ - p - 5

i

, , j , , , i , , ,

0.4

0.6

0.8

1.0

X2 FIG.

5. E m i s s i v e

power

on the upper

surface

from

MTSVD

regularization:

p = 1, 3, a n d

5. ebl - - 1.0, qTI(Xl) - - 16X 2 -- 16Xl -- 6; H / L = 0 . 5 , ex = 0 . 8 , m = n = 30.

6. Tikhonov Regularization Solution of ill-posed problems began after the pioneering work of A. N. Tikhonov in the 1960s [2-4]. His method of regularization considerably broadened the bounds of the effective practical solution of ill-posed problems in the physical sciences. The main idea of Tikhonov regularization is to introduce a side constraint in order to stabilize or regularize the problem. Usually, this side constraint is linear and allows the inclusion of an initial estimate, Xo, if available. The side constraint involves a norm that can be defined as a,X/'i(X)

--

ILi 9 (x - x0)l 22,

(18)

where Li approximates the ith derivative operator and x0 is an initial (biased) solution estimate. For i = 0, the identity operator is assumed. The Tikhonov regularization method minimizes the functional ~ ( x ) , which is defined as 7

9 ~(x) -lAx

- b[~+ Z i=o

7

e~ff~i(x) -- l A x - b l ~ + Z

~

9 ( x - x0)l~.

(19)

i=o

Therefore, the Tikhonov solution, x,~, is a function of the regularization parameter a. According to Eq. (19), for larger a, the the solution is more

19

INVERSE DESIGN OF THERMAL SYSTEMS

regularized in the sense of minimization of the side constraint, but the residual is also larger. A small oL has the reverse effect. In this aspect, the selection of c~ resembles the choice of the TSVD and MTSVD regularization parameter p by a compromise of the size of the residual and the level of regularization of the solution. Thus, the selection of o~ is an important part of the inverse solution and must be made carefully. Selection of the initial estimate x0 is less critical, although it can be used in design to provide a better final solution. A choice of x0 = 0 introduces no bias. Minimization of Eq. (19) with respect to x implies that 7

9 '~(x) - 2A 7" 9 (A 9 x -- b) -~- 2 Z

OLi2LiT 9 L i 9 ( x - x0).

(20)

i=0

At the minimum, ~'~ should be zero, thus Eq. (20) can be rearranged as 7 A T 9 A 9 x -~- Z OL/2Ll 9 L i 9 x i=0

A T 9 b -~- Z OL~LT 9 L i 9 X0' i=0

(21)

which has the form A~; 9x = b~;.

(22)

For example, what is called standard Tikhonov regularization is the case in which the series is terminated at ~, = 0, where L0 is the identity matrix I. That is, (A T 9A + ~2oI) 9x - A r 9b + ~2xo. Another OL0 = OL1 = 0,

(23)

example is second-order regularization, for which ~, = 2, leaving oL2 as the only regularization parameter:

2 2T 9 L 2 ) e x -- A r 9 b + o~2L 2 2r 9 L2 9 xo. (A r 9 A + o~2L

(24)

Note that, in both cases, the original system of equations is modified. In the second-order regularization, the new matrix of coefficients is A~2-2 T ATe A + oLzL 2 9L2, whereas the vector of independent values becomes 2 T b~2 -- A T 9 b + otzL 2 9 L2 9 xo. A s c a n b e inferred, f o r a larger r e g u l a r i z a t i o n

parameter oL, the system will be modified and stabilized more by operator matrices L. When o~ is adequately chosen, the system of equations becomes more stable while keeping the most important information of the original system. In this way, the system of equations can be solved by any conventional matrix solver (as Gaussian elimination or Gauss-Seidel iteration) to lead to a solution that is still accurate. Figure 6 shows the emissive power on the upper surface as obtained from second-order Tikhonov regularization, Eq. (24). Three values of the

20

F R A N C I S H. R. FRANC~A

22

,

,

,

!

.

.

.

.

.

.

!

9

,

,

!

/ i \.~ .........

...............

18

.... /- . . . . . . . . . . . . . . . . . . . . . . . . . . . I

9 o-

\

, .

/ix

i

; .................

i .........

"

i

.% . . . . . . . . . . . . . . . . \ /

0

~\

,

I

,

.

i

20

eb2

,

_.

ri,, /.

ET AL.

( .......

i ......

er

.~ .....

e_ek4D-e

'~.. . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

.%'.'"%

~"/

I

I

'

I

]6

14

o--

oc = 0.05

-~--

..

c~ = 0 . 0 2

o

i

12

'

!

0.0

n

l

0.2

I

l

I

l

0.4

I

I

0.6

I

n

i

n

i

i

0.8

1.0

x2 F I G . 6. E m i s s i v e p o w e r o n t h e u p p e r s u r f a c e f r o m T i k h o n o v and ~.

Ebl = 1.0,

qTl(Xl)

= 16X 2 -- 16Xl -- 6;

H/L

r e g u l a r i z a t i o n : ot = 0.02, 0.05,

= 0.5, el = 0.8, m = n = 30.

regularization parameter were considered: et = 0.02, ct = 0.05, and oL ~ c~. The choice of these values led to the solutions that were the closest to the solutions from the MTSVD scheme presented in Fig. 5. This indicates a close relation between the Tikhonov method and the SVD-based methods. In fact, it was verified that Tikhonov solutions also resembled those of the MTSVD for zeroth-order and first-order regularizations. The major difference found is that Tikhonov solutions have a continuous dependence on et, whereas TSVD and MTSVD solutions are dependent on the discrete number of singular values, p. Another noteworthy aspect of the Tikhonov solution is the case where ot ~ c~. When a large value for oL is chosen, the side constraint becomes dominant in the minimization of the functional ~ ( x ) . Therefore, in second-order regularization, where the side constraint relates to the second derivative of the components of vector solution x, the solution is the uniform vector that gives the least residual lap. x - b I. To complete the regularization procedure, the regularization parameter ct has to be determined. There are several ways to find the optimal value: the L curve [5], the discrepancy principle [6], or generalized cross-validation [5].

INVERSE DESIGN OF T H E R M A L SYSTEMS

21

The L curve is a convenient graphical tool for the analysis of discrete ill-posed problems. It is a plot of the seminorm used as a side constraint, Li. (x - x0), versus the corresponding residual norm, [A-x - b[, for different values of the regularization parameter o~. In this way, the L curve displays the compromise between minimization of both quantities. This curve is very important because it divides the first quadrant into two regions. Hansen [6] indicated that it is impossible to construct any regularized solution that corresponds to a point below the L curve; therefore, any regularized solution either lies on or above the L curve. When plotted on a log-log scale, the curve almost always has a characteristic L-shaped appearance with a distinct comer separating the vertical and the horizontal part of the curve. This comer corresponds to the near optimal value for the regularization parameter cx. Unfortunately, the present study has found that the solutions corresponding to this corner may have undesirable oscillations; in some circumstances, some components of the solution vector become negative. Because these components represent the outgoing radiative energy, which is defined as positive, negative values are unacceptable on physical grounds. For larger values of the regularization parameter, the solution has less or no oscillations, but at the expense of a larger residual error. Thus, the optimal solution predicted by the L curve may present the best tradeoff between meeting the side constraint and providing a low residual norm, but this solution may not be suitable or acceptable as a design solution on physical grounds. To determine the value of the regularization parameter o~ for standard Tikhonov regularization, equations like (23) or (24) must be solved for several values of cx. This process is time-consuming and can be avoided by using the singular value decomposition of matrix A. The first step is to determine the singular value decomposition for A, as in Eq. (8). Then, a solution for a fixed oLcan be computed by introducing the filters f., which, for the case where L0 = I, can be evaluated using 2 0.i

(25)

Thus, the regularized solution for a fixed or, x~, can be computed by

n

x~ : ~ - ~ uT" b V i - i= 1

O'i

s 0.i

(u T. b)vi.

(26)

i= 1 0.2 _Jr_OL2

After the SVD of A is obtained, additional solutions can be computed by only recomputing the filters J~ for different values of o~ and the matrix multiplications involved in Eq. (26).

22

FRANCIS H. R. FRAN(~A E T AL.

7. Conjugate Gradient Method The conjugate gradient method is an iterative technique for producing regularized solutions that avoids the explicit decomposition of the matrix A. Matrix decompositions such as SVD are time-consuming for large matrices, but are an inherent part of TSVD, MTSVD, and Tikhonov methods. Therefore, the conjugate gradient method is often chosen for multidimensional problems. Hansen [5] points out that the operations used in conjugate gradient regularization (CGR) readily lend themselves to parallelization. The classical conjugate gradient method was originally derived for the case where the coefficient matrix is positive definite and symmetric. For a generic matrix A, the method can still be applied by premultiplying both sides of Eq. (6) by A T. The new coefficient matrix A T. A becomes positive definite and symmetric; for this case, a common and stable implementation of the method, known as the CGLS algorithm, consists of the following steps: 9 guess: xo; r o - b - A . x o ; Po = AT. ro; 9 for k _> O, follow the iterative steps" etk - IAT. rk [2/ [ A . p k 12

(27)

Xk+l - x~ + oL~pk

(28)

rk+l -- rk -- a~A- pk

(29)

[3k _ [AT.

2

AT"

2

Pk+l _ A T. r~+~ + [3kpk

(30) (31)

The sequence of Eqs. (27) through (3 l) describes the formation of a series of orthogonal residuals rk, and A-orthogonal vectors pj"

ri'rj-Pi'(A'pj)--

1 if 0 if

i -j

iCj"

(32)

It follows from these properties of the CG algorithm that, for an infinite precision computation, the residual will be exactly zero after n iterations (n being the size of the square matrix A T. A). In ill-posed problems, where singular values decay to very small values, the convergence rate using machine precision requires many more iterations than n. Preconditioning of A is often used to improve the convergence of forward problems, but for ill-posed problems there is no point in improving the condition of A because the fully converged solution is not interesting because it is affected by instabilities related to the high-order terms of the SVD. In fact, in most

INVERSE DESIGN OF THERMAL

23

SYSTEMS

cases, just a few iterations are needed to achieve a desirable inverse solution. After that, the solution becomes very unstable, presenting steep oscillations. The regularization parameter of the CG method is the number of iterations K of the steps described earlier (0 < k < K). If the initial guess for the solution vector is x0 - 0, then the 2-norm of the vector solution, Ixl, increases with the number of iterations k, whereas the 2-norm of the residual, Irl, decreases with k. This monotonic behavior of both Ixl and Irl is essential as a stopping rule for regularization of the CG iterations. The CG method often produces iteration vectors in which the spectral components associated with the large eigenvalues tend to converge faster than the remaining components. In connection with discrete ill-posed problems, the same behavior is observed when the CG scheme is applied to the normal equation A T. A - x - A T. b. Because the eigenvalues of A T. A are simply cr2, this means that the SVD components associated with the larger singular values tend to converge faster than the remaining terms. Therefore, the CG algorithm has a regularizing effect that resembles the TSVD. This is confirmed in Fig. 7 which presents the CG solutions for three different

22

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9

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= .......................................................

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x2 FIG.

7. Emissive power on the upper surface from CG regularization: K - -

ebl = 1.0, q r l ( X l ) = 1 6 x 2 -- 1 6 x l -- 6; H / L

= 0 . 5 , el = 0 . 8 , m = n = 30.

1,2, and

3.

24

FRANCIS H. R. FRAN~A E T AL.

numbers of iterations: K = 1, 2, and 3, which are coincident with the TSVD solutions having p = 1, 3 and 5, respectively. The errors for p = 1, 3, 5 are thus 4.39, 0.872, and 0.715%, respectively. Setting the number of equations equal to or larger than 4 leads to solutions having negative emissive power, which are of no practical interest as design solutions. 8. Additional Comments on Regularization Methods

The previous solutions exemplify the application of some different available regularization methods. As seen, there is a clear relation among the methods in the sense that similar results and levels of precision can be obtained by each. The MTSVD and the Tikhonov present more flexibility than the TSVD or conjugate gradient method, as additional constraints can be imposed easily on the solution. However, the additional constraints are imposed by means of minimization functions, and so when a given physical constraint is intended to be imposed on the solution, it is first necessary to know what is the corresponding minimization function, which may not be easy to formulate. However, as shown in Section VI,C, it is possible to impose additional constraints on the solution even when applying the TSVD and CG methods, which are more straightforward regularization methods. In the discussed examples, it is found that imposing more constraints to the problem can also cause the system of equations to become more stable in the sense that the singular values do not decay to such small values as those of the unconstrained problem. In this case, truncation of the smallest singular values may become unnecessary in order to recover acceptable solutions. Another aspect of regularization concerns grid independence. The validation of any numerical solution requires a grid independence study. It is of interest to learn how the change of grid resolution can affect an inverse solution, because, for a given grid size, the solution also depends on the regularization imposed on the system. In other words, it is necessary to verify how the change in the grid resolution affects the choice of the regularization parameters. As an example, consider the TSVD regularization for the problem of Fig. 1 for two grid resolutions: m = n - - 30 and 40. Figure 8 compares singular value spectra for these two grid sizes, indicating that they are coincident for the large values of cri, and depart only for the smaller values of cri. Considering that only the large singular values are inserted in the linear combination of Eq. (12), it is reasonable to expect that the solutions for m = n = 30 and 40 will be the same if the same regularization parameter p is chosen. Figure 9 confirms that the solutions for the two grids, when the samep is used for both, are consistently coincident. The same can be said for the Tikhonov and CG solutions, which indicates that the same regularization

INVERSE

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;

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FIG. 8. Singular values from the singular value expansion of matrix A for the problem of Fig. 1 for grid resolutions m = n -- 30 and 40.

parameter, ot and K, respectively, will lead to a similar solution for grid resolutions that do not significantly differ from each other. For very different grids, this is not true. In fact, specifying an unusually coarse grid can be thought of as a means of regularizing the problem. The system of equations becomes less ill-conditioned on a very coarse grid, while the accuracy of the solution worsens. In general, for sufficiently resolved grids, the regularized problem is robust with respect to grid choice (above a sufficiently accurate grid), and the same regularization parameters can be used for different resolutions to lead to the same shape of the solution. As a final remark, it is interesting to realize that this problem allows a number of different solutions that satisfy the design specification within an accuracy of 1.0%. This is a typical feature of inverse design. However, it is also typical of inverse design to find situations where n o solution is found to satisfy the two boundary conditions imposed on the design surface. For this example, if both temperature and heat flux are required to be uniform on the bottom surface of Fig. 1, it is not possible to find an emissive power distribution on the upper surface that provides the uniform conditions

26

FRANCIS H. R. FRAN(~A ET AL.

I.

~'

20

.........

eb2 18

!

~~...~.

)r/ .........

"~q,/

..........

ri

"

:

i. . . . . . . . . . . . . . . . . .

; ......

i ................. !~~:X O ~

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-

~...

........

................. !

........ ~~

.............i.................i.................i . ......... --I,-14

0.0

0.2

p=

:

1, m = 30 ----o---- p = l ,

p=3 m=30 = 51 m = 3 0

0.4

--v---0--

0.6

t

m=40

p=3 m = 4 0 ....~" p = 51 m. = 4 0

0.8

1.0

x2 FIG. 9. Emissive power on the upper surface from TSVD for m = n = 30 and 40. ebl = 1.0, qrl(Xl) = 16X 2 -- 16Xl -- 6; H / L = 0.5, zl -- 0.8.

with such accuracy because of the effect of the ends of the enclosure. However, even for this case, when no solution exists, inverse design is an important tool because it is a fast, safe way to verify whether a given design set can be achieved within a specified accuracy. B. OTHER APPROACHES

Various methods are available for treating ill-conditioned problems. We have concentrated here on TSVD, MTSVD, the Tikhonov method, and the CGR approach. Other alternatives include inverse Monte Carlo methods [8], the application of neural nets to determine the input that achieves the desired output [9], and the use of optimization approaches, such as the simulated annealing algorithm [9-11] or Levenberg-Marquardt multivariate optimization [12]. There is considerable room for investigation of the application of these and other methods to inverse design. It remains an open question whether optimization of a series of forward solutions might provide a more efficient solution of certain design problems than a single inverse solution.

INVERSE DESIGN OF THERMAL SYSTEMS

27

C. LITERATURE REVIEW

Now that the common methods for treating inverse design of thermal systems with significant radiation have been reviewed, we can proceed to examine some of the available literature. In the heat transfer field, the extensive work on inverse analysis of conduction problems by Beck et al. [13] has laid the groundwork for many solution techniques that can be applied to more general inverse problems. More recently, inverse convection heat transfer problems have been considered by Huang and Ozisik [14], Hsu et al. [15], and Park and Chung [16]. They considered the inverse problem of finding internal conditions in the system from measurement quantities. Inverse problems in radiation heat transfer can be classified into problems seeking (1) inverse property values, (2) inverse measurements, (3) inverse boundary values, (4) inverse heat sources, and (5) inverse geometries. The first category relates to the inverse analysis to determine the thermal properties of the system from the knowledge of some output conditions, e.g., measurements on the boundary. Some examples of this approach can be found in the papers by Dunn [17], Matthews et al. [18], Wu and Mulholland [19], Lin and Tsai [20], Subramaniam and Mengii? [21], Tsai [22], Hendricks and Howell [12], Jones et al. [23, 24], McCormick [25], and Kudo et al. [26]. In the second category, Li and Ozisik [27], Siewert [28, 29], Li [30, 31], Linhua et al. [32], Liu et al. [33-35], and Yousefian and Lallemand [36] have considered how to reconstruct the temperature profiles in a medium from measurement of the radiation intensity exiting the boundaries. Ruperti et al. [37] extended the analysis to consider coupled radiation and conduction. In this case, an important aspect to be analyzed is the effect of the random data uncertainties on the solution. Most of these cases were solved by applying the conjugate gradient method to minimize the error between calculated incident radiation fluxes and experimental data. Inverse design can involve the three last categories: inverse boundary, heat source, and geometry problems. In thermal inverse boundary design, the conditions on one or more surfaces are to be determined to satisfy the two constraints imposed on the design surfaces, usually temperature and heat flux. Inverse heat source design deals with finding the heat source generation in the medium such that the two specifications imposed on the design surface are satisfied. Inverse geometry design aims at finding some geometric characteristics of the system such that the two conditions prescribed on the design surface are attained. A complete solution for the latter problem has not yet been attempted, although some results are presented in Section IV,A,2. Noting the complex role that geometry plays on radiation exchange, one should expect this to be a very difficult problem.

28

FRANCIS H. R. FRANt~AE T AL.

Although the inverse design and measurement problems are both described by an ill-posed set of equations, which can in principle be treated by similar regularization methods, there are some fundamental differences between them concerning uniqueness and existence of the solution, as discussed earlier. D. STATE OF THE ART IN INVERSE DESIGN

The first work in inverse design considered the problem of finding the conditions on one of the surfaces of a two-dimensional rectangular enclosure so that the two conditions on the design surface were satisfied. Oguma et al. [7] developed a modified inverse Monte Carlo technique to solve an inverse design of an enclosure having black walls and no medium within. The method is different from the forward/inverse and reverse Monte Carlo methods and is based on first evaluating the incident intensity at the design surface and then using it to predict the conditions on the surfaces where no condition is imposed. The accuracy of the solution was found to be sufficient for practical designs. Due to the characteristics of the Monte Carlo method, the problem did not result in a system of linear equations, as happens in the other approaches. However, the solution is based on a time-consuming iterative procedure. Erturk et al. [38] used Monte Carlo to determine geometric exchange factors as a preprocessing step for an inverse design problem, and then solved the resulting equation set by the conjugate gradient regularization approach. Harutunian et al. [39] solved the same problem but the system of integral equations was discretized by the finite difference method, resulting in an ill-condition system of equations. Morales et al. [40] extended the problem to include a participating medium with a given uniform temperature. Both works employed MTSVD to regularize the system of equations. By using MTSVD, they were able to find solutions with physical meaning and adequate level of" smoothness, which still presented satisfactory accuracy. Franqa and Goldstein [41] used the Jacobi and Gauss-Seidel iterative methods to solve the ill-conditioned system of equations, with the former providing the more stable and accurate results. It was possible to obtain accurate and well-behaved solutions over a range of parameters. The limitation of the Jacobi scheme is that it requires the system to have the same number of unknowns and equations, and the regularization role of the number of iterations for the method is not as efficient and flexible as it is for the conjugate gradient scheme. Matsumura [42] and Matsumura et al. [43] used MTSVD along with the READ method [44] for computing exchange areas for the Hottel zone method to solve more complex geometries. One practical problem solved was the temperature distribution needed for heaters to obtain a uniform heat

INVERSE DESIGN OF THERMAL SYSTEMS

29

flux and temperature on the material on the bottom of an industrial furnace. The geometry of the enclosure was changed, and for each geometry the inverse solution was applied until the conditions on the design surface were satisfied. Morales [9] carried out a detailed study of inverse boundary design techniques, concentrating in particular on a comparison of the MTSVD and Tikhonov methods for two- and three-dimensional enclosure problems with an absorbing-emitting medium with a known uniform temperature. The solutions of the regularization methods presented comparative levels of smoothness and accuracy. Kudo et al. [45] used the TSVD scheme to find the heat source distribution in the medium necessary to satisfy the prescribed conditions on all the surfaces of a two-dimensional rectangular enclosure. The first example was constructed so that the numbers of unknowns and equations were the same. Then, uniform heat flux and temperature were imposed on the surfaces, and the entire medium region was left unconstrained. As a result, the number of equations became smaller than the number of unknowns. To obtain uniform temperature and heat flux on the surfaces, it was verified that most of the heat source should be in the corners of the enclosure. The solution presented undesirable oscillations, which were smoothed by truncating some of the smallest singular values of the system of equations. Considering a similar problem, Fran~a et al. [46], analyzed two methods of solution: complete and reduced formulations. In the first one, the system of equations included the unknown heat sources, whereas in the former the heat sources were calculated only after the system of equations is solved for the medium temperature distribution. The reduced formulation was more advantageous in the sense that the ill-conditioned system of equations had a smaller size, making the SVD decomposition less time-consuming. The inverse design was used to find the heat source in the medium necessary to attain uniform temperature and heat flux on the surfaces. Although the trends were similar in both works, the presence of oscillations in the heat source, as found in Kudo et al. [45], was not observed by Fran~a et al. [46], even when all the singular values were kept. Jones [47] considered the design of a two-dimensional rectangular oven to cure the coatings on long metallic strips, located on the side wall. The power input in the system was provided by a cylindrical heater located in the center of the enclosure. The objective of the analysis was to verify what some of the geometric parameters of the enclosure (distance of the metallic strips to each other and to the heater) should be such that the temperatures of the metallic strips were uniform. The problem was first solved by finding the geometric parameters from a forward analysis based on the assumption that the irradiation and radiosity on all the surfaces were uniform. Then inverse

30

FRANCIS H. R. FRAN(~A E T AL.

analysis was employed to test the obtained geometric parameters. It was verified that those geometric parameters provided physically unrealistic results, indicating that some key assumptions of the forward analysis were flawed. In all of these papers, radiation was the sole heat transfer mechanism, and the medium was either participating or gray, so that the resulting system of equations was linear. In most practical situations, however, the resulting system of equations is nonlinear, as happens with combined heat transfer problems and with nongray media and surfaces. In these cases, the system of equations will be both nonlinear and ill-conditioned. Franga et al. [48] considered a participating medium that was nongray, and consequently, the spatially averaged radiative properties of the medium were temperature dependent, making the problem nonlinear. To solve the nonlinear problem, an iterative procedure was adopted so that, at each iterative step, TSVD was applied to the new calculated system of equations. Even though the singular values changed from iteration to iteration (due to the modifications on the matrix of coefficients), the convergence was achieved by keeping the same regularization parameter p in all the iterations. Considering the same geometry as Kudo et al. [45] and Fran~;a et al. [46], Fran~;a et al. [49] presented an inverse heat source determination combining radiation and conduction heat transfer modes. To deal with the extreme nonlinearity of the problem, the energy equation was formulated in terms of the radiation terms (the dominant mode for the cases studied), treating the conduction terms as pseudo-source terms, calculated from the conditions found in the previous iteration. At each iteration, a system of linear equations was solved by TSVD regularization. The work also proposed a way to impose a shape constraint in the unknown heat source distribution to simulate an expected spatial distribution of the heat sources due to a physical mechanism that governs the process, as, for instance, the diffusion and combustion of chemical species. This was achieved by relating the unknown heat sources to each other by means of an imposed shape factor, assumed to be known previously. Lan and Howell [50] examined various approaches to formulating the inverse radiation-conduction problem to avoid convergence difficulties in the nonlinear equation set. Fran~;a et al. [51] considered the inverse boundary design where there was a developed laminar flow of a participating medium between the two parallel plates that form the enclosure. The heat transfer was governed by combined radiation, convection, and conduction. The combined heat transfer problem is described by a system of nonlinear equations, which is expected to be illconditioned due to the inverse analysis. The system of equations was solved by an iterative procedure in which the basic set of equations relates the

INVERSE DESIGN OF THERMAL SYSTEMS

31

design surface directly to the heater, and all the other terms were found from the conditions of the previous iteration. By doing so, the ill-posed part of the problem was isolated for a more effective treatment using TSVD regularization. Reviews of inverse design are presented in Morales et al. [52] and Fran~a et al. [53, 54].

IV. Inverse Design of Linear Systems Dominated by Radiative Transfer In this section, systems in which radiation is the sole heat transfer mode are considered. The equations describing these systems are linear or may have weak nonlinearity due to the effect of temperature-dependent properties. These problems are therefore formulated by an ill-conditioned system of linear equations as indicated by Eq. (6). The discussed regularization methods can then be applied readily. A. SYSTEMS WITH SURFACE RADIATIVE EXCHANGE

First, consider systems with radiative exchange among multiple surfaces where the medium between the surfaces is transparent and does not participate in the radiative exchange process. 1. Two- and Three-Dimensional Results

The three-dimensional results shown in Fig. 10 show a rectangular enclosure with a heater in the upper left section. The imposed conditions are a uniform temperature and prescribed total flux on the bottom of the enclosure, with the sides and top insulated. The required heater temperature distributions are found by inverse analysis. The calculation was performed on an IBM RS6000 computer system and required 2 to 5 min of CPU time using MTSVD. This example shows the power of inverse design in the design process, as using forward solutions to determine the correct temperature distributions for achieving the desired conditions on the bottom surface would be extremely tedious. More discussion of this work is in Section V,D. 2. Determination of Geometry by Inversion

Inverse design should be a useful tool for finding the geometry of the system boundaries necessary to best meet design constraints in thermal systems where radiant heat transfer dominates. Such a problem may require specifying the constraints to be imposed on the design such as the minimum surface

32

FRANCIS H. R. FRAN(~A ET AL.

FIG. 10. Temperature profiles on the interior surfaces of a radiantly heated process furnace by inverse analysis. All surfaces are adiabatic except for the upper half of the left end (the heater) and the bottom surface, which is at uniform temperature and given heat flux distribution. Heater temperature profiles are for cases when the bottom surface emissivity is varied.

area for the enclosure, the minimum surface area for the furnace heaters, and the minimum furnace volume. The inverse geometry problem requires inversion of Eq. (1) for the problem of Fig. 1 modified for gray walls:

ebl(Xl) -- qTI(Xl) -- Ii ~;1

qo2(xz)K(Xl, x2)dx2,

(33)

2=0

where qo2 is the dimensionless radiosity of surface 2. If surface 2 is allowed to slant at an arbitrary angle with respect to surface 1, the configuration factor dFi_j = Ki_j dxj between the design surface 1 and the radiating surface 2 is unknown. The configuration factor depends on the system geometry and is normally a function of at least two variables involving the orientation and distance of surface i with respect to surface j. In the previous inverse problems, the geometry was fixed so that dFi_j is known and the unknown in the problem was the scalar quantity qo(Xj). The additional degree of freedom in unknown geometry problems requires imposition of additional constraints.

INVERSE DESIGN OF T H E R M A L SYSTEMS

33

To solve this problem, a search routine was implemented that optimized the match between the prescribed conditions of net heat flux and temperature on the design surface and that found by inverse solution allowing the geometry to vary. This approach successfully solved some test problems in two-dimensional geometries. The algorithm is based on the simulated annealing algorithm outlined in Goffe et al. [10] and Corana et al. [11]. Consider a square enclosure (Fig. 11, solid line) with the following conditions: 2, ~;top = 0 . 5 9 tl~ft = 1, Cleft = 1.0 9 tright = 1, 8right ~- 1.0

9 ttop =

9 tbottom ----- 1, ~;bottom - - 1 . 0

For the conditions shown, the forward solution predicts the heat flux on the bottom surface shown by the solid line in Fig. 12. This heat flux is then prescribed along with the other prescribed temperatures and emissivities of the enclosure, and the geometry of the enclosure is allowed to vary by allowing the position of the ends of the top surface to float along the y direction. The inverse search program was then imposed to determine the orientation of the top surface that minimizes the difference between imposed and calculated heat flux on the bottom surface. As would be expected, the best predicted match occurs when the top surface returns to its original position, forming the square enclosure. A more difficult inverse variable geometry problem is now posed. The emissivity of the right-hand surface in Fig. 11 is changed to 0.5. With the original square geometry, an asymmetric heat flux must then result.

t2 ........................... ..............

Base case ;.;:"-"~ ........... j i. i.

............................C a s e 1 ............. Case 2

i i. i. i. i

tl

i.

tl

tl FIG. 11. Possible geometries t h a t satisfy the prescribed radiative flux on the b o t t o m surface. Solid line for enght = 1.0; d o t t e d a n d d a s h e d lines for gright "- 0.5.

34

F R A N C I S H. R. F R A N ( ~ A

ET AL.

The inverse search algorithm was used to find the position and orientation of the top surface that provides the closest match to the same imposed initial conditions on the b o t t o m surface [i.e., t b o t t o m ---- 1, 8 b o t t o m = 1, qT, bottom (X1)] from the base case of Fig. 12. The right-hand and left-hand ends of the top surface were allowed to move only along the y axis. The optimum orientation is shown by the dotted line in Fig. 11, and the match between imposed and computed radiative flux on the bottom surface is shown by the dashed line (case 1) in Fig. 12. The top tilts so that the length of the left-hand black surface increases, providing more radiation to the right-hand end of the b o t t o m surface and making the heat flux on that surface comparable with the prescribed value. The match shown provides the m i n i m u m total residual error integrated over the bottom surface. Next, both the x and y coordinates of the connection between the righthand wall and the top surface were allowed to float while the left-hand corner of the top surface was fixed. The optimum enclosure orientation was found to be as shown by the dashed line in Figure 11, and the comparison between imposed and predicted radiative flux on the bottom surface is shown by the dashed line in Fig. 12 (case 2). The right-hand surface for these

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flux

35

INVERSE DESIGN OF THERMAL SYSTEMS

conditions is found to tilt into the enclosure while the top now tilts up to the right. This orientation gives a much better match to the prescribed bottom surface heat flux. The percentage error between prescribed and predicted heat flux is shown in Fig. 13 for the two cases. The case where the upper-right corner location is allowed to float in two dimensions is found to provide a solution that is accurate within 1% at all locations on the bottom surface. The simulated annealing algorithm used to generate Figs. 12 and 13 is an optimization technique that requires iterative solution of forward radiative transfer algorithms rather than the inversion techniques discussed previously (TSVD, MTSVD, Tikhonov, etc.)

3. Revisit of Existence and Uniqueness The behavior of the variable geometry problem solved earlier provides another viewpoint to the question of the uniqueness and existence of solutions to inverse problems. Figure 11 shows two quite different geometries that provide heat flux distributions on the design surface that are quite close to the prescribed distribution. The two geometries were obtained by imposing

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.

. . 0.8

13. P e r c e n t a g e e r r o r in l o c a l r a d i a t i v e flux f o r e x a m p l e p r o b l e m s .

i

36

FRANCIS H. R. FRAN•A E T AL.

different constraints on the allowable geometries. Thus, the designer can control the form of the desired solution by imposing particular constraints. Such constraints may be practical limitations imposed by available headroom or volume, available low-cost structural components, etc. However, the presence of multiple acceptable solutions in the sense of meeting the imposed thermal conditions implies that the designer should be quite careful in applying constraints on the design so that the widest range of acceptable final designs can be investigated.

B. RADIATIVE SYSTEMS WITH PARTICIPATING MEDIA

1. Design of Heat Source Placement The designer of a thermal system commonly faces the problem of finding the conditions within a participating medium such that two specifications on the design surface are attained. Figure 14 illustrates this problem for a twodimensional rectangular enclosure of length L and height H. The wall surfaces are gray emitters and absorbers having emissivity e. The enclosure is filled with a gray homogeneous participating medium. The heater source (HS) region is the portion of the medium where there is heat generation. Outside this region, there is no heat generation; the medium is in radiative equilibrium. The objective is to find a heat source distribution in the HS region that satisfies the prescribed temperature and heat flux on the design surfaces.

FIG. 14. Two-dimensional radiative enclosure. The HS region corresponds to the region in the medium where the heat source distribution is to be found from the inverse solution.

37

INVERSE DESIGN OF T H E R M A L SYSTEMS

To find the heat source in the HS region from the two conditions imposed on the surfaces, it is necessary to set and solve the energy balance in the medium and on the surfaces. Considering that heat transfer is assumed to be governed solely by radiation, the energy conservation in the medium and on the surfaces reflects the balance between thermal radiation and heat generation. The zonal method is applied in the discretization of the continuous form of the energy equations. Figure 15 presents the division of the medium and surfaces into F volume and N surface zones. The darker region corresponds to the HS region, composed of a total of FHS zones. All the zones have uniform size to make the calculation of the direct exchange less time-consuming by taking advantage of geometric symmetry of the elements. The discrete form of the energy equation in the medium is given by

k

SkGi,jqo, k d- ~ Gk, lGi,jt4,k,l - 4ATt4,i,j + ATSG, i,j -- O. k,l

(34)

In Eq. (34), the dimensionless size of each zone is Ax = aAX, whereas the dimensionless heat source generation is given by sa - Q6/a. Eb.ref. SG and GG are surface-to-volume and volume-to-volume direct-exchange areas per unit of depth normalized by the size of each zone element, AX. For an element i on the surface, the total heat flux corresponds to the radiative heat flux, so _p/t

--

FIG. 15. Division of the two-dimensional enclosure into medium and surface zones of uniform size.

38

FRANCIS H. R. FRAN(TA

qT, i - qo, i - Z

ET AL.

SkSiqo, k -- E Gk'lSitg,4 k, l' k k,l

(35)

where SS and GS are the surface-to-surface and volume-to-surface directexchange areas per unit of depth normalized by the size of each zone element, AX. The radiosity of the surface element relates to the imposed heat flux and temperature by t4 qo, i -- w,i

(1 - - q--T ,~;)

i.

(36)

In conventional design, the temperature on the surfaces and the heat source distribution in the medium are usually specified. The problem consists of finding the heat flux on the surfaces. To define the system of equations, the radiosity of each surface zone is first expressed in terms of the emission and reflection of radiation:

q~ -- e't4'i + (1-- e') ( ~

SkSiq~

+ Z

Gk'lSit4'k'l)

(37)

The system of equations is formed by writing Eqs. (34) and (37) for each of the F medium zones and each of the N surface zones, respectively, making a total of N + F equations. There are N unknown radiosities on the surface zones, and F unknown emissive powers (the fourth power of the temperature) in the medium zones, which gives a total of N + F unknowns. In forward problems, not only are the number of unknowns and the number of equations always the same (N + F), but also the system of equations is well conditioned in the sense that the singular values of matrix A do not decrease to very small values as happens for ill-conditioned systems. In inverse design, both the temperature and the heat flux distributions are imposed on the surfaces, whereas the medium zones in the HS region are not constrained by any thermal condition. In fact, inverse design aims at finding the heat source distribution in the HS medium zones from the two specifications on the surfaces. The medium zones located outside of the HS region automatically have one condition imposed, which is no internal energy generation (sG = 0). Another aspect to be considered in the inverse problem is that the radiosity of the surfaces can be calculated directly from Eq. (36), as both the heat flux and the emissive power are known. The inverse problem can be described by either the reduced or the complete systems of equations, depending on whether the system of equations includes or excludes the equations for the elements in the medium where the heat source is unknown. The complete system of equations includes the energy balance for all the elements of the system. Equation (35) is written for all the N surface

INVERSE DESIGN OF THERMAL SYSTEMS

39

zones, and Eq. (34) is set for all the F medium zones, making a total of N + F equations. All the conditions (emissive power, heat flux, and radiosity) are known on the surface zones. There are F unknown emissive powers in the medium, plus FHS unknown heat sources in the medium zones located in the HS region, making a total of F + FHS unknowns. Therefore, the number of unknowns and the number of equations are the same only when N = FHS. When an equation is written for the medium zones located in the HS region, one unknown is added into the system, the heat source generation in those zones. If this equation is eliminated from the system, the unknown heat source is also taken out, as it appears only in the eliminated equation. In the reduced formulation, only the equations for the medium zones that are not in the HS region are included in the system, in addition to the equations for the surface zones. The unknowns correspond only to the F medium emissive powers. Once the system is solved for the emissive power in the medium, the heat source can be found from Eq. (34). The system contains N equations for the surface elements, plus F - FHS equations for the medium zones that are not in the HS region, making a total of N + F - FHS equations. Again, the number of equations and the number of unknowns are the same only when N = FHS. To illustrate the application of the aforementioned procedure, the following example is considered. The system is a two-dimensional square enclosure ( H / L = 1.0) having optical thickness equal to Zn = a H = 1.0. The emissivity of the surfaces is e = 0.9. As shown in Fig. 15, the HS region is located in the center of the enclosure and corresponds to a square region having half the lateral size of the enclosure. The enclosure is divided into a uniform grid mesh, containing a total of F = 16 x 16 = 256 medium zones, and N = 4 x 16 = 64 surface zones on the walls. The total number of HS medium zones is FHS = 64. First, a forward problem is solved to provide a benchmark to compare with the inverse solution. The temperature of the walls is uniform and equal to tw = 1.0 (i.e., Tref = Tw). The HS region has a uniform heat source generation equal to sa = 48.0; outside this region, the medium is in radiative equilibrium, which means that sa = 0. This completes all the information necessary for the solution of the forward problem. Figure 16 presents the heat flux on the walls for half of the bottom surface, which is the same for all the other surfaces because of the symmetry in the problem. The direction of the energy is from the medium to the surfaces, and so the net heat flux on the walls has a negative sign. The maximum heat flux occurs at the center of the surfaces, where the effect of the medium is the greatest, and then it decreases towards the corners, where the effect of the cold walls become more relevant. Figure 16 also compares solutions obtained with a more refined grid,

40

FRANCIS

4.5

4.0

. . . .

I

.

.

H. R. F R A N ( ~ A

.

.

.

.

.

.

ET AL.

I

'

i

'

i

.

.

.

.

.

.

.

.

................. ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilliiiiiiii!iI iiiiiiiiiiiiiiii

-qT

2.0

........

..........i................. i........

....

. . . . . . . . . . . . . . . ~................. i ................. i ................. ; ................ :

1.5

FIG.

16. F o r w a r d

. . . . . . . . . . . . 0.0 0.1 0.2

solution:

i , , , , 0.3

heat flux on the bottom

16 • 1 6 ( N = 6 4 ) a n d 2 4 x 2 4 ( N = 9 6 ) . P r o b l e m

surface

conditions:

.... 0.4

0.5

for the two grid resolutions:

tw = 1.0,

SG = 4 8 . 0 , H / L = 1.0,

~H --- 1.0, ~ = 0.9.

resulting in a division of the enclosure into 24 x 24 elements (which gives N = 96 elements on the surfaces). The agreement between the two solutions indicates that the 16 x 16 grid is adequate and is retained for the next calculations. An inverse problem can be proposed that considers the same physical conditions in the enclosure (H/L = 1.0, ~/~ = 1.0, ~ = 0.9). The HS region has the same location as shown in Fig. 15. A uniform temperature, tw - 1.0, and the dimensionless heat flux qT of Fig. 16 are imposed on the surfaces. The medium elements located outside the HS region are in radiative equilibrium (SG = 0). The inverse problem consists of finding the volumetric energy generation in the HS medium elements that satisfies the two conditions on the design surfaces. Because all the conditions are kept the same, one solution for the problem is a uniform heat source, sa = 48.0, in the HS region. According to the previous discussion, the inverse problem can be formulated by (1) the complete formulation, where the system of equations contains the energy balance for all the elements of the enclosure, and (2) the reduced formulation, which eliminates equations related to the HS medium elements. For the present grid division (F = 256, FHs = 64, N = 64), the coefficient matrices of both the complete and the reduced systems of equa-

41

INVERSE D E S I G N OF T H E R M A L SYSTEMS

tions are square, the number of equations and unknowns is the same, because FHS - - N . For complete formulation, the dimensions of matrix A are m = n - 320. For the reduced formulation, m = n = 256. Although the proposed inverse problem is formulated by a square system of linear equations, any attempt to solve the problem using conventional matrix solvers will not succeed to provide the expected solution, sG = 48.0, in the HS medium zones. The explanation for this is found in Fig. 17, which presents the singular values for the complete and reduced formulations. In both cases, the singular values decay to very small numbers, of the order of 10 -18. Two forms of decay are seen: a continuous one, typical of the discretization of ill-posed problems (as for a Fredholm integral equation of the first kind), and a distinct gap between the singular values, between 10 -1~ and 10 -18, indicating that there are a number of rows or columns that are nearly a linear combination of the others. After the SVD of matrix A is performed, a solution can be computed readily by just using Eq. (10) and keeping all the singular values of the linear combination. Figure 18 shows the components xi of vector x for the reduced formulation when no regularization is applied. As seen, the components present steep oscillation between large positive and negative absolute

lO~

~r-m"'

. . . . . .

n ....

, ....

, ....

n ....

u ....

...................... +.........-~.~..--.-.....i............ i............ i...........

..............................................

J............ [ ............ i ...........

10-3 ..........

(~i

i ...........

iiilil Iiiiiiii)i!iii!i.,

10 "9

............ i..~ ....... t

:

,ol, 9

lO-lS

.... 0

i .... 50

.

i .... 100

t

.o

.

i .... 150

i,,,, 200

,,,I 250

.... 300

350

i

F I G . 17. Inverse s o l u t i o n : s i n g u l a r v a l u e s o f m a t r i x A for c o m p l e t e (m = n = 320) a n d r e d u c e d (m = n - 256) f o r m u l a t i o n s . P r o b l e m c o n d i t i o n s : t w - 1.0, H / L 1.0, x n - 1.0, = 0.9.

42

F R A N C I S H. R. FRAN(~A E T AL.

2000

"

"

9. . . . . . . . . . . . . . .

0 Xi -1000

...

-2000

....

_

-3000

: I

0

FIG.

18. C o m p o n e n t s

50

I

I

I

I

100

: I

I

i

I

9 I

150

i

i

,

i

i i

200

i

t

i

I

250

o f v e c t o r s o l u t i o n x for s o l u t i o n w i t h n o r e g u l a r i z a t i o n , p - - n .

R e d u c e d s y s t e m o f e q u a t i o n s : m = n = 256. P r o b l e m c o n d i t i o n s : tw -

1.0, H / L

= 1.0, z a =

1.0, ~ = 0.9.

numbers with alternating signals. In the reduced formulation, the components ofx correspond to the unknown emissive power in the medium, t4, which must be a positive number. Therefore, the solution with no regularization cannot be accepted, even though the vector b is in the range of A, in the sense that b was obtained first for the forward problem from an imposed x. The reason for this relates to some singular values of the present case being smaller than the machine precision (10 -14 to 10-16), and so the round-off error becomes critical. If there were no round-off error, the result for this inverse problem would be exactly s6 = 48.0 for the medium elements in the HS region. TSVD regularization is applied to eliminate all the singular values that are smaller than 10 -1~, just before the gap. For the reduced and complete formulations, respectively, this corresponds to p = 252 and p -- 316. Then, the vector solution can be computed by Eq. (12) for both formulations. The resulting heat source generation distribution in the HS region is presented in Tables I and II for the reduced and complete formulations, respectively. Because of the symmetry in the problem, the results are presented only for the elements in the darkest part of the HS region, as seen in Fig. 15. In both cases, because the heat source s6 is close to the original value of 48.0, the truncation of the singular values smaller than the computer precision does not prevent recovering a solution close to that originally imposed.

43

INVERSE DESIGN OF T H E R M A L SYSTEMS

TABLE I

HEAT SOURCE SG IN MEDIUM FROM THE REDUCED SYSTEM OF EQUATIONS, m -- n -- 256 a i=5

6

7

8

j = 5

48.001

48.006

48.010

48.005

6

48.006

47.997

47.963

47.993

7

48.010

47.963

48.002

48.004

8

48.005

47.993

48.004

48.037

a P r o b l e m w i t h no p e r t u r b a t i o n : ~ = 0.9, z H = 1.0,

H/L =

1.0.

T S V D solution: p = 252.

Tables I and II show that the solutions for the reduced and complete system of equations are not exactly the same, even though they involve the same set of energy balance equations, albeit not solved in the same way. The explanation for this is that TSVD regularization seeks an approximate solution that minimizes a norm of the vector solution x, which is not the same for the two formulations. In the reduced system, the vector of unknowns is formed by only the emissive power of the medium, whereas in the complete system, the vector also contains the heat sources in the HS medium elements. Minimizing the two vectors does not lead necessarily to the same solution. So far, only the case with no perturbation has been considered; that is, the physical conditions of the forward and inverse problems were kept the same so an exact solution (the input of the forward example) was known to exist. In a real design problem, however, a solution is not known a priori. It would be very fortuitous for the vector b to be numerically in the range of matrix A in the sense that only the singular values below computation precision need

T A B L E II

HEAT SOURCE SG IN MEDIUM FROM THE COMPLETE SYSTEM OF EQUATIONS, m -- n = 320 a i-5

6

7

8

j = 5

48.000

48.000

48.001

48.000

6

48.000

48.000

47.998

47.999

7

48.001

47.998

48.000

48.000

8

48.000

48.000

48.000

48.002

a p r o b l e m w i t h no p e r t u r b a t i o n : e = 0.9, z u = 1.0, T S V D solution: p = 316.

H/L =

1.0.

FRANCIS H. R. F R A N C A ET AL.

44

to be truncated. In general, it is necessary to eliminate more singular values such that unrealistic, impractical solutions are avoided. A way to study such problems is by perturbing the problem. For instance, the emissivity of the surfaces is changed from 0.9 to 1.0; all the conditions are kept the same as before, including the temperature and heat flux specifications on the surfaces. In this case, the uniform heat source generation in the HS region is not expected to be a solution. Changing the emissivity of the surfaces only causes a perturbation on vector b. Matrix A is the same as for the unperturbed problem, including the same singular value spectrum of Fig. 17. Then TSVD regularization is applied to obtain the solution vector x. For the reduced formulation, solutions with a physical meaning (i.e., no negative emissive power in the medium) occur only when p is set equal to or smaller than 207, in a total of 256 singular values. This corresponds to a minimum singular value, ( Y m i n , of about 4.0 x 10 -4. Under this regularization, the obtained heat source is no longer uniform, but presents a peak in the corner of the HS region and decreases toward the center, as depicted in Table III. The accuracy of the inverse solution is verified by means of the evaluation of the relative error ~, Eq. (13), between the imposed heat flux and the one obtained from the heat source distribution of Table III. Figure 19 presents the relative error (not the absolute value; the sign is retained) along the surface wall, as well as the imposed and the inverse solution heat fluxes. As seen, the relative error is kept below 1.0% for most of the length of the surface, with the exception of the region close to the corner of the enclosure, where the wall-to-wall effect becomes important. In this example, the average and the maximum relative errors are 0.586 and 1.42%, respectively. In the complete formulation, it is possible to find a solution that is physically acceptable only when p is maintained equal to or below 271 (also corresponding to l Y m i n - - 4.0 x 10-4). The solution for this regularization is presented in Table IV. Comparing Tables III and IV, the difference

T A B L E III HEAT SOURCE SG IN MEDIUM FROM THE REDUCED SYSTEM OF EQUATIONS, m = n = 256 a

j = 5 6 7 8

i=5

6

7

8

120.530 77.530 44.155 25.134

77.530 58.114 42.631 33.650

44.155 42.631 37.778 34.406

25.134 33.650 34.406 33.677

a p r o b l e m with p e r t u r b a t i o n : T S V D solution: p = 207.

e = 1.0, x/4 = 1.0, H / L = 1.0.

45

INVERSE DESIGN OF T H E R M A L SYSTEMS

4.0 !

9

................ i ................ i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....

3.5

,~

: .......

2.40

relative e

i 1.20

3.0

o.oo

-qr

................ i ............

!................ i ................. ~.............

: .............................

!................. ; ................ i . . . . . . . . . . . . . . .

~ (%)

2.5

2.0

........

~

.....

-

imposed inverse solution

-1.20

!i ................ ! ................ i:

"'! . . . . . . . . . . . .

-2.40

1.5 0

0.1

0.2

0.3

0.4

0.5

X

F I C . 19. Comparison between the imposed and the inverse solution heat flux. Reduced system of equations. TSVD solution: p = 207. Physical conditions: H / L = 1.O, x n = 1.0, = 1.0. Problem with perturbation on e.

between the solutions for the complete and reduced formulations becomes more apparent, although the trend presented for both formulations is the same. The accuracy of the complete formulation solution, verified by using Eq. (13), was found to be nearly the same as for the reduced formulation, shown in Fig. 19. This gives an advantage to the reduced formulation because considerable effort can be saved for the SVD decomposition of a smaller matrix. For the present calculation, the SVD decompositions of

T A B L E IV

HEAT SOURCE

IN MEDIUM FROM THE COMPLETE SYSTEM OF EQUATIONS, m = n = 320 a

SG

i--5 j = 5

6

7

8

118.250

76.290

40.613

19.698

6

76.290

60.160

44.206

34.326

7

40.613

44.206

41.346

38.268

8

19.698

34.326

38.268

38.554

aproblem with perturbation: e = 1.0, XH = 1.0, solution: p = 271.

TSVD

H/L=

1.0.

46

FRANCIS H. R. FRAN(~A E T AL.

matrix A for the reduced formulation (m = n = 256) takes less than half the computational time of the complete formulation (m = n = 320). The difference is expected to be even greater for grids with a larger number of elements. Many processes require that both the temperature and the heat flux be uniform on the surfaces, as for the furnace design described in the beginning of this chapter. Inverse analysis can be used to determine the distribution and rate of firing in the boiler that is able to satisfy these conditions on the surfaces. To study this problem, the geometry and thermal conditions of Fig. 14 are considered. The aspect ratio is L / H = 1.0, the optical thickness is ~/-/= 1.0, and the surface emissivities are e = 0.9. On the surfaces, the dimensionless temperature and heat flux are specified to be tw = 1.0 and qT = -3.4, respectively. If the HS region occupies the same position indicated in Fig. 15, then the only change to the system of equations occurs in vector b. Using the reduced formulation (m = n = 256), the singular values of matrix A are the same as those of Fig. 17. The solution can be computed by Eq. (12) by means of TSVD regularization by eliminating the linear combinations related to the smaller singular values, as performed for the previous examples. By doing so, it was found that even reducing p to 199 (as employed in the previous cases) did not prevent the solution vector from presenting some negative values for medium emissive power, making the result physically invalid. A further decrease in p introduces such a modification of the original system as to make the solution invalid because of the large resulting error. Therefore, no useful solution can be found. As observed before, there are sets of specification on the design surfaces that, under the geometric and physical constraints of the enclosure, allow no solution within some acceptable degree of error with respect to the specifications. The inverse solution is a safe way to inform the designer that some of the enclosure geometric or thermal specifications must be changed to achieve a desirable design solution. Using conventional trial-and-error techniques would require a number of unsuccessful guesses before realizing that there is probably no solution for that particular design set. For the problem considered here, the difficulty in meeting both uniform temperature and heat flux on the surfaces relates mostly to the corners, where the wall-to-wall effect is dominant. One alternative modification to obtain a solution is increasing the size of the HS region to include the entire enclosure, not only the square shown in Fig. 15. In this case, all the medium elements are HS elements, so that Fns = F -- 256. Using the reduced formulation, the system of equations is formed only by the energy balance on the surface elements; no equation is written for the medium elements, as none of

47

INVERSE DESIGN OF T H E R M A L SYSTEMS

them are in radiative equilibrium. It follows that the number of unknowns of the system (the emissive power in the medium) is n - F = 256, whereas the number of equations is rn = N + F - Fus = 64. The problem is singular and allows an infinite number of solutions. Following the discussion on SVD, the solution that has the smallest norm can be selected by just computing the SVD of A and then using Eq. (12). For this, decomposition can be applied directly to A resulting in the singular values presented in Fig. 20. Of the total 256 singular values, only 64 are not zero, which are the ones shown in Fig. 20. The SVD solution can be computed by Eq. (12) eliminating all the terms related to the null singular values. The emissive power in the medium is calculated, and then Eq. (34) is employed to find the heat source in the medium. Figures 21 and 22 show the resulting temperature and heat source distribution in the medium for e - 0.9. As seen, to achieve uniformity on both temperature and heat flux on the surfaces, it is necessary to have the largest heat source close to the corners to balance the strong interaction between the cold surfaces in this region. As a result, the medium temperature is the largest close to the corners. A similar result was obtained

100

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-1

....

Gi 10 -2

10-3

i

i 10-4

. . . .

~ . . . .

10

l l J , i l l ,

20

30

. . . . . .

t

40

50

. . . .

l

60

. . . .

t

70

i FIG. 20. Singular values o f m a t r i x A for u n i f o r m t e m p e r a t u r e and heat flux problem. R e d u c e d f o r m u l a t i o n , rn = 64, n = 256. P r o b l e m conditions: tw = 1.0, q r = - 3 . 4 0 , H / L = 1.0, T ~ / = 1.0, ~ = 0.9.

48

F R A N C I S H. R. F R A N t ~ A ET AL.

F I G . 21. D i m e n s i o n l e s s m e d i u m t e m p e r a t u r e f o r u n i f o r m h e a t flux o n t h e s u r f a c e s . R e d u c e d f o r m u l a t i o n , m = 64, n = 256. T S V D : p -- m = 64. P r o b l e m c o n d i t i o n s : tw = 1.0, qT" = - 3 . 4 0 ,

H / L = 1.0, x n = 1.0, e = 0.9.

in Kudo et al. [45], but the present solution does not exhibit the oscillations found in that work. Because only the null singular values were truncated, the solution of Figs. 21 and 22 can be taken as exact, except for round-off errors. The TSVD solution selects the smallest norm solution vector x, which is formed by the emissive power distribution in the medium. If high temperatures are to be avoided in the enclosure, then the TSVD seems to be a good choice for the solution. A point to be recognized is that the heat source distribution in the medium was determined from the radiative exchange relations in the enclosure, with no other restriction being imposed on the heat source. In fact, in combustion processes, the heat source distribution results from chemical species diffusion, which obeys independent laws. Section VI discusses imposing a shape constraint on the heat source.

49

INVERSE DESIGN OF THERMAL SYSTEMS

F I G . 22. D i m e n s i o n l e s s h e a t s o u r c e f o r u n i f o r m h e a t flux o n t h e s u r f a c e s . R e d u c e d f o r m u l a tion,

H/L

m = 64, n = 256.

TSVD:

p = m = 64.

Problem

conditions:

tw =

1.0, qT -

-3.40,

= 1.0, ZH = 1.0, ~ = 0.9.

V. Design of Thermal Systems with Highly Nonlinear Characteristics In the cases considered so far, the heat transfer process was governed solely by thermal radiation. All the system radiative properties (emissivity and absorption coefficient) were assumed to be uniform and independent of the thermal conditions. In this case, the numerical discretization of the energy conservation leads to a system of linear equations. In most situations, however, the heat transfer involves temperaturedependent properties and other mechanisms than thermal radiation, such as conduction and convection. Both the first and fourth powers of the unknown temperatures arise in the numerical discretization of the system, resulting in a highly nonlinear system of equations. The same is valid for the case where the thermal properties are dependent on the unknown temperatures. Modeling

50

FRANCIS H. R. FRANt~A E T

AL.

a nongray medium where the absorption coefficient is dependent on the wavelength of radiation is a special case because the local mean radiative properties then depend on the temperature.

A. TECHNIQUES FOR TREATING INVERSE NONLINEAR PROBLEMS The inverse solution methods described previously are all valid for systems of linear discrete equations, and the mathematical basis and proofs are based on the assumption oflinearity. As noted earlier, interesting engineering design problems, particularly those involving mixed mode heat transfer, are highly nonlinear because of the fourth-power dependence of radiation coupled with first-power temperatures and their derivatives through the advection and conduction terms. There is little guidance from the mathematical literature on inverse solution of these nonlinear problems. Here, the authors' experience is presented in formulating and solving nonlinear inverse problems. As with the forward solution of nonlinear problems, inverse solutions are also almost always iterative. Linearization of equations describing a thermal system provides an obvious approach to solving inverse problems, as the linear solvers outlined earlier can be applied iteratively. However, care must be used in this approach. Some linearization procedures may cause the form of the resulting linearized equations to introduce solution-dependent terms into the coefficient matrix A; i.e., A = A(x). This means, for example, that expensive operations such as SVD would have to be performed at each iteration step in a TSVD, MTSVD, or Tikhonov solution or a new A(x) and AT(x) at each iteration in the CGR method, making the inverse solution quite costly. Thus, wherever possible, it is desirable to arrange the linearized equation into the form A. x = b(x)

(38)

by placing the x-dependent terms on the right-hand side of the equation and leaving the coefficients in A as constants. The highly nonlinear nature of mixed mode heat transfer relations causes well-known convergence difficulties even for forward solutions. Relaxation factors must often be introduced to prevent divergence of the solutions during iterative solution, and the same is true for inverse solutions. Because of the dependence of b(x) terms on the solution vector x, nonlinear problems present some further convergence problems over linear problems. Examination of the TSVD decomposition shows the reason. Equation (15) now becomes

P u~" b(x~-') x~= Z i=1

vi, ~

(39)

INVERSE DESIGN OF THERMAL SYSTEMS

51

where the solution vector xp at iteration n depends on the b(xp -1) that is computed at the previous iteration. For small singular values cri, fluctuations in b(xp -1) from iteration to iteration will cause convergence difficulties, as the fluctuations are magnified greatly when divided by these small singular values. Even for conditions on the design surface that are generated from forward solutions, it is often not possible to recover the conditions used in the forward solution, as was the case for linear problems. Within these constraints, there is still wide latitude in solution approaches, as well as a number of potential pitfalls in arranging the solution algorithm, as will now be shown. For either forward or inverse solutions, the system of equations must be arranged in such a way that the guessed terms are not the dominant information of the system. For instance, if radiation is the dominating heat transfer mechanism, then the conduction and convection terms should form the guessed part of the system, not the contrary. In matrix representation, the system of equations becomes A ( x n - 1 ) . Xn --

b(xn-1),

(40)

where matrix A and vector b are dependent on the unknown vector x. The convergence of the iterative scheme is viable only if the components of the linear combination of Eq. (39) corresponding to the smaller singular values are truncated. As seen, in previous solutions, the truncation of these terms was also necessary to guarantee inverse solutions with physical meaning and acceptable smoothness. The next sections present the application of this procedure for the solution of some nonlinear inverse problems. B. INVERSE DESIGN OF SYSTEMS WITH NONGRAY MEDIUM

The gray medium assumption is not a realistic approximation for real gases in engineering applications. Incorporating the dependence of the radiative properties of the medium with respect to the wavelength is usually a difficult task, considering that the complex contribution of each wavelength band must be integrated across the total spectrum. When the medium temperature is unknown, the problem becomes nonlinear, as the amount of radiant energy within each wavelength band that is being integrated depends on the unknown temperature. An example of inverse design of a system containing a nongray medium is presented here. Figure 23 presents a two-dimensional rectangular enclosure, where the interior is filled with a participating medium, whose absorption coefficient is dependent on the wavelength. The walls are gray emitters and absorbers, and the total emissivity e is uniform on the walls. On the bottom wall, both temperature and heat flux distribution are imposed; the net heat

52

FRANCIS H. R. FRAN(~.A ET AL.

FIG. 23. Geometry of inverse design for nongray participating medium case.

flux is zero on the side and upper walls. The problem consists of determining the temperature in the medium such that the two constraints on the bottom surface are both attained. This is a simple model of a gas-fired furnace: on the bottom surface, the desired temperature distribution is specified, and an independent energy balance on this surface sets the necessary radiative heat flux; the remaining surfaces are insulated. The rate of firing in the gas and the resulting gas temperature profile that provide the two conditions on the design surface are the parameters to be computed. For simplicity, the temperature of the medium is assumed to be one dimensional, tg-- tg(y). The solution of the medium temperature distribution relies on the solution of the energy balance applied on the surfaces and in the medium. Employing zonal discretization of the continuous domain, the most suitable method to incorporate the wavelength dependence of the medium radiative properties is the weighted sum of gray gases. Figure 24 presents the division of the enclosure into N surface zones and F medium zones. As the medium temperature is only y direction dependent, the medium zones are formed by slabs. All the surface zones have the same dimension AX, which is also the height of each medium slab. The number of design surfaces, located on the bottom, is ND. The energy balance for a surface zone element (where q r = qR) is given by qo, k = qT, k +

SjSkqo,j + j=l

Gv, Skt 4,~.

.

(41)

y*=l

In Eq. (41), SS and G;~ are, respectively, the dimensionless surface-tosurface and gas-to-surface directed-flux areas, given by

INVERSE DESIGN OF THERMAL SYSTEMS

53

FIG. 24. Division of the enclosure into volume and surface zones.

SJ Sk --

Gy.Sk-

(42)

Ce, i, l w,j i=0

l=1

~

/

Ce, i, lTlg,-~1 (G~/.Sk)i,

i=0

l=1

(43)

where the coefficients C e account for the spectral properties of the medium in the sum of gray gases model [55] and (SS)i and (GS)i a r e determined for each gray gas window, having absorption coefficient ai, that forms the real gas. The final set of equations can be developed from an energy balance on each medium zone element:

F N 4 v __ ~ay, aTl4,7, 4"CLtg, ' + ~ SjGvqo,j + T.LSG ~ )

T*=l

(44)

j=l

4 in which sc, r - q,~/(aoTre f) is the dimensionless volumetric heat generation in the medium element T, zL is the optical thickness based on the length of the medium slabs, aL, and a is the absorption coefficient for the medium element T, based on the weighted contribution of each gray gas window: Ill

9

a-

~ i=0

Ce,i, lT~,-~1 ai.

(45)

l=1 )

The dimensionless gas-to-gas and surface-to-gas directed-flux areas, GG ) and SG, are given by

aT. aT -- ~ i=0

)

Ce, i, lZlg,---T1 (Gv. Gv) i l=1

(46)

54

F R A N C I S H. R. F R A N ( ~ A

SjGy- ~

ET AL.

Ce,i, l w,j

i=o

(SJGy)i,

(47)

1=1

where the dimensionless gas-to-gas and surface-to-gas direct-exchange areas, (GG)i and (SG)i, are calculated from the absorption coefficients ai corresponding to each gray gas of the sum of gray gases model. For the inverse problem considered here, two conditions are imposed on the ND design surface zones; tw, k and qT, k. The radiosity qo, k in the design surface zones can be computed directly from the knowledge of the heat flux and the temperature, making the number of unknown radiosities equal to N - ND 9However, two unknowns are present in each medium zone, t 4G, ~, and sa,~, adding 2F more unknowns to the problem. The total number of unknowns is 2F + N - ND. The number of equations is N + F, corresponding to the energy balance applied to each zone of the system. Thus, the unknowns and equations will be the same only when ND = F. In this case, because the enclosure is square and the dimension of each zone is the same, the aforementioned relation is verified, making the number of unknowns and equations the same. This condition is not necessary though because the regularization methods are able to deal with systems having a different number of unknowns and equations. The resulting system of equations can be represented by A(x). x = b(x),

(48)

where the vector x contains the unknowns of the inverse problem: the radiosity of the adiabatic surface zones, the gas emissive power, and the volumetric heat source in the medium zones. The matrix of coefficients, A, and the independent vector, x, are dependent on the unknown temperatures because of the temperature dependence of the directed flux areas relations, so the system of equations is nonlinear. Moreover, the system is expected to be ill-conditioned because of the inverse analysis. Thus, this problem appears to present serious difficulties for inversion because the matrix A depends on the solution for temperature through the exchange areas. This is the type of problem formulation that should be avoided if possible, but must be faced here. Following the discussion on nonlinear problems, an iterative procedure is employed for the solution of this problem. First, the medium temperature and the adiabatic surface radiosity distributions are guessed so that matrix A and vector b can be calculated. Then, matrix A is regularized using the TSVD scheme, and a new solution vector x is guessed. The procedure is repeated until convergence is achieved; i.e., when the change on x between the two last iterations falls below a prescribed value.

INVERSE DESIGN OF THERMAL SYSTEMS

55

First, a forward problem is solved to provide a known benchmark solution that can be used to evaluate the inverse technique. A square enclosure is considered, L = H. As shown in Fig. 23, the side and top surfaces are adiabatic so that qr = 0. The bottom surface is at an uniform temperature of tw = 1.0, having an emissivity of 0.5. (The emissivity of the adiabatic surfaces does not appear in the governing equations when radiation is the only mode of heat transfer.) In the proposed forward problem, the medium within the enclosure has a known temperature distribution that varies only with y = Y / H (0 < y < 1), according to:

to(y) = 4 - (2y - 1)2.

(49)

As an example of nongray medium, the gas is a mixture of carbon dioxide (0.1 atm), water vapor (0.2 atm), and nitrogen (0.7 atm), which is a common ratio for the products of stoichiometric combustion of methane at a total pressure of 1.0 atm. Smith et al. [55] have evaluated the polynomial coefficients Ce, i as well as the absorption coefficients ai of each of the three gray gases windows that represent the total spectrum of the gas. They are presented in Table V. The use of coefficients requires the knowledge of a reference temperature, which is taken as Trey = 600 K. The length of the enclosure is L = 1.0 m. As only one condition is assigned to the surfaces and to the medium, this is a typical forward problem, which is formulated by a well-conditioned system of linear equations. Figure 25 presents the dimensionless net heat flux on the design surface. Two uniform grids were used, having N = 120 and 160 surface zones (30 and 40 zones in each surface), corresponding to F - 30 and 40 medium zones, respectively. According to Fig. 25, the grid with 120 surface zones and 30 medium zones provided accurate results when compared to the finer grid and so is kept in all subsequent results. Because the coefficients of the weighted sum of gray gases model depend on unknown temperatures, an iterative solution was necessary. All the unknown dimensionless temperatures were assumed to be 1.0, matrix A and vector b were

TABLE V COEFFICIENTS FOR THE WEIGHTED SUM OF GRAY GASES MODEL a i 1 2 3

ai(m -1)

Ce, i, 1 x 101

0.12603 1.9548 39.570

6.508 -0.2504 2.718

Ce, i, 2 X 104

-5.551 6.112 -3.118

Ce, i, 3 X 107

3.029 -3.882 1.221

Ce, i, 4 X 1011

5.353 6.528 -1.612

aMixture of carbon dioxide (0.1 atm), water vapor (0.2 atm), and nitrogen (0.7 atm) (Smith et al. [55]).

56

FRANCIS H. R. FRAN(~A ET AL.

20

.0""

/

19

--qT 18

17

9

NN-

120 160

16 0.0

0.1

0.2

0.3

0.4

0.5

X

F I ~ . 25. H e a t flux on the design surface for 120 and 160 surface zones. ~ = 0.5, L = 1.0m,

Trey = 600K.

determined from the relations for the directed flux areas, and the system was solved. The new calculated temperatures were used for a new evaluation of the coefficients. This iterative process was carried out until convergence was achieved. No numerical relaxation was necessary. Consider the following inverse problem. The enclosure geometry and properties are the same as described previously. The side and top surfaces are adiabatic. On the design surface, two conditions are required: uniform temperature equal to tw = 1.0 and the same heat flux shown in Fig. 25, for half the length of the enclosure due to symmetry. The objective is to find the medium temperature distribution that will provide this result. Again, an iterative procedure is invoked: all the unknown dimensionless temperatures were guessed to be 1.0. The grid is kept the same, with 30 zones in each surface and in the medium. The number of medium and design surface zones is the same, so the number of equations equals the number of unknowns. Due to inverse analysis, the system of equations is expected to be ill-conditioned. According to the previous discussion, the regularization of inverse nonlinear problems is necessary to allow convergence and to obtain smooth solutions. With this purpose, the TSVD scheme is applied. Figure 26 presents the medium temperature obtained for the different regularization parameters p. Solutions for p greater than 130 are not physically acceptable

INVERSE DESIGN OF THERMAL SYSTEMS

57

3.6 3.2 2.8

tg

_

~t~ ~ .

_'2,~.

2.4

- . - forward -

2.0 "~t

~p= 130 --~- p = 125

1.6 1.2 0.0

X

-,,-- p = 122p = 121

0.2

0.4

0.6

0.8

1.0

Y FIG. 26. Predicted medium temperature distribution for different values of p. ~ = 0.5, L = 1.0 m, Trey -- 600K.

(i.e., in some of the medium zones, the emissive powers were found to be negative). For p equal to 130, the temperature obtained is close to the original parabolic profile of Eq. (49), which is the exact solution of the problem. Decreasing p, the medium temperature becomes more and more distant from the original profile, especially for p equal to 121. Although these profiles are different from the benchmark solution, the ultimate verification is the calculation of the error defined by Eq. (13). Table VI presents the maximum and average errors, "[max and "[mean, for different regularization parameter p. While the error of the solution for p equal to 121 is rather large, reaching a maximum of 5%, solutions for p -- 122 and 125 lead to errors smaller than 1.0%, which is usually very satisfactory for inverse design problem. Because matrix A changes at each new iteration, it is interesting to study the effect of the iterations on the singular values. Figure 27 presents the singular values O"i o f matrix A for different iterations. As seen, they decrease steeply for i greater than 120, down to 10-17, a clear indication that the system of equations is highly ill-conditioned. Second, the spectrum of variation

T A B L E VI M A X I M U M AND AVERAGE ERRORS FOR D I F F E R E N T V A L U E S OF

P

"/max(~176

121 122 125

7mean (~176

5.057 0.537 0.133

a .__ 0.5, L = 1.0 m,

pa

2.250 0.184 0.129 Tref

--

600 K.

58

F R A N C I S H. R. F R A N • A

ET AL.

1.0E+02

1.0E-03

i

i ..................... :................................................................ i

a ~ ............... Jt

O

~

8

Q

1.0E-08

.......................................................................................................

1.0E-13

...............................................

%

9 1st i t e r a t i o n

~

• 2nd iteration

................... ~ ....

o last iteration

.h

1.0E-18 0

30

60

90

120

F I G . 27. S i n g u l a r v a l u e s f o r d i f f e r e n t i t e r a t i v e s t e p s o n t h e s o l u t i o n .

150

~ = 0.5, L = 1.0m,

Tref = 6 0 0 K .

of 13"i does not change considerably from one step to the other. This is an important observation, as the same regularization parameter p can be applied in all the iterations for a specified minimum singular value. To find the singular values of Fig. 27, it was necessary to regularize matrix A at each iteration, so that acceptable values of the coefficients of A were obtained at each iteration. For the singular values in Fig. 27, p was kept equal to 130. In the example just given, all the conditions of the inverse problem were the same as the original forward problem so a benchmark solution was known. It is interesting to investigate how the inverse solution behaves when a perturbation is applied to the original system by changing the values of e, L, and Tref. Consider the emissivity of the bottom wall to be 0.6 instead of 0.5. All the other parameters are kept the same, including the two conditions on the design surface. The problem is to determine the temperature distribution in the gas that satisfies this system. Again, the iterative procedure using TSVD regularization was applied, and the solutions for different values of p are

INVERSE DESIGN OF THERMAL SYSTEMS

59

3.6

3.2

2.8

t

tg 2.4

I I I

2.0

I I t

1.6

p=

124

p=

122---

p-

121 I I

1.2 0.0

0.2

0.4

0.6

018

1.0

Y FIG. 28. Predicted medium temperature for different values of p. e = 0.6, L = 1.0m, 600K.

Tref =

shown in Fig. 28. In this case, using p larger than 124 led to results with no physical meaning, and they are not presented. Again, the medium temperature profile is sensitive to p. Table VII shows the maximum and average errors for the three cases, p = 121,122, and 124, indicating that the two last cases can satisfy the problem within an acceptable accuracy of about 1%.

T A B L E VII MAXIMUM AND AVERAGE ERRORS FOR DIFFERENT VALUES OF

pa

P

~max(%)

~/mean (%)

121 122 124

3.362 0.162 0.113

1.686 0.115 0.100

a

___ 0.6,

L = 1.0 m, Tre f = 600 K.

60

FRANCIS H. R. F R A N ~ A E T AL.

Figure 29 presents the medium temperature distribution along y for wall emissivities ranging from 0.6 to 1.0. The regularization parameter p was kept equal to 122 in all cases, as greater values ofp led to unphysical solutions for the highest emissivities and smaller values o f p led to solutions with increasing errors. It can be seen that the medium temperature increases close to the design surface as the emissivity is raised and decreases at points more distant from the surface. (It is interesting to observe that the amount of energy generated in the entire medium is the same in all cases, as expected from the global energy balance on the enclosure.) The solutions for emissivities smaller than 0.8 present a steep oscillation close to the wall and may seem to be inadequate solutions if smoothness is required. It is important to note that the pure radiation heat transfer itself cannot impose continuity or smoothness in the temperature due to the lack of diffusion or advection of the thermal conditions. Table VIII presents the errors of the solutions for the different emissivities; in all cases, the average error is below 1.0%.

3.2

2.8

2"4 " ~ S-s 77,

[

I

I

I

I

I

I

I

I t

i I

-~~

-

'

-

~

~

... _. ._ . . . _ ---_-. _ _

I I

tg 2.0

I I I

e =0.6 1.6

....

e = 0.7

. . . .

---~:=0.8 ~ e = 0 . 9

1.2

....

e=l.0 I I

0.8

I 0.0

0.2

0.4

0.6

0.8

1.0

Y FIG.

29. M e d i u m

L = 1.0m,

Tref

--

temperature

600K.

distributions

for different wall emissivity e. p - - 1 2 2 ,

61

INVERSE DESIGN OF THERMAL SYSTEMS

T A B L E VIII MAXIMUM AND AVERAGE ERROR FOR DIFFERENT DESIGN SURFACE EMISSIVITIES a 7max (%)

7mean ( % )

0.537 0.161 0.714 1.299 1.861 2.409

0.184 0.115 0.246 0.411 0.577 0.741

0.5 0.6 0.7 0.8 0.9 1.0

aL--

Tref =

1.0m, p = 122,

600 K.

Choosing the emissivity of the wall to be 0.9, a case where the medium temperature in Fig. 29 is smooth, the length L of the square enclosure was changed from the original case, 1.0 to 5.0 m. The results are presented in Fig. 30 indicating a trend of decrease in the medium temperature with the

2.6

I~_

I

I

I

I

I

t

I

I

I

i

t

t

aI

,_ _ I

2.5 -1- - - - ~ . -' [ ~ I

Ix.

[

,

~

i

2.44 .......

',- - - - ~ - ~

I

I

,

,

t

I

--

L - 1.0 m L=2.0m -'_~._

L=3.0m _

L - 4.0 m

.

,0m

tg 2.3

.

.

.

.

.

.

.

.

11

2.1"1" 2.0 0.0

0.2

0.4

0.6

0.8

1.0

Y FIG. 30. M e d i u m temperature for different lengths L of the square enclosure, s = 0.9, p = 122, Tref = 600K.

62

FRANCIS H. R. FRANt~A

ET AL.

T A B L E IX

MAXIMUM AND AVERAGE ERRORS FOR DIFFERENT HEIGHTS H a

H(m)

7max (~176

1.0 1.5 2.0 2.5 3.0

]rave (~176

1.862 2.463 2.684 2.728 2.715

a = 0.9, p = 122,

0.577 0.766 0.822 0.827 0.820

Tref = 600K.

increase on L. Table IX presents the errors of the solutions, whose maximum values have the order of 2.0%, but the average values are kept below 1.0%. Similarly, the reference temperature was changed from 600 to 800 K, and the results are presented in Fig. 31. With exception of the region close to the design surface, the effect of the reference temperature on the medium

2.8

2.7

T r e f = 600 K 2.6

T r e f = 750 K T r e f = 700 K

tg

T r e f = 750 K

2.5

T r e f = 800 K

2.4

2.3

2.2 0.0

0.2

0.4

0.6

0.8

1.0

Y

FIG. 31. Medium temperature for different reference temperatures Tref. e = 0.9, p = 122, L = 1.0m.

INVERSE DESIGN OF THERMAL SYSTEMS

63

TABLE X MAXIMUM AND AVERAGE ERRORS FOR DIFFERENT REFERENCE TEMPERATURES Trey a

Trey (K)

"/max (%)

"/ave (%)

600 650 700 750 800

1.862 1.767 1.662 1.548 1.423

0.577 0.549 0.516 0.482 0.469

% --- 0.9, p -- 122, H = 1.0m.

temperature is small. The maximum and average errors decrease with the increase of the reference temperature and are indicated in Table X.

C . INVERSE HEAT SOURCE DESIGN COMBINING RADIATION AND CONDUCTION

Similar to Section IV,B,1 on heat source placement, the inverse problem investigated here finds the her source distribution in the participating medium that satisfies both the specified temperature and the heat flux distributions on the surfaces of a two-dimensional rectangular enclosure. In this case, however, the heat transfer is governed by combined radiation and conduction, a problem known to be highly nonlinear. To solve such a system, an iterative procedure is applied, where the energy balance is solved in terms of the relations for thermal radiation, the dominant heat transfer process, while the terms describing the conduction mode are guessed. Once the thermal conditions are found, the conduction terms are reevaluated and then inserted into the energy balance for a new calculation. This iterative solution has gained the advantage that, at each step, a system of linear equations is to be solved. Because of the inverse nature of the prescribed problem, this system is expected to be ill-conditioned, requiring regularization tools. The same configuration shown in Fig. 14 is considered here: a twodimensional enclosure containing a gray participating medium. The problem consists of finding the heat generation in the HS region that satisfies the two conditions imposed on the design surfaces: the temperature and the heat flux. This problem is described by a system of nonlinear equations due to the combined radiation-conduction heat transfer process. Inverse analysis starts with the energy balance being applied to the medium and the surfaces of the enclosure. For the case where the heat transfer involves radiation and conduction heat transfer, but not convection, the energy balance for the medium can be expressed by

64

FRANCIS H. R. FRAN(TA ET III

kV2Tg 4- QR

AL.

Ill

+ Q a - O,

(50)

where the first and second terms correspond to the heat transferred by conduction and radiation, and the last is the volumetric heat source. For medium elements not located in the HS region, the heat generation is simply zero, Q ~ - - 0 . The numerical discretization of Eq. (50) is attained here through the division of the grid into control volumes to account for the conduction term and zones to account for the radiative heat exchange, as shown in Fig. 32. Each control volume and radiation zone occupy the same spatial position and have uniform size. The discrete form of the energy balance becomes k

Skai,jqo, k + ~ Gk, lGi,jt4,k,l - 4A'ct4, i,j q- A'~SG, i,j k,l

(51)

-~ (jx, i,j + --jx, i,j-) -q- (jy, i,j + --jy, i,j-),

where j x andjy are the conductive heat fluxes crossing the boundaries of each medium element (i, j), as shown in Fig. 33. They are approximated by a second-order finite-difference approximation. For the y direction, this gives Jy, i,j -- 4 N c R tg, i,j-1 -- tg, i,j

(52)

A~ j+. . - 4NcR

y, t,j

tg, i,j -- tg, i,j+ 1 9 AT

(53)

Similar relations can be written for conductive heat fluxes in the x direction.

Fz G. 32. Two-dimensional enclosure divided into control volumes and radiation zones. Each control volume and zone occupy the same position in the enclosure and have uniform size.

65

INVERSE DESIGN OF THERMAL SYSTEMS

FIG. 33. Conduction heat fluxes crossing the boundaries of the control volume or radiation zone in the medium.

The conduction-radiation parameter number is defined as NcR = ka/ 4crTr3ef. This term arises in problems of heat transfer combining the conduction and radiation mechanisms and gives a measure of the relative importance of conduction in comparison to radiation. Larger values of NcR correspond to increased importance of the conduction mechanism. For an element on the boundary, the heat flux must account for both radiation and conduction mechanisms. From the combination of the zonal and control volume formulations, the total heat flux, qr = qR + qc, can be found by qT, i - qo, i - Z

4

SkSiqo, k -- Z Gk'lSitg, k, 1 -tk k,l

qc, i,

(54)

where qc is the dimensionless conduction heat flux on the surface element. The radiosity, emissive power, and radiative heat flux (which is obtained by qr - qc) on a surface element are related by qo, i _ t4w, 1 - -(1 ~- ~) (qr,

i - qc, i).

(55)

Radiosity can also be expressed in terms of the emission and reflection of thermal radiation:

q~ - e't4'i -+-(1-- e') ( Z

SkSiq~ -+-Zk,l Gk'lSil4'k'l)

(56)

The conductive heat flux qc is directly related to the conductive heat fluxes jx and jy. For a surface element in the bottom of the enclosure, as shown in Fig. 34, they are related by qc, i --Jy.i,l"

(57)

66

FRANCIS H. R. FRAN(~A E T AL.

F I c . 34. Conductive heat flux in the interface of a bottom surface and a medium element.

Applying a second-order approximation, the conductive heat fluxes at the surface-medium interface can be determined by jy, i, l - - 4Ncr 8twl,j + ta, i, 2 - 9tG, i, 1 3 ZX'c

(58)

Equivalent relations can be found for the elements located on the top and side surfaces. The system of equations comprises Eqs. (51) and (54), which contain both the medium dimensionless emissive power, t4, and temperature, t, as the unknowns. To solve this nonlinear problem, the conduction mechanism is guessed such that the system of equations becomes linear in the emissive power. This procedure is affordable, as the radiation mechanism is expected to be dominant in this problem, which is ensured by choosing Ncr< 1. Once this linear problem is solved, the conduction heat fluxes jx and jy are reevaluated from the knowledge of the temperature field, and the calculations are rerun until convergence of the medium emissive power distribution. In inverse design, the temperature and the heat flux are known on the boundaries, and no condition but the location is known about the Fns (heat source) medium elements located in the HS region. The zones that are not located in the HS region have no heat source generation, and therefore a condition is already specified, s6 = 0. To solve this problem, the reduced formulation, as described in Section IV, B,1, is used. The system of equations is formed by writing Eq. (51) for the F-Fns medium elements that are not in the HS region and Eq. (54) for the N boundary elements, making a total of F-Fns + N equations. The unknowns of the reduced formulation correspond only to the F emissive powers in the medium. Therefore, the number of unknowns and equations is the same only when N = Fns. The following iterative approach is employed:

INVERSE DESIGN OF THERMAL SYSTEMS

67

(1) Initially, a temperature distribution is assumed and the conductive heat fluxes are neglected, jx = jy = O. (2) The conductive heat flux distribution on the surfaces, qc, is calculated from Eqs. (57) and (58). Then, the radiative heat flux distribution, q r - qc, is determined. (3) The radiosity of each boundary element is calculated from Eq. (55). (4) The system of equations, formed by Eqs. (51) and (54), is solved. (5) After the system is solved for the emissive power (and thus the temperature) in the medium, the conductive heat fluxes jx and jy are found from Eqs. (52), (53), and (58). (6) Return to step 2 and repeat until convergence is achieved. The iterative procedure is repeated until the thermal conditions do not change within some prescribed tolerance between the iterations. Once the emissive powers in the medium and the radiosities on the boundary elements are found, Eq. (51) is applied for each HS medium element to determine the required heat source. The problem is described by a system of nonlinear equations of the following type: A - x = b(x).

(59)

The independent vector b contains the conductive heat fluxes, which in turn depend on the unknowns of the problems (the medium temperature). In this way, b is dependent on x. The matrix of coefficients is formed of the direct exchange areas, which are independent of the sought parameters. Applying singular value decomposition to A to find the solution at every iterative step i gives __E_ n u I. b(x/-1)

xi -- ~_~ i=1

Vi"

(60)

O'i

If the system contains small singular values O'i, then any small change in vector b between two iterations will be amplified by these singular values. This makes convergence even more difficult, unless regularization (elimination of the components related to the smallest singular values) of the system is used. Figure 35 presents a two-dimensional square enclosure, H / L = 1.0, where the optical thickness is ZH = a H = 1.0, the emissivity of the surfaces is e = 0.9, and the conduction-radiation parameter is NCR = 0.1. The HS region with volumetric sources is located in the center of the enclosure, as shown in Fig. 35. It is instructive to start the analysis by first solving a forward problem and then using the inverse formulation to check if the original case can be

68

FRANCIS H. R. FRAN(~A ET AL.

FIG. 35. T w o - d i m e n s i o n a l square enclosure divided into u n i f o r m size m e d i u m a n d surface zones. H / L = 1.0,~ = a H

= 1.0, e = 0.9, NcR = 0.1.

recovered. For this forward design, the surface dimensionless temperature is uniform and equal to tw = 1.0, and the dimensionless heat source in the HS region is uniform and equal to sa = 48. For the numerical solution, the enclosure is subdivided into a 16 x 16 uniform grid, resulting in N = 64, F = 256, and FHs = 64, as shown in Fig. 35. Figure 36 presents the radiative, conductive, and total heat fluxes on half of the bottom surface. (Due to the symmetry of the problem, all the other surfaces present the same distribution.) The signs of the heat fluxes are negative because the direction of the energy transfer is from the medium to the surfaces. As seen in Fig. 36, thermal radiation is the dominant heat transfer mode, as expected for the small conduction-radiation parameter chosen, NcR = 0.1. The conduction mechanism, however, is not negligible, corresponding to about 30% of the net heat flux for the center surfaces. In such cases, convergence of the solution is possible by means of underrelaxation of the unknown emissive power and of the conductive heat flux. From one iteration to the next, only 10% of the information was diffused. For a convergence of the medium emissive power within a tolerance of 1.0 x 10 -6, a total of 121 iterations was necessary. The accuracy of the solution was verified by comparing it with the solution obtained by a more refined grid, 24 x 24, as shown in Fig. 36. The good agreement of the solutions indicates that the 16 x 16 grid provides a solution with sufficient accuracy and so is kept in all the remaining calculations. An inverse problem can be considered by imposing the uniform dimensionless temperature of tw = 1.0 and the net total flux of Fig. 36. The

69

INVERSE DESIGN OF THERMAL SYSTEMS

5.0

f

|

|

|

|

~.

I

|

|

|

total (N--

--~--

|

I

64)

rad. ( N = 6 4 )

,

|

.

.

|

|

I

i

|

l

|

|

|

|

]"1 .~

,

total (N = 96) rad. ( N = 9 6 )

.....

-t

_.1

4.0

i !

i

..............................................................

3.0

. . . .

. . . .

-qT" ...

2.0

"

-0

"

9

0.0

F I G . 36. F o r w a r d 1:/~ = 1.0, ~ = 0.9,

:

.......

'

0.0

two grid resolutions:

"

..............

1.0

'

~ . * ' "

'

~

: ................

.....: ....... '

l

0.1

.i .

'

'

'

I

.

.

.

.

.

"

.

..

.... ,- ........ .,* ....... ~ ....

; . . . . . . . . . .....,,~--.~-'.'. . . . . . . . . . . . . .

; ...............

:.:

:.

l

0.2

i! I

I

I

'

I

0.3

,

,

,

,

I

0.4

,

,

,

,

0.5

s o l u t i o n : t o t a l , c o n v e c t i v e , a n d c o n d u c t i v e h e a t flux o n t h e s u r f a c e s f o r t h e 16 x 1 6 ( N --- 64) a n d 24 x 2 4 ( N = 96). tw = 1.0, $ 6 = 48.0,

NcR =

H/L

= 1.0,

0.1.

problem now is to determine the heat source distribution in the HS region that can satisfy these two conditions. The HS region occupies the same region as in Fig. 35, and all the physical conditions of the problem are the same: ~/4 = 1.0, ~ = 0.9, H/L = 1.0 and NCR = 0.1. The uniform heat source distribution equal to sa = 48.0, as imposed originally, is therefore the exact solution for this problem. The inverse problem is dealt with by using the iterative solution methodology described earlier. Because the reduced formulation is being used, unknowns of the system correspond to the emissive power distribution in the medium. For the given grid resolution, the number of equations and the number of unknowns are the same (m = n = 256), as FHs = N. However, any attempt at solving this system using a conventional matrix solver will not succeed. The reason for this can be found by applying the SVD decomposition on the coefficient matrix A. Figure 37 shows the singular values ~r; of matrix A, which are as small as 10 -18. Inserting these singular values into Eq. (60) results in a solution completely dominated and corrupted by roundoff errors. This difficulty can be overcome by regularizing matrix A by the TSVD method to eliminate the singular values below computer precision. For the pure radiation problem of Section IV,B,1, it was possible to keep singular values as small as 10-l~ in the linear combination of Eq. (60) in the direct inversion of the forward problem, where b is in the numerical range of

70

F R A N C I S H. R. F R A N t ~ A

. . . .

I

. . . .

I

. . . .

I

.

.

.

ET

.

.........................................................

10 o

...........

: . . . . . . . . . . . . . .

: . . . . . . . . . . . . . .

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

AL.

:1

. . . .

I

. . . .

":............................ 9. . . . . . . . . . . . . .

:

. . . . . . . . . . . . .

.i

............~..............!..............~.............. . .............~..............I 10 -3

............ ............

:, . . . . . . . . . . . . . . i ..............

i". . . . . . . . . . . . . . i ..............

i .............. .. . . . . . . . . . . . . . .

i ............ : ............

i .............. i ..............

i -;

'r . . . . . . . . . . .

.. . . . . . . . . . . . . . .

! ..............

i ..............

! ............

.. . . . . . . . . . . . . . .

-;

10 -6

r. . . . . . . . . . . .

~. . . . . . . . . . . . . .

9. . . . . . . . . . . . . .

. ..............

: .............

~. . . . . . . . . . . . . .

]

1~

r ....................................................................... [" .........................................

9 ............ i

r"

lO-t2

"1

: .............

' .............................

............

"I

r ...................................................................... 9............. i

[.

............ I.

r

............

10-15

i r . . . . . . . . . . . ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 .

,

,

."

I

. . . .

I

. . . .

'. . . . . . . . . . . . . . .

!

I

. . . .

: .............. :~

i

I

.

.

.

.

.

.

.

b. . . . . . . . . . . . : .......... I .

9

lO-/S 0

50

100

150

200

250

300

F I G . 37. S i n g u l a r v a l u e s o f t h e c o e f f i c i e n t m a t r i x A (m = n = 256) o f t h e s y s t e m o f e q u a t i o n s o f s t e p 4.

tw

= 1.0,

s6

= 48.0,

H/L

-

1.0, TH = 1.0, e = 0.9,

NcR

= 0.1.

matrix A. However, in the iterative solution of the combined radiationconduction problem, using such small singular values simply does not permit convergence. As suggested by Eq. (60), any small change on b will cause a considerable amplified change on x, which would make numerical convergence unattainable. Indeed, for the solution to converge (still keeping the same under-relaxation factor of 10%), the minimum singular value had to be kept as O'mi n - - 10 -3 (corresponding to p = 204). It was necessary to complete 281 iterations for the convergence of the medium emissive power within a maximum relative error of 10 -6. The heat source distribution in the HS region obtained for this case is presented in Table XI. (Due to symmetry, only the values of the heat source in the bottom quarter of the HS region are presented in Table XI.) The obtained solution differs considerably from the original case, s a - 48.0, presenting larger values in the outer part of the HS region and smaller values toward the center. However, the final check of the inverse design is to use the calculated heat source distribution as the input of a forward problem to find the heat flux on the surfaces (keeping the same surface temperature, tw1.0). Equation (13) was used to find the arithmetic average and the maximum errors, which w e r e 7 a v g - 0-074% , and 7max = 0.220% for this regularization, p = 204. Even though the obtained heat source distribution

71

INVERSE DESIGN OF T H E R M A L SYSTEMS

T A B L E XI HEAT SOURCE SG IN MEDIUM FROM THE T S V D

SOLUTION O'MIN ----- 10 -3

(WITH P ---- 204) a

i=5 j = 5 6 7 8

70.449 61.916 74.612 82.063

6

7

8

61.916 28.557 29.599 30.665

74.612 29.599 28.090 27.882

82.063 30.665 27.882 27.052

a S y s t e m o f e q u a t i o n s : m = n = 256. e - 0.9, z/4 = 1.0,

H/L

= 1.0,

NcR= 0.1.

differs from the imposed value of 48.0, it satisfies the conditions on the design surface with a small error. Other solutions can be attempted by reducing the number of singular values. For instance, Table XII presents the solution when the minimum singular value is set as Ormin z 1 0 - 2 (with p = 197). The heat source generation has a greater peak close to the corner of the HS region when compared to the case where O'min ~-- 10 -3 (with p = 204). The solution of Table XII can be verified by calculating the arithmetic average and the maximum errors on the heat flux, 7avg ~ - 0 . 3 3 9 % and 7max - - 0 . 6 8 9 % , which is satisfactory if an error of 1.0% is acceptable. These results indicate the presence of different solutions that satisfy the problem within some prescribed precision. There is, however, a limit in the number of singular values that can be truncated. For instance, if the minimum singular value is taken as O'min --- 10 - 1 (with p = 192), the maximum error of the solution reaches a value as high as 35%. Conjugate gradient regularization (CGR) can also be used for the illconditioned system of equations of step 4 of the iterative scheme. As with TSVD, whose regularization parameter p was kept the same in all the iterations, the regularization parameter K (which refers to the number of iterative steps of the CG algorithm itself) is kept the same in all the iterative steps. An

TABLE XII HEAT SOURCE SG IN MEDIUM FROM THE T S V D (WITH a - -

i=5 j = 5 6 7 8

SOLUTION O'MIN -- 10 -2

197) a

6

7

8

114.870 69.011 67.314

69.011 29.834 28.589

67.314 28.589 26.810

66.905 28.108 26.197

66.905

28.108

26.197

25.493

a S y s t e m o f e q u a t i o n s : m = n = 256. e = 0.9, xH = 1.0,

H/L

= 1.0,

NcR= 0.1.

72

F R A N C I S H. R. FRAN(~A

ET AL.

T A B L E XIII HEAT SOURCE Sc IN MEDIUM FROM THE C G SOLUTION WITH K---- 6 a i=5 j = 5

6

7

8

114.010

68.833

67.453

67.237

6

68.833

29.802

28.608

28.160

7

67.453

28.608

26.837

26.232

8

67.237

28.160

26.232

25.525

aSystem

of

equations:

m = n = 256.

e = 0.9,

~ / - / = 1.0,

H / L = 1.0, NcR = 0.1.

initial guess for K can be made by inspecting the CG solutions for the pure radiation problem (Section IV,B,1), where K = 6 and 11 provided solutions that were similar to the TSVD method. Table XIII shows the heat source generation distribution using K = 6. As seen, the solution is close to the one obtained by the TSVD for 7min = 10 -2 (with p = 197), showing the relation between the two methods. The arithmetic average and the maximum errors of the solution of Table XIII a r e ]tavg = 0.339% and q/max = 0.689%. The same underrelaxation was used, oL - 0.1, requiring a total of 115 iterations to converge. It is expected that increasing K may result in the heat source generation of Table XI, which relates to the maximum value for p that allowed the convergence of the solution. Table XIV presents the solution for K = 10, which has a similar distribution to the results in Table XI. The and 7max ~ - 0 . 2 9 7 % . However, the errors of this solution a r e 7avg - 0 . 1 6 6 % convergence of the solution for K = 10 was substantially more difficult to achieve, requiring an underrelaxation factor as low as 0.01 and about 1341 iterations. None of the aforementioned solutions recovered the uniform heat source generation in the HS region. The reason for this was that most of the smaller singular values had to be truncated for the iterative scheme of Eq. (60) to

TABLE XIV

HEAT SOURCE S~ IN MEDIUM FROM THE C G SOLUTION WITH K -- 10 a i=5

6

7

8

j = 5

71.856

61.187

74.246

82.366

6

61.141

28.699

29.645

30.688

7

74.198

29.643

28.152

27.949

8

82.326

30.689

27.950

27.134

aSystem o f equations: m = n = 256. e = 0.9, x/4 = 1.0, H / L = 1.0,

NcR= O.1.

INVERSE DESIGN OF THERMAL SYSTEMS

73

converge. The use of a smaller (o~ <<0.1) underrelaxation parameter was attempted in order to retain more singular values, but none of the attempts was successful. Although this may be considered a drawback in the inverse solution for use in a measurement problem, inverse design is mostly concerned with finding a heat source distribution that satisfies the two conditions on the design surface. The TSVD and CG methods seek the solution that tends to minimize the error in the norm of the solution vector x, which is formed by the F emissive powers of the medium elements. This explains why the peak of heat source distribution is located closer to the surfaces, for this keeps the medium temperature lower than it would be if the peak were located in the center of the enclosure. Section VI demonstrates how to impose additional constraints on the heat source generation such as an expected distribution of the heat source generation. D . INVERSE BOUNDARY DESIGN COMBINING RADIATION AND CONVECTION

Figure 38 shows a schematic view of a two-dimensional channel formed by two parallel surfaces having length L and separated by a distance of H. A gray participating medium flows between the two plates, with a fully developed laminar velocity profile. The design surface and the heater are centered on surfaces 1 and 2. The lengths of the design and of the heater surfaces are L9 and LI-I. All the physical properties are assumed to be constant. Surfaces 1 and 2 are diffuse, gray emitters and absorbers, and their emissivities are indicated by 81 and c2. The inlet and outlet of the enclosure are modeled as porous black surfaces at the same temperature as

FIG. 38. Two-dimensional enclosure formed by two parallel plates. The thermal conditions on the heater, on the top, are sought to satisfy the prescribed conditions on the design surface, on the bottom.

74

FRANCIS H. R. FRANCA ET AL.

the inlet and outlet bulk temperature of the flowing medium, designated as tel and te2. The elements of surfaces 1 and 2 not located on the design surface or on the heater are assumed to be adiabatic. The process is at steady state. The inverse design problem is to find the total heat flux distribution on the heater that is able to provide the specified temperature and heat flux on the design surface. The formulation presented here provides a useful design tool for devices such as industrial drying and processing ovens. To find the heat flux distribution to be imposed in the heater, it is necessary to solve the energy balance in the medium and on the surfaces of the enclosure. In this problem, the heat transfer interactions in the medium involve radiation, conduction, and convection mechanisms, but there is no heat source generation in the medium. The energy balance is given by tt!

~7 . ( p c p V T g - k V T g )

-

QR.

(61)

The flow is assumed fully developed from the entrance, so there is no component of velocity in the Y direction. It is also assumed that the conduction mechanism is negligible in the X direction, where the temperature gradient in the medium is expected to be small in comparison to the one in the Y direction. Under these conditions, the energy balance in the medium becomes ~Tg ~2Tg ,,, pCp U - ~ - k ~ y2 - QR"

(62)

The numerical solution of the integro-differential system of equations can be accomplished by applying the zonal method for the radiation mechanism and the control volume approach for the conduction and convection modes. Figure 39 shows the division of the enclosure into control-volumes and zones

FIG. 39. Grid division of the enclosure. The grid is uniform in each of the directions, but A y # AX. To make the computation of the direct-exchange areas less expensive, A Y/AX is an integer.

75

I N V E R S E D E S I G N OF T H E R M A L SYSTEMS

of radiation, which are spatially coincident and have uniform size in the X and Y directions to make calculation of the direct-exchange areas more efficient. It is expected that the temperature gradient in the Y direction will be greater than in X direction, so a greater refinement in the grid is set in the Y direction, A y < AX. For the discrete domain, the energy balance for the medium, in dimensionless form is (jx, i,j + --L,i,j-)mT, y + (jy, i,j + --jy, i,j-)A'Cx = SR, i,jmy, ymg.x,

(63)

wherejx andjy correspond, respectively, to the advective and conductive heat fluxes across the boundaries of the control volume (similar to the conductive energy fluxes of Fig. 33). The convective heat flux is calculated according to the upwind approximation, where the temperature in the boundary of each control volume is assumed to be equal to the temperature of the center of the upstream neighbor control volume. Therefore, jx, i,j-

=

j x , i,j § - -

4NcR Pr Re Ujtg, i-l,j

(64)

4NcR Pr Re uj tg, i,j,

(65)

where Pr is the Prandtl number of the medium; Re is the Reynolds number based on the bulk velocity, Um, and the channel height, H; and u is the dimensionless velocity, u = U~ Um. The conductive heat flux is determined by a second-order approximation: jy, i,j

_

-- 4NcR

jy, i,j + -- 4NcR

tg, i,j-

1 --

tg, i,j

(66)

my.y

(67)

tg, i , j - tg, i,j+l AT, y "

In Eq. (63), sR accounts for the "radiation heat source," i.e., the total radiative energy received by the medium element minus the emitted radiation. It is given by SR, i,j

=

1

, t4,k, l AT.y _(--4A~'yt4, i,J -+- Zk,l Gk, lGi j \

(68)

+ Z SkGi,jqo, k + SelGi, jt41 + Se2Gi,jte42}, / k where Sel Gi,j and Se2Gi,j a r e the dimensionless direct exchange areas between the medium, element i,j and the inlet and outlet reservoirs, respectively.

76

FRANCIS H. R. FRAN(~A ET AL.

The same analysis for the heat interaction on the surfaces, as presented for the combined radiation-conduction problem, Section V,C, is valid here. The heat flux on a boundary element must account for both radiation and conduction: qT, i -

qo, i -

Z

SkSiqo, k -

Z

k

Gk, lSit4,k, l

k,/ 4

(69)

4

-- S e l S i t e l - Se2gite2 4- qc, i,

where S e l S i and element i and the The radiosity, qR = q r -- q c ) on

Se2S i a r e the direct-exchange areas between the surface inlet and outlet of the enclosure. the emissive power, and the radiative heat flux (i.e., a surface element are related by

qo, i

_

t4

w, i

(1

-

--qR,

~)

(70)

i.

Radiosity can also be expressed in terms of the emission and reflection of thermal radiation:

qoi t4w,i 4- (1 -

e)

4)

SkSiqo, k + Z Gk'lSitg, k, l k,l

"

The conductive heat flux q c corresponds to the conductive heat flux jy on the wall. For a surface element in the bottom of the enclosure they are related by qCl, i = ~v, i, 1- .

(72)

Applying a second-order approximation, the conductive heat fluxes at the surface-medium interface can be determined by jy, i, 1- -- 4Ncr 8twl,j 4- tg, i, 2 - 9tg, i, 1. 3A~y

(73)

An equivalent relation can be found for the elements located on the top surface. Equations (63) through (73) contain all the information necessary for the solution of the problem. They form a system of nonlinear equations, as the unknowns include both the dimensionless temperature t and the emissive power t4 of the medium and wall elements. In Section IV,C, the solution of the system employed an iterative approach, where the emissive powers were chosen as the primary unknowns of the system, whereas the temperatures were taken from the previous iterations. The reason for this choice was based on the knowledge that the thermal radiation from the medium accounted for

INVERSE DESIGN OF T H E R M A L SYSTEMS

77

most of the energy on the surfaces (low conduction-radiation parameter, NcR). In the design problem considered here, a mixed situation is involved. On the one hand, the medium should not be optically thick (large ~/-/), because in this case the heater would present a small effect on the thermal conditions of the design surface, and the proposed design would be flawed from the very start. If optically thick, the convection and conduction mechanisms are likely to be dominant in the heat transfer in the medium. As a consequence, it is preferable to solve the energy balance for the medium in terms of the temperature t, guessing the radiative heat source sR from the conditions of the previous iteration. Inserting the relations for the conductive and convective heat fluxes into Eq. (63) and rearranging it gives

4NcR [Pr R e u j ( t g ,

i,j - tg, i _ l , j ) A T y

k

(74)

_ (2tg, i,j -- tg, i,j-1 -- tg, i,j+l) A~x] _

A~y

J

SR, i,jAY, xA'~y.

The energy balance for the medium elements neighboring the two walls must be changed to incorporate the modified conductive heat flux relations as in Eq. (73). On the other hand, the heat transfer from the heater to the design surfaces should be dominated by thermal radiation. Therefore, the system of equations should be formulated to relate the design surface directly to the heater. The primary unknown is the radiosity (or the emissive power for black surfaces) of the heater elements, whereas the other unknown thermal conditions in the enclosure are guessed from the previous iteration. An attractive aspect of this procedure is that the ill-posed part of the problem, which arises from the fact that the design surface contains two conditions while the heater has no condition at all, is separated from the remainder of the problem. This allows a more effective regularization and solution. The system of equations linking the design surface to the heater is formed by writing Eq. (69) for all the elements on the design surface and rearranging it to form a system of equations in terms of the unknown radiosities of the heater elements (the radiosity for the elements in the adiabatic region of surface 2 are included among the guessed terms, as shown here): ZSkSiqo2, k-heater

k -- qol,i - qRl,i - Z SkSiqo2, k k-adiabatic 4 -- Z Gk, lSitg,4 k, l -- Se 1Sitel4 - Se2Site2" k,l

(75)

In Eq. (75), all the terms in the right-hand side are either known or guessed from the previous iteration. Equation (75) is written for all the ND design

78

FRANCIS H. R. FRAN(~A E T AL.

surface elements, while there are NI-Z unknown radiosities. For the system to have the same numbers of equations and unknowns, N D must be equal to N/~. The medium emissive power could be taken as an unknown of the problem as well, but this would make the systems of equations much larger, which in turn makes the regularization much more time expensive. Another advantage of using Eq. (75) instead of (69) to calculate the medium temperature is that it forms a sparse system of equations, which requires less computational effort.

1. I t e r a t i v e P r o c e d u r e

The known conditions of the problem are the temperature and heat flux on the design surface elements, the heat flux on the adiabatic regions of surfaces 1 and 2 (i.e., q r = 0), and the inlet medium bulk temperature. The solution is achieved by the following iterative steps. 1. As the initial guess to start the solution, the medium temperature is assumed to be uniform and equal to the known inlet bulk temperature, tg = te~; and the radiosity of the elements located in the adiabatic 4 regions of surfaces 1 and 2 are guessed as tel. 2. The outlet medium bulk temperature, te2, is calculated from the medium temperature distribution at the enclosure exit at X = L. 3. The conductive heat fluxes on the surface elements, q c , are determined by Eqs. (72) and (73). 4. The radiosities of all elements of the design surface are calculated from Eq. (70), using the specified temperature twl and the radiative heat flux, given by q1~1 -- qT1 - q c l . 5. Solve the system of equations formed by writing Eq. (75) for the elements on the design surface. 6. Once the radiosity of each element on the heater surface is determined, the emissive powers of these elements are calculated by first applying Eq. (69) to determine qT2, and then Eq. (70). 7. From the knowledge of the radiosity on each element of the wall, the radiative heat source in the medium elements is found from Eq. (68). Next, the energy balance, Eq. (75), is written for all of the F medium elements, forming a system having F equations and F unknowns on the temperature of the medium elements. This system is sparse and well conditioned and is solved by the iterative Gauss-Seidel method; 8. The radiosity of the elements located in the adiabatic regions of surfaces 1 and 2 is calculated from the application of Eq. (69). 9. If convergence has not been achieved, the process returns to step 2 and is repeated.

79

INVERSE D E S I G N OF T H E R M A L SYSTEMS

2. Convergence Criter& and Relaxation In the iterative procedure just presented, step 5 involves the solution of an ill-conditioned system of equations, as will be shown later. In this system, the coefficient matrix A is formed by the direct-exchange areas, which are constant throughout the iterative calculation. The right-hand side depends on the unknown radiosities, i.e., b = b(x). The nonlinear problem is again expressed as A . x = b(x).

(76)

Therefore, this nonlinear problem is similar to the one discussed in Section V,A, which is solved iteratively as described by Eq. (40). The convergence is verified by calculating the maximum change in the radiosity of the heater elements between two successive steps: 7max - - m a x

qo2, i, at step (k) - - qo2, i, at step ( k - l ) qo2, i, at step (k)

< 71im,

(77)

where 71imis an imposed value. For convergence of the solution, it is necessary to apply underrelaxation to some of the parameters in most of the cases. The unknown radiosities are underrelaxed by a factor of oLaccording to qo2, i, used at step (k) ---- qo2, i, at step ( k - l ) -+- ot(qo2, i, at step (k) - - qo2, i, at step ( k - l ) ) .

(78) It is also necessary to underrelax the medium temperature tg, the conductive heat fluxes jx and jy, and the radiative heat flux q R 2 - - q T 2 - qc2 on surface 2.

3. Results and Discussion For the system shown in Fig. 38, uniform temperature and heat flux are imposed on the design surface, equal to twl = 1.0, and qrl = - 1 6 . 0 (the negative sign of qrl indicates that heat is transferred from the enclosure to the design surface). The elements on surfaces 1 and 2 that are not located on the design surface or on the heater are adiabatic, i.e., qr = 0. A participating medium enters the enclosure with a uniform temperature of tg = 1.0. The fully developed laminar flow has an average Reynolds number of Re = 2000. The Prandtl number is Pr - 0.69, and the Stark number is N c R = 7.60x 10 -4. The optical thickness of the enclosure is ~n = a = 0.2. This set of thermal properties for the medium is representative of a mixture of air and water vapor, as found in drying systems. All the walls have the same emissivity: ~1 = ~3 = 0.8. The dimensions of the enclosure and of the design surface are, respectively, L / H = 5.0 and LD/H = 3.0. A shorter length of the

FRANCIS H. R. FRAN(~A ET AL.

80

design surface was chosen to minimize the effect of the ends of the enclosure, which are cooler than the heater. The proposed inverse design is applied to find the necessary heat flux distribution on the heater for different dimensionless lengths LI-I/H and different TSVD regularizations. For numerical solution of the problem, the x-y grid is formed by 50 and 20 uniform size elements. This results in AX/H = 0.10, A Y/H = 0.05, and AX/A Y = 2. The accuracy of this grid will be verified.

a. Solution for the Same Number of Equations and Unknowns, L t t = Lo Because the dimensionless size of the design surface is 3.0, there are ND = 30 design surface elements. As discussed previously, a total of 30 relations as Eq. (75) can be written. If the size of the heater is also set equal to LD/H = 3.0, the number of unknown radiosities on the heaters is also N/-/-- 30, making the number of unknowns and equations the same. The components of matrix A are the direct-exchange areas relating the elements on the design surfaces and heaters. Once the matrix A is formed, the singular value decomposition is performed to generate the singular values cri, which are presented in Fig. 40. As can be seen, the singular values decay continuously to a magnitude as small as 10 -9, which confirms the ill-posed nature of the system of equations in step 5.

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i FXG. 40. Singular values of the coefficient matrix A (m = n = 30) of the system of equations of step 5 for LH/H = 3.0. L/H -- 5.0, LD/H = 3.0, ~/4 = 0.2, ~l = E2 - - 0 . 8 , Re = 2000, Pr = 0.69, NcR= 7.60 x 10 -4.

INVERSE DESIGN OF T H E R M A L SYSTEMS

81

TSVD regularization is necessary to smooth the solution, as well as allow convergence of the iterative solution of Eq. (75). An optimum regularization was achieved in earlier solutions by taking the minimum singular values at about O'min = 1 0 - 3 . Using this information, the first choice for the regularization parameter was p = 10, found by checking the singular values of Fig. 40. Solving the problem for this p, some components of vector x converge to negative numbers, which is not physically acceptable, as they stand for the radiosity of the heater elements. Other solutions are attempted by decreasing p until the vector x contains only positive values. This occurs for p = 5. The radiosity and the corresponding total heat flux distributions on the heater (1.0 < x2 <_ 4.0) are presented in Fig. 41. As seen, the radiosity and the total heat flux present the same oscillating profile, although the total heat flux to be imposed in the heater has some negative values (the radiosity must remain always positive for the solution to be physically acceptable). As expected, the maximum heat flux on the heater occurs on the left side to compensate for the fact that the enclosure is cooler at the entrance than it is at the exit. Because of the regularization of the problem, the solution of Fig. 41 is not expected to be exact. To verify it, the obtained heat flux is imposed on the

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X2/H FIG. 41. R a d i o s i t y a n d total h e a t flux r e q u i r e d on the h e a t e r for L H / H = 3.0. T S V D regularization: p = 5. L / H = 5.0, L D / H = 3.0, ~H = 0.2, el = ~2 = 0 . 8 , R e = 2000, Pr = 0.69, NCR = 7.60 • 10 -4.

82

FRANCIS H. R. FRAN~A E T AL.

heater, and for the uniform temperature on the design surface, the heat flux on the design surface is calculated and compared to the imposed heat flux by Eq. (13). The arithmetic average and maximum errors for this solution are high: ~ a v g = 1.26% and ~ m a x - - 5.26%. Another unsatisfactory aspect of this solution is the presence of a negative heat flux in some of the elements, which is not an interesting design for a heater. For the convergence of the iterative solution, relaxation factors of a = 0.50, 0.10, and 0.05 were used for the medium temperature and for the heater elements' radiative heat flux and radiosity, respectively. The convergence of the solution required 273 iterations to achieve an error smaller than 7max : 1.0 • 10-6, requiring lm 42s of CPU time (IBM PowerPC 233 MHz). b. Solutions for Different Sizes of the Heaters LH Figure 41 indicates that to keep the design surface at uniform heat flux and temperature, the heat flux should be the greatest for the elements closer to the extremities of the heater. This can be an indication that if the heater is "stretched" toward the two ends of the enclosure, where a greater heat flux is needed in the heater, the peak heat flux can be reduced. (An alternative approach would be to shorten the size of the design surface.) For instance, the dimensionless size of the heater, LH/H, can be set as 3.4, 3.8, 4.2, and 4.6 instead of 3.0. The first implication of this concerns the change of the number of unknowns. Keeping the same grid resolution as before, the numbers of unknowns when LH/H is 3.4, 3.8, 4.2, and 4.6 are, respectively, 34, 38, 42, and 46. The number of equations, which is related to the number of elements on the design surface, is still 30. This makes the number of unknowns greater than the number of equations (m < n), and the system is underdetermined. The singular values of matrix A for these four cases are presented and compared to the case where LH/H = 3.0 in Fig. 42. There is no significant change in the singular value spectrum for the five cases presented. Only the 30 largest singular values are presented in Fig. 42, as the remaining are zero for the system having more than 30 unknowns. The TSVD method can be applied in exactly the same way as before for the underdetermined systems by keeping the p largest singular values in the iterative solution of Eq. (40). Elimination of the terms related to zero singular values introduces no additional error to the solution. For the cases where LH/H is 3.4, 3.8, 4.2, and 4.6, it is possible to use p larger than 5 and still recover a solution having physical meaning (nonnegative radiosities). Figure 43 presents the heat flux on the heater setting p equal to 6 for the different values of LH/H: 3.4, 3.8, 4.2, and 4.6. Figure 43 shows that the increase in the size of the heater smoothes the heat flux distribution, decreasing the peaks of heat flux. This is possible because the area of the heater increases, and to maintain the same energy input in the enclosure, high values of heat flux are not necessary. However, by increasing

I N V E R S E D E S I G N OF T H E R M A L SYSTEMS

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p = 6. L / H

84

FRANCIS

ET AL.

H. R. F R A N ( T A

the length of the heater to LI-I/H = 4.6, there is a negative heat flux at the right side of the enclosure, which is not a useful heater design. For a given LI-I/H, increasing p from 6 to 7 adds one more oscillation to the predicted heater heat flux distribution. Figure 44 shows the solutions for different values of Lt~/H when the regularization parameter is set to p = 7. As for the previous case, the increase in the heater length from Lt-I/H = 3.4 to 4.6 smoothes the heat flux distribution, making the solution more attractive if peaks in the heat flux are not desirable. Increasing the number of retained singular values from 6 to 7 causes the heat flux distribution to present one more oscillation than in Fig. 43. Considering that the problem allows different solutions, as indicated on Figs. 43 and 44, it is interesting to establish a set of criteria to select one or some of them. As discussed previously, solutions that require negative heat flux on the heater are not interesting designs. This eliminates the solutions and obtained for LI-I/H= 3.0 with p = 5, LI-I/H=4.6 with p = 6 , LH/H = 3.4 with p - 7. Due to the need for regularization of the inverse problem, the maximum (~max) and average (~avg) errors must always be checked to validate or eliminate a given solution. Another aspect to be considered for this particular problem is the thermal efficiency of the system. It might be suspected that heaters of greater length lose more heat to the

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INVERSE DESIGN OF THERMAL SYSTEMS

85

TABLE XV AVERAGE AND MAXIMUM RELATIVE ERRORS, AND EFFICIENCY FOR DIFFERENT LH / H a

LH/H 3.4 3.8 4.2 4.6

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2.182 1.319 0.925 0.762

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a T S V D regularization: p = 6. L/H = 5.0, LD/H = 3.0, ~/4 = 0.2, el - e2 = 0.8, Re = 2000, Pr = 0.69, NcR = 7.60 x 10 -4.

reservoirs at the ends of the enclosure, decreasing the efficiency of the system. The efficiency 1"1of the heating process is defined as 11 -

energy required on the design surface energy input in the heater

.

(79)

Tables XV and XVI present the maximum and the average errors for each solution of Figs. 43 and 44, as well as the efficiency of the heating process. Keeping the regularization parameter p constant, the average and the maximum errors of the inverse solution decrease as LI-I/H is increased from 3.4 to 4.6. The efficiency of the heating process decreases as LrI/H increases from 3.4 to 4.6, from 61 to 58%, because the increment in the size of the heater is balanced by the smoothing of the heat flux peaks. Considering that this decrease in the efficiency may not be a high price to pay for a smoother heat flux on the heater, then the solutions for LI-I/H = 4.2 with p - 6 and 7, and LI4/H = 4.6 with p = 7 can be chosen as the most interesting. If a minimum number of oscillations is preferred, then the solution for solutions for LI-I/H - 4.2 with p = 6 can be selected.

TABLE XVI AVERAGE AND MAXIMUM RELATIVE ERRORS, AND EFFICIENCY FOR DIFFERENT LI4/H a

LI-1/H 3.4 3.8 4.2 4.6

')tmax (%)

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0.773 0.335 0.152 0.091

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60.1 59.1 58.2 57.5

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Lz)/H

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86

F R A N C I S H. R. F R A N ~ A ET AL.

c. Grid Independence Study The validation of a numerical solution requires a grid independence study. It is of interest to learn how the change of resolution can affect the inverse solution because the solution depends on the regularization imposed on the system for a given grid size, as shown in the previous results. In the example of Section III,A,8, it was found that keeping the same regularization parameter p between two different grids led to a similar distribution in the emissive power of the upper surface. To verify whether this remains valid for the problem considered here, the grid is now divided into 30 and 75 uniform elements on the x and y directions, with AX/H = 0.0667, A Y/H = 0.0333, and AX/AY = 2. For comparison with the previous grid, 20 x 50, the case LI-I/H--4.2 is considered. For the 30 x 75 grid, the design surface and the heater are divided into 45 and 63 elements. The system of equations of step 5 consequently will have 45 equations and 63 unknowns on the radiosities of the heaters. Once matrix A is formed, it is decomposed by means of SVD. Figure 45 compares the singular values of matrices A for the two grid resolutions, 20 x 50 and 30 x 75. As seen in Fig. 45 the singular values for the two grids are coincident in the higher values (lower i), which are the ones retained in the regularization of Eq. (63). This is an indication that the same regularization parameter p can be used in both cases to generate a similar result. In fact, Fig. 46 shows that the heat

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87

INVERSE DESIGN OF T H E R M A L SYSTEMS

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fluxes on the heater for these two grids, both obtained by setting p present very good agreement.

6,

d. Using a Smaller Number of Heating Devices A practical aspect concerns the implementation of the inverse solution itself. For solution with LI-I/H = 4.2 and p = 6, the designer must specify individual heating devices along the heater such that the distribution shown in Fig. 46 is attained. Within the constraint of the grid resolution of 50 • 20, a total of 42 heating devices, each one having the heat input required by the curve, should be installed. The design may become more practical if a smaller number of heating devices were employed, even at the expense of accuracy. In Fig. 47, the heat flux distribution on the heater is approximated by seven heaters with uniform heat input. The five heating elements in the center have uniform size, A L I ~ / H = 0.7, and the elements on the left and right have size ALI-I/H = 0.3 and 0.4, respectively, making a total size of L I 4 / H = 4.2. The amount of energy to be provided by each heating device was determined by simply averaging the energy of the elements that form it, found from the solution with the grid resolution of 42 elements. Such a solution leads to the heat flux on the design surface shown in Fig. 48, which is compared to the imposed uniform heat flux, as well as the one obtained when all of the

88

FRANCIS H. R. FRAN(~A E T AL.

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INVERSE DESIGN OF T H E R M A L SYSTEMS

89

42 element devices are employed. The avei-age and maximum errors of the solution with seven heater elements, each one with uniform heat input, are ~/max = 1.711% and 7avg =0.337%. The greater error is observed near the ends of the design surface, where the absolute value of the heat flux is smaller than the specified one. A better solution could be attempted by increasing the energy input of the heating elements close to the ends of the enclosure. An alternative procedure to solve for fewer heating devices could be attempted by dividing the heater into zones of larger dimension, as the use of a coarse grid is in itself an effective regularizing procedure, as the singular values of the system of equations show a less intense decay. With this grid resolution, the problem would be formulated by a system having more equations than unknowns (i.e., less zones on the heater than on the design surface), which could be solved by the application of SVD decomposition of matrix A to find the least-squares solution. Finally, the uniform heat flux that is sought on the devices would be simply the heat flux of the equivalent zone. The problem with this scheme is that all thermal properties, as temperature and radiosity, would have to be assumed uniform, an inaccurate approximation for the large zones. A better procedure is to keep the same grid resolution and consider that every device is formed by a number of zone elements. Then, the imposed heat fluxes on the zones located in the same heating device are the same, letting the temperature and radiosity vary. Section VI,B discusses how to incorporate the uniform heat flux constraint in the system of equations. e. Parametric Study of the Thermal Conditions Because of the many physical parameters involved in this problem, it is interesting to study their effect on the thermal design. The basic case for study is the heater of size

LH/H=4.2 Figure 49 presents the temperature distribution on the two surfaces of the enclosure, as well as the bulk medium temperature along the enclosure. The discontinuity in the temperature on the surfaces occurs due to the change of boundary conditions between adiabatic and nonadiabatic regions. On surface 1, the design surface is kept at the uniform temperature of tw~ = 1.0. The adiabatic portion of surface 1 is at a relatively high temperature because of the radiation that it receives from the heater. To balance the incident radiation, it must be at a higher temperature than the medium so that it can release the energy by thermal radiation and convection. The temperature of the heater, on surface 2, shows an oscillatory behavior that is a consequence of the oscillations required on the heater surface. The adiabatic part of surface 2 is at a lower temperature than on surface 1 because it is not facing the heater directly. The medium temperature increases steadily,

90

FRANCIS H. R. FRANI~A E T AL.

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X/H

FIG. 49. Temperature distributions on surfaces 1 and 2, and medium bulk temperature. TSVD scheme: p = 6 . L / H = 5 . 0 , L D / H = 3 . 0 , L H / H = 4.2,?/-/= 0.2,el =~2 = 0 . 8 , R e = 2000, Pr = 0.69, N c R = 7.60 • 10 -4.

although the rate of increase varies along the enclosure, achieving the maximum rate in the region where the heater and the adiabatic lower surface temperatures have the greatest values. The exit temperature reaches the value of 1.95, which is the temperature of the porous black body that models the exit reservoir. Because this portion of the enclosure is at a higher temperature, there is a smaller required power to the heater to keep the design surface at uniform temperature and heat flux. Figure 50 shows the effect of increasing the optical thickness of the medium from ~/-/= a H - 0.1 to 0.6. In case of the drying system, this could be caused by the increase in the concentration of water vapor in the air-water mixture. Due to the increase of a, the conduction-radiation parameter changes accordingly, NcR = ka/4crT3ref. As seen in Fig. 50, the increase in the optical thickness elevates the required heat input on the heater at the entrance of the enclosure because the medium absorbs more of the energy. In turn, this causes the medium exit temperature to increase with the optical thickness. For ~/_/= a H - 0.1 and 0.6, for instance, the exit temperature reaches the value of 1.72 and 2.06, respectively. As a consequence, at the enclosure exit the amount of energy that the medium radiates to the design surface increases with optical thickness because of the higher temperature and emissive power of the medium, lowering the required power input to the heater. Due to this balancing effect, the thermal efficiency of the system, as

91

INVERSE DESIGN OF T H E R M A L SYSTEMS

50.0

40.0 I]-............ 'a". x~~............ .!. . . . . . . . . . . . . . .

a H = 0.2 ..........

30.0

qT2

20.0

10.0

-10.0 0.0

1.0

2.0

3.0

4.0

5.0

X2/H F I c . 50. T o t a l h e a t flux r e q u i r e d on the h e a t e r for different optical thickness 1:/-/= a l l . T S V D regularization: p = 6. L / H = 5.0, L D / H = 3.0, L n / H = 4.2, el = ~2 = 0 . 8 , R e = 2000, Pr = 0.69, NcR = 7.60 x 10 -4.

defined by Eq. (79), changes little with the optical thickness, remaining at about 59.4% for the four cases. The effect of the emissivity of the design surface on the power input to the heater is presented in Fig. 51 for three different emissivities: ~1 - 0 . 6 , 0.8, and 1.0. As indicated in Fig. 51, the required power input increases for lower design surface emissivities to compensate for the smaller absorption of the wall (e~l - ~1). At the exit of the enclosure, however, the power input to the heater is slightly smaller for the case where the emissivity is the lowest because the highest temperature of the heater causes the medium to exit at a higher temperature, which in turn heats the design surface more. For the case where ~1 = 0.6 and 1.0, the exit temperature reaches the values of 2.15 and 1.79. The thermal efficiency increases with the design surface emissivity. For the emissivities of ~1 = 0.6, 0.8, and 1.0, the thermal efficiencies are 1"1= 53.9, 59.4, and 63.5%, respectively. Figure 52 presents the effect of the flow on the heat input for four different Reynolds numbers within the range of laminar flow: Re = 1000, 1500, 2000, and 2500. As seen, the increase in the Reynolds number from 1000 to 2500 causes an increase in the required heat input, especially near the enclosure exit, because the medium temperature at the exit decreases for the higher

92

F R A N C I S H. R. FRAN(~A ET AL.

50.0

40.0

30.0

qT2

20.0

10.0

0.0 :

--I0.0

. . . .

i , ,

0.0

i ,

,

1.0

I

; ~

,

,

,

2.0

I

,

,

,

~

. . . .

3.0

4.0

5.0

X2/H F I G . 51. T o t a l h e a t flux r e q u i r e d o n t h e h e a t e r for d i f f e r e n t d e s i g n s u r f a c e e m i s s i t i v i t y el. T S V D r e g u l a r i z a t i o n : p = 6. L / H -- 5.0, Lz)/H = 3.0, LI-I/H = 4.2, z / - / = 0.2, R e = 2000, P r = 0.69, N c R = 7.60 • 10 -4.

50.0

. . . . . . . . . . . . . . . .

40.0

i ....

~9

.............................................................

R e = 1000. Re-

. ~ . 30.0

:. . . . . . . . . i. . . . . . . . . . . . . . . . .!. . . . . .

........ -

!

iiii

10.0

-"

*.

"

9 - - R e. = 2500"

i

i

A

9

1500

R e = 2000

.

o.o

.

.

.

.

.

.

.

.

.

.

~ ....

-10.0

. . . . . . 0.0

1.0

~ ,

I

,

~ ,

2.0

,

I 3.0

,

,

~ ~

. . . . 4.0

t 5.0

Xz/H F I G . 52. T o t a l h e a t flux r e q u i r e d o n t h e h e a t e r for d i f f e r e n t R e y n o l d s n u m b e r Re. T S V D r e g u l a r i z a t i o n : p -- 6. L / H = 5.0, L D / H = 3 . 0 , L H / H = 4.2, ?/4 = 0.2, Cl = e2 -- 0.8, P r = 0.69, NcR = 7.60 x 10 -4.

INVERSE DESIGN OF THERMAL SYSTEMS

93

Reynolds number, and so the medium contributes a smaller amount of energy to the design surface. For Re = 1000 and 2500, the exit temperatures are 2.02 and 1.89. As a consequence, the thermal efficiency of the system decreases with the increase in the Reynolds number. For Re = 1000, 1500, 2000, and 2500, the thermal efficiencies are r I = 67.8, 63.1, 59.4, and 56.0% , respectively. The same TSVD regularization is employed in all cases, keeping p = 6. For all the solutions presented earlier, the maximum error on the design surface heat flux, ~max, is below 1.0%.

VI. Imposing Additional Constraints on the Regularized Solution As discussed in Section III,A, a regularized solution is constructed by constraining the solution with respect to a given criteria, which is usually a minimization of the size of the solution and residual vectors. The MTSVD and Tikhonov methods are more general methods in the sense that different minimization criteria can be applied through a specified functional. However, the designer may wish to find a solution that satisfies some physical conditions that cannot be achieved easily by means of a functional. For instance, consider the heat source design described in Sections IV,B and V,C. There may be specific physical laws that govern the heat source distribution that need to be incorporated into the regularized solution. This section presents two examples of imposing additional constraints to the inverse solution. This is done by inserting new relations into the system of equations. This procedure allows methods such as TSVD and conjugate gradient to be applied even if they are not capable of imposing additional constraints other than the least-square solution. The procedure is used first in a heat source problem where the heat source is expected to obey a given spatial distribution and then in determination of a smaller number of heaters with uniform power, as discussed in Section V,D,3,d. A. ENERGY GENERATION SHAPE CONSTRAINT In a furnace, for instance, the heat source is provided by burners located in designated positions in the enclosure. Once the fuel is injected into the furnace, the heat source distribution obeys particular laws related to the diffusion and combustion of the chemical species. Figure 53 shows a possible example of a heat generation distribution. The continuous distribution can be approximated by a discrete distribution, as shown in Fig. 53, by relating the heat source in the vicinity of the center of heat source (burner location) by

94

FRANCIS H. R. FRAN~A ET AL.

FIG. 53. Heat source distribution in the vicinity of the center of the heat source.

SG, i+ 1,j -- fs " SG, i,j,

(80)

where the shape factor fs is assumed to be known previously from the expected shape distribution of the heat source. Similar relations can be applied for the other elements that are located in the neighborhood of the center of the heat source. In the reduced system of equations, heat sources are not among the unknowns of the system of equations, which contain only the emissive power in the medium. To insert the shape constraint information, Eqs. (75) and (51) (see Section V,C) can be combined to form a new equation to be added to the system: 4A~y4,i, j - ~

SkGi,jqo, k -- ~ Gk, lGi,jt4,k,l k,l

k

-+- (jx, i,j + --jx, i,j-) + (jy, i,j + --jy, i,j-) -I"

-fs 14A,x,4/+l,j-E l

SkGi+l,jqo, k -- ~ Gk, lGi+l,jt4,k,l

k

k,t

+ ( jx, i,j + -- jx, i,j ) + ( jy, i,j + -- jy, i,j -

-)].

(81)

As shown in Fig. 53, those elements not located in the immediate vicinity of the heat source center are assumed to be in radiative equilibrium, but it is possible to apply Eq. (81) to as many elements as needed to extend the heat source region. For the solution of this problem, the procedure of Section V,C is employed. The system of equations is formed by writing Eq. (54) for the N wall elements, Eq. (51) for each element located outside the HS region (setting

INVERSE DESIGN OF THERMAL SYSTEMS

95

sa = 0), and Eq. (81) for each element in the vicinity of the center of heat

source. Designating the number of center of heat source elements by FHS, then there are N + F - FHS equations because no equation is written for these elements. The unknowns of the system correspond to the F medium emissive powers. Once they are found, Eq. (51) is used to determine the heat source in the center-of heat source elements, and Eq. (81) is employed to find the heat source in the elements in the vicinity of the center of heat source elements. The results of Section V,C present heat source distributions in the HS region satisfying the two boundary conditions on the walls. However, an inspection of the heat sources shows that the given distribution can be difficult to obtain in practice. The reason for this is due mostly to the fact that no physical constraint was imposed on the heat sources. As they are, they just satisfy the energy balance equations. Suppose that instead of a continuous HS region, burners were to be located in specified locations, as shown in Fig. 54. There are a total of eight burners, and their locations correspond to the centers of heat sources, as indicated in Fig. 53. To simulate the functional distribution of sa, the shape relations can then be applied to the neighboring medium elements. For example, consider that (1) the neighbor elements located side by side to the center of the heat source are related by a factor f~ = 0.30 and (2) the elements located on the diagonal of the center of heat source are related by a factor fs = 0.09. These factors are imposed conditions and should be known from the expected or desirable shape distribution of the heat source. For this case, the number of equations is N + F - F H s = 64+ 2 5 6 - 8 = 312, whereas the number of unknown medium emissive powers

FIG. 54. Location of the eight burners in the two-dimensional enclosure, FHS--8. H / L = 1 . 0 , ~ - - a l l = 1 . O , e = 0 . 9 , N c n =0.1.

96

FRANCIS H. R. FRANOA

ET AL.

is F = 256. Therefore, because the number of equations is greater than the number of unknowns, the system of equations is overdetermined. This system cannot be solved to give an exact solution, but the SVD decomposition can still be applied to matrix A to find the least-squares solution of the problem. The singular values of the system are presented in Fig. 55. It can be seen that the singular values do not degenerate to such small values as for the problem in Section V,C. In fact, all the singular values are larger than 1.0 • 10 -3. An important consequence of this is that no singular value needs to be truncated to avoid round-off error effects or to assure convergence of the iterative solution (keeping ot = 0.1 and 71im = 10-6). Figure 56 presents the resulting heat source distribution in the enclosure. As can be seen, nearly all the heat source is located in the central elements (locations 3, 4, 5, and 6 in Fig. 54). The error of this solution can be estimated by using this heat source distribution as the input of the forward problem and checking the calculated net heat flux on the walls against the imposed one. The maximum error of the solution is still below 1.0%: 7max = 0 . 5 6 0 % . In this case, the error does not arise from the truncation of the singular values (as none of them had to be truncated to allow convergence of the solution), but from the fact that an overconstrained system of equations was solved.

"l 10-1 .................................................. ............. ~

............. q

(Yi

10-2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-3 0

. . . . . . . . . . . . . . . . . . . . . . . . 50 100 150 200 250 i

300

FIG. 55. Singular values o f coefficient matrix A for the o v e r c o n s t r a i n e d problem: m = 312, n = 256. H / L = 1.0,~/4 = 1.0, c = 0.9, NcR = 0 . 1 .

INVERSE DESIGN OF THERMAL SYSTEMS

97

F I ~ . 56. Dimensionless heat source in the elements shown in Fig. 54. SVD or minimumsquare solution, tw = 1 . O , H / L = 1.0,~/4 = 1.0, e = 0.9, NcR = 0 . 1 .

As most of the heat source is concentrated in the elements in the center of the enclosure (locations 3, 4, 5, and 6), it is interesting to solve the problem considering that the burners are located only on these four positions. The problem can be solved as before, but a brief inspection of the problem reveals that the heat sources on these elements have to be the same due to the symmetry of the problem. In addition, the energy conservation in the enclosure implies that the sum of the heat generated in the enclosure must be equal to the sum of the total heat flux on the surfaces. F r o m this, it is possible to find that the heat source on each of the center of heat source elements must be s6 = 300. For the elements located side by side and by the diagonal of the center of heat source elements, the heat sources should be sG = 0.30 x 300 = 90 and s6 = 0.09 x 300 = 27, respectively, as shown in Fig. 57. Using this heat source distribution as the input to find the heat flux on the surfaces, which is compared to the imposed heat flux in Fig. 58, it is found that the arithmetic average and m a x i m u m errors are still very satisfactory: 7avg = 0.4025% and Irmax - - 0 . 7 1 2 % . It is interesting that such an overconstrained problem can be solved with a small error, but this result

98

FRANCIS H. R. FRAN(~A ET AL.

FIG. 57. Dimensionless heat source in the elements located only in positions 3, 4, 5, and 6. H / L = 1.0,z~/= 1.0, e = 0.9, NcR =0.1.

cannot be taken for granted. For instance, consider that the four burners were positioned in locations 1, 2, 7, and 8 and that the previous heat source distributions were set to these elements. The maximum error of this design would be as high as 30.0%, as indicated in Fig. 58. The satisfactory solution obtained when the burners were located in the central elements (3, 4, 5, and 6) indicates that a useful way to design an enclosure with a minimum number of burners would be first solving the problem for a given set of burners and checking the locations where the heat sources are the greatest. Then, a new solution can be attempted by only allowing these burners and verifying whether the resulting, error is within the specified tolerance. B. IMPOSING A REDUCED NUMBER OF UNIFORM HEAT FLUX DEVICES

In the discussion of Section V,D,3, instead of the continuous heat flux distribution in the heater, the designer may be more interested in finding a set of devices, each one with uniform power input, that is able to satisfy the two conditions on the design surface. In the example presented in

99

INVERSE DESIGN OF THERMAL SYSTEMS

5.0

'

'

'

|

|

|

|

|

'

'

'

'

|

'

|

|

I

|

|

|

|

o

4.0

....................

~,,,

v""

::..-.........,. i

!

;

::

: ""*.

3.0 --qT 2.0

.............. i....i!iii"i"i"i'i...... :::

1.0 .................... i .................. i ...... ---

.... :

,

i

0.0 0.0

|

i

i

i

0.1

i

i

i

I

I

i

i

0.2

i

J

0.3

i

i

i

i

i

0.4

i

!

i

0.5

X

FIG. 58. Heat flux on the design surfaces for cases where the burners are located in positions 3, 4, 5, and 6, a n d i n positions 1,2, 7, and 8. H / L = 1.0,~H = 1.0, e = 0.9, NcR =0.1.

Section V,D,3, the power to each of the seven heating devices into which the heater was divided was obtained by averaging the heat flux distribution from the inverse solution across the equivalent region. Simply dividing the grid into uniform zones having the size of the desired heaters would lead to a solution with poor accuracy, as the assumption of uniform radiosity and emissive power across the entire heating device would be poor. A better procedure is to impose uniform power input within each zone that forms a particular heating device. The radiosity and temperature distributions are then nonuniform within each zone and are found from the inverse solution. For instance, considering two surfaces zones i and j in the same heater, as seen in Fig. 59, the following constraint is imposed: qT2, i - - q T Z , j .

(82)

In this problem, the inverse solution is achieved by the solution of the system formed by Eq. (75), where the unknowns are the radiosities of the zones on the heater. To add the information in Eq. (82) to the system of equations, and so imposing the uniform heat flux on each heating device, it is necessary to derive a relation between the radiosities from the constraint imposed on Eq. (82). This is given by first decomposing the heat flux into its radiative and conductive components:

100

FRANCIS H. R. FRAN(~A E T AL. uniform heat flux device

i j FIG. 59. Division of the heater system shown in Fig. 38 into devices having uniform heat flux input. Each device is subdivided into a number of surface zones.

qR2, i q- q c 2 , i - - q R 2 , j q- q c 2 , j .

(83)

The radiative heat flux is related to the radiosity and irradiation by (84)

q R 2 , i ~ qo2,i -- qi2,i,

where the irradiation of zone i is determined from: qi2, i -

Z skgiqo,,k + Z k - surface 1 k, l

Gk,tgit4,k,t +

S e l g i t e l4 if- S e 2 g i t e 24"

(85)

Equation (82) is then combined with Eqs. (83) and (84) for both the surface zones i and j to give qo2, i -- q o z , j --- qi2, i -- q i z , j -- ( q c 2 , i -- q c z , j ) .

(86)

Finally, Eq. (86) is written for every pair of zones that is contained in the same device and is added to the system of equations. This assures that the heat flux will be uniform across each zone after inverse solution. For each heating device, the number of equations to be added equals the number of zones that form it less one. Note that, in Eq. (86), the right-hand side, which forms the components of vector b, is unknown, but this does not add much additional difficulty to the problem in the sense that these same terms must be recalculated in any case for each new step of the iterative procedure to update the components vector b, as described in Section V,D,1. As an example, consider the problem of Section V,D,3 as depicted in Fig. 47. The heater surface (LI4/H- 4.2) is divided into a total of seven elements, where the five elements in the center have a uniform size of ALI-I/H = 0.7, and the elements on the left and right ends have sizes of ALI-I/H - 0.3 and 0.4, respectively. With the grid resolution of 50 x 20 elements, the entire heater is divided into 42 zones, which is the same number of unknown radiosities to be calculated from the system of equations. However, 30 equations are written for the design surface zones, and 35 relations like Eq. (86) are added to impose the uniform heat flux condition in each heating device. Therefore, the system of equations, having 65 equations against 42 unknowns, is overconstrained so the TSVD scheme can be used to provide the least-squares solution for the problem.

101

INVERSE DESIGN OF THERMAL SYSTEMS

Figure 60 presents the singular values of matrix A when the uniform heat flux constraint in the heating devices is added (under the nonuniform size label). The number of singular values is equal to the number of unknowns of the problem, i.e., 42. As seen, the singular values do not decay steeply to the small values (lower than 10-7) that the unconstrained problem present, as indicated in Fig. 45. Instead, the smallest singular value for this problem is close to 10-2 and is the same order as the minimum singular value retained in previous solutions for the iterative solution to converge. Table XVII presents the heat flux on the heating devices for different regularization parameters, p = 42, 41, 40, and 39, as well as the heat flux obtained from averaging the heat flux from the inverse solution (presented in Fig. 47). Table XVII also indicates the error and thermal efficiency of each solution. As seen, it is possible to solve this problem without truncating any singular value of the solution, p = 42. The heater power reaches a peak in the first heating device, and then decreases smoothly until the last device, where it raises again. The error of the solution is 7max = 0.089% and 7min--0"020O'/o. It is interesting that the solutions for p = 41 and 40 are

1.0E+02

1.0E+01

1.0E+00

(Yi

I

1.0E-01

-7

1.0E-02 r

1.0E-03

1.0E-04 0

I

. . . . .

i---

n o n u n i f o r m size

!~ . . . .

u n i f o r m size

I'~

I

I

I

I

q

r

I

I

I

I

I !

I !

10

20

30

40

50

FIG. 60. Singular values o f the coefficient m a t r i x A for the p r o b l e m where u n i f o r m heat flux is i m p o s e d on the h e a t i n g device elements. L / H = 5.0,Ln/H = 3.0, z/-/ -- 0.2, ~1 = e2 = 0.8, Re -- 2000, Pr -- 0.69, NcR = 7.60 • 10 -4.

102

FRANCIS H. R. FRAN~A

ET AL.

T A B L E XVII HEAT FLUX ON HEATING DEVICES FROM THE CONSTRAINED SOLUTION (p - 39 to 42) AND FROM AVERAGE SOLUTION a

Heating element

ALn/H

Size p -- 42

p = 41

p = 40

p = 39

Average heat flux

0.3 0.7 0.7 0.7 0.7 0.7 0.4

52.719 20.317 19.125 18.493 16.530 12.898 17.188

22.57 32.63 14.54 22.03 13.84 17.55 10.72

23.16 32.45 14.51 22.18 13.54 18.29 9.45

15.4 29.3 24.3 12.3 21.5 17.9 2.5

23.736 31.418 15.395 22.026 13.491 19.045 7.630

0.020 0.089 57.2

0.316 1.114 58.9

0.326 1.069 59.0

1.654 4.564 60.5

0.337 1.711 59.2

1 2 3 4 5 6 7 ~avg (%) ~max (%)

rl (%)

aHeating elements have nonuniform size. Heating elements are counted from left to right. L/H = 5.0, LD/H = 3.0, xn = 0.2, Cl = e2 = 0.8, Re = 2000, Pr = 0.69, NcR= 7.60 • 10 -4.

similar to the average heat flux of the unconstrained inverse solution. The errors in these inverse solutions also have the same order of magnitude, with maximum and average values of about 1.0 and 0.3%, respectively. The fact that the heat flux has an oscillatory behavior for these solutions is not a drawback, as a continuous solution is not necessary for the independent heating devices. The solution for a regularization parameter o f p - 39 leads to a larger error, which makes it less satisfactory. This trend will intensify if more singular values are truncated from the solution. Because thermal efficiency decreases with the increase in p (which results in greater accuracy), some decision is needed for the choice of the heat input on the heater. In the solutions shown in Table XVII, the heater was divided into nonuniform elements. The reason for this was that it allows the average heat flux to better reproduce the oscillations in the heat flux so the two end heaters had smaller sizes. Considering seven heating devices of uniform size, ALn/H = 0.6, the solution procedure can be repeated to determine the heat flux on each device. The singular values of this problem, as shown in Fig. 60, are close to those of the nonuniform size heating devices, having a minimum value of about 10 -2. The solutions for different regularization parameters (p - 42, 41, 40, and 39) also follow the same trends in terms of accuracy and thermal efficiency. Note that the average solution, calculated from averaging the continuous solution in the location of each device, has a larger error than the average solution for nonuniform devices. The possible reason for this is

103

INVERSE DESIGN OF THERMAL SYSTEMS

TABLE XVIII HEAT FLUX ON HEATING DEVICES FROM THE CONSTRAINED SOLUTION (p = 39 to 42) AND FROM AVERAGE SOLUTIONa Heating element

Size

ALI4/H 0.6 0.6 0.6 0.6 0.6 0.6 0.6

"/avg (%) 7max (%) 1] (%)

Average heat flux

p = 42

p = 41

p = 40

p -- 39

37.185 17.237 20.136 17.870 17.333 12.891 15.928

31.63 24.61 14.07 23.97 11.25 20.34 10.71

28.56 28.02 12.49 24.00 12.88 16.93 13.70

19.8 32.0 20.5 12.3 20.9 21.2 5.5

28.262 26.417 15.170 22.332 13.105 19.256 10.684

0.020 0.089 57.7

0.332 0.733 58.6

0.371 1.277 58.6

1.788 5.114 60.5

0.450 2.562 59.2

aHeating elements have uniform size. Heating elements are counted from left to right. = 5.0, LD/H = 3.0, ~n = 0.2, ~l = E2 = 0 . 8 , Re = 2000, Pr = 0.69, Ncn= 7.60 • 10 -4.

L/H

that the average heat flux in Table XVIII does not follow the same oscillations of the continuous solution shown in Fig. 47, which can be reproduced only if the heaters have a nonuniform size. The heat fluxes shown in Tables XVII and XVIII were obtained from a solution of overconstrained problems, which were solved in the least-square sense through the TSVD scheme. Therefore, all equations of the systems, including Eqs. (82), which impose constant heat flux in each device, can only be solved within some degree of approximation. The heat flux values in Tables XVII and XVIII are shown to as many decimal digits as remain unchanged for all the zones that compose the heating device. For p = 42, the heat fluxes are coincident up to three digits, which decays to two digits for p = 41 and 40, and one digit for p = 39. C.

F I N A L REMARKS ON IMPOSING ADDITIONAL CONSTRAINTS

The previous two sections discussed how to impose additional features to the inverse solution. Although the nature of the two problems considered was different, the proposed procedure is essentially the same. The additional characteristics are written as an algebraic relation that is then expressed in terms of the unknowns of the problem. In the heat source problem, the heat source distribution was expressed in terms of the unknown medium emissive power; in the inverse boundary problem, the uniform heat flux on each heater was derived in terms of the unknown heater zone radiosities.

104

FRANCIS H. R. FRANt~A E T

AL.

A first aspect of this method is that adding the new relations to the system of equations will probably cause the problem to become overconstrained, i.e., a system having more equations than unknowns. Such problems cannot be solved exactly, but an exact solution is not expected in general for an inverse design, which relies on regularization methods. Another important characteristic of the two considered problems is that the additional equations helped stabilize the problem in the sense that the singular values did not decay to the very small values found in the unconstrained problems. This can be verified by comparing Figs. 37 and 55 and Figs. 45 and 60. A major implication of this is that a satisfactory solution can be obtained without any regularization in the sense that none of the singular values are truncated from the solution. In this case, the error of the inverse solution is solely related to the fact that the problem is overconstrained. If imposing additional constraints can eliminate the use of truncation of the singular values, does it mean that regularization methods, whose discussion dominated most of the present work, can become superfluous after all? The answer is very likely n o . First, it should be noted that the overconstrained problem itself requires special methods of solution, which also can be regularization methods. In fact, the TSVD scheme was used in the two problems discussed earlier. Second even the problem with additional constraints allows regularization in the sense that fewer singular terms from the matrix of coefficients decomposition can be used to compute the vector solution and still render satisfactory answers. This is shown, for instance, in Tables XVII and XVIII. The nonregularized problem (p = 42) is the least efficient of the solutions, and so might not be the preferable one, even though it has the best accuracy. Inverse design, with all its inherent difficulties, often allows multiple acceptable answers. Not applying regularization methods would probably mean that a number of potentially satisfactory and/or unexpected solutions will not be considered.

VII. Conclusions A. THE CASE FOR INVERSE DESIGN

The examples of inverse design present some characteristics that are of great significance to a designer of a thermal system in which radiative transfer plays an important role. First, it is observed that multiple solutions to design problems are generated by the inversion technique and that the number of acceptable solutions grows as the allowable deviation from the imposed conditions is relaxed. Second, the characteristics of the multiple inverse solutions can be quite different from those that intuition and experi-

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ence might initially suggest. Third, these unexpected characteristics can be used to find practical designs that accurately meet the design requirements while moving toward design simplicity and practicality. These three factors are particularly apparent in the case of the heater designs for the duct with flow, where the complete inverse design led to simplified designs with fewer heating devices. Even these simple designs meet the imposed design constraints very well, but would be very difficult to obtain by trail-and-error solution. The solution of the ill-conditioned system of equations arising in inverse design requires some expertise in regularization techniques. However, a still greater level of expertise and judgment may be required to find a thermal system design that satisfies the two imposed conditions on the design surface through trial and error. Moreover, inverse design has the potential to become a useful, attractive feature in future CFD codes, which could be used by designers without much experience in inverse techniques. In such codes, the generation of different solutions from different regularizations could be implemented as a built-in algorithm (different results can be obtained by just setting different values of the regularization parameter, such as p in the TSVD, K in the conjugate gradient scheme, or ot in the Tikhonov approach). Then, for a given specification, the designer would be given a number of solutions, with one to be selected based on precision and practicality. Finally, one cannot claim that inverse techniques are inherently preferable to forward technique in all cases, or vice versa. As discussed extensively in this work, inverse techniques become most attractive when two conditions are specified on some boundaries of a thermal system. In such a case, the forward solution would rely on crude trial-and-error attempts, whereas inverse design is flexible enough to provide a number of valid answers. It is true that inverse design is still a recent technique in comparison to the forward/conventional technique; at this point, the problems that have been solved by inverse analysis do not possess the level of complexity that the conventional technique can handle. However, recent years have seen a great interest and development of inverse techniques, as indicated by the literature review and examples presented in this work, and further research can extend the range of applicability of the inverse design even further. B. AREAS FOR FURTHER RESEARCH

The major characteristics of inverse methods applied to the design of thermal systems with radiative transfer have been laid out in the preceding pages. There remain unanswered questions and areas of research for inverse analysis and design. These include in particular the question of existence;

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i.e., when the designer specifies a design set for a particular problem, does a physically acceptable solution exist that will satisfy this set within a given error tolerance? It would be useful to have a priori tests that answer this question before undertaking a detailed inverse analysis. Such tests might be based on requiring adherence within some tolerance to the first and second law; however, inverse problems by their very nature preclude straightforward checks of this kind, as the conditions on at least one boundary are unknown at the time of problem specification. These checks themselves result in inverse problems. Creative approaches to the question of practical existence present a potentially valuable research area. Some work in this direction has been reported by Landl and Andersson [56]. As discussed in Section Ill,A, there is much room for research and application of the available techniques to invert an ill-conditioned system of equations. For the simple examples presented in this chapter, it was noted that such techniques as TSVD, MTSVD, Tikhonov, and conjugate gradient lead to similar results, even though they impose different constraints to the regularized system. How this observation will apply to more complex problems is still an open question. The application of regularization techniques to nonlinear problems is also little explored. Here, the solution of inverse nonlinear problems was achieved by regularizing the matrix of coefficients and then using iterative techniques that have been used in forward problems. To optimize numerical convergence, the iterative scheme must be established in such a way that the terms of the dominant heat transfer mechanism form the matrix of coefficients of the system of equations, letting the other terms (related to the less important heat transfer modes) to be estimated and incorporated in the independent vector. The examples shown in this work were developed for the case of dominant thermal radiation. However, if conduction and/or convection mechanisms are dominant, different inversion techniques will probably need to be developed, as the structure of the matrix of coefficients is altered greatly. For instance, in formulating the channel flow problem in Section V,D, it was observed that an efficient solution could be obtained by separating the problem such that the terms describing the most important energy transfer affecting the design surface were determined by the inverse solution, whereas other terms were "lagged" in the iterative process. This approach reduced the computational load in the inverse solution greatly, as the ill-conditioned and the well-conditioned parts of the problem were separated from each other, allowing regularization (which imposes an error) to be applied only when needed. Such a procedure can be adapted to other nonlinear problems still to be solved.

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References 1. Hansen, P. C. (1998). "Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion." SIAM. Philadelphia. 2. Tikhonov, A. N. (1963). "Solution of Incorrectly Formulated Problems and the Regularization Method," Soviet Math. Dokl., 4, 1035-1038; Engl. trans. 1963, Dokl. Akad. Nauk. SSSR, 151, 501-504. 3. Alifanov, O. M. (1994). "Inverse Heat Transfer Problems," Springer-Verlag, Berlin. 4. Alifanov, O. M., Artyukhin, E. A., and Rumyantsev, S. V. (1995). "Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems." Begell House, New York. 5. Hansen, P. C. (1992). Numerical tools for analysis and solution of Fredholm integral equations of the first kind. Inverse Problems 8, 849-872. 6. Morizov, V. A. (1984). "Methods for Solving Incorrectly Posed Problems." SpringerVerlag, New York. 7. Oguma, M. and Howell, J. R. (1995). "Solution of Two- Dimensional Blackbody Inverse Radiation by an Inverse Monte Carlo Method." Proc. 4th ASME/JSME Joint Symposium, Maui, March. 8. Fausett, L. (1994). "Fundamentals of Neural Networks: Architecture, Algorithms and Applications," Prentice Hall. 9. Morales, J. C. (1998). "Radiative Transfer within Enclosures of Diffuse Gray Surfaces: The Inverse Problem." PhD dissertation, Department of Mechanical Engineering, The University of Texas at Austin, May. 10. Goffe, Ferrier and Rogers (1994). Global optimization of statistical functions with simulated annealing. J. Economet. 60, 65-100. 11. Corana, A., Marchesi, M., Martini, C., and Ridella, S. (1987). Minimizing multimodal functions of continuous variables with the "simulated annealing" algorithm. A CM Transact. on Mathemat. Software 13, 262-280. 12. Hendricks, T. J., and Howell, J. R. (1994). "Inverse Radiative Analysis to Determine Spectral Radiative Properties Using the Discrete Ordinates Method." Proc. 10th International Heat Transfer Conference, Vol. 2, pp. 75-80, Brighton, Aug. 13. Beck, J. V., Blackwell, B., and St. Clair, Jr. (1995). "Inverse Heat Conduction: Ill-Posed Problems." Wiley-Interscience, New York. 14. Huang, C. H., and Ozisik, M. N. (1992). Inverse problem of determining unknown wall heat flux in laminar flow through a parallel plate duct. Numerical Heat Transfer, A 21, 55-70. 15. Hsu, P.-T., Chen, C.-K., and Yang, K.-T. (1998). A 2-D inverse method for simultaneous estimation of the inlet temperature and wall heat flux in a laminar circular duct. Numerical Heat Transfer, A 34, 731-755. 16. Park, H. M., and Chung, O. Y. (1999). An inverse natural convection problem of estimating the strength of a heat source. Int. J. Heat Mass Transfer 42, 4259-4273. 17. Dunn, W. L. (1983). Inverse Monte Carlo solutions for radiative transfer in inhomogeneous media. J. Quant. Spectrosc. Radiat. Transfer 29, 19-26. 18. Matthews, L. K., Viskanta, R., and Incropera, F. (1984). Development of inverse methods for determining thermophysical properties of high-temperature fibrous materials. Int. J. Heat Mass Transfer 27, 487-495. 19. Wu, W. J., and Mulholland, G. P. (1989). Two-dimensional inverse radiation heat transfer analysis using Monte Carlo techniques. In "Heat Transfer Phenomena in Radiation, Com-

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27.

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33. 34. 35.

36.

37. 38.

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bustion and Fires" (R. K. Shah, ed.), ASME Publication HTD-Vol. 106. National Heat Transfer Conf., Philadelphia. Lin, J.-D., and Tsai, J.-H. (1991). "Comparison of P 1 and S-P Two Flux Approximations in Inverse Scattering Problems," ASME Paper 91-WA-HT-13, ASME, New York. Subramaniam, S., and Mengfiq, M. P. (1991). Solution of the inverse radiation problem for inhomogeneous and anisotropically scattering media using a Monte Carlo technique. Int. J. Heat Mass Transfer 34, 253-266. Tsai, J.-H. (1993). Inverse scattering problem with two flux methods. Int. Comm. Heat Mass Transfer 20, 585-596. Jones, M. R., Curry, B. P., Brewster, M. Q., and Leong, K. H. (1994). Inversion of lightscattering measurements for particle size and optical constants: Theoretical study. Appl. Opt. 33, 4025--4041. Jones, M. R., Tezuka, A., and Yamada, Y. (1995). Thermal tomographic detection of inhomogeneities. J. Heat Transfer, 117, 969-975. McCormick, N. J. (1997). Analytical solutions for inverse radiative transfer optical property estimation, In "Proc. ASME Heat Transfer Division," Vol. 3, pp. 367-371. Kudo, K., Kuroda, A., Ozaki, E., and Oguma, M. (1997). Estimation of absorption coefficient distribution in two-dimensional gas volume by solving inverse radiative property value problem. In "Radiative Transfer-II: Proc. Second Int. Symp. on Radiation Transfer" (M. P. Mengu~;, ed.). Begell House, New York. Li, H. Y., and Ozisik, M. N. (1992). Identification of the temperature profile in an absorbing, emiting, and isotropically scattering medium by an inverse analysis. J. Heat Transfer 114, 1060-1063. Siewert, C. E. (1994). An inverse source problem in radiative transfer. J. Quant. Spectrosc. Radiat. Transfer 50, 603-609. Siewert, C. E. (1994). A radiative-transfer inverse-source problem for a sphere. J. Quant. Spectrosc. Radiat. Transfer 52, 157-160. Li, H. Y. (1994). Estimation of the temperature profile in a cylindrical medium by inverse analysis. J. Quant. Spectrosc. Radiat. Transfer 56, 755-764. Li, H. Y. (1997). Inverse radiation problem in two-dimensional rectangular media. AIAA J. Thermophys. Heat Transfer 11,556-561. Linhua, L., Heping, T., and Qizheng, Y. (1999). Inverse radiation problem of temperature field in three-dimensional rectangular furnaces. Int. Comm. Heat Mass Transfer 117, 524-526. Liu, L. H., Tan, H. P., and Yu, Q. Z. (2000). Inverse radiation problem in axisymmetric free flames. J. Thermophys. Heat Transfer 14, 450-452. Liu, L. H., Tan, H. P., and Yu, Q. Z. (2001). Inverse radiation problem of sources and emissivities in one-dimensional transparent media. Int. J. Heat Mass Transfer 44, 63-72. Liu, L. H. (2000). Simultaneous identification of temperature profile and absorption coefficient in one-dimensional semitransparent medium by inverse radiation analysis. Int. Comm. Heat Mass Transfer 2% 635-643. Yousefian, F., and Lallemand, M. (1998). Inverse radiative analysis of high-resolution infrared emission data for temperature and species profiles recoveries in axisymmetric semi-transparent media. J. Quant. Spectrosc. Radiat. Transfer 60, 921-931. Ruperti, N. J., Raynaud, M., and Sacadura, J. F. (1996). A method for the solution of the coupled inverse heat conduction-radiation problem. J. Heat Transfer I18, 10-17. Erturk, H., Ezekoye, O. A., and Howell, J. R. (2000). Inverse solution of radiative heat transfer in two-dimensional irregularly shaped enclosures. In "Proc. 2000 IMECE", ASME HTD-Vol. 366-1, pp. 109-116.

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39. Harutunian, V. Morales, J C., and Howell, J. R. (1995). Radiation exchange within an enclosure of diffuse-gray surfaces: The inverse problem. In "Proc. ASME/AIChE 1995 National Heat Trans. Conf.," Portland. 40. Morales, J. C., Harutunian, V., Oguma, M., and Howell, J. R. (1996). Inverse design of radiating enclosures with an isothermal participating medium. In "Radiative Transfer I: Proc. First Int. Symp. on Radiative Heat Transfer" (M. Pinar Menguc, ed.), pp. 579-593, Begell House. 41. Fran~a, F., and Goldstein, L. (1996). "Application of the Zoning Method in Radiative Inverse Problems." Brazilian Congress of Engineering and Thermal Sciences, ENCIT 96, Florianopolis, Brazil. 42. Matsumura, M. (1997). "Optimal Design of Industrial Furnaces by Using Numerical Solution of the Inverse Radiation Problem," MS Thesis, Dept. of Mechanical Engineering, The University of Texas at Austin, May. 43. Matsumura, M., Morales, J. C., and Howell, J. R. (1998). "Optimal Design of Industrial Furnaces by Using Numerical Solution of the Inverse Radiation Problem." Proc. 1998 Int. Gas Research Conf., San Diego. 44. Yang, W.-J., Taniguchi, H., and Kudo, K. (1995). Radiative heat transfer by the Monte Carlo method, In "Advances in Heat Transfer," (J. P. Hartnett and T.F. Irvine, eds.), Vol. 27. Academic Press, San Diego. 45. Kudo, K., Kuroda, A., Eid, A., Saito, T., and Oguma, M. (1996). Solution of the inverse radiative load problems by the singular value decomposition. In "Radiative Transfer-I: Proc. First Int. Symp. on Radiation Transfer" (M.P. Mengug, ed.), pp. 568-578. Begell House, New York. 46. Franga, F., Lan, C., and Howell, J. R. (1999). "Inverse Design of Energy and Environmental Systems." Invited keynote lecture, Proc. US-Republic of South Africa Workshop on Energy and The Environment, Durban, RSA, June, 1998, and in Energy and the Environment" (A. Bejan, P. Vadasz, and D. Kroger, eds.), 65-74. Kluwer Academic, Dordrecht. 47. Jones, M. R. (1999). Inverse analysis of radiative heat transfer systems. J. Heat Transfer 121, 481-484. 48. Fran~a, F., Oguma, M., and Howell, J. R. (1998) "Inverse Radiative Heat Transfer with Nongray, Nonisothermal Participating Media" (R. A. Nelson, T. Chopin, and S. T. Thynell, eds.), 1998 IMECE, pp. 145-151. 49. Franga, F., Ezekoye, O. A., and Howell, J. R. (1999). "Two-Dimensional Inverse Heat Load Problem For Combined Radiation and Conduction." Proc. 1999 IMECE, Nashville, November. 50. Lan, C. H., and Howell, J. R. (1998). "Numerical Bifurcation and Chaos in a OneDimensional Nonlinear Combined Mode Problem and the Inverse Application" (R. A. Nelson, T. Chopin, and S. T. Thynell, eds.), 1998 IMECE, pp. 109-115. 51. Franqa, F. R., Ezekoye, O. A., and Howell, J. R. (2001). Inverse boundary design combining radiation and convection heat transfer. J. Heat Transfer 123, 884-891. 52. Morales, J. C., Matsumura, M., Oguma, M., and Howell, J. R. (1997). "Computation of Inverse Radiative Heat Transfer Within Enclosures," Proc. 1997 ASME National Heat Transfer Conference, Baltimore. 53. Franga, F., Morales, J. C., Oguma, M., and Howell, J. R. (1998). Inverse design of radiating systems dominated by radiative transfer. In "Radiative Transfer II:-Proc. Second Int. Symp. Radiative Heat Transfer" (M. Pinar Mengtig, ed.), Begell House. 54. Franqa, F., Morales, J. C., Oguma, M., and Howell, J. R. (1998). Inverse radiation heat transfer within enclosures with participating media. In "Proc. 1998 1lth Int. Heat Trans. Conf.," (J. S. Lee, ed.), vol. 7, pp. 433-438, KyongJu, Korea, August.

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55. Smith, T. F., Shen, Z. F., and Friedman, J. N. (1982). Evaluation of coefficients for the weighted sum of gray gases model. J. Heat Transfer 104, 602-608. 56. Landl, G., and Anderssen, R. S. (1996). Non-negative differentially constrained entropylike regularization. Inverse Problems 12, 35-53

ADVANCES IN HEAT TRANSFER, VOL. 36

Advances in Temperature Measurement

P. R. N. CHILDS Thermo-Fluid Mechanics Research Centre University Of Sussex Brighton, United Kingdom BN1 9QT

Abstract

The need for temperature measurement is ever present in science and industry from requirements for monitoring processes, in the management of quality control and research. Temperature can be measured by means of direct contact between the medium of interest and the measuring device or by remote observation of a temperature-dependent parameter. The range of devices with which temperature can be measured is extensive, not surprisingly because most physical parameters exhibit a dependency on temperature. In recent years the dominant position of liquid-in-glass and bimetallic thermometers, thermocouples, and resistant temperature detectors has been challenged as the common choice for temperature measurement by infrared thermometers and an increasing array of other noninvasive techniques. This chapter outlines the principal techniques available for the measurement of temperature, describing the physical phenomena exploited, the temperature range of use, equipment required, and typical applications. Recent trends in requirements for traceability and quantification of uncertainty, as well as developments in the areas of instrumentation capability and technique, are described. 9 Elsevier Science (USA). I. Introduction

A.

TEMPERATURE

SCALES

Temperature can be defined as the degree of hotness or coldness of a body. It is the property that determines whether a system is in thermal equilibrium 111 ISBN: 0-12-020036-8

ADVANCES IN HEAT TEANSFER, VOL. 36 Copyright 2002, Elsevier Science (USA). All rights reserved. 0065-2717/02 $35.00

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with other systems [1]. If the temperature of two bodies in thermal contact with each other is the same, then there will be no net transfer of thermal energy between the two bodies. Quantitatively, temperature is defined from the second law of thermodynamics in terms of the rate of change of entropy with energy. In order to allow the assignment of numerical values to bodies at different temperatures, some form of temperature scale is necessary. The unit of the thermodynamic temperature scale is the kelvin with the symbol K. This is defined in terms of the interval between absolute zero, 0 K, and the triple point of pure water, 273.16 K. The kelvin is defined as the fraction 1/273.16 of the temperature of a system exhibiting the triple point of water. Other temperature units are in common use, including Celsius, Fahrenheit, and Rankine temperature scales. Conversion equations for these are given as t = T - 273.15

(1)

1.8t + 32

(2)

To F =

To R - - To F _qL 459.67.

(3)

The thermodynamic temperature scale is defined by means of theoretically perfect heat engines. These are not practically realizable, and the International Temperature Scale of 1990, denoted by ITS-90 (see Preston Thomas [2]), was set up to be a practical best approximation using available technologies to the thermodynamic temperature scale. Its range of application extends from 0.65 K up to the highest temperature practically measurable using Planck's law of thermal radiation. The ITS-90 is believed to represent thermodynamic temperature to within -t-2 mK from 2 to 273 K, +3.5 mK at 730 K, and +7 mK at 900 K (one standard deviation limit; see Mangum and Furukawa [3]). The ITS-90 is constructed using a number of overlapping temperature ranges. This leads to some ambiguity, albeit small, in the true value of a temperature but allows greater flexibility in the use of the scale. The ranges are defined between repeatable conditions using a variety of specified materials at their melting, freezing, and triple points. The ITS-90 was adopted by the International Committee of Weights and Measures in 1989. It was developed under the auspices of the Metric Treaty and the associated consultative committees, BIPM (Bureau International des Poids et Mesures), CIPM (Comit6 International des Poids et Mesures), and CCT (Comit6 Consultatif de Thermom6trie) and has been adopted almost universally. The principal role of the ITS-90 in practical measurement is to allow a means of traceability. Traceability, in the context of temperature, is the process by which a measurement made using a temperature-measuring device can be related to the ITS-90. It is unlikely that a thermometer utilized in defining the ITS-90 would actually be used to measure the temperature of a system of interest. Instead, some other thermometer would be used that may

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ITS-90 )

National Standards Laboratory

Laboratory Standards

(

3~

(Accreditation Service)

ustomer evices )

FIG. 1. Recommended practice to ensure traceability of the measurement of temperature to the international temperature standard, ITS-90.

have been calibrated against another device, which has itself been calibrated using guidelines specified in ITS-90. The schematic given in Fig. 1 illustrates this process. The chain between the thermometer in use and the ITS-90 may, in practice, have more links than those shown in Fig. 1, with each link tending to increase the uncertainty associated with the measurement. B. OVERVIEW OF METHODS

Temperature cannot be measured directly. Instead, its measurement requires the use of a transducer to convert a measurable quantity to temperature. The terms sensor and transducer are used commonly in discussions on instrumentation. The moniker sensor is used here to describe the temperature measuring device as a whole, whereas the term transducer is used to define the part of the sensor that converts changes in temperature to a measurable quantity. Examples of temperature sensors and the associated transducer property used and the measured quantity are listed in Table I. As can be seen from Table I, a wide variety of sensors are available and within each type listed a further selection is usually available. C. SELECTION CRITERIA

The selection of a specific sensor for a temperature measurement can become a complex activity requiring consideration of a number of aspects, including uncertainty, temperature range, thermal disturbance, level of contact, size of the sensor, transient response, sensor protection, availability, and cost.

1. Uncertainty The difference between a measurement and the true value of temperature is known as the error. The term uncertainty is used to quantify confidence in

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TABLE 1 EXAMPLESOF TRANSDUCERPROPERTIESAND PHYSICALPROPERTYMEASUREDFOR SOME TEMPERATURESENSORS

Sensor

Transducer property

Physical quantity measured

Liquid-in-glass thermometer Constant volume gas thermometer Constant pressure gas thermometer Bimetallic strip Thermocouple Platinum resistance thermometer Thermistor Transistor Capacitance thermometers Noise thermometers Quartz thermometers Paramagnetic thermometers NMR thermometers Thermochromic liquid crystals Thermographic phosphors Heat sensitive crayons and paints Pyrometric cones Infrared thermometer Thermal imager Schlieren Interferometry Line reversal Absorption spectroscopy Emission spectroscopy Rayleigh scattering Raman scattering CARS (Coherent anti-Stokes Raman scattering) Degenerative four wave mixing Laser-induced fluorescence Speckle methods Acoustic thermography

Thermal expansion Thermal expansion Thermal expansion Thermal expansion Seebeck effect Electrical resistance Electrical resistance Electrical resistance Electric permittivity Johnson noise Vibration Vibration Vibration Reflection Fluorescence Chemical Chemical Thermal radiation Thermal radiation Density Density Density Absorption Emission Elastic scattering Inelastic scattering Scattering Scattering Emission Interference Sound velocity

Length Pressure Volume Length Voltage Resistance Resistance Voltage Capacitance Power Time Inductance Inductance Color Intensity Color Shape Radiation intensity Radiation intensity Wavelength Interference fringes Color Wavelength Wavelength Wavelength Wavelength Wavelength Wavelength Time Length Time

the measurement indicated by a sensor. For instance, a sensor may be supplied with a 95% confidence interval uncertainty of + 1.5~ This would mean that, provided the device does not disturb the temperature distribution in the medium of interest, measurements made will be within 1.5~ of the true value 95% of the time or 19 times out of 20. The term uncertainty is used in preference to accuracy, as the latter should generally be reserved for qualitative use.

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2. Temperature Range The range of application, defined as the span from the minimum temperature of operation to the maximum temperature for a particular measurement technique or for a specific instrument, is a critical parameter in selection. Some types of measurement techniques, such as thermocouples are suitable for use over a wide temperature range, in this case from approximately - 2 7 3 to 3000~ but this can only be achieved using several different types of thermocouples, as no single thermocouple is suitable for the entire range. Figure 2 illustrates the approximate temperature range capability for a variety of methods.

3. Thermal D&turbance The installation of a measurement probe or sensor in a medium of interest, unless its temperature already matches the medium, will distort the temperature distribution as heat is transferred between the application and the instrument. If a sensor is installed in an application, permanently, then unless the thermal properties of the instrument and those of the application are the same, such as specific heat capacity, thermal conductivity, and density, then the thermal distribution in the application will be modified in comparison to the undisturbed case.

4. Level of Contact The principle of temperature measurement is to allow the measurement transducer to reach an acceptable level of thermal equilibrium with the medium of interest. This can be achieved by direct contact between the medium of interest for which the temperature is required and a measurement probe containing the transducer. This kind of instrumentation can be classified as invasive or contact instrumentation. In this case, heat transfer will be predominantly by means of conduction and possibly convection. Heat transfer by radiation can also be exploited to enable temperature measurement. In this case, the sensor can be located some distance from the application, provided there is an optical path between the two. Measurement where the sensor is located some distance from the application without causing significant disturbance can be classified as noninvasive.

5. Sensor Size The size of a measurement probe determines the level of thermal disturbance caused, the speed of response, and the volume for which an indication

liquid in glass thermometers bimetallic strips gas thermometry thermocouples RTDs thermistors semiconductor ICs noise thermometry capacitance thermometry quartz thermometry paramagnetic and NMR thermometry pyrometric cones thermographic phosphors thermochromic liquid crystals infrared thermometry schlieren, shadowgraph and interferometry line reversal absorption spectroscopy emission spectroscopy Rayleigh scattering Raman scattering CARS laser induced fluorescence acoustic thermography speckle methods -2;3 I ' ' ' ~ ' ' '2~0'

II

I

' '4~0'

' '6~0'

' '800'

' 'ld00'

' '12100' ' '14~)0' ' '1~00' ' '18100' ' '2~00' ' '22100. ' '24100' ' '2~00' ' '28100' ' '3d00

Temperature (%)

FIG. 2. Approximate ranges of capability for a variety of temperature measurement techniques.

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117

of temperature is given. Often a point measurement is required, in which case the smaller the probe volume the better. The speed of response is a function of the specific heat capacity, mass of the probe, thermal boundary conditions, and the speed of response of the associated processing system. In addition to the size of the probe, the size of the processing system should also be considered to ensure that it can fit in the available space appropriately and that any display is of a sensible size for the application.

6. Transient Response In many applications, the temperature cannot be considered constant and varies appreciably with time. In such circumstances if a temperature probe is installed in close thermal contact with an application, then the temperature indicated by the probe will follow that of the application but with a phase lag and amplitude difference. It is normally desirable to minimize the phase lag and temperature difference, which can be achieved most expediently by minimizing the size of the measurement probe. Even if the temperature of an application is constant and then if a measurement probe is brought into thermal contact with it, it will take a period of time for the temperature indicated by the probe to approach that of the application. The speed of response of a sensor is dependent on the thermal properties of the probe containing the transducer, its size, and the thermal boundary conditions and also on the speed of response of the associated processing system and display. Transient heat transfer can occur due to any combination of conduction, convection, and radiation. In the case of a measurement probe exposed to a convective boundary condition, the probe can often be modeled by the lumped capacitance method, and the speed of response of the system is governed by the time constant, ~ = p Vcp/hAs. For a time constant calculated in this way at a time equal to ~, 3~, and 5~, the output of the sensor will have reached a level of 63.2, 95, and 99%, respectively, of the input conditions, in this case the temperature of the application.

7. Sensor Protection Sensors can be damaged by reducing or oxidizing atmospheres, high or low temperatures, exposure to low or high pressures, impact, and vibration. A given application should be studied in order to identify whether the sensor, including the probe and the sensing and display elements, will operate adequately. In certain applications it will be necessary to isolate the probe from the environment, which can often be achieved by use of thermowells and protection tubes. The availability of these components is widespread and the

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range is extensive. Details of appropriate materials to be used for a wide range of environments are listed, in the case of thermocouples, elsewhere [4].

8. Availability The availability of a temperature measurement solution should be considered. If a ready replacement is required, then an off-the-shelf system may be the only solution precluding the use of some of the more advanced noninvasive methods or exotic material solutions. Some methods, coherent anti-Stokes Raman scattering (CARS), for example, can require some months of setup and commissioning and expert guidance prior to use. Such solutions are not normally available for instant use. Other classes of instrumentation, such as standardized thermocouples, thermistors, some PRTs, and infrared thermometers, are available readily and, if damaged, a replacement can be sourced easily.

9. Cost Cost is the ever-present driver in the majority of business-related activities. Is a measurement necessary, will it be used, and does the cost of the one-off or ongoing measurement warrant the initial and recurrent investment? These questions are common sense but always necessary. The cost of infrared methods, for example, relative to resistance thermometry and thermocouples has reduced dramatically during the last two decades. One example is the use ofnoncontact infrared handheld devices to monitor food temperatures. These methods are now typically just two or three times the first cost price in comparison to resistance temperature detectors, but can be less expensive when considering the lack of need to sterilize or replace measurement probes and the time reqlfired to take the measurement. An additional advantage is that infrared techniques reduce the probability of bacteria contamination in this case.

II. Measurement Methods

A. INVASIVEMETHODS

1. Liquid-in-Glass Thermometers Liquid-in-glass thermometers do not require an external power supply and can be relatively inexpensive. Use of a glass stem means that they are relatively stable in a wise variety of chemical environments. Their disadvantages include fragility and the lack of remote logging capability.

ADVANCES IN TEMPERATURE MEASUREMENT

l 19

Liquid-in-glass thermometers exploit the higher expansion of liquids with temperature in comparison with that of solids. Although there is a large variety of liquid-in-glass thermometers, there are some common features, including the bulb, stem, thermometric liquid, and temperature-indicating scale. In the case of a solid stem thermometer, the bulb is usually a thin glass container with 0.35- to 0.45-mm-thick walls holding a themometric liquid such as mercury, ethanol, pentane, toluene, or xylene. The choice of the liquid depends on the desired temperature range, which for mercury, ethanol, pentane, toluene, and xylene are -35-510~ -80-60~ -200-30~ - 8 0 to 100~ and - 8 0 to 50~ respectively. The bulb is connected to a glass stem that contains a small-bore capillary tube. The space above the thermometric liquid can be evacuated or filled with an inert gas. As the temperature of the liquid in the bulb rises, the liquid will expand and some of it will be forced up the capillary. The temperature of the bulb is indicated by the position of the top of the meniscus, in the case of mercury, against markings engraved on the stem. The range of usefulness of liquid-in-glass thermometers is from approximately -196 to 650~ although no single instrument is capable of use across the whole range because of the limitations of the thermometric liquids. Liquid-in-glass thermometers can be calibrated at a number of fixed points and a scale subsequently applied to a stem supporting the capillary tube to indicate the range and value of temperature. The uncertainty of an industrial glass thermometer depends on the actual device with values ranging from 0.01 to 4~ (see, e.g., [5]). Uncertainty levels of as little as 0.005~ can be achieved with laboratory glass thermometers. A variety of considerations and error sources exist in the use of liquid-in-glass thermometers. 1. Nonuniformities in manufacture of the capillary bore, which if the thermometer has been calibrated at fixed points will mean that the thermometer may indicate correctly at these values but not in between. 2. Changes in the glass structure known as secular change (see Liberatore [6]). 3. Temporary depression of the temperature reading while the glass structure responds to the thermal conditions (see Van Dijk et al. [7]). 4. Immersion effects to account for differences in the temperature of the thermometric liquid in the bulb and along the capillary (see Nicholas and White [8]). 5. Pressure effects expanding or contracting the bulb [5]. 6. The tendency for a thermometric liquid to adhere to the capillary tube, known as stiction, and indicate a false reading. 7. Construction faults and malfunctions.

120

p.R.N. CHILDS

General use of liquid-in-glass thermometers is reviewed by Ween [9], Wise [10], Nicholas and White [8], and Nicholas [11]. A number of standards are available for liquid-in-glass thermometers [5, 12-22]. Mercury-in-glass thermometers are increasingly being replaced by relatively inexpensive resistance devices, which have cost and environmental advantages; by more expensive infrared devices, giving a digital readout and noninvasive measurement; or by thermally sensitive paint devices, which give an obvious visible indication of temperature. 2. Bimetallic Thermometers

Bimetallic thermometers consist of two strips of materials that are bonded together, which on heating will deform due to a mismatch of the coefficient of linear expansion between the two materials. If one end is fixed, then a needle mounted on the other end can be used to indicate the temperature against a calibrated scale. In order to maximize bending of the assembly, materials with significantly different coefficients of expansion can be selected. Table II lists some of the materials used in bimetallic thermometers. As the name implies, metals are usually used, but in principle, any dissimilar TABLE II PROPERTIES OF SELECTED MATERIALSUSED IN BIMETALLICELEMENTSa

Material

Density

Young's modulus

Heat capacity

Coefficient of thermal

Thermal conductivity

(kg/m 3)

(GPa)

(J/kg K)

expansion 10-6K -1

(W/m K)

2700

61-71

896

8954 7100 19300 7870 8906 10524 7304 4500 19350 8000 2340 2328 2300 3100 2200

129.8 279 78.5 211.4 199.5 82.7 49.9 120.2 411 140-150 113 130-190 150-170 304 57-85

A1

Brass Cu Cr Au Fe Ni Ag Sn Ti W Invar(Fe64/Ni36) Si n-Si p-Si Si3N4 SiO2

aAfter Stephenson

et

383.1 518 129 444 446 234.0 226.5 523 134.4 703 700 770 600-800 730

al. [23]; data from Meijer [24, 25].

24 19 17 6.5 14.1 12.1 13.3 19.1 23.5 8.9 4.5 1.7-2.0 4.7-7.6 2.6 3.0 0.5

237 386 94 318 80.4 90 419 64 21.9 163 13 80-150 150 30 9-30 1.4

ADVANCES IN T E M P E R A T U R E M E A S U R E M E N T

121

materials could be utilized and the moniker bimaterial thermometer is sometimes used. The use of ceramics and semiconductors to form the differential strip, for example, has been demonstrated by O'Connor [23]. The general equations for defining the curvature of a bimetallic strip are developed in Timoshenko [24] and have been reviewed by Stephenson et al. [25]. A number of standards are available for bimetallic materials and thermometers [28-34]. The bimetallic strip can be coiled in a spiral or helical configuration in order to provide increased sensitivity for a given volume and this form is used commonly in dial thermometers. Dial bimetallic thermometers tend to be rugged devices and have the advantage that they do not need an independent power supply. A variety of options are typically available for attaching the instrument to an application, including magnetic bases and clips. The uncertainty of typical commercial bimetallic thermometers is 1 to 2% of the full scale deflection with an operating range o f - 7 0 to 600~ The theoretical limit of operation is from about -270~ to the elastic limit of available materials. A long length and thin section can be used for the bimetallic element in order to provide high sensitivity. A device with a sensitivity of 0.0035~ and a repeatability of 0.027~ was demonstrated by Huston [35]. Although bimetallic thermometers can be read easily, can be used to both as an indicator of temperature or as an actuator, are relatively inexpensive, and do not require an independent power supply, they are subject to drift; measurements are usually relatively uncertain in comparison to say thermocouples and industrial platinum resistance thermometers and cannot provide a remote indication of temperature. 3. Manometric Thermometry

Manometric thermometry refers to techniques that utilize the measurement of pressure in order to determine temperature. There are two principal categories: gas thermometry and vapor pressure thermometry. Gas thermometry is based on the ideal gas law: p V = n~T,

(4)

where p is pressure (N/m2), V is volume (m3), n is number of moles of the gas [n = m / M (m is mass, M is molar mass)], ~ is the universal gas constant (=8.31 J/mol K), and T is temperature (K). By assuming values for the quantity of mass and for the gas constant, the temperature is obtained from a measurement of pressure and/or volume. The basic components of a gas thermometer are enclosures to contain the gas sample of interest under carefully controlled conditions and a flow circuit to allow the pressure to be measured. Although gas thermometers

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P.R.N. CHILDS

to manometer pressure diaphragm <

by-pass valve

connecting tube I

capillary

@ cryostat . . . .

FIG. 3. Principal components of a constant volume gas thermometer.

can be used at temperatures up to 1000 K, the principal application has been in cryogenics, and the gas enclosure, commonly known as the bulb, is usually located within a cryostat. Figure 3 illustrates the principal components of a constant volume gas thermometer. The range of application of gas thermometers is from a few kelvins to approximately 1000 K. Gas thermometers are not generally available commercially and component parts are usually sourced from a variety of manufacturers and a system assembled to produce a gas thermometer, although some companies, such as those specializing in cryostats, will assemble a gas thermometry system for a client. Gas thermometry tends to be a specialist activity and is usually confined to standards laboratories and cryogenic applications. Direct use of Eq. (4) requires knowledge of the gas constant. In order to reduce uncertainty arising from uncertainty in the gas constant, a number of methods have been devised that eliminate the need for knowing it operating on the principle of maintaining either a constant pressure or a constant volume and/or a constant bulb temperature. Such techniques include absolute PV isotherm thermometry, constant volume gas thermometry, constant pressure gas thermometry, constant bulb temperature gas thermometry, and two bulb gas thermometry. Unfortunately, real gases do not behave exactly according to the ideal gas equation. The nonideal nature or real gases can be modeled using the virial equation

(~T B(T) C(T) D(T) p=

~+

V2

+

V3

+

V4

) + .......

,

ADVANCES IN TEMPERATURE MEASUREMENT

123

where B ( T ) , C ( T ) , and D ( T ) are the second, third, and fourth virial coefficients. Virial coefficients have been evaluated for a number of gases at various temperatures (see, e.g., Aziz et al. [36], Berry [37], Kemp et al. [38], Matacotta et al. [39], Steur et al. [40], Steur and Dureiux [41], Astrov et al. [42], and Luther et al. [43]). Manometric thermometry depends on the determination of pressures and, in some cases, volumes of the enclosures used. A variety of corrections are made commonly to reduce uncertainty in the measurement, including 9 accounting for the dead space in connection tubes (see Berry [37]), 9 compensating for thermal expansion of the gas bulb (see Pavese and Steur [44], Mangum and Furukawa [3]), 9 accounting for the difference in density of the gas at different levels in the pressure sensing tubes (see Berry [37], Pavese and Steur [44]), 9 a thermomolecular pressure correction to account for temperature differences along the pressure-sensing tube (see Guildner and Edsinger [45], Berry [37], McConville [46], Weber and Schmidt [47]), 9 accounting for the absorption of impurities in the gas (see Berry [37], Pavese and Steur [44], Gershanik et al. [48]). The uncertainty of gas thermometry is a function of the care taken and the temperature range. Pavese and Steur [44] reported an uncertainty of 0.5 mK for measuring the temperature between 0.5 and 30 K. The saturation vapor of a pure substance above its liquid phase varies only with temperature and is known with low uncertainty for a number of cryogenic liquids, including helium 3, helium 4, hydrogen, neon, nitrogen, oxygen, argon, methane, and carbon dioxide. Measurement of the vapor pressure can therefore be used to determine temperature, and this technique provides a method with good sensitivity and requires relatively simple equipment in comparison to other competing techniques, such as noise thermometry (see Section II,A,8). Equations quantifying the relationship between pressure and temperature for a variety of cryogenic liquids are listed in Pavese [49], and vapor pressure thermometry is described in detail by Pavese and Molinar [50]. 4. Thermocouples

Thermocouples remain one of the workhorses of temperature measurement. They are in widespread use, with applications from -272~ to over 2000~ and are available as off-the-shelf devices. In its simplest practical form, a thermocouple can consist of two dissimilar wires connected together at one end with a voltage measurement device across the free ends. A net emf due to the Seebeck effect will be indicated by the voltmeter, which is a function

124

v.R.N. CHILDS

of the temperature difference between the join and the voltmeter connections. The merits of thermocouples are their relatively low cost, small size, rugged nature, versatility, reasonable stability, reproducibility, reasonably low uncertainty, and fast speed of response. Although platinum resistance thermometers have lower uncertainty and are more stable and thermistors are more sensitive, thermocouples are generally a more economical solution and their temperature range is greater. In addition thermocouples are more attractive for multiple point measurement applications, and a large number of data acquisition systems are available for thermocouples. The main disadvantage of thermocouples is their relatively weak signal; approximately 4.1 mV at 100~ for a type K thermocouple. This makes their reading sensitive to corruption from electrical noise. In addition, their output is nonlinear and requires amplification, their calibrations can vary with contamination of the thermocouple materials, cold-working and temperature gradients. The theory and use of thermocouples have been widely reported, and the reader is referred to Kerlin [51] and Ref. [4] for details of their practical use and to Pollock [52] for a specialist overview of thermoelectric phenomena. 5. Resistance Temperature Devices

The resistance of a material is a function of temperature, and this phenomena is exploited by a class of instruments known as resistance temperature detectors (RTDs). Types of RTD include platinum resistance thermometers (PRTs), thermistors, and a variety of semiconductor-based instruments. RTDs are used in a wide range of applications. Some types, such as PRTs, can have very low uncertainty, better than 0.002 K [3], whereas others, such as thermistors and some semiconductor-based devices, can be inexpensive in comparison to say thermocouples. Some RTDs involve the measurement of resistance across a length of material, usually metal but germanium and carbon are used in some cryogenic (<273 K) applications. Any metal could, in principle, be used, but stability considerations normally limit the choice to platinum, copper, and nickel. Copper and nickel are useful in the ranges -196-120~ and - 2 1 2 350~ respectively [53] and have the merit of low price in comparison to platinum. Platinum is, by comparison, more chemically stable and tends to be the preferred material for the majority of metal-based RTDs. PRTs are used to define part of the ITS-90, and its use is considered in Section II,A,5,a. Carbon, ruthenium, and germanium are used in some cryogenic applications; these devices are described in Section II,A,5,b. Thermistors and semiconductor devices are described in Sections II,A,5,c and II,A,6, respectively.

ADVANCES IN TEMPERATURE MEASUREMENT

125

a. Platinum Resistance Thermometers Platinum resistance thermometers consist of a length of platinum, connection wires, and a means of measuring the resistance. The resistance element may be coil of fine wire or a track of platinum deposited on a surface. The design of the resistance element depends on the requirements of the thermometer. Most materials tend to expand and contract with temperature, which induces strain in a material. This can affect a PRT in a number of ways. Expansion of a track of platinum can alter the cross-sectional area available for conduction and therefore alter the resistance. In addition, repetitive strain cycling can modify the microscopic structure of the platinum and therefore the resistance. A further compounding effect is any variation in the electrical properties of the insulation and connection wires. Platinum resistance thermometers fall into a number of broad categories. 9 Standard platinum resistance thermometers (SPRTs), which are used for low uncertainty measurements. 9 Industrial platinum resistance thermometers (IPRTs), which are used for practical laboratory and industrial use. 9 Secondary SPRTs, which are designed for laboratory environments and limited temperature ranges while still providing low uncertainty measurements. The relationship between resistance and temperature can be approximated by Rt = Ro(1 + at),

(6)

where Ro is resistance at 0~ (f}), Rt is resistance at temperature t (1~), a is the temperature coefficient of resistance (~ and t is temperature (~ The temperature coefficient of resistance, a, is calculated from oL=

RlOO -

Ro

100 ~ C x R0'

(7)

where R100 is resistance at 100~ (f}). Much of the original work on resistance thermometry was undertaken by Siemens [54] and Callendar [55-57]. Callendar found that the resistance of platinum could be described fairly accurately by a quadratic in the form Rt = Ro(1 + At + Bt2),

(8)

where A and B are constants. This equation has traditionally been written in the form given in Eq. (9), which simplifies the calculations necessary to determine the calibration constants oL and g. Equation (9) is known as Callendar's equation.

126

P.R.N. CHILDS

Rt

-

Roll-+-~t+~8(~O0)(1- 1--~0)]'

(9)

where 8 is a constant. For temperatures below 0~ additional terms are needed and the temperature resistance characteristic is defined relatively accurately by the Callendar-Van Dusen equation [Eq. (10)], which was the basis of the now superceded International Practical Temperature Scales of 1927, 1948, and 1968:

Rt -- R0(1 + At + Bt 2 + C ( t - 100)t3).

(10)

Here C is a constant and is zero above 0~ Typical values for the coefficients in Eq. (10) for a standard PRT and an industrial PRT are listed in Table III. To represent the resistance temperature relationship of a given SPRT by a polynomial with low uncertainty, the necessary resulting degree would exceed the number of fixed points available. As a result, the behavior of an SPRT is defined in the ITS-90 using a reference function that defines the intrinsic functional dependence of the resistance, R, on the temperature, T, and a deviation function that expresses the difference between the calibration for an individual PRT and the reference function. Standards [58-60] have been produced for PRTs. Values for resistance and the corresponding temperature can be interpolated readily in order to determine a given reading. Two classes of uncertainty are defined for IPRTs: class A devices where the uncertainty is within +(0.15 + O.O02/t/) and class B devices with an uncertainty +(0.3 + 0.005/t/). The design of a PRT is aimed at ensuring that the sensing element responds to the temperature of the application while being unaffected by other environmental factors, such as corrosive liquids and gases, vibration, pressure, and humidity. The most significant concern in the design of a PRT is possible strain of the platinum element due to mechanical shock or thermal expansion. In its simplest form, a resistance thermometer could T A B L E III TYPICAL VALUES FOR CONSTANTSIN CALLENDARVAN DUSEN EQUATION FOR SPRT AND I P R T a Constant A B C a p

SPRT 3.985 x 10-3/~ - 5 . 8 5 x 10-7/~ 2 4.27 x 10-12/~ 3.927 • 10-3/~ 1.11814

aData from Nicholas and White [8].

IPRT 3.908 • 10-3/~ - 5 . 8 0 x 10-7/~ 2 4.27 x 10-12/~ 4 3.85 • 10-3/~ 1.1158

ADVANCES IN TEMPERATURE MEASUREMENT

127

comprise a coil of wire mounted on an electrical insulating support. Such a design would be sensitive to shock and vibrations, and inadvertent knocks could cause sections of the wire to flex and strain, resulting in work hardening of the wire, which introduces defects to the crystal structure and increases the resistance. A possible solution is to mount the coil on a solid former, but this kind of design would be sensitive to differential expansion between the former and the coil and work hardening of the platinum with subsequent uncertainties in the resistivity. PRT design is thus a compromise between mechanical robustness and minimization of strain. The three principal types of SPRTs are capsule, long stem, and high temperature. Although the PRT is specified for realizing the ITS-90, no single instrument can cover the whole range. Capsule-type SPRTs should be used for low temperatures to 13.8 K. Lead wires should be kept short and are usually welded to thin copper of constantan extension leads. The principle of the design is to minimize thermal conduction errors along any connections. The sensor typically has a nominal resistance of 25 1~ at 0~ and thermal contact is achieved by means of a helium gas fill with a fraction of oxygen to aid heat transfer at a pressure of 30 kPa at room temperature and a platinum sheath. In order to produce a resistance of 25 1), about 60 cm of 0.07-mm-diameter coiled wire is necessary. The capsule, which is usually less than 50 mm long, must be totally immersed in the medium of interest. This is normally achieved by inserting the gauge in a thermowell filled with a suitable high thermal conductivity grease or alternative substance. For low uncertainty measurements, capsule thermometers should not be used above 30~ They are, however, capable of measuring temperatures up to 250~ if reduced lifetime and higher uncertainty are acceptable. For use at higher temperatures, SPRTs need longer stems so that the seal is close to room temperature in order to avoid significant thermal gradients, and the accepted interpolation instrument for the temperature range -189.352-419.527~ is the long stem SPRT. These devices consist of platinum wire wound onto a former, which is contained within a glass or quartz sheath. In traditional designs, see, for example, Le Chatelier and Boudouard [61], the former or mandrel used to support the platinum wire is made from thin mica sheet in the form of a cross. The edges of the mica cross are cut with fine teeth to allow location of the platinum wire. The grooves form a double helix so that one-half of the wire is wound onto the former, leaving alternate grooves empty, and the wire is then wound back located in the vacant grooves. The result is known as bifilar winding and has the advantage that the electrical fields in adjacent wires approximately cancel minimizing inductive effects. The total resistance is commonly designed to be 25.5 1) at 0~ as this provides a convenient resistance change of approximately 0.1 I~/~ Mica is bound with water and will deteriorate and flake at high temperatures,

128

P.R.N. CHILDS

which drive off the water [62]. Phlogopitic mica used in quartz tube designs is limited to use below 600~ and ruby mica using Pyrex tubes is limited to operation below 500~ Glass or quartz sheaths are used to limit conduction along the stem. Thermal radiation, however, can be transmitted along the stem, and room lighting, for example, can raise the indicated temperature for the triple point of water by 0.2 mK [8]. The quartz tubes of most thermometers are clear at the hot end to enable visualization of the element and the rest of the element is roughened by sand blasting or is coated with a black material to minimize the transmission of thermal radiation. SPRTs can be used to measure temperature with very low uncertainty and are used to define part of the International Temperature Scale of 1990 between the triple point of hydrogen, 13.8023 K, and the freezing point of silver, 1234.93 K. An uncertainty of +2 x 10 -3 K at the lower end of the scale and +7 x 10-3 K at the upper end of the scale is possible [3] For operation at elevated temperatures, SPRTs (sometimes referred to as high-temperature SPRTs or HTSPRTs) must be designed to resist contamination. Although relatively stable, platinum does oxidize [63-64] and undergoes crystal growth [65]. Oxidization ultimately limits the upper useful temperature for PRTs and also the long-term stability and uncertainty for a particular thermometer. Mica cannot be used at sustained temperatures above about 600~ and high-purity quartz is usually used as an alternative. Unfortunately, the insulation properties of substances reduce with temperature, and the quartz mandrel serves as a shunt resistance across the platinum windings, introducing an uncertain error source. Examples of high-temperature capability PRTs are presented by Li Xumo [66], Sawada and Mochizuki [67], and Strouse et al. [68] with measurements made to 1100~ Secondary SPRTs are designed for laboratory environments and can withstand more handling than an SPRT, although the sensing element is still mounted so that it is strain free and mechanical shocks can cause damage. Secondary SPRTs are manufactured using less expensive materials, such as reference grade high-purity platinum wire, and are intended for use over a limited temperature range, normally -200 to 500~ They can be used to produce measurements with an uncertainty as low as +0.03~ The delicate strain-free designs of an SPRT would not survive the shock and vibration encountered in some industrial environments. The IPRT typically comprises a platinum wire encapsulated within a ceramic housing or a thick film sensor coated onto a ceramic surface. The actual sensing element of an IPRT, can be further protected from the environment by a metal sheath. In the design of an IPRT the constraint for mounting the platinum wire so that strain is minimized is relaxed in favor of producing a robust device. Most designs have just two wires sealed within the sensor body, as this reduces the risk of short circuits and makes the device more

A D V A N C E S IN T E M P E R A T U R E M E A S U R E M E N T

129

robust. The ceramic encapsulation used should be selected so that its thermal coefficient of expansion closely matches that of platinum and the former over the temperature range of the PRT. The ceramic material used for encapsulation can include compounds and impurities that react with the platinum and change the temperature coefficient of the transducer. The uncertainty achievable for an IPRT is generally of the order of +0.01 to +0.2~ over the range 0 to 300~ (see Hashemian and Petersen [69]). The platinum-sensing element can also be deposited as a thin or thick film on an insulating substrate such as alumina. For a thick film sensor, the surface area can be of the order of 25 mm 2 with an active area of 5 mm 2. They are mass produced with each sensor individually trimmed to produce the desired resistance. The thick film structure produces a device that is robust and capable of operating in applications experiencing minor shock or vibration. They are noninductive and therefore insensitive to stray electrical fields. Thick film PRTs are normally designated to class B of IEC-751 [70]. An innovative form of temperature measurement is the application of thin film PRTs to polyamide sheet [71]. Massed arrays of PRTs can be formed on a polyamide sheet substrate by evaporating or sputtering platinum followed by depositing gold leads using a magnetron. The resulting sheet is flexible and can be glued to the geometry of interest. In order to obtain a measure of temperature using an RTD, the resistance must be measured. Simplistically, this can be achieved by passing a current through the sensing resistor and measuring the potential across it. If the current is known, then the ratio of potential to current provides a measurement of the resistance from Ohm's law and the temperature can be determined from the resistance temperature characteristic. However, even if an uncertainty of only 1~ is desired, the resistance must be measured to better than 0.4% and, as a result, must be undertaken with care [8]. There are two basic methods for measuring resistance used commonly in platinum resistance thermometry-potentiometric methods and bridge methods-and these are reviewed by Nicholas and White [8], Connolly [72], and Wolfendale et al. [73].

b. Rhodium Iron, Doped Germanium, and Carbon Resistors A number of types of R T D are suitable for cryogenic applications, including platinum, rhodium iron, doped germanium, and carbon. Although used to define the ITS-90 between 1234.93 and 13.81 K, the sensitivity of platinum falls off below about 20 K. The rhodium iron alloy has a similar resistance temperature characteristic to platinum at temperatures above 30 K. Below 30 K the sensitivity drops to a minimum between 25 and 15 K and then rises again, giving a R T D with good sensitivity at low temperatures (see Rusby [74] and Schuster [75]). Doped germanium resistors are available with a relatively wide temperature range for cryogenic applications, from 0.05 to 325 K, and

130

P.R.N. CHILDS

typically comprise the semiconductor encapsulated in a 3-mm-diameter, 8.5mm-long cylinder (example cited is the GR-200A from Lakeshore Cryotronics). Certain carbon resistors produce stable RTDs. The insulating skin of the resistor can be removed, and the resistance element can be potted in a protective capsule to prevent moisture absorption [76-79]. c. Thermistors Thermistors consist of a ceramic semiconductor for which the resistance is sensitive to temperature; they were originally named after a shortened form of a term thermally sensitive resistor [80]. Modern thermistors are usually mixtures of oxides of chromium, nickel, manganese, iron, copper, cobalt, titanium, other metals, and doped ceramics. They are manufactured by sintering particles in controlled atmospheres. Thermistors can have either a positive temperature coefficient (PTC) or a negative temperature coefficient (NTC). A typical resistance temperature characteristic is 1 I}/0.01~ To a first approximation the temperature can be determined from the relationship given in Eq. (11) [81]: Rr-

Ro exp I - B ( 1

- ~o)],

(11)

where R0 is the resistance at To and B is a constant for the particular thermistor material. The resistance characteristic of a thermistor expressed by Eq. (11) is negative and nonlinear. This can be offset if desired by using two or more matched thermistors packaged in a single device so that the nonlinearities of each device offset each other (see Beakley [82], Sapoff and Oppenheim [83], and Sapoff [84]). Thermistors are usually designated by their resistance at 25~ with common resistances ranging from 470 I'Z to 100 kl). The high resistivity of thermistors normally negates the need for a four-wire bridge circuit. In order to produce a lower uncertainty fit for the resistance temperature characteristic of a thermistor, a polynomial can be used with the degree used depending on the temperature range and the type of thermistor. Equation (12) is adequate for most systems: A1 A2 A3 l n R r - Ao +-~- +-~5 + T---g.

(12)

Here A0, A1, A2 and A3 are constants. Alternatively, Eq. (13) can be used where the second-order term has been neglected [85]: 1 - - = bo + b l R r + b3(ln R T ) 3.

T

Here oLo, O~1 Or2, and

O~3 are

constants.

(13)

ADVANCESIN TEMPERATUREMEASUREMENT

131

6. Semiconductor Devices A forward voltage across a transducer junction can be used to generate a temperature-sensing output proportional to absolute temperature. Semiconductor-based temperature sensors are used in an increasing proportion of temperature measurement applications related to the need to monitor integrated circuit temperatures. Temperature sensors based on simple transistor circuits can be incorporated readily as part of an integrated circuit to provide onboard diagnostic or control capability. The majority of semiconductor junction sensors use a diode connected bipolar transistor. If the base of the transistor is shorted to the collector, as illustrated in Fig. 4, then a constant current flowing in the remaining pn junction (base to emitter) produces a forward voltage that is proportional to absolute temperature. This can be modeled by Eq. (14) and, in practice, has a temperature coefficient of approximately - 2 mV/~ Ve-(~)

ln(~) ,

(14)

where: k is Boltzmann's constant (1.38 x lO-23j/K), Tis temperature (K), q is charge of electron (1.6 x 10 -19 C), 1F is forward current (A), and Is is junction's reverse saturation current (A). Sophisticated circuitry can be added to a transistor-based temperature sensor in order to improve linearity, precision, and specific control features, such as high and low set points. Self-contained sensors are available comprising a miniature integrated circuit enclosed in a standard electronic package, such as a TO99 can, a TO-92 plastic molding, or a DIP plug. There are a large number of types of semiconductor temperature sensors because of their versatility as a temperature sensor and their use in control.

constant ~.~ current ~"~1 IF source T [collector baselP

~/~n

VF o

mitter ~

Fxa. 4. If the base of a transistor is shorted to the collector, the output is proportional to the absolute temperature

132

P.R.N. CHILDS

For temperature sensing only, small signal semiconductors, such as 2N2222 and 2N2904 transistors, are common choices. Integrated temperature sensors, such as the LM35 produced by National Semiconductors, provide a linear output voltage of 10 mV/~ for the range - 4 0 to 110~ The TMP01 programmable temperature controller produced by Analogue Devices gives a control signal from one of two outputs when the device is above or below a specific temperature that can be set by external resistors, which can be selected by the user. Semiconductor temperature sensors can be incorporated within a larger integrated circuit to produce a smart sensor that can control regions of the circuitry on a chip at predetermined temperatures, [86]. The typical temperature range for semiconductor-based temperature sensors is approximately - 5 5 to 150~ The temperature range for some smart integrated circuits can be extended to 185~ The advantages of transistor-based semiconductor IC temperature sensors are their linearity, simple circuitry, good sensitivity, reasonable price, and ready availability. Because they are a high impedance, current-based unit, they can be used for remote sensing, requiring just a twin copper cable for connection purposes. Suitable digital voltmeters, some scaled in temperature units, are available for use with silicon transistor sensors and these frequently incorporate the necessary power supplies to operate the sensors. A disadvantage is the need to add extra circuitry and for calibration to attain a reasonably low uncertainty. 7. Diode Thermometers

The forward voltage drop across a p-n junction increases with decreasing temperature, providing a phenomena that can be exploited for the measurement of temperature. For certain semiconductors, the relationship between voltage and temperature is almost linear, and in silicon this occurs between 400 and 25 K with a sensitivity of approximately 2.5 mV/K. Commonly used semiconductors for thermometry are Si and GaAs with the addition of certain elements for stability. The output for Si diodes is lower than that for GaAs diodes, but they have better stability and are less expensive. Generally, rectifying diodes are used and these can be potted in a small container. Diode thermometers are available commercially e.g., the 1.25-mm diameter and 0.75-mm-long DT-420 device from Lake Shore Cryotronics. Zener diodes have also been used to indicate temperature [87]. The merits of diode thermometers include price, a simple voltage temperature relationship, a relatively large temperature range (1 to 400 K), no need for a reference junction, relatively high sensitivity, uncertainty lower than +50 mK [88], and simplicity of operation with a constant current source and a digital voltmeter. Errors in the indicated temperature can occur if the

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133

supply current is not a true DC but has an AC component due to noise induced in the circuit as a result of improper shielding, electrical grounds, or ground loops. The instrumentation should be electrically shielded and appropriate grounding techniques used in order to minimize noise (see Morrison [89] for guidance on grounding and shielding). The current supply for the diode should have a single ground, generally located at the voltmeter that then requires a floating current source. Currents between 1 and 100 I~A can be used. Generally, however, the current needs to exceed 10 p~A in order to overcome noise problems. The choice is a compromise, as higher currents can cause problems associated with self-heating at very low temperatures. 8. Noise Thermometry

The thermal motion of atoms and molecules and charge carriers within an electrical conductor will generate a broad band electrical noise [90, 91]. The noise power generated is called Johnson noise and is given by power = 4k T A f ,

(15)

where k is Boltzmann's constant (J/K), T is absolute temperature (K), and Af is frequency band (Hz). The thermal noise power generated in a resistor is usually very small e.g., 7 I~V over 2-rrMHz for a 1MI) resistor at 1K and 0.4 p~V over 100 kHz for a 100 1~ resistor at 1000~ Sensitive instrumentation is therefore necessary. In principle, the temperature range of application is from a few millikelvins to over 2500~ The circuitry used depends on the temperature range and has been reviewed by Kamper [92], Blalock and Shepard [93], and White et al. [94]. Because the amplifier can contribute a noise equal or greater than that of the sensing resistor, a cross-correlation technique can be used involving measuring the voltage and current signals separately and integrating only the correlated part of the signals within the specified bandwidth. In addition, random fluctuations about the mean value in the noise power itself mean that a long averaging period, generally over 10 min, is necessary [95]. For measurements at temperatures less than 1 K, two types of absolute noise thermometer are reported by Soulen et al. [96]. Both types measure the noise voltage generated by a resistor utilizing a superconducting quantum interference device (SQUID). In one of the types, the sensing resistor is inductively coupled to the SQUID and in the other, resistive SQUID (RSQUID), the resistor is connected directly across a Josephson junction. The use of an RSQUID is reported by Menkel et al. [97] for temperatures up to 4.2 K and by MaeFarlane et al. [98] for the range 10 to 50 K. Above

134

p.R.N. CHILDS

273.15 K there are two principal options. The first, a comparison method, compares the measured open circuit Johnson noise voltage, at the unknown temperature, across the sensing resistor to that of a reference resistor at a known temperature. In the second the temperature is obtained by measuring the open circuit Johnson nose voltage and the short circuit Johnson noise current (see Blalock and Shepard [93]). 9. Capacitance Thermometers

The electric permittivity of some materials such as strontium titanate is highly dependent on temperature over a certain range. A practical sensor can be formed by encapsulating samples of the material, with a capacitance of a few nanofarads, and lead out wires to a bridge circuit. Capacitance thermometers provide good sensitivity below 290 K, but the output voltage becomes irreproducible after thermal cycling so recalibration against another type of sensor is necessary, say on cooling. Capacitance thermometers do, however, exhibit virtually no magnetic field dependency and are therefore useful as control devices in high magnetic fields where other types of device may fail or produce erroneous signals [99]. Penning et al. [100] reported the independence of reading in a field up to 20 T for a (Pbo.45Sno.55)zPzSe6 crystal in the temperature range 25 mK to 5 K. Glass capacitance sensors are widely used in cryogenic applications as a means of temperature measurement [101]. The permittivity-temperature characteristics of a number of materials have been reported (see Bazan and Matyjasik [102], Lawless [103], Li et al. [104], Penning et al. [100, 105], Strehlow [106, 107], Wiegers et al. [101], BoutardGabillet et al. [108], Mohandas et al. [109], and Rotter et al. [110]). Capacitance temperature sensors are available commercially in a 3-mm-diameter, 8.5-mm-long package from Lake Shore Cryotonics Inc. 10. Paramagnetic and Nuclear Magnetic Resonance Thermometry

Nuclear magnetic resonance is a quantum mechanical phenomenon exhibited by atoms having an odd number of protons or neutrons. For many materials, the magnetic susceptibility varies inversely with temperature, as defined by the Curie law for nuclear spin susceptibility: C • -~,

(16)

where • is the susceptibility, C is Curie constant, and T is absolute temperature (K). The procedure used in paramagnetic thermometry involves placing an appropriate sample of material between the coils of a mutual inductance

ADVANCES IN T E M P E R A T U R E M E A S U R E M E N T

| 35

bridge or ratio transformer (see Cetas and Swenson [111] and Quinn [112]). The sample must be a paramagnetic material such as cerium magnesium nitrate (CMN), manganous ammonium sulfate (MAS), chromic methyl ammonium (CMA), or gadolinium sulfate octahydrate. CMN is the most common and is useful at temperatures from a fraction of a kelvin up to 4.2 K and in particular for the measurement of 3He and 4He temperatures (e.g. see Rusby and Swenson [113], Greywall and Busch [114-115] and Mohandas et al. [116]). The development of PdMn and PdFe, allowing high resolution measurements in the range 1.5 K to 3 K with application to measurements in outer space is reported by Klemme et al. [117]. A mutual inductance bridge can be used to determine the ratio between two mutual inductors, one containing the sample of interest and the other acting as a reference value. Examples are reported by Durieux et al. [ 118], Cetas and Swenson [111], and Schuster [119]. The reproducibility of this method can be better than 0.5 mK below 50 K with an uncertainty of 1 mK in the range of 18 to 54 K [120] and 50 IxK at temperatures up to 2 K [119]. The uncertainty depends on installation, bridge circuitry, and the stability and uncertainty of the associated constants.

11. Quartz Thermometers The natural frequency of quartz crystals is a function of the dimensions and density, which are in turn dependent on temperature. A measurement of the natural frequency can therefore be used as a measure of temperature. The frequency temperature characteristic of a quartz crystal can be modeled by [121]

A f -- fo Z

a,,(T - To)".

(17)

The performance of a LC cut quartz resonator-based digital thermometer operating over the temperature range - 8 0 to 250~ has been reported by Benjaminson and Roland [122], that for a YS cut crystal operating over the range 10 to 400 K by Agatsuma et a/[121,123], and that for a FC cut crystal over the range 1.5 to 10 K by Suter [124]. Quartz thermometers are highly stable devices with divergence of a few thousandths of a degree Celsius per month after the initial aging process. A resolution of 10 lxK is possible with a logging time of 100 s [122]. 12. Pyrometric Cones, Thermoscope Bars, and Bullers Rings A number of deforming temperature and heat rate indicating devices have been developed, known as pyrometric cones, thermoscope bars, and Bullers

136

P.R.N. CHILDS

rings, for use in kilns and furnaces. Pyrometric cones, also known as Harrison, Seger, or Orton cones, are slender trihedral pyramids manufactured from mixtures of various mineral oxides [125]. On heating the effects of gravity cause the inclined cone to bend progressively as the material softens. Pyrometric cones do not provide an exact indication of temperature. Their deformation is a function of both the rate of heating and the temperature cycle and can therefore be used to provide an indication of process completion. Cones vary in shape, size, and method of use according to manufacturer. They are an economic means of identifying firing completion and can be distributed at a number of locations in a kiln to indicate local conditions. An alternative is the use of thermocouples, RTDs, or infrared thermometers, but these have a higher initial cost. One manufacture, Orton Ceramics, provides cones suitable for temperatures in the range of 590 to 2015~ Determination of performance equivalents between different manufacturers is standardized in A N S I / A S T M C24-79 [126] and is reported by Fairchild and Peters [127] and Beerman [128]. The practical use of pyrometric cones is described by McGee [129] and in manufacturers' guidelines. Deforming devices known as thermoscope bars and Bullers rings are also sometimes used to indicate temperature in the firing process. Thermoscope bars comprise a rod of material, which are mounted in a special stand. The bars soften and deform with temperature and this deformation can be measured and compared with a look-up table supplied by the manufacturer to provide an indication of temperature. The range of operation for the temperature measurement of thermoscope bars is between 590 and 1525~ in intervals of about 15 to 30~ Bullers rings are precision dust-pressed rings that shrink linearly with temperature rise. This contraction can be measured using a gauge. The temperature range capability for Bullers rings is between approximately 960 and 1400~ The usage of thermoscope bars and Bullers rings is outlined elsewhere [130]. B. SEMI-INVASIVEMETHODS Some temperature measurement scenarios permit modification of a surface of interest with a thermally sensitive material that can be observed remotely. These techniques are classed as semi-invasive and include heatsensitive paints, thermographic phosphors, thermochromic liquid crystals, and variety of crayons and labels. Some of the materials available provide a means of monitoring temperature continuously, whereas others merely provide a means of identifying that a certain temperature has been reached or exceeded. The latter are known as peak temperature indicators.

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1. Peak Temperature Indicating Devices A number of products are available, including a variety of heat-sensitive crayons, pellets, labels, and paints, that provide an indication that a particular temperature has been attained or exceeded. On heating above some critical temperature the material used for the indicator melts, fuses, or changes composition, providing a permanent record that a given temperature has been reached or more likely exceeded. One method of application of fusible indicating material is the use of crayons. These are available from a number of suppliers, including Tempil, and can be used to draw on the component of interest to indicate, for example, whether it has been heated to the correct temperature. Applications include identifying temperatures in metal treatment and welding. Self-adhesive labels consisting of a temperature-sensitive indicator sealed under a transparent window are also widely available. The indicating material permanently changes color from say a light gray to black at the rated temperature, which is printed on the label. Some labels include multiple indicators, sensitive to a range of temperatures, to enable greater resolution of the peak temperature. Labels are a particularly convenient means of identifying whether a component such as an electronics package has experienced an excessive temperature or in medical applications to validate that a process temperature has been attained. The limitations of peak-indicating temperature materials include poor performance in regions of high thermal gradient, their inability to indicate the peak temperature reached above the rated temperature, and the labor requirement in their application and observation.

2. Temperature-Sensitive Paints The same kind of fusible material used in peak temperature indicating crayons is available in paint form. One of the demanding applications for paints is the indication of peak temperatures on turbine blades [131]. Early work on turbines involved the use of proprietary paints from Thermindex, Faber Castel, and Tempil. The development of a range of multichange paints is reported by Bird et al. [132]. Some paint formulations are available that provide a reversible indication of peak temperature. One principle exploited is to drive water off a colorful salt, thus changing its color. On cooling the salt can reabsorb water vapor from the atmosphere and revert to its initial color state. The Chromonitor brand, for example, changes color from red to deep maroon in the temperature range 65 to 71~ before reverting to its original color on cooling. Applications for this form of paint include, as an example, hair curling tongs.

138

1,. R. N. CHILDS

Another class of temperature indicating paints exists known as parametersensitive paints (PSPs). These paints are sensitive to a number of physical parameters, including pressure and temperature, and, in the case of the latter, are known as temperature-sensitive paints (TSPs). The paints comprise luminescent molecules and a polymer binder material that can be dissolved in a solvent. TSPs can be applied by brush or spray and as a the paint dries, the solvent evaporates to leave luminescent molecules embedded in a polymer matrix. The principle exploited by TSPs is photoluminescence where a probe molecule is promoted to an excited state by the absorption of a photon of appropriate energy in a fashion comparable to thermographic phosphors. The difference is that the intensity of the luminescence is related to temperature by photophysical processes known as thermal and oxygen quenching [133], and the intensity of luminescence is inversely proportional to the temperature (see Donovan et al. [134] and Cattafesta et al. [135]). Examples of TSPs include rhodamine dyes and europium thenoyltrifluoroacetonate. The technique has been used in turbomachinery testing and heat transfer experiments [136-138]. Cattafesta et al. [135] reported an average uncertainty of +0.3~ for several sample applications of rhodamine dyes operating in the temperature range of 0 to 95~ and using an industrial grade charged couple device camera. 3. Thermographic Phosphors

Thermographic phosphors can be used to indicate temperatures from cryogenic levels (see, e.g., Cates et al. [139] and Simons et al. [140]) to 2000 ~ Thermographic phosphors are materials that can be excited by energy absorption and subsequently emit light in a temperature-dependent fashion. A phosphor-based thermometry system will generally comprise a source of excitation energy, a method of delivery of energy to the target, a fluorescing medium bonded to the target, optical, detector, and data acquisition, and analysis systems. A typical layout for a thermographic phosphor temperature measurement system is illustrated in Fig. 5. Before excitation, the electronic levels of a phosphor are populated in their ground state. They can be excited by the absorption of energy by electromagnetic radiation such as visible or UV light, X or ? rays, particle beams such as electron, neutron, or ion beams or by an electric field. After absorption of energy the atomic configuration of the material may not remain excited but return to its initial or some intermediate state (Fig. 6). Inorganic phosphors such as Y203:Eu and La202S:Eu tend to be used in thermometric applications. The intensity of emitted light is an inverse function of temperature. In the case of Europium-based phosphors experiencing continuous illumination, this is given by Eq. (18)[142]:

ADVANCES IN TEMPERATURE MEASUREMENT

[ nitr~ laser

139

"~(~mirror

~il

lens data acquisition ] ] system

I detector I

stationary or moving target

FIG. 5. Use of thermographic phosphors to measure temperature (after Allison and Gillies [141]).

IT - [aj -k-ajAe-A E/kT] -1 ,

(18)

where IT is intensity, aj is probability rate, A is factor related to aj, E is energy (J), T is temperature (K), k is Boltzmann's constant (J/K), and acT-s is charge transfer state rate. When selecting the type of phosphor consideration should be given to the temperature range of interest (see, e.g., Allison and Gillies [141] and Ballico [95]). Table IV lists the useable ranges for a limited number of phosphors. Sensitivities of 0.05~ and an uncertainty of 0.1 to 5% of the Celsius temperature reading are possible [141]. The chemical compatibility of the phosphors with the surface of interest and the surrounding atmosphere should also be considered. Most thermographic phosphors are ceramics and therefore relatively inert. Bonding of the phosphor to the application

con( luction band l fluorescence tm

excitation

impuritylevel

---~ intermediate state valence band

FIG. 6. Schematic of the energy levels of a phosphor.

140

e.R.N. CHILDS T A B L E IV USEABLE RANGES OF SELECTED PHOSPHORSa Phosphor

Useable temperature range (~

Y203:Eu LazO2S:Eu Nd:Glass Nd:YAG A1203:Cr Cr-YAG

500 0 - 50 0 100 - 100

to to to to to to

1000 100 350 800 500 300

aAfter Ballico [95].

can be achieved by mixing the phosphor slurry with epoxy, paint, or glue and then brushing or spraying the mixture onto the surface of interest. For harsh environments subject to, for example, high mechanical shock or large thermal gradients, chemical bonding by vapor deposition, RF sputtering, and laser ablation can also be considered. Thermographic phosphors have been used for a variety of applications, including determining the temperature of flat plates in supersonic flows [143], wind tunnel models [144, 145], turbine blades and components [146151], curved surfaces [152], surface temperature fields [153], color TV screens [ 154], textiles during microwave drying [ 155], and tumors [ 156]. The specialist skills required by the technique limit its general use.

4. Thermochromic Liquid Crystals Thermochromic liquid crystals are materials that exhibit significant changes in color over discrete temperature bands. At a particular temperature the liquid crystal material selectively reflects incident light within a certain band of wavelengths. As the temperature rises, a thermochromic liquid crystal mixture will change from colorless, black against a blackened background, and pass through the visible spectrum of colors-red to orange to yellow to green to blue to violet-before turning colorless again. The process of color change is reversible with cooling. Colors displayed by a thermochromic liquid crystal layer can be related to temperature by a calibration process. Pure crystals deteriorate rapidly with age and exposure to ultraviolet light. Their life can be extended by encapsulation of the crystals within a polymer coating [157]. Thermochromic crystals are available commercially in the form of water-based slurry. This can be sprayed onto a dark or blackened surface using an airbrush (see, e.g., Baughn [158], Farina [159], and Roberts

ADVANCES IN TEMPERATURE MEASUREMENT

141

and East [160] for practical guidance). They are also available as a preformed layer on a blackened substrate of mylar or paper. The temperature at which a liquid crystal formulation begins to display color is called its red-start temperature. The range of temperature over which the crystals display color is often referred to as the color play bandwidth or the temperature event range. The range of temperature over which a liquid crystal displays a single color is called the isochrome bandwidth, which can be as little as 0.1 ~ The value of the red-start temperature and the color play bandwidth can be controlled by selecting appropriate cholesteric estors and their proportions. The red-start temperature and color play bandwidths for a selection of thermochromic liquid crystals are listed in Table V. A number of approaches have been adopted in determining the value of temperature from color in the use of thermochromic crystals. For instance, in the use of forehead clinical thermometers, human eyesight and judgement are used comparing the displayed color to a look-up table or identifying the most significant illumination in a panel. Cameras can also be used to capture the images produced by liquid crystals. One approach is based on the red, green, and blue (RGB) color system used in domestic video cameras. The color signal measured by the camera is encoded into a composite signal PAL (Phase Alternation Line) or NTSC (National Television System Committee) format and recorded onto videotape or in computer memory. An alternative is to use a color index approach based on RGB signals. The hue signal in the hue, saturation, and intensity color definition is a monotonic function of the crystal temperature and is independent of the local illumination strength. Hue, saturation, and intensity are defined by Eqs. (19)-(23) [161]: H

-

3-~

90 -

tan-

+

0, G > B ] 180, G < B '

TABLE V RED-START TEMPERATURESAND COLOR PLAY BANDWIDTHSFOR A RANGE OF THERMOCHROMIC LIQUID CRYSTALSAVAILABLEFROM HALLCREST INC. Red-start temperature (~ -30 0 30 60 90 120

Minimum color play bandwidth (~ 2 1 0.5 1 1 1

Maximum color play bandwidth (~ 30 25 25 20 20 20

(19)

142

e.R.N. CHILDS

where F

2R-G-B G-B

F-R

for G C B

(20)

for G - B.

(21)

Saturation and intensity are defined as S-

1 D

I -

min (R, G, B)-

R+G+B

3

.

(22)

(23)

The frames can be analyzed digitally using a computer and frame grabber and data converted from the RGB color system to values of hue, saturation, and intensity. Comparison of hue values with calibration results gives the surface temperature for each pixel location (see, e.g., Camci et al. [162], Farina et al. [159], and Hay and Hollingsworth [163]). The uncertainty in temperature measurement using liquid crystals depends on the experimental conditions and the image processing system. Simonich and Moffatt [164] reported a calibration uncertainty of 0.25~ using mercury vapor lamp illumination. The speed of response of thermochromic liquid crystals depends on the viscosity, and Ireland and Jones [165] demonstrated a typical response time constant of 0.003 s. Applications of thermochromic liquid crystals have included forehead and fish tank thermometers, novelty stickers, turbine heat transfer experiments (e.g., Campbell and Molezzi [166], Ou et al. [167], Hoffs et al. [168], and Wang et al. [169]), jet engine nacelles, and vehicle interiors [170]. C. NONINVASIVEMETHODS

Certain temperature-dependent phenomena, such as the emission of radiation, scattering, and luminescence, can be observed remotely. Remote observation is noninvasive to the medium of interest, which is often referred to as the target. Noninvasive methods are attractive for a variety of reasons and applications, including targets in motion, fragile targets, remote targets, unsteady temperature applications, harsh environments, and temperature distribution required. Noninvasive methods of temperature measurement have undergone significant development, hand in hand with the advances in semiconductor fabrication technology and the availability of lasers as a source of excitation. Use of infrared thermometers is, for example, now commonplace. Tempera-

ADVANCES IN TEMPERATURE MEASUREMENT

143

ture measurement techniques based on infrared radiation are reviewed in Sections II,C,1 and II,C,2, those based on the index of refraction in Section II,C,3, absorption and emission spectroscopy in Section II,C,4, line reversal in Section II,C,5, scattering in Sections II,C,6-II,C,9, laser-induced fluorescence in Section II,C,10, light polarization in Section II,C,11, the speed of sound in Section II,C,12, and speckle methods in Section II,C,13. Some of the techniques, such as laser-induced fluorescence (Section II,C,10), are highly specialized and expensive, requiring extensive expertise and highcost capital equipment. This type of method is not available off the shelf and its use should be justified carefully in terms of the value of the data it would produce.

1. Infrared Thermometry Infrared thermometers are now widely available and although once very expensive, even at the low end of the market, they are now available in forms where the cost is the same order of magnitude as, for example, a thermocouple indicator. Infrared thermometers measure the thermal radiation emitted by a body due to its temperature and have found applications from cryogenic temperatures to over 6000 K. Any body will emit thermal radiation due to its temperature with the quantity rising with increasing temperature. The energy emitted over the electromagnetic spectrum due to temperature by a black body can be modeled by Planck's law [Eq. (24)]. The characteristics of infrared radiation are that the energy radiated reduces with temperature but the wavelength distribution shifts toward those with longer wavelengths. E~,b

-

C1 )~5[exp(Cz/)~T)

, -

-

(24)

1]

where: Ez, b is spectral emissive power for a black body (W/m3), C1 is first radiation constant (3.7417749 • 10-16 Wm 2 [171]), 9~is wavelength (m), C2 is second radiation constant (0.01438769 m K [171]), and T is absolute temperature (K). Integration of Eq. (24) over all wavelengths gives the total thermal radiation emitted by a black body:

Eb = ~rT4,

(25)

where cr is the Stefan-Boltzmann constant, which is equal to 5.67051 • 10-8 W / m 2 K 4 [171]. An infrared system contains three basic elements: source, propagation medium, and measuring device (Fig. 7). The infrared measuring device itself

144

P.R.N. CHILDS

FIG. 7. Infrared temperature measurement system.

may comprise an optical system, a detector, processing circuit, and display. The purpose of the optical system is to focus the energy emitted by the target onto the sensitive surface of the infrared detector, which usually converts the energy into an electrical signal. There are a number of types of infrared thermometer, including those sensitive to all wavelengths, classed as total radiation or broadband thermometers, and those sensitive to radiation in a specific band of wavelengths, classed as spectral band thermometers, ratio thermometers, and thermal imagers. Unfortunately, the radiation emitted by a target does not match the ideal modeled by Eqs. (24) and (25) and is instead a function of both surface temperature and surface properties. The surface property limiting the quantity of radiation is called the emissivity, ~, and is the ratio of the electromagnetic flux that is emitted from a surface to that emitted from a black body at the same temperature. Emissivity is a function of the dielectric constant and subsequently its refractive index and is generally wavelength dependent. Quantification and accounting for the effects of emissivity in undertaking a temperature measurement using an infrared thermometer are usually necessary (see Corwin and Rodenburgh [172] and Madding [173]). Target surfaces can be classified into three categories: black bodies, gray bodies, and non gray bodies. Information about the emissivity of a wide range of materials is documented in the reference texts edited by Touloukian et al. [174-176] and also tends to be available from infrared thermometer suppliers. The medium across which the radiation is transmitted can also attenuate the radiation due to absorption, scattering, and turbulence with absorption and emission dependent on the energy structure of the constituent molecules. In addition to its main constituents, air typically contains a proportion of water vapor and CO2 along with a number of trace molecules. The strong dipole moment and light hydrogen atoms in water vapor result in strong and broad absorption bands. In addition, water vapor is an asymmetric top,

ADVANCES IN T E M P E R A T U R E M E A S U R E M E N T

145

which results in an irregular spectrum. The properties of water vapor and CO2, along with other substances, act to absorb a proportion of the radiation emitted from a target at specific wavelengths and even make air opaque at some frequencies. This means that using Eq. (24) without modification to account for the radiation not transmitted would lead to an error. Information concerning the transmissive properties of air, which vary with distance and local composition, has been published by Yates and Taylor [177] and is also modelled in a number of software packages, such as LOWTRAN [178-181], MODTRAN, HITRAN, and FASCODE. These codes are maintained by the Geophysics Directorate at Hanscom Air Force Base, Masschusetts, and are reviewed in Smith [182]. When dealing with other transmission mediums or specific particulates, such as smoke, the transmission characteristics for the system concerned must be identified in order to allow for the temperature reading to be interpreted correctly. A way around the problem of nontransmission of thermal radiation is the use of spectrally sensitive detectors. These operate across a specific waveband only and a detector can be selected to match the characteristics of a given application. For example, there are a number of wavelengths, known as windows, near 0.65, 0.9, 1.05, 1.35, 1.6, 2.2, 4, and 10 txm where the transmittance is very high and air is effectively transparent to thermal radiation. A large number of detectors have now been identified that are sensitive to thermal radiation, such as PbS, PbSe, InSb, HgCdTe, and PbSnTe (for further data, see Rogatto [183]). As a general rule, point-sensing instruments used for measuring hot targets operate at shorter wavelengths (e.g., 0.91.1 txm), whereas those for cooler objects operate at longer wavelengths (35 txm or 8-14 p,m). Further complications arise from the use of optics and windows between the target and the detector, as materials used for lenses and windows also have their own spectral characteristics and only transmit a proportion of the radiation at specific wavelengths. Because an infrared measurement system comprises an optical system and detector, as well as the medium of transmission, an infrared sensor should only be operated when the spectral range over which a target transmits, over which the media transmits and over which the detector will operate all overlap. A further compounding problem affecting a temperature measurement using an infrared thermometer is background radiation. An infrared thermometer will read any radiation incident on the detector whether it is emitted, transmitted, or reflected from a target. If a surface is not a perfect absorber of incident radiation, then a proportion of the incident radiation can be reflected, which can distort the indicated temperature. A number of strategies are available to minimize such effects, involving repositioning the detector or using screens as illustrated in Fig 8. The practical use of infrared thermometers is described by Kaplan [184].

146

P.R.N. CHILDS

FIG. 8. Some strategies to minimize the effects of background incident radiation (after Ircon

[1851).

When an infrared thermometer is aimed at a target it collects energy within a collecting beam. The shape of this beam is determined by the optical system and the detector and is typically conical. The cross section of the collecting beam is called the field of view, which determines the spot size: the area on the target over which a temperature measurement is made. In principle, it is possible to side step the requirement of knowing the surface emissivity by using a class of infrared thermometers called ratio thermometers, which are also known as ratio pyrometers, dual wavelength, or two-color pyrometers. These measure the radiation emitted from a surface around two fixed wavelengths. The ratio of the radiation emitted is given by R

-

-

e(~l ))~2(e 5 C2/k2 T -- 1). F . ( ~ 2 ) ~ ( e C2/kl T - 1)

(26)

ADVANCES IN TEMPERATURE MEASUREMENT

147

If the emissivity is not spectrally sensitive and is near unity the emissivities will cancel and, provided the target temperature is low, then T -

C2(~1 - ~ 2 ) / ~ 1 ~ 2

lnR

.

(27)

~

An optical fiber can be used to channel thermal radiation into a narrow wavelength band from the location of interest or a convenient viewing point to a measurement sensor. This allows the detector to be located remote from the target and is particularly useful in monitoring combustion processes. Optical fiber based devices are useful for temperatures in the range of 100 to 4000~ The use of this technique is reported by Dils [186], Saaski and Hartl [187], Sun [188], Ewan [189], Zhang et al. [190], Grattan and Zhang [191], and Krohn [192]. 2. Thermal Imaging

Infrared thermometry principles can be used to measure the spatial distribution of temperature on a target surface, which is known as thermal imaging or thermography. Applications of thermal imaging are extensive, ranging from plant condition monitoring [193-197], materials testing [198], process control [199-201], energy losses in structures [202, 203], and surveillance [204, 205]. The merits of thermal imaging include those of infrared thermography combined with the ability to resolve spatial temperature distributions. As for infrared thermography, developments in semiconductor technology and improvements in product design have resulted in the wider availability of thermal imaging devices, lower cost products, and products with various features, including portable devices with liquid crystal displays, as illustrated in Fig 9. Thermal imagers typically comprise an optical system, a detector, processing electronics, and a display and are extensions of infrared thermometer technologies combined with some form of scanning optics. Thermal imagers do not require any form of additional illumination in order to operate, which makes them highly attractive in military and surveillance applications. However, it should be noted that military and surveillance thermal imagers tend to be optimized to produce an image rather than quantitative information on the distribution of temperatures. It is possible to make use of a single detector with some form of scanner to transmit the radiation signal from specific regions of the optical system to enable a two-dimensional image of the temperature distribution to be built up, (Fig. 10). The disadvantage of single detector scanners is the trade-off between the speed of response of the instrument and the signal-to-noise

148

P.R.Y. CHILDS

FI~. 9. The FLIR Systems PM 675 thermal imager. Photograph courtesy of FLIR Systems Ltd.

ratio of the detector. The detector typically needs to be cooled and operated at performance limits in order to achieve the desired time response. Multidetector scanners comprising a linear detector array or a two-dimensional array of detectors, known as a staring array (Fig. 11), enable the temporal to spatial burden to be reduced. Cooling of the detectors can be achieved in a

focussing optics

x-scanner

. "~~"-.

i

s

~

~ linear "-~

etector array

I

FIc. 10. Two-dimensional scanning for a small detector array or single element detector (after Runciman [206]).

ADVANCES IN TEMPERATURE MEASUREMENT

149

staring detector l ~ ~ ray

ocuss ng 9

optics ~

I

I

.~~~

.~.~

I

j. s,

FI6. 11. A staring array for a thermal imager without scanning (after Runciman [206]).

number of ways, such as the use of liquid nitrogen, Peltier coolers, or using a miniature Sterling engine. The optimum wave band for a thermal imager, as for most other infrared thermometers, is dictated by the wavelength distribution of the emitted radiation, the transmission characteristics of the atmospheric environment between the imager and the target, and the characteristics of the available detector technology. The optical window for air between 1.1 and 2.5 txm is referred to as the short wavelength infrared imaging band (SWIR), that between approximately 2.5 and 7 txm as the medium wavelength imaging band (MWIR) with a notch at 4.2 Ixm due to C02 absorption, and that between 7.5 and 15 Ixm as the long wavelength imaging band (LWIR). Further bands beyond 15 txm are classed as far infrared (FIR) and very long wave infrared (VLWIR). The emissivity of most naturally occurring objects and organic paints is high in the long wave infrared but is lower and more variable in the medium wave infrared. Metallic surfaces tend to have lower emissivity in either band. As such, use of a thermal imager to provide quantitative information for the temperature distribution, particularly of a surface comprising different materials, has be managed carefully. Without correction for local emissivity values, the thermal imager will assume a default value and apply this to the whole image. However, basing the choice of thermal imager on emissivity alone ignores attenuation in the transmission medium. Fog particles attenuate radiation by scattering with the magnitude dependent on the particulate size [205]. If moderately sized

150

e.R.N. CHILDS

particulates are present, scattering affects the MWIR region more than the LWIR, and a LWIR system can generally provide better range performance. A number of performance parameters are used for thermal imagers, allowing different systems to be compared (see also Runcimann [206] and Holst [207, 205]). 9 Thermal sensitivity [noise equivalent differential temperature (NEDT)] refers to the smallest temperature differential that can be detected and depends on the optical system, the responsivity of the detector, and the noise of the system. 9 Spatial resolution defines the smallest quantity that can be discerned and is often quantified by Airy disk size. 9 Minimum resolvable temperature (MRT). 9 Minimum detectable temperature (MDT). Thermal imagers remain expensive devices, although their cost is falling. Prices range from over $100,000 (U.S. year 2000 prices) for high-performance military imagers to a few thousand dollars for uncooled imagers. The price reflects the performance, ruggedness, and image processing capability. Some compact imagers can be readily handheld, whereas other systems are designed to be mounted on a platform weighing as much as 100 kg. Table VI provides an indication of the specification for a small selection of the thermal imagers currently available (a more extensive list is available in Kaplan [184]). During the 1980s, the MWIR platinum silicide detetector and quantum well-infrared photodetector (QWIP) became available, followed more recently in the 1990s by uncooled and Sterling engine-cooled detector arrays, making batterypowered rugged portable devices viable. The uncertainty associated with the temperature measurement is specific to the device, but typical figures are +2 K or • full-scale output (see, e.g., Runciman [206]). Modern thermal imagers can be particularly easy to use, needing simply to be aimed at the target of interest and the image captured by pressing a button on the camera. Some devices allow the image to be streamed continuously to memory on board the camera or via a communications link.

3. Index of Refraction Methods The index of refraction can be used as a measure of temperature in certain compressible gases, and techniques based on this principle, including schlieren, shadowgraph, and interferometric methods, have been used to measure temperatures between 0 ~ and 2000~ The index of refraction of a fluid is normally a function of the thermodynamic state. If the pressure of the system can be considered constant, and the density of the gas modeled by

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TABLE VI SOME SPECIFICATIONSFOR A SELECTIONOF COMMERCIALLYAVAILABLETHERMAL IMAGERSa Manufacturer

Resolution

Detector

Description

Mitsubishi IR M700

801 • 512 pixels

PtSi

Uses a Stirling engine to provide continuous cooling and therefore continuous operation. Size 128 • 250 x 131 mm. Mass 5 kg

Mitsubishi IR G600

512 x 512 pixels

PtSi

Suited to mounting on gimbals and turrets. Size 185 x 210x 210mm. Mass 6.8 kg

FLIR Systems ThermaCAM PM 695

320 x 240 pixels

Bolometer, spectral response 7.5-13 I~m

This is an uncooled hand-held or tripod-mounted thermal imaging camera. Size 220 x 133 x 140mm. Mass 2.3 kg. Temperature range - 4 0 to 2000~

Compix PC2000

244 x 193 pixels

PbSe 3-5 Ixm

PC-based thermal imaging system. Thermoelectrically cooled detector. 17 to 150 ~ 5-240, 17 to 1000 ~ Size 140 x 430 x 110mm. Mass 2kg

Indigo Systems Merlin-Mid

320 x 256 pixels

Indigo Systems Merlin-Long

320 x 256 pixels

Quest Integrated Inc TAM Model 200

12.5 Ixm

InSb 1-5.5 lxm

Camera may be operated by a button panel or via a PC-based graphical interface application. Mass 602.8 kg. Size 140 x 127 x 249 mm Cool-down start time less than 10 min at 30 ~ Temperature range 0 to 350 ~ or 300 to 2000 ~

QWIP (quantum well-infrared photodetector). Spectral response 8 to 9.2 Ixm

A long wave infrared camera system. Mass < 2.8 kg. Size 140 x 127 x 249mm. Temperature range 0 to 2000 ~

2.4--5.5 Ixm

A microthermography system for identifying temperature distribution on a microscopic scale

aAfter Childs [208].

the ideal gas equation,

then the temperature

of the gas can be determined

[209]:

T-

ni, o - 1 p To, n i - 1 Po

(28)

152

P.R.N. CHILDS

where rti is refractive index at a particular location, p is pressure (Pa), p0 is pressure at a reference condition (Pa), and To is temperature at a reference condition (K). The various techniques involve monitoring the variation of refractive index with position in a transparent medium in a test section through which light passes. Each of the techniques, however, involves the measurement of a different quantity. The schlieren method involves determination of the first derivative of the index of refraction, the shadowgraph method measures the second derivative, and interferometers measure the differences in the optical path lengths between two light beams. A schlieren system is designed to enable the angle of deflection of a beam of light from a source that has traveled through the test section to be measured. A typical system for this is illustrated in Fig. 12 based on optical lenses, although it is also possible to use focusing mirrors. The light is collected by the second lens, and the resulting image can be projected onto a screen to allow the angle to determined. Alternatively, a camera focused on the center of the cross section can be located after the knife edge. If the index of refraction is assumed to vary in the y direction only, then the angular deflection of a light beam is given by -

1 I~ni~y _ ~-dz,

(29)

hi, air

where ni, air is the refractive index of ambient air. Once the angle of deflection is known, the refractive index can be evaluated and hence the temperature determined. A shadowgraph system, as illustrated in Fig. 13, enables the second derivative of the index of refraction, Eq. (30), to be determined, and the displacement of the disturbed beam rather than the angular deflection is measured. The displacement tends to be small and therefore difficult to measure and, as a result, the contrast, Eq. (31), is used.

FIG. 12. The schlieren method.

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FIG. 13. Optical path for a shadowgraph system.

a2P ---

130

ni, o -

~y2

(30)

a2ni 1 ~y2 "

The contrast between light and dark patterns is given by

AI

Zsc f C[ p~2T ai-------7 T ~y2

Ii - -- hi,

2p (~_~yT)21 ~- --T-5

dz,

(31)

where C is the Gladstone Dale constant [= (n - 1)/p]. Interferometers measure differences in optical path lengths between two light beams, one of which has passed through the test section and the other bypassing it, as illustrated in Fig. 14. If the two light beams from a monochromatic light source pass through media with an equal index of refraction variation, the recombined beam should be uniformly bright. If, however, the temperature in the test section is elevated in comparison to the bypass beam, with a corresponding difference in the index of refraction, then there will be a difference in the optical path lines and the recombined light beam will exhibit interference fringes with bright and dark patterns. If the temperature distribution can be assumed to be two dimensional and the variation of the index of refraction occurs perpendicular to the light beam, then the fringe shift, q~, can be expressed by Eq. (32) [209]: L ( n i -- ni, O) -

q~

,

)Vo

(32)

154

p.R.N. CHILDS

Fic. 14. Schematicof a Mach-Zehnder interferometer.

where L is length of the test section (m),)~ is wavelength of the monochromatic light source (m), ni is the refractive index field to be determined, and ni, 0 is the reference refractive index in reference beam 2. The temperature distribution can be evaluated from Eq. (33), assuming that the index of refraction is a function of density only, that the density of the gas can be evaluated by the ideal gas law, and that the pressure in the test section is constant [209]: ( )~o~ V(x, y) -

k , p C L M ~p +

1 )-I

(33)

where ~ is the universal gas constant (J/mol K), p is absolute pressure of the test section (Pa), and M is molecular weight. Holographic interferometers utilize the same principle as an interferometer with the exception that the fringe pattern between the object beam and the reference beam is shown on a hologram plate. The principal application of index of refraction methods is the determination of temperatures in combustion processes (see Kosugi et al. [210], Kato and Maruyama [211], Schwartz [212], and Saenger and Gupta [213]).

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4. Absorption and Emission Spectroscopy

Emission and absorption spectroscopy is useful in the determination of temperatures in gases and flames. An atom or molecule will emit electromagnetic radiation if an electron in an excited state makes a transition to a lower energy state and the band of wavelengths emitted from a particular species is known as the emission spectrum. Emission spectroscopy involves measurement of the emission spectrum. This can be undertaken with an atom cell, a light detection system, a monochromator, and a photomultiplier detection system (see Metcalfe [214]). Atoms with electrons in their ground state can absorb electromagnetic radiation at specific wavelengths, and the corresponding wavelengths are known as the absorption spectrum. Absorption spectroscopy relies on measurements of the wavelength dependence of absorption of a pump source, such as a tuneable laser due to one or more molecular transitions. In order to determine the temperature from absorption and emission spectroscopy, it is necessary to fit the spectrum observed to a theoretical model. This normally requires knowledge of molecular parameters, such as oscillator strength and pressure-broadened line widths. The temperature can then be evaluated from the ratio of the heights of two spectral lines using the Boltzmann distribution. The typical uncertainty for these techniques is normally about 15% of the absolute temperature. (Wang et al. [215] reported an uncertainty of 6%.) The use of emission spectroscopy for temperature measurement in flames and gases is reported by Clausen and Bak [216], Malyshev and Donnelly [217], Gicquel et al. [218], Hall and Bonczyk [219], and Uchiyama et al. [220] and for absorption spectroscopy by Cheskis et al. [221], Dhanak et al. [222], Mallery and Thynell [223], and Martin et al. [224].

5. Line Reversal

The line reversal technique involves comparison of the spectral emission of a sample with a comparison continuum. An optical system, as illustrated schematically in Fig. 15, is used to allow the viewing of both a comparison continuum at a known temperature and the medium of interest. If the temperature of the gas of interest is less than that of the brightness comparison continuum, then the spectral line indicated by the spectrometer will appear in absorption, i.e., dark against the background. If the temperature of the test gas is higher than the comparison continuum, then the spectral line will appear in emission or bright against the background (see Gaydon and Wolfhard [225] and Carlson [226]). The temperature of the target gas can be determined by adjusting that of the brightness continuum in a controlled manner until a reversal of brightness occurs. The temperature

156

P.R.N. CHILDS

FIG. 15. Measurement of gas temperatures by line reversal.

of the sample can then be assumed to be equal to that of the comparison continuum. The line reversal method is useful in the temperature range of approximately 1000 to 2800 K with an uncertainty in the order of +15 K. Applications have included combustion chambers [227], flames [228-230], rocket exhausts, and shock waves. 6. Rayleigh Scattering

Scattering is the term used for the absorption and emission of electromagnetic radiation by atoms, molecules, and small particles. Rayleigh scattering is the elastic scattering of light by molecules or very small particles, typically less than about 0.3 txm in size. The intensity is proportional to the total number of particles, N, and the irradiance Ic, IR -- CILN Z

XiO'i'

(34)

i

where C is a calibration constant for the optical system, Ic is laser irradiance (W/m2), N is the number of particles, xi is the mole fraction, and ~i is the effective Rayleigh scattering cross section of each species. Substitution for the number of particles using the ideal gas law gives IR -- CIL p~-~,~R, V

(35)

where ~rR -- ~

Xi~i.

(36)

i

If the Rayleigh cross section is kept constant, the temperature of the probed volume can be found. The calibration constant, C, can be determined by measuring a reference temperature under known conditions and the

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157

measured temperature related to this, assuming that the probe volumes for the reference and test conditions are equal, by T =

ILp~rlR

To.

(37)

It~,ref Po ~ref Iref

Rayleigh spectra can be observed using continuous wave and pulsed lasers to excite the flow, and a typical setup for making measurements using scattering methods is illustrated in Fig. 16. In Rayleigh scattering the collected signal will typically be a factor of 109 smaller than the pump signal, making it susceptible to corruption by other processes, such as Mie scattering, optical effects, and background radiation. Any particle will radiate so the test section must usually be free from contaminant particles, such as soot, and because of this the application of Rayleigh scattering is usually limited to clean flames. In order to analyze spectra, it is normally necessary to know the individual concentrations of the species in the flow. The range of usefulness for Rayleigh scattering is from 293 to 11,000 K [231,232]. Temperature measurements with uncertainty levels of less than 3% are reported by Yalin and Miles [233] and + 3.2 K in a free jet with a temperature from 150 to 170 K by Forkey et al. [234] and are discussed by Namer and Schefer [235] and Otugen [236]. Applications of Rayleigh scattering have included plasmas [232, 237], combustor flames [238-240], sooting flames [241], vapors [242], air jets [243], and supersonic flows [234, 244].

FIG. 16. Schematicof a typical setup for the measurement of temperature using scattering methods (after Iinuma et al. [245]).

158

P.R.N. CHILDS

7. Raman Scattering

Raman scattering is a useful technique for determining the temperature of crystals, films, and gases. If a molecule is promoted from the ground state to a higher state by incident radiation, it can either return to the original state, which is classed as Rayleigh scattering as described in the previous section, or can occupy a different vibrational state, which is classed as Raman scattering. This latter form of scattering gives rise to Stokes lines on the observed spectra. Alternatively, if a molecule is in an excited state, it can be promoted to an even higher unstable state and then subsequently return to its ground state; this process is also classed as Raman scattering and gives rise to anti-Stokes lines on the observed spectra. Raman scattering involves inelastic scattering of light from molecules. There are two basic methods for determining temperature by means of Raman scattering. One is referred to as the Stokes-Raman method and the other as the Stokes to anti-Stokes ratio method. The Stokes-Raman method is based on measurements of density of the unreactive species assuming uniform pressure and ideal gas law conditions. The Stokes to anti-Stokes ratio method involves measurements of the scattering strengths of the Stokes to anti-Stokes signals of the same spectral line. The temperature can then be determined using the Boltzmann occupation factors for the lines concerned [246]. This method is generally only suitable for temperature measurement at temperatures associated with intense combustion due to the relative weakness of the anti-Stokes signal [247]. The setup illustrated in Fig. 16 shows the typical arrangement for undertaking temperature measurements using this technique. The range of application of Raman scattering for temperature measurement is from a few kelvins (e.g., Li et al. [248] and Maczka et al. [249]) to 2230~ The uncertainties associated with the technique are discussed by Laplant et al. [250] and are of the order of 7%; Karpetis and Gomez [251] reported an uncertainty of +75 K. Applications have included crystal temperatures [252], film temperatures [253], reactive flows [254], flames [251, 255], observation of atmospheric conditions [256], and sprays [257].

8. Coherent Anti-Stokes Raman Scattering ( C A R S )

Coherent anti-Stokes Raman scattering is a useful noninvasive technique for the measurement of local temperatures in gases, flames, and plasmas and the temperature range 20~ to about 2000~ Laser beams are used to stimulate Raman scattering through the third order susceptibility of the molecules in a gas flow or a sample. The signal generated is produced in the form of a coherent, laser-like beam that can be separated physically and

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FIG. 17. Schematic of a typical CARS (coherent anti-Stokes Raman scattering) gas temperature measurement system.

spectrally from sources of interference. The emitted light contains information about the population distribution within the rotational energy levels of the target molecule and its intensity is proportional to the density. The population distribution is proportional to temperature. In a typical CARS system, as illustrated schematically in Fig. 17, a N d : Y A G laser (532 nm) is split into three beams, two of which are focused on a small volume (< 0.5mm 3) within the target. The third beam is used to pump a broad band dye laser, referred to as the Stokes laser, at a frequency ~2 with a wavelength of about 606 nm. This beam is also focused onto the target volume. The pump and probe beam frequencies are selected so that the difference O~l- ~2 is equal to the vibrational frequency of a Raman active transition of the irradiated molecules so that a new source of light is generated within the medium, as indicated schematically in Fig. 18. Because the signal appears on the high-frequency side of the pump beam, i.e., an antiStokes spectrum, and because it is observable only if the molecular vibrations are Raman active, this mechanism is called coherent anti-Stokes

FIG. 18. Schematic of the pump and probe beams and the generated beam.

160

v.R.N. CHILDS

Raman scattering [258]. Because temperature is related to the rotational state of molecules, anti-Stokes lines increase in intensity with increasing temperature. CARS is a nonlinear Raman technique so the scattered signal is not linearly related to the input laser intensity. Because the interaction is nonlinear, high laser powers are necessary, which can be attained using pulsed lasers, typically 200 mJ in 10 ns; 20 MW. More recent applications of CARS have made use of the XeC1 excimer laser in place of the Nd:YAG laser, as an XeC1 excimer laser can provide increased flexibility in the laser repetition rate, which can be moderated to match a periodicity in an application, such as, for example, engine speed. Precise alignment and focusing of the three target beams and the detector are necessary. This can limit the effective range of the technique to about 0.5 m from the last focusing lens. Nitrogen is a common target species for temperature measurement using CARS, as N2 molecules tend to be stable and therefore less likely to be involved in reactions. Examples of such systems are reported by Eckbreth [259], Leipertz et al. [260], and Jarrett et al. [261]. Either narrowband or broadband CARS is used depending on the nature of the flow. Narrowband CARS requires the use of a narrowband laser, and fine wavelength tuning must be undertaken in order to match an excitation wavelength to obtain a sufficient output signal. Broadband CARS utilizes a broadband laser and does not need to tuned to a specific excitation peak. The accuracy of broadband CARS is lower than narrowband, but the technique is capable of performing more rapid measurements. Applications of CARS have included flames [262, 263], internal combustion engine in-cylinder flows [264], combustion and plasma diagnostics [265, 266] jet engine exhausts [267], lowpressure unsteady flows [268], and supersonic combustion [269]. The uncertainty associated with CARS is usually specified at around 5%. Merits associated with CARS are the capability to measure temperature noninvasively and at high temperatures. Disadvantages of the technique are its complexity, requiring specialist skills to set the system up and use it, and its high price, up to several hundred thousand U.S. dollars at (year 2000 prices). A further difficulty arises from the requirement to match theoretical energy levels, especially at pressures higher than atmospheric, to the observed band structure. 9. Degenerative Four Wave Mixing

Degenerative four wave mixing (DFWM) is a technique similar to CARS that is also capable of measuring flame temperatures. In DFWM, two highpower pump laser beams are used to intersect with a weaker probe beam, all at the same frequency. Two of the input beams interfere to form a grating and the third is scattered by the grating, producing a fourth beam. The four

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wave mixing process results in an instantaneous stimulated emission and so a stable excited state is not required. D F W M is thus less sensitive to predissociation and quenching processes in comparison to say LIF (Section II,C,10). The output signal produced is a coherent beam and can be filtered easily from background noise. Advantages of D F W M over CARS are that phase matching conditions are satisfied, the process is Doppler free, beam aberrations are lower, and signal levels are higher. A complication of this method, as for CARS, is that severe temperature gradients in a gas cause variation in the index of refraction that can result in significant deflection of the laser beams, making it difficult to set the system up with intersecting beams for all conditions. The use of D F W M for measuring flame temperatures is reported by Smith and Astill [270], Herring et al. [271], Lloyd et al. [272], and Ju et al. [273]. 10. Laser-Induced Fluorescence ( L I F )

For species with concentrations below about 100 parts per million, the density is not high enough to produce a strong enough scattered signal and temperature measurement attempts based on the Raman process are inappropriate. In these situations it may be possible to use LIF. When an atom or molecule has been excited, it will tend to return to its ground state by decreasing its energy level. This can be achieved by a number of routes: absorption of an additional photon, which may excite the molecule to even higher states; inelastic collisions, producing rotational and vibrational energy transfer as electronic energy transfer and quenching; internal or half collisions, which may lead to dissociation or predissociation; and spontaneous fluorescence. Fluorescence is the emission of light that occurs between energy states of the same electronic spin states. Fluorescence can be induced by a number of methods. LIF is the optical emission from molecules that have been excited to higher energy levels by the absorption of laser radiation. Lasers tend to be the preferred means of inducing fluorescence due to their ability to reach high temporal, spatial, and spectral resolutions. The typical lifetime of fluorescence is between 10-l~ and 10-Ss at a wavelength equal to the excitation source, referred to as resonant fluorescence, or at a wavelength longer than the excitation source, referred to as fluorescence. Although resonant fluorescence produces a larger signal, it is susceptible to laser light interference. In the LIF technique the excitation laser is tuned to a frequency that causes a specific species to fluoresce, e.g., NO, SiO, OH, N2, and 02. Two different strategies are available: one using two laser beams at different excitation wavelengths, the two-line method, and the other a single laser beam.

162

P.R.N. CHILDS

The two-line method uses a pair of excitation wavelengths in order to produce two fluorescence signals corresponding to two distinct lower states of the same species. If the two signals have the same upper state, then any difference in quenching is avoided. This method can be applied to molecules, e.g., OH, 02, and NO, or to atoms such as In, Th, Sn, or Pb, which may occur naturally in the medium or can be seeded into it. Uncertainties of 5% can be achieved, but the requirement for two lasers and two ICCD cameras makes the technique expensive. The measurement of flame temperatures using this technique is reported by Dec and Keller [274] and for the temperature of an argon jet by Kido et al. [275]. Use of a single laser beam requires a constant or known mole fraction of the species to excite [276, 277]. This can be achieved by using premixed gases of known concentrations or by seeding the flow with a specific species. The range of application of LIF is extensive, between 200 and 3000 K. LIF has been applied extensively to combustion measurements in flames [240, 278, 279], in cylinder flows [280], and in diesel sprays [281]. 11. Ellipsometry

Ellipsometry is an optical technique involving recording differences in phase jumps for different light polarizations [282]. The surface under study can be illuminated by a light beam, (Fig. 19), with known parameters of the polarization ellipse. The polarization ellipse parameters change after reflection, which can be recorded by the optical system. Ellipsometric thermometry has been reported by Tomita et al. [283], Kroesen et al. [284], Hansen et al. [285], and Magunov [286], with applications to 2000 K.

/x

polariser ~ compensator I

]

target FIG. 19. Ellipsometry(after Magunov [282]).

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12. Acoustic M e t h o d s

The speed of sound is a function of temperature, and this phenomena can be exploited to enable the measurement of temperature in gases, liquids, and solids. In an ideal gas the speed of sound is related to temperature by c = v/TRT,

(38)

where T is the isentropic index, R is the characteristic gas constant (J/kg K), and T is temperature (K). The speed of sound can be measured by a pulse echo or pulse transit time technique [95] where a transducer on one side of the test section of interest is used to generate a pulse, which travels across the test section and is monitored by a microphone on the opposite side at a known separation distance. For a homogeneous gas, the time taken for the pulse transit is 1/x/~ along the signal path and from this the temperature can be evaluated. If the temperature in the region of interest cannot be considered constant, then a number of additional detectors can be installed to determine the weighted average for different paths within the test section. One of the difficulties with this technique is that the speed of sound varies with the composition of the gas. For example, if a number of gases or particulates are present, then these will distort the temperature reading. Acoustic methods have been used to measure temperatures to 17,000 K [287], but at these temperatures, gas density is low and the speed of sound is very high and the time delay becomes dominated by propagation through cooler regions. It can, as a result, be difficult to determine the average value. In liquids, the speed of sound is related to the bulk modulus by

(39) where K is the bulk modulus (N/m2). As there is no means of predicting the variation of bulk modulus and density with temperature, the variation of the speed of sound with temperature must be calibrated against another standard. Data for the variation of the speed of sound for a number of different liquids are tabulated in Lynnworth and Carnevale [288]. One of the applications of acoustic thermometry has been the detection of changes in ocean temperature reported by Forbes [289]. This was achieved by receiving low-frequency sounds (below 100 Hz) transmitted across an ocean basin. In solids the speed of sound is related to Young's modulus for the material by

164

e.R.N. CHILDS

c

(4o)

where Ey is Young's modulus (N/m2). Because solids support both compression and shear, a number of wave types can be utilized for thermometry. The solid of interest itself can be used as the sensor or, alternatively, a wire, e.g., in close thermal contact with the solid, can be used as the sensor and the speed of sound the wire used measured instead. Acoustic methods have been used to monitor temperature in rapid thermal processing where an electric pulse across a transducer generates an acoustic wave guided by a quartz pin [290]. This results in the generation of Lamb waves, which propagate across the medium. Lamb waves are a type of ultrasonic wave propagation where the wave is guided between two parallel surfaces of the test object. Temperatures can be measured from 20 to 1000~ (with a proposed limit of approximately 1800~ with an accuracy of +5~ [291]. Measurement of internal temperatures in steel and aluminum billets has been reported by Wadley [292].

13. Speckle Methods

Speckle photography can be used for gas temperature measurements and provides a line of sight average temperature gradient in any direction. Two sheared images of the object are superimposed to produce an interference pattern using a diffractive optical element as a shearing device [293]. The range of application of speckle photography is approximately from 20 to 2100~ Speckle photography was used by Farrell and Hofeldt [294] to examine a cylindrical propane flame at gas temperatures up to 2000~ The uncertainty associated with speckle methods is approximately 6% of the Celsius reading. Speckle shearing interferometry can be used to calculate the temperature distribution in a gaseous flame [295, 296]. This provides the line of sight average temperature gradient in the direction normal to a line connecting the two apertures of the imaging system. The contours seen are at a constant temperature gradient in one direction only. The temperature range of application is from 0 to 1200~ and the uncertainty level achieved is +0.15% of full scale. The use of this method for flames has been described by Shakher et al. [297] and Nirala and Shakher [298]. A review of speckle photography and its applications is given by Erf [299].

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III. Conclusions

This chapter has reviewed the scope of a large number of techniques currently being used and under development for temperature measurement. In terms of recent advances and developments in the subject of temperature measurement, five areas may be highlighted as significant, namely: 9 requirements that measurements be made with traceability to the ITS-90, 9 use of uncertainty analysis methods to quantify the bounds of uncertainty associated with a measurement, 9 increased capability arising from microfabrication methods, 9 a proliferation of noninvasive techniques, and 9 increased analysis capabilities allowing, for example, real-time indication of measurements and the mapping of temperatures from sparse measurements and accounting for the distortion of a measurement due to local conditions. Developments are likely to continue, especially in the areas of noninvasive measurement and integration of temperature sensors in silicon and in the development of databases and analysis techniques to augment either remote sensing or sparse sensing. In the consideration of temperature measurement, a number of sources of information are highly helpful. These include a number of review articles on the subject, including Valvano [300], Webster [301], Liptak [302], Childs et al. [303], and Rubin [77-79], concentrating on cryogenic applications (normally taken as temperatures less than the freezing point of water). Books on the subject include Quinn [112] concentrating on fundamental considerations relating to standards laboratories; Kerlin and Shepard [304] providing an accessible account, particularly for the inexperienced user of temperature sensors; Bentley [305-307] providing a wide series of overviews; McGee [129] giving an introduction to temperature measurement; Nicholas and White [8] providing insight to the subject of traceable measurement practice; and Childs [208] giving an extensive introduction to temperature measurement practice. In addition, the proceedings of the symposium series entitled Temperature, Its Measurement and Control in Science and Industry [308-312] and the proceedings of Thermosense [313] and Tempmeko [314] provide an excellent source of specialized articles on the subject.

166

P.R.N. CHILDS

Nomenclature aj an As B B(T) c Cp C

C(T) C1 C2 D(T) E Er E~,b f fo h H I /F It Is

Ir k K L m M n ni

probability rate constants surface area of measurement probe (m 2) constant (K) second virial coefficient (cm3/mol) speed of sound in an ideal gas (m/s) specific heat at constant pressure (J/kg K) Curie constant, the Gladstone Dale constant, a calibration constant third virial coefficient (cm6/mol 2) first radiation constant (3.7417749x 10-16Wm 2 [171]) second radiation constant (0.01438769m K [171]) fourth virial coefficient (cm9/mol 3) energy (J) Young's modulus (N/m 2) spectral emissive power for a black body (W/m 3) frequency at T (Hz) frequency at To (Hz) heat transfer coefficient (W/m 2 K) hue intensity forward current (A) laser irradiance (W/m 2) junction's reverse saturation current (A) intensity Boltzmann's constant (= 1.380658• 10-23j/K[171]) bulk modulus (N/m 2) length of the test section (m) mass (kg) molecular weight number of moles of the gas refractive index at a particular location

ni, air ni, 0 N p po q R R Ro R t

Rr R100 S t T To V xz y z a • ~ e ~ ~, p tr

trR O"i

~ Af

refractive index of ambient air reference refractive index in reference beam 2 number of particles pressure (N/m 2) pressure at a reference condition (Pa) charge of electron(1.6 • 10-19C) universal gas constant (-- 8.314510 J/mol K [171]) characteristic gas constant (J/kg K) resistance at 0 ~ resistance at temperature t (f~) resistance at temperature T (12) resistance at 100 ~ (11) saturation temperature (~ temperature (K) temperature at a reference condition (K) volume (m 3) mole fraction coordinate (m) coordinate (m) temperature coefficient of resistance o r ~ -1) (lql'~-1 ~ susceptibility constant emissivity fringe shift wavelength (m) density (kg/m 3) Stefan-Boltzmann constant and is equal to 5.67051 • 10-SW/m2K 4 [171] Rayleigh cross section (m2/sr) effective Rayleigh scattering cross section of each species (m 2/sr) time constant (s) frequency interval (Hz)

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ADVANCES IN HEAT TRANSFER, VOL. 36

Swirl Flow Heat Transfer and Pressure Drop with Twisted-Tape Inserts

RAJ M. MANGLIK Thermal Fluids and Thermal Processing Laboratory Department of Mechanical Industrial and Nuclear Engineering University of Cincinnati Cincinnati, Ohio 45221, USA

A R T H U R E. BERGLES Department of Mechanical Aerospace, and Nuclear Engineering Rensselaer Polytechnic Institute Troy, New York 12180, USA

Abstract

An extended review of the application of twisted-tape inserts in tubular heat exchangers and their thermal-hydraulic performance is presented. Twisted tapes promote enhanced heat transfer by generating swirl or secondary flows, increasing the flow velocity due to the tube partitioning and blockage, and providing an effectively longer helical flow length. Depending on the tape-edge to tube-wall contact, some fin effects may also be present. Their usage in both single-phase and two-phase (boiling and condensation) flows is considered, and heat transfer and pressure drop results from different investigations are presented. The characteristic features of swirl-induced heat transfer enhancement, nature of swirl flows and their scaling, and development of predictive correlations for heat transfer coefficients and friction factors (or pressure drop) are discussed. Also, some aspects of the use of geometrically modified twisted-tape inserts, as well as compound application with other enhancement techniques, are briefly discussed. 9 2002, Elsevier Science (USA).

183 ISBN: 0-12-020036-8

ADVANCES IN HEAT TRANSFER, VOL. 36 Copyright 2002, Elsevier Science (USA). All rights reserved. 0065-2717/02 $35.00

184

RAJ M. MANGLIKAND ARTHURE. BERGLES I. Introduction

A. GENERAL BACKGROUND Heat transfer is perhaps the most important, as well as the most applied, process in virtually every industrial, commercial, and domestic activity. The conversion, utilization, and recovery of energy invariably entail a heat exchange process. Common examples are fuel combustion-products heating of water for steam generation, steam heating of viscous process media, air and liquid cooling of automobile engines, and waste-gas heating of various fluid streams. A striking example of an industry where heat transfer plays a dominant role is the process industries (e.g., chemical, petrochemical, food, plastics and rubber, and pharmaceutical). Effective heat exchange is critical to the process efficiency and it significantly influences the economics of the design and operation of the plant. By one estimate [1], the chemical and process industry consumes 34.6% of the total annual primary energy usage, of which almost 50% is utilized for process heating. In terms of capital expenditure, heat exchangers for this sector of the industry accounted for over 950 million dollars in 1991, i.e., 55% of the estimated total heat exchanger market in the United States [2]. In process heat exchange applications, a variety of complex, highly viscous liquids are involved. These fluids more commonly undergo a heating, cooling, or phase-change process while flowing through tubular type exchangers. Because of their viscous nature, they tend to have flow rates that are in the laminar regime, with inherently low heat transfer coefficients. These conditions generally represent the dominant thermal resistance in a shell-and-tube heat exchanger and adversely affect its size and capital and operating costs. The goals of energy conservation require the use of more efficient heat exchangers. With increased thermal performance of such equipment, substantial fuel, material, and cost savings can be made and, at the same time, the consequent environmental degradation can be reduced. As documented by Bergles [3] and Webb [4], a large number of techniques have been developed to enhance heat transfer. They reduce the thermal resistance by promoting higher convective heat transfer coefficients with or without surface area increases. The application of these techniques can thus result in reduced heat exchanger size, increased capacity of an existing exchanger, decreased pumping power requirements, or reduced approach temperature differences. The latter is particularly important to foods, pharmaceutical, and plastic media, where thermal degradation of the end product can be avoided. In fact, relatively higher performance improvements can be achieved in viscous fluid, low Reynolds number flows than in highly turbulent flows.

SWIRL FLOW HEAT TRANSFER

185

This is of considerable significance to the process industry where primarily viscous liquids are handled and produced. Heat transfer enhancement techniques are broadly classified as passive and active techniques, and their different subcategories are listed in Table I. Passive techniques generally use surface modifications or incorporate an additional device into the system. Except for extended surfaces that increase the heat transfer area, these devices promote higher heat transfer coefficients by essentially disturbing or altering the existing flow behavior. They tend to generate "well-mixed" flows with considerably sharper wall-temperature gradients than normal. The thermal gains, however, are accompanied by increased pressure drops. Active techniques require the addition of external power in order to bring about the desired flow modification and improve the rate of heat transfer. Compound enhancement entails the usage of two or more passive and/or active techniques together. There is an enormous database of technical literature on enhancement of heat transfer (now estimated at over 8000 technical papers and reports), which has in the past been variously referred to as augmentation and intensification, among other terms. It dates back to 1861, the classical study by Joule [5], when the first attempt to enhance heat transfer coefficients was reported. This collection of literature has been periodically recorded in several bibliography reports [6-8], and its growth now urgently requires that a digital library be established [9]. The role and impact of these techniques in conserving energy largely depend on the type of heat exchanger, flow conditions, and the technique itself. Passive techniques, however, have found greater usage, and several different criteria for establishing their enhancement capabilities in any given application are given by Bergles [3] and Webb [4].

1. Swirl Flows Of the several enhancement techniques used in thermal processing, energy conversion, and other applications, a very attractive and much utilized method is to generate swirl flows in tubes and ducts. Swirl-flow devices are a class of passive techniques and, as highlighted in Table I, consist of a variety of tube inserts, geometrically varied flow arrangements, and duct geometry modifications. A schematic representation of each of these three swirl-generating methods is given in Fig. 1. Examples of tube inserts include twisted tapes, axial cores with screw-type windings, helical vane inserts, static mixers, and periodically spaced propellers. Tangential fluid entry (as in cyclone separators), periodic tangential injection, and inlet swirl vanes are some different flow arrangements that produce swirl in ducts. Tubes of noncircular cross sections (elliptic and rectangular), which are helically

186

RAJ M. MANGLIK AND ARTHUR E. BERGLES

TABLE I CLASSIFICATION OF HEAT TRANSFER ENHANCEMENT TECHNIQUES AND CHARACTERIZATION OF PASSIVE SWIRL FLOW GENERATING METHODS

Passive techniques Treated surfaces Rough surfaces Extended surfaces Displaced enhancement devices I Swirl flow devices Coiled tubes Surface tension devices Additives for fluids

If

Tube/duct inserts Altered flow arrangements and radial-inlet swirlers Tube/duct geometry modifications

Active techniques Mechanical aids Surface vibration Fluid vibration Electrostatic fields Suction or injection Jet impingement

Compound enhancement Rough surface with a twisted-tape swirl flow device, for example.

twisted about their central axis, are some examples of duct geometry modifications that induce swirl flows. Furthermore, tube rotation, either around its own axis or a lateral axis, is an example of an active swirl generation technique. This review, however, is restricted to the use of twisted-tape inserts (passive technique). They are perhaps the most widely used swirl-generating devices, with a broad range of application in both single-phase (heating/cooling of gases and liquids) and two-phase (refrigerant, water-steam, and organic liquid-vapor systems) forced convection. Also, there is a considerable body of literature on their application, and the associated flow physics and design issues have attained a fair degree of maturity. Information on some other swirl flow devices, as well as active methods, can be found elsewhere [3,6-8,10].

2. Twisted Tapes Most swirl-flow techniques have been found to promote significant improvements in heat transfer, although a penalty for the concomitant increase in pressure drops is also incurred. In tubes fitted with swirl-flow devices, the heat transfer enhancement occurs primarily due to the fluid agitation and

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mixing induced by the cross-stream secondary circulation that is generated in the helical fluid motion. In the case of twisted-tape inserts, apart from their longitudinal vortex-generating characteristics, the tape thickness, helical flow path, and fin effects of the tape surface also influence the thermalhydraulic performance. Perhaps the most attractive feature of twisted tapes is the relative ease with which they can be manufactured. A thin metallic strip, of a width equal to the tube inside diameter, with some fit allowance, is twisted into a constant-pitch helix. As shown in Fig. 2, the geometrical characteristics are described by the 180 ~ twist pitch H, tape thickness 8, and tape width w ~ tube inside diameter d; usually the severity of tape twist is referred to by a dimensionless twist ratio, y = H i d . Depending on the application, tube diameter, and tape material, inserts of very small twist ratios can be employed--generally, the smaller the twist ratio the greater the heat transfer enhancement. Also, they can be made in a variety of metallic (e.g., carbon steel, stainless steel, yellow brass, monel, and nickel), and nonmetallic materials (e.g. ceramic and plastic). Twisted tapes are typically inserted in tubes with a snug-to-loose fit, and, as mentioned earlier, the enhanced thermal-hydraulic performance is primarily attributable to swirl flow mixing, though there may be some contributions of an increased effective flow length (or residence time), increased flow velocity in the partitioned duct, and tape fin effects [3, 11].

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FIG. 3. Typical usage of twisted-tape inserts in a shell-and-tube heat exchanger (courtesy of Brown Fintube Company).

These devices are readily available commercially, and one manufacturer 1 has been marketing them since the mid-1960s for different shell-and-tube and double-pipe heat exchanger applications. In the commercial literature, twisted tapes are often referred to as flow "turbulators," "retarders," or "mixers." Typical liquid service applications are for heating (or cooling, as the case may be) of glycols, crude oil, lube oil, resins, polymers, and many other Newtonian as well as non-Newtonian viscous process fluids. Some examples of gas flow systems, where twisted tapes provide effective enhancement, are fire-tube boilers, domestic water heaters, waste heat recovery units, and process gas cooling exchangers. A very frequent usage of these devices is to retrofit existing heat exchangers in order to upgrade their performance; if employed in a new exchanger, for the same heat duty, significant size reduction can be achieved. The ease with which the inserts can be placed (and removed) in tubes of a multitube bundle, as depicted in Fig. 3, makes them 1Brown Fintube Company, Houston, Texas.

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RAJ M. MANGLIK AND ARTHUR E. BERGLES

useful in fouling situations as well. Unlike the enhancement techniques that involve permanent surface modifications (integral fins, ribbed tubes, etc.), tape inserts can be removed readily to facilitate periodic cleaning. This, of course, presumes that the tapes have not been "cemented" to the tube by the fouling deposit. B.

HISTORICAL OVERVIEW

In one of its earliest application more than a century ago in 1896, Whitham [12] recorded what is perhaps the first scientific experiment on a practical use of twisted tapes. Inserts with y - 15 were loosely fitted in a 74.6 KW boiler, consisting of forty-four 101.6-mm-inner-diameter and 6.096-m-long horizontal tubes. Fuel savings of as high as 18% were obtained under peak operating conditions. A similar application was reported more than 50 years later by Kirov [13]. In this instance, tapes of two different twist ratios (y = 2.55 and 5.11) were used, resulting in fuel savings of 7 to 10%. Furthermore, the first U.S. patent on twisted tapes can be traced to 1930 when Patent No. 1,770,208 was assigned to J. Kemnal [14,15]. Its intended application was to improve the gas-side heat transfer in a vertical, multitube, waste heat recovery air heater. The flueways of many modern, gas-fired, domestic water heaters are also fitted with twisted-tape inserts (or some modifications thereof) to enhance the gasflow heat transfer coefficients [7]. Beginning in the early 1960s, and driven by the rapid developments in nuclear power generation, the application of twisted tapes was extended to water flows. The primary need was to improve the performance of boilers and nuclear reactors, and several studies have recorded such efforts (see, e.g., Brevi et al. [16] and Gambill [17]). For this two-phase flow problem, twistedtape inserts have been found to be beneficial for almost all flow boiling situations, and the critical heat flux for subcooled boiling of water can be increased by up to 100% over smooth tube flows [3]. A decade later, in the 1970s, investigations dealing with viscous process fluids and twisted tapes inserts for flows in the laminar regime began to be reported in the literature (Hong and Bergles [18], Donevski and Kulesza [19], and several others [11]). The relative heat transfer improvements in such conditions are substantially greater than that in turbulent flows [20,21], and the use of twisted-tape inserts has now been extended to highly viscous media [22] that include non-Newtonian fluids [23,24] as well. Again, responding to energy conservation challenges in the 1970s, twisted tapes began to be used to enhance heat transfer in refrigerant evaporators and condensers [3,8]. An example application in a compact tube-fin heat exchanger is illustrated in Fig. 4. Several modifications of twisted-tape inserts have also been used in various studies reported in the literature. They range from short inlet inserts [25],

SWIRL FLOW HEAT TRANSFER

191

FXG. 4. Typical usage of a twisted-tape insert in tube-fin type compact heat exchangers used for refrigerant evaporators (courtesy of the Trane Company).

to punched or perforated tapes and bent-strip turbulators [26, 27], to intermittently spaced inserts [28, 29]. The primary motive in these variations is to reduce the pressure drop penalty while maintaining, if not improving upon, the thermal performance of a full-length insert. Some compound enhancement schemes involving twisted tapes have also been considered [30-32]. However, compound usage is perhaps best suited where one form of enhancement preexists in the system (e.g., rotating channels of electric machines [33]). The primary usage of a twisted-tape insert to enhance in-tube heat transfer coefficients can be grouped into the following three primary application modes:

1. Full-length twisted-tape inserts, when the insert occupies the entire heated (or cooled) length of the heat exchanger tube. 2. Modified twisted-tape inserts, which include a short length of tape placed in the tube inlet, variations in tape geometry and/or surface, and interspaced short-length inserts. 3. Compound enhancement with twisted tapes, when tape inserts are used along with one, or more, other enhancement techniques. Bergles et al. [6, 7] and Jensen and Shome [8] have produced exhaustive bibliographies of the worldwide enhancement literature. A chronological and annotated listing of the literature on single-phase flows with twistedtape inserts is presented in Ref. [11], and the compilation is representative of the historical development and the current status of technical information on

192

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

twisted tapes. The current imperatives have been to identify the phenomenological behavior, understand the involved mechanisms, and develop design correlations that have generalized applicability [20, 21, 34-39]. Most of the pioneering work has been concerned with in-tube gas flows, primarily for fire-tube boiler and and hot-water heater applications. An early but comprehensive study of twisted-tape inserts was reported by Royds [40], who conducted detailed experiments to test the "prevailing idea that the spiral motion given to the gas flowing through the tube." A large number of air-cooling tests were made with inlet temperatures ranging from 494 to 811 K, and nine different twisted tapes with a wide range of twist ratios (2.36 <_ y < 19.24) that were covered with soot to simulate actual firetube boiler conditions. Heat transfer enhancement of up to 1.4 times over smooth tube flows was obtained with smaller twist-ratio tapes. Subsequently, Colburn and King [41], Kirov [13], and Evans and Sarjant [42] also reported experimental results for cooling of hot air flows, whereas Koch [43] and Kreith and Margolis [44] considered heating of air. Evans and Sarjant [42] recognized that at high temperatures there will be a radiation contribution from the inserts and attempted to separate the radiation and convection effects of twisted tapes. Up to a 25% radiation contribution of the total heat transfer was estimated with gas temperatures ~811 K. In a few studies, efforts have been made to characterize the swirl flow or secondary circulation structure by measuring the axial velocity profiles. Notable among some of the earlier works are the studies by Smithberg and Landis [45] and Seymour [46]. In the former investigation, air-flow velocity distribution at the exit of a test section containing a twisted tape with y = 5.15 was measured using hypodermic static and impulse probes. Seymour [46] employed a radioactive gas tracing method to obtain secondary flow patterns in the cross section of a tube with a y = 4.76 twisted tape. Axial velocity profiles were also measured by a thermistor anemometer, and the patterns were similar to those given by Smithberg and Landis [45]. The turbulent flow field was essentially found to consist of double-helical longitudinal vorticies. The more recent flow visualizations for laminar flows of air reported by Manglik and Ranganathan [47] also confirmed these observations. Furthermore, isothermal friction factor data of Seymour [46] displayed a "smooth" transition from laminar to turbulent flows, particularly with tapes of small twist ratios. A similar trend has been observed in several other studies [19, 21,40, 48], and the implication is that secondary motion tends to suppress and delay the onset of turbulent fluctuations. A large number of experiments have been conducted with turbulent flows of water in tubes containing twisted tapes (see, e.g., Refs. [16, 21, 49-52]). As indicated previously, a majority of the early work was intended to provide single-phase baseline data for the study of flow boiling (and condensation to

SWIRL FLOW HEAT TRANSFER

193

a lesser extent), and much of it was for the turbulent flow regime [3, 8, 11, 39]. Depending on flow rates and twist ratios, 20 to 50% enhancement in heat transfer has been obtained in comparison with smooth tube flows, on a fixed pumping power 2 basis. Also, several Nusselt number and friction factor correlations have been devised. Hong and Bergles [18] reported the first set of experimental data for twisted tapes in laminar flows of viscous liquids. Based on heat transfer measurements for water and ethylene glycol flows in electrically heated tubes, a correlation for predicting Nusselt numbers in fully developed swirl flows is given. Marner and Bergles [22, 53] have reported data for in-tube flows of polybutene 20 and ethylene glycol in double-pipe heat exchangers with a y = 5.39 twisted-tape insert. Data for polybutene (1000 < Pr < 7000) suggest that in highly viscous flows, swirl does not set in and the enhancement is simply due to the duct partitioning. Also, in data for both liquids, cross-stream temperature-dependent viscosity variation needs accounting to reconcile heating and cooling effects, and there is a general lack of agreement with the Hong and Bergles [18] correlation for uniformly heated tubes. Extended data for constant-temperature tubes reported by Manglik and Bergles [20,21] corroborate this further, which essentially dismantles the previously held contention that entrance effects, fluid property variations, and boundary conditions have little effect in swirl flows. In fact, even buoyancy-driven mixed convection can play a role in weakly swirling flows with large y tape inserts [20, 54, 55]. Nazmeev [56], Manglik et al. [23], Dasmahapatra and Raja Rao [24], Sivkumar and Raja Rao [31], and a few others [11] have investigated the heat transfer enhancement in laminar flows of non-Newtonian liquids. Most of this work has been restricted to purely viscous liquids whose rheology can be represented by the power-law constitutive relationship. It has generally been found that after accounting for non-Newtonian effects, data tend to agree well with correlations for Newtonian liquids [23, 24]. Singh and Manglik [57] and Etemad and Mujumdar [58] have computationally modeled the limiting case of laminar flow of pseudoplastic fluids in a semicircular duct (y = oo, 8 = 0). However, there are relatively few data available, and more investigations are needed to fully characterize the effects of twisted-tape inserts on enhanced heat transfer in non-Newtonian fluids. Particular attention needs to be given to viscoplastic (yield stress type shear thinning and shear thickening fluids) and viscoelastic behaviors, as they are encountered quite commonly in a large number of chemical process fluids. There have been very few attempts to computationally model the swirl flow heat transfer enhancement induced by twisted tapes. The study 2This figure of merit and other performance evaluation criteria are addressed later in Section V.

194

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

reported by Date and Singham [59] and Date [60], where numerical solutions for the uniform heat flux boundary condition have been presented, is perhaps the first such attempt. There was, however, a computational error in the reported Nusselt number results in this work, which was subsequently increased (corrected) by a factor of two based on the experimental findings of Hong and Bergles [18]. Furthermore, the main focus of their numerical study was to calculate global results ( f and Nu) and develop a correlation. This largely excludes the advantage of computational simulations in extracting local information about the flow structure and convection behavior. The numerical modeling reported by DuPlessis and Kr6ger [61, 62] has similarly engaged only in the development of predictive or correlating expressions for f and Nu. Computational results for laminar forced convection in the limiting case of a straight tape insert (y = oc) with zero thickness (8 = 0) have been given by Hong and Bergles [63] for the UHF condition and by Manglik and Bergles [64] for the UWT condition. For the latter case, the effects of tape thickness or flows in circular-segment ducts have also been considered by Manglik and Bergles [65]. They have analyzed both the thermal entrance region and fully developed flows and have shown that considerable enhancement can be obtained by simply partitioning the tube, and by the fin effects of the insert. In a more recent study [66], the character of swirl flows induced in the low Reynolds number regime by zero-thickness twisted-tape inserts has been identified. The computational results corroborate the features seen in flow visualization studies [47], i.e., a single helical vortex at low Re and/or large y grows into two counterrotating helical vortices with increasing Re or decreasing y. This promotes considerable cross-stream fluid mixing and perhaps is the dominant enhancement mechanism in the fully developed swirl-flow regime. In two-phase flow applications, twisted tapes have commonly been used for enhancing flow-boiling heat transfer [3, 8, 39]. Depending on the tapetwist ratio, heat transfer coefficients that are up to 90% larger than those in smooth tubes for the same flow conditions have been obtained by Jensen and Bensler [38]. They observed that the higher performance tended to be at high qualities with small (or tighter) twist-ratio tapes. Likewise, in horizontal refrigerant evaporators with R-12, heat transfer has been found to be enhanced by almost 3.5 times that in a plain tube per unit pumping power for some optimum twist ratio, heat flux, and mass flux conditions [67]. The greatest benefit, however, is obtained in the enhancement of critical heat flux (or the suppression of CHF conditions) when the centrifugal-type force imposed by the helical flow pushes the liquid droplets from the core of the two-phase flow to the wall, thereby delaying dryout [3, 68]. Gambill et al. [68] have reported up to 100% increase in CHF when compared to empty smooth

SWIRL FLOW HEAT TRANSFER

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tubes, at the same mass flux, in subcooled boiling of water. Royal and Bergles [69, 70] experimentally investigated the use of twisted-tape inserts for in-tube steam condensation and found a 30% improvement in the heat transfer coefficient; the pressure drop, however, also tended to be rather large. Later, Luu and Bergles [71,72] studied the application of twisted tapes to tube-side condensation of R-113 and obtained similar results. Recent research activity has also focused attention on several variations of full-length tapes. The more promising embodiment is a short length of twisted tape placed in the inlet of the duct. The inlet tape section is long enough to initiate swirl, which then decays in the subsequent empty-tube section. Depending on the twist ratio, the swirl-enhanced heat transfer is sustained through most of the decay region where the absence of the insert significantly reduces the pressure drop penalty. For a fixed pumping power constraint, Klepper [25] found up to 7% enhancement in heat transfer over full-length tapes in an experimental study with turbulent airflows. In several other studies [73, 74], theoretical analyses of decaying turbulent swirl flows have been presented. The analytical predictions obtained by Algifri and Bhardwaj [74] from a series solution of simplified Navier-Stokes equations are in good agreement with the experimental data of Klepper [25]. Zozulya and Shkuratov [75] have considered short tapes of variable twist pitch to reduce the pressure drop but maintain the same heat transfer performance as full-length twisted-tape inserts. An extension of the decaying swirl concept is to use short lengths of twisted tapes intermittently along the tube length. This, in principle, allows the decaying swirl to be reinitiated successively, thereby enhancing the heat transfer with relatively less pressure drop in comparison with fulllength inserts. Burfoot and Rice [28] and Saha et al. [29, 76], among a few others, have investigated the performance of intermittently placed twisted-tape elements in laminar and turbulent flow regimes. In the Saha et al. [29, 76] studies, single-pitch (180 ~ twist) long elements were connected at regular intervals by a thin rod whose ends were slotted to fit the tape. While up to 1.47 times improvement in heat transfer was obtained in laminar flows on the basis of constant pumping power, there was no significant advantage in turbulent flows. In yet another modification of twisted tapes, the use of punched or slit-edged inserts does not appear to provide any advantage over plain full-length tapes [26, 77]. Inserts made of strips with S-shaped cross sections have been proposed in a recent patent (U.S. Patent No. 4,700,749 [78]), where it is claimed that such tapes reduce the pressure drop, but no supporting data have been presented. With renewed interest in waste heat recovery and energy conservation, the application of twisted-tape inserts to enhance heat transfer in hot gas flows

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RAJ M. M A N G L I K AND A R T H U R E. BERGLES

has been revisited. The recent work has special emphasis on high temperature waste heat recovery, where, along with the swirl-promoted convection heat transfer improvements, there are substantial tape surface radiation effects that provide additional enhancement [7]. Davidson [79] and Beckermann and Goldschmidt [80] have reported the same order of magnitude contributions due to convection and radiation in the 50% increase in the heat transfer performance of turbulent hot gas (Tb ,-~ 1 5 0 - 700~ when compared to that in an empty tube. In a more comprehensive investigation by Watanabe et al. [81] with high temperature ( ~ 1000~ gas flows and tape and tube emissivity of about 0.5, the radiation effects are seen to increase the heat transfer coefficient by as much as 1.5 times that of the convective coefficient, which, in turn, with a twisted-tape insert is about 3 times that of smooth tube flows. Additional gains can perhaps be obtained by improving the tape and tube surface emissivity. In fact, a patent has been issued (U.S. Patent No. 4,559,998 [82]) for a twisted tape that supposedly enhances radiation heat transfer. Armstrong and Bergles [83] have tested twisted tapes in SiC tubes for waste heat recovery exchangers in corrosive high temperature environments. Likewise, Junkhan et al. [27] have considered twistedtape and bent-strip turbulator inserts for fire-tube boilers. In both these studies [27, 83], however, low temperature airflows were used to define the convection heat transfer enhancement. Yamada et al. [84] have described an interesting and promising modification for cross-flow shell-and-tube recuperators. Twisted cross-tape inserts are used along with a radiation plate on the shell side to enhance both convective and radiation contributions to heat transfer. From this overview it is amply evident that a twisted-tape insert is an effective heat transfer enhancement device in a variety of applications that range from laminar viscous liquid (Newtonian and non-Newtonian) flows, to high-temperature turbulent gas flows, to two-phase (boiling and condensation) flows. The continuing present-day research activity [34, 66, 85-89] further testifies to their beneficial utility. The relatively higher enhancement in viscous liquid flows makes twisted tapes particularly attractive for process heat exchanger applications, where their usage should provide effective thermal processing, as well as energy conservation. Likewise, high-performance and close temperature control benefits can be obtained in cryogenic, refrigeration, and human comfort systems, which are also energy-intensive applications. Their geometric simplicity, which entails low manufacturing costs, and availability in several metallurgies are additional incentives. However, for effective practical usage, generalized guidelines are needed for designing and evaluating the performance of heat exchangers with twistedtape inserts, which are presented in Section II for single-phase flows and Section III for two-phase flows.

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II. Single-Phase Flow and Heat Transfer

Perhaps the most critical measure of the research literature on twistedtape inserts, and its relative maturity, is the availability of generalized and reliable correlations for predicting the thermal-hydraulic performance. They are the primary design tools for the practicing engineer and vital technology transfer constituents, where heat exchanger designs have to cater to a broad spectrum of operating services that involve different fluids, flow rates, and heating or cooling duties. Quite frequently, off-design operating performances also have to be predicted and established. This implicitly requires a fair understanding of the tape-induced swirl-flow structure and its scaling, an ability to model the complex convection behavior, and a broad experimental database. These issues for single-phase forced convection in circular tubes with full-length twisted tapes are addressed in the following subsections. A. ENHANCEMENT CHARACTERISTICS 1. Heat Transfer and Pressure Drop Data

Experimental results for enhanced heat transfer and the associated friction loss have been reported extensively in the literature. They include data for gas, water, and highly viscous liquid flows in both laminar and turbulent flow regimes. The enhanced performance is generally attributed to swirl-flow mixing, increased effective flow length, increased flow velocity, and tape fin effects. The largest benefits of using twisted-tapes inserts, however, appear to be obtained in viscous liquid laminar flows [11,90, 91]. This is not surprising, as turbulent flows are inherently characterized by well-mixed, highly convective flows, and tape-induced swirl would tend to have relatively less impact on further flow agitation. Experimental heat transfer coefficient data for laminar flows of ethylene glycol (97 < Pr < 191) and water (3 < Pr < 8) has been presented by Hong and Bergles [18, 92]. Two different full-length, snug-fitting insulated tapes of twist ratios y - 2.45 and 5.08, and (g/d) - 0.0454, were used in an electrically heated tube, which simulates the uniform wall heat flux condition (UHF) in tubular heat exchangers, either directly for heating of a liquid or with equal heat capacity rates (C* - 1) for two fluid streams. Their results are essentially for fully developed swirl flow conditions, as seen by the weak dependence of Nu on the reduced thermal length or Graetz number in Fig. 5. It is evident that in addition to tube partitioning, 3 represented by the 3The tube partitioning alone, with no fin-effect contributions from a straight tape insert, is seen to provide heat transfer enhancement in Fig. 5 when compared with the laminar flow performance of uniformly heated (UHF) smooth circular tubes [93].

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numerical results [63] for the limiting case of an insulated straight-tape (y - oo) insert, the tape-twist ratio y, flow Re, and fluid Pr have a significant influence on the heat transfer enhancement. The latter effects are depicted in Fig. 6, where it is seen that the tighter tape twist produces higher swirlenhanced heat transfer coefficients. Marner and Bergles [22, 53] have reported heat transfer coefficient data for laminar flows of polybutene 20 (1000 < Pr < 7000) [22] and ethylene glycol (20 < Prl00) [53]. A twisted-tape insert with y - 5 . 3 9 and (8/d) = 0.053 was used in a circular tube that was maintained at a uniform wall temperature (UWT). Their data, graphed in Fig. 7, could not be correlated with the Hong and Bergles [18] results, thereby indicating that wall boundary conditions influence the heat transfer performance [94]. Furthermore, as mentioned earlier in the preceding section, it is evident from Fig. 7 that tapeinduced swirl flows are not fully established in the more viscous liquid (polybutene 20) and that enhancement is simply due to tube partitioning, flow acceleration, and thermal entrance effects. The agreement of these data with the numerical results [64] for an insulated straight ( y - oo) insert

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0

@

Hong and Bergles [18] data (8/d) - 0.0454 9 o 9 []

1

101

I

I

I

I

I

I III

I

1

10 2

I

I

I

Ethylene glycol, y = 2.45 Ethylene glycol, y = 5.08 Water, y = 2.45 Water, y = 5.08

I 111

10 3

I

I

I

1

i

~ ii

10 4

Re FIG. 6. The effect of tape-twist ratio y on fully developed swirl-flow heat transfer in uniformly heated (UHF) tubes.

suggests that fin effects are somewhat negligible with snug-to-loose fit tapes, which is further supported by data of Manglik and Bergles [90] and is shown in the subsequent discussion. Ethylene glycol data, however, exhibit a fully developed swirl flow behavior where the heat transfer coefficient depends on Re, Pr, and y. Also, temperature-dependent fluid viscosity variations are significant, despite the helical vortex mixing promoted by the twisted tape, and a Sieder and Tate [95] type of viscosity-ratio correction accounts for its influence. Another set of data for heat transfer in isothermal (UWT) tubes reported by Manglik and Bergles [11,90] provide further insights for the convective mechanisms at play in the presence of twisted-tape inserts. Three different twisted tapes (y = 3.0, 4.5, and 6.0) were employed with ethylene glycol (68 < Pr < 100) and water (3.5 < Pr < 6.5) as test fluids. These results for snug-fitting tapes with ( B / d ) = 0.0228, which are primarily for fully developed swirl flows where Nu ~ ~(Gz), are graphed in Fig. 8. In addition to further demonstrating that a viscosity-ratio factor is needed to account for fluid heating and cooling effects, even under swirl flow conditions, they show a rather smooth transition from laminar to turbulent flows. This has

200

RAJ M. MANGLIK AND ARTHUR E. BERGLES

I

I

t

I

I

I I I I

r

I

I

I

J

Marner and Berlges [22, 53] data: y = 5.39,

9 o 9 D

102

Z

_

I

r

I

I

I I

polybutene 20 (heating) polybutene 20 (cooling) ethylene glycol (heating) ethylene glycol (cooling)

y=oo, 8=0 insulated tape [64] ~

~

~

../.~-

~

101 - ~ ~I

I

I

I lli

I

oo .......... _. . . . . . "'"

oo

.................

_

~

Uniform temperature _ circular tube [93] (UWT)

........... . ......... i,-.'"["

101

[ I I[

(tiM) - 0.053

I

102

I

1

i

i Ill

I

103

I

I

J

-i

Ili

104

Gz FIG. 7. Thermal entrance, heating and cooling, tube partitioning, and tape-generated swirl flow effects on the mean Nusselt number in a uniform wall temperature circular tube.

also been observed in water data reported by Nair [48] for y = 3.3 and 4.8 and by Ishikawa and Kamiya [96] for y = 1.5. 4 Of course, tapes with smaller twist ratios produce higher Nusselt numbers for the same flow conditions. As found earlier with results from Marner and Bergles [22], the contention that heat transfer enhancement due to fin effects of snug- to loose-fitting tapes is negligible is once again shown in Fig. 9. Here data for twisted-tape inserts with y = 6.0 but two different tape width to tube diameter ratios (w/d = 0.89 and 0.96) are graphed and, as can be seen, there is virtually no difference between the two sets of Nusselt number results. Similar observations have been made by Fujita and Lopez [86] for turbulent flows of liquid R-113 using two snug-fitting tapes with y -- 10 made of stainless steel and Teflon. The turbulent water flow heat transfer data of Lopina and Bergles [52] with insulated and uninsulated tapes, however, suggest some fin effects. It may be noted that in instances where the tape is made of metallic material of very high thermal conductivity and is mechanically attached to the tube wall, it could act as a fin and provide additional active heat transfer 4This is perhaps the tightest twist-ratio tape that can be produced, although, as noted previously, typically y ~ 2 is the practical limit for most industrial manufacturing by mechanically twisting fiat metal strips [97].

SWIRL FLOW HEAT TRANSFER

103

i

i

r

i ' ll'l

~

'

.....

201

I

Manglik and Bergles [901 data, y = 3.0, 0S/d)' = 0.02'3' ' o Ethylene glycol (heating) Ethylene glycol (cooling) ~ Water (heating) v~ v Water (cooling) o~s~_ =l.

e~ s S ~

"~ 102 ~ C , ~ r

Z

~ ~

~ V ~

R e < 1 0 4

n = 0.14 R e > 104 n - 0.18 (heating) n - 0.30 (cooling)

~7 v

10 ] 102

i

i

i

i i iii1

1

i

1

1

11

103

i

i

i

i

1 1

104

105

Re 103 Manglik and Bergles [90] data, y o Ethylene glycol (heating) o Ethylene glycol (cooling) ~ Water (heating) v Water (cooling)

4.5, (~5/d) = 0.023

v v~

:a.

,~ 102 ~ R e < 104 n =0.14 Re > 104 n - 0.18 (heating) n = 0.30 (cooling)

Z ~ ~W ' ~ 101 102

i

i

i

~ ~ ~

i i i ill

i

i

i

i [ 11]

103

1

i

]

i

i

1

104 Re

103

,

p

,

r ,T,I

. . . .

i

Manglik and Bergles [90] data, y ~ Ethylene glycol (heating) o Ethylene glycol (cooling) Water (heating) v Water (cooling)

"

6.0, (~5/d) = 0.0*23 '_z

--t

102 Z

Re < 104

--

n = 0.14 Re>

101

~

102

J

L j j ~ l

I

I

103

I

(heating)

n = 0.30

(cooling)

i

L J Ittl

104

104

n = 0.18

i

i

J i i ii

105

Re FIG. 8. E n h a n c e d heat transfer in viscous liquid swirl flows in circular tubes maintained at u n i f o r m wall temperature ( U W T ) with twisted-tape inserts ( y = 3.0, 4.5, and 6.0).

202

RAJ M. MANGLIK AND ARTHUR E. BERGLES

10 3

,

I

I

I

I 1TII

I

I

I

1

I

I I I I

I

I

I I I_

Manglik and Bergles [90] Water data (3.5 < Pr < 6.5) y - 6.0,

(6/d)

-

0.023

Loose fitting tape

o

~o

Snug fitting tape

F m. :a. 1 0 2 ~ ~

_

Z

Re < 104 n = 0.14 Re > 104 n = 0.18 (heating) n = 0.30 (cooling)

101

i

101

i

1

i

I i Ill

i

I

102

I

I

i

,

Ill 103

I

I

1

I

I

I II

104

Re FIG. 9. Effect of tape fit (tape-edge to tube-wall clearance) inside a uniform wall temperature ( U W T ) circular tube on the heat transfer.

area. However, in most practical applications, the tapes are snug to loosely fitted inside circular tubes. This is dictated primarily by the mechanics of inserting and removing the tapes from tubular heat exchangers (see Fig. 3) to facilitate cleaning and/or retrofitting. That tube partitioning and flow acceleration from a straight-tape ( y = c~, ~ / d = 0.051) insert result in enhanced heat transfer is also evident in turbulent gas flows. This is seen from Smithberg and Landis [45] data for air in Fig. 10a, where up to 17% higher Nu are obtained with a y = ~ tape insert. Again, higher heat transfer coefficients are obtained with a tighter (smaller y) tape, which is similar, although of a lesser magnitude, to the behavior seen in laminar flows. The Nusselt number is up to two times higher with y = 1.81 than that in a smooth tube. This order of enhancement is seen further in the more recent air heating data of Armstrong and Bergles [83] for y = 2.46,4.0, and 5.85 and ( g / d ) = 0.0215, graphed in Fig. 10b. Comparable performances are also found in data for other gas flows reported by Fujita and Lopez [86] (R-113 vapor), Kidd [99] (nitrogen), and Bolla et al. [100] (helium), as well as for liquid flows reported by Ibragimov et al. [49] (water), Lopina and Bergles [52] (water), and Fujita and Lopez [86] (R-113 liquid), among several others [11]. Another noteworthy feature of the

203

SWIRL FLOW HEAT TRANSFER

' I

4 -f

Air, Pr--0.7

i

F

!

i

'

3

o~

2

o ,o~ =

, 11

-

o

_

t

9

-

i

r

Armstrong and Bergles [83] Air (Pr = 0.71) 9 y = 2.46 y=4.0

9 y = 5.85 9

A

102

~ Z

~

t

h

4 3 2

101 104

~

8

"5" z

4

tube !' Gnielinski [98] _

5

Smithberg and Landis [45] * y = 1.81 y = 2.17 [] y = 11.0 o y--oo ~

t

1

4

i

t

Smooth tube Gnielinski [98]

101

L ~ ~

7

105

I

ill

1

I

[

I

I I 1

105

104

Re

Re

(a)

(b)

FIG. 10. Enhanced heat transfer performance of twisted-tape inserts in turbulent flow of air.

experimental database is that even in turbulent flows, which tend to have a greater well-mixed swirl flow character when twisted tapes are used, temperature-dependent fluid property variations in the axial flow cross section have to be accounted for in the heat transfer. This is usually incorporated as the classical power-law type bulk-to-wall ratio of temperature for gases [27, 45, 81, 83,99] and viscosity for liquids [21,45, 49], with different exponents for fluid heating and cooling [11,21]. The typical pressure drop penalty for the enhanced heat transfer promoted by twisted-tape inserts is depicted in Fig. 11. Here, isothermal Fanning friction factor data for tapes with (g/d)= 0.023, y = 3.0 and 6.0 are graphed. Also included are the predictions for a zero-thickness (8 = 0), straight-tape (y = c~) insert given by [21, 65] f-

(42.23/Re) 0.146Re -~

Re _< 2300 Re_>4000'

as well as results for an empty, smooth circular tube. The higher frictional loss, which increases with y and Re, is apparent. For example, with y = 3.0, the friction factor is 3.9 times that for a smooth tube when Re ~ 100 and 7.5 times higher when Re ~ 2000. The performance of a y = 4.5 tape insert lies between those for y = 3.0 and 6.0 [90] and is not graphed in Fig. 11 for sake

204

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

I

I

I

I

I I I1[

I

I

1

I

I t11[

I

M a n g l i k a n d Bergles [90],

~td?~o_.

o

[-",,~q~,-

I

I

I IIL

= 0.023

-

y=3.0

[]

",,-~o ~

(k/d)

1

y=6.0

-

. . . . y = o% ~5 = 0 [21, 65]

-

, ~, .,~ 9 10_ 1

I-

E

\

"',, B

Cboo

---

10-2

-

10_ 3

102

f = {0.15635 In(Re/7)} -2

~

i

J

i

i iill

I

I

103

I

I

I llll

I

104

-

I

I

1 l llJ

105

Re FIG. 11. V a r i a t i o n o f isothermal friction factor with flow R e y n o l d s n u m b e r a n d tape-twist ratio.

of clarity. Results reported by Ishikawa and Kamiya [96] for y = 1.5, 0.119 are plotted in Fig. 12. Similar behavior is also evident in friction factor data of many other investigators [20, 21, 34]. The influence of twist and thickness of the tape insert is clearly discernible from Figs. 11 and 12. The higher friction loss resulting from the increased flow velocity due to the tube blockage (8 > 0) and increased effective flow length (y << c~) is seen in the low Reynolds number (Re ~ 100) region, where all data lie well above the results for a simply partitioned tube (y = c~, 8 = 0). The onset of swirl flows, induced by the helical twist of the tape insert, is identifiable by the change in slope from the ( f R e ) = C behavior of laminar flows. For example, on the f - Re coordinates of Fig. 11, with y - 6.0, a change in slope occurs at Re ~ 650 and significantly higher friction factors are obtained for Re > 650. The onset of swirl is also dependent on the twist ratio, as seen from the results for y - 3.0, where the change in slope occurs at Re ~ 350. Flow agitation due to the helical secondary fluid motion generated by the twisted tape-insert results in higher

(8/d)-

SWIRL FLOW HEAT TRANSFER

10-0 I

I

I

1 I I I II]

<> ~

v v

"

,,,

1 1 I I II I

205

I

I

I

t

I

o

y = 1.5.(k/d) = 0.119 [96]

I lilt._

-

. . . . y = ,~, 5 = 0 [21, 65]

~%

-

10-1

10-2

f = {0.15635 In(Re/7)}-

10-3 I 102

I

i

i

I IIIII

I

1

103

I I 11111

I

104

-

I

1

I

ilil

105

Re FIG. 12. Isothermal friction factor data for an extremely tightly twisted tape ( y = 1.5) reported by Ishikawa and Kamiya [96].

wall shear stresses and, hence, higher friction factors. The more severe the tape twist, the earlier the onset of swirl flow, and friction factors increase with decreasing y and increasing Re. Another interesting feature of data for y < c~ in Figs. 11 and 12 is the absence of the characteristic discontinuity in the f - Re plot, which signals the onset of turbulent flow. Data exhibit a smooth continuity as the flow rate increases to transition from the laminar to turbulent flow regimes, suggesting that the swirling secondary fluid motion tends to suppress turbulence and delay the onset of flow instabilities. Although not presented here, delayed turbulent transition has been observed in the results of several other investigators as well [11,46]. 2. Swirl Flow Structure

While much of the literature has focused on the quantification of heat transfer enhancement and evaluation of thermal-hydraulic benefits, a few

206

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

studies have attempted to identify the tape-induced swirl flow structure and the associated enhancement phenomena, A tape insert alters the flow field in a circular tube in several different ways. The blockage and partitioning of the flow cross section by the finite-thickness tape increase the axial velocity and the wetted perimeter. The partitioned, helically twisting duct also provides a longer effective flow path and imposes a curvature-induced transverse force on the axial flow to produce secondary circulation. Of these, the dominant flow mechanism in most applications is the generation of swirl, which causes a transverse fluid transport across the tape-partitioned duct cross section [20, 34, 45-47, 101]. This promotes greater fluid mixing and higher heat transfer coefficients; of course, the associated friction penalty also increases. Smithberg and Landis [45] and Seymour [46] were perhaps the first to measure and visualize velocity profiles in the turbulent or high Re regime in circular tubes with twisted-tape inserts and to characterize the swirl behavior. The measurements made by Seymour [46], using radioactive gas tracing and a thermistor anemometer, for turbulent airflows with a y - 4.76 tape insert are depicted in Fig. 13. Figure 13a shows the cross-stream secondary circulation at Re - 3.1 x 105, which clearly indicates a double-vortex structure. The axial velocity contours for Re = 6.2 x 104 are graphed in Fig. 13b, and the characteristic dual peaks near the cores of the two crosscirculation cells are evident. The axial velocity measurements of Smithberg and Landis [45] are qualitatively similar and implicitly suggest a two-cell secondary fluid motion, as seen from Fig. 14a. This is also evident from Fig 14b, which gives the typical axial velocity contours presented by Donevski and Kulesza [101]. Similar observations were made by Lopina and Bergles [102] in their peripheral static pressure measurements in turbulent single-phase water flows with y - 3.6. Their results are reproduced in Fig. 15, along with their assessment of the secondary flow pattern in the presence of a twisted-tape insert. These observations once again suggest a double-vortex swirl behavior that is superimposed on the helical axial primary fluid motion. However, it should be noted that most of the measurements depicted in Figs. 13-15 are for high Reynolds number flows, where turbulence eddies and inherently well-mixed cross-stream fluid motion have a comparable, if not a dominant, effect on the convective behavior. Tape-induced secondary circulation, as discussed next, has a more profound influence in the laminar flow regime. In laminar flows, using smoke-injection techniques, Manglik and Ranganathan [47] have characterized the tape-induced swirl structure. Their photographic results for laminar flows in the range 119 < Re _< 1003 with two different twist-ratio tapes (y = 4.32 and 3.53) are depicted in Fig. 16, which clearly show the cross-stream swirl flow patterns. The centrifugal-type force

SWIRL FLOW HEAT TRANSFER

4 INJECTION o SUCTION

I I

VIEWED F R O M U P S T R E A M TO D O W N S T A E A M

207

M

(a) TAPE TWIST 5

6

Co) FIG. 13. Measured tube-side airflow distribution in the turbulent regime with twisted-tape inserts reported by Seymour [46]: (a) secondary flow with y = 4.76 and Re - 3.1 x 105 and (b) constant axial velocity contours for y = 4.76 and Re = 6.2 x 104.

that the tape surface curvature imposes on the bulk axial flow in the helically partitioned duct essentially produces this secondary circulation. With increasing axial flow rates (Re = 223 ~ 1003), as seen in Fig. 16a for y = 4.32, the intensity of the swirl motion increases, which is characterized by the visually observed stronger fluid circulation. In fact, the initial single circulating cell breaks up into two asymmetrical, counterrotating vortices. The magnitude of the second cell, which sits on the right corner of the primary swirling core, further increases to produce a well-mixed fluid flow behavior. Similar flow patterns are obtained with a tighter tape twist (Fig. 16b,

208

RAJ M. MANGLIK AND ARTHUR E. BERGLES

X~ %o~

I

!

l -

%\!i \ k t~

i~

.4.oI#It ~\ ,

....

I,,~ARROWS INDI(~ATE SE1NSEOF " TAPE TWIST IN FLOW DIRECTION (OUT OF PLANE SHOW)

,,11

1

(a)

(b) FIG. 14. Axial velocity measurements for turbulent flows in circular tubes with twisted-tape inserts: (a) Smithberg and Landis [45] for y = 5.15 and Reh -- 1.4 >< 105, and (b) Donevski and Kulesza [ 101 ].

y = 3.53). The extent of swirl is clearly seen to grow with increasing flow rates (or Re) and decreasing twist ratio y. The numerical simulations of Manglik and You [66] further verify this tape-induced secondary flow structure. They have considered laminar flows in circular tubes with twisted-tape inserts of negligible thickness. The computational model also ignores the tape-edge leakage and tape-surface curvature that are usually present in practical applications. Their results for secondary flow velocity distribution, represented by the vector field in the partitioned flow cross section, are presented in Fig. 17. The initial single-cell circulation at low Re is seen to grow and develop into two cells of helical counterrotating vortices as fully developed swirl is established with increasing Re (Fig. 17a). As depicted in Fig. 17b, this feature of the tape-generated

SWIRL FLOW HEAT TRANSFER

209

/

\

r

I I

" ,~%

//" 'N\

=~ "

lu

,i,

/

/'

Peripheral pressure

f <~

Tape Direction of tape twist Primary Flow "Vortex Mixing" induced by secondary flows

FIG. 15. Twisted-tape-induced secondary circulation and circumferential static pressure variation in turbulent flows with y - 3.6 [102].

Direction of ~

tape twist

FIG. 16. Smoke streaks depicting swirl-flow patterns in laminar flows in circular tubes with twisted-tape inserts (~/d = 0.021) [47]: (a) y = 4.32 and (b) y -- 3.53.

210

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

D i r e c t i o n o f t a p e twist

y = 3.0, R e - 200

y - 12.0, R e - 800

y = 3.0, R e = 600

y - 6.0, R e - 800

,~\

I

9

y = 3.0, R e = 1000

y - 3.0, R e - 800

(a)

(b)

FIG. 17. C o m p u t a t i o n a l s i m u l a t i o n o f swirl flows in circular tubes w i t h t w i s t e d - t a p e inserts o f negligible thickness (8 = 0) [66]: (a) v a r i a t i o n with R e for fixed twist ratio ( y = 3) a n d (b) v a r i a t i o n w i t h y for fixed R e = 800.

swirl is obtained with an increasing severity of tape twist (or decreasing y) for a fixed flow Re as well. The limited numerical simulations of DuPlessis [103] also show a similar secondary circulation character. Manglik and Bergles [20] have shown that this twisted-tape-generated fluid circulation in laminar flows can be scaled by a swirl parameter Sw[ = ~(Res, y)]. The values of Sw for the different flow conditions are also included in the legends of Figs. 16 and 17, and the development of this dimensionless parameter and its validity are discussed in detail in Section IIB1. Swirl flow, initially characterized by a single vortex that encompasses much of the partitioned flow cross section, is seen to set in

211

SWIRL FLOW HEAT TRANSFER

when Sw ~ 70. With increasing Re, or decreasing y, this circulation grows and breaks up into double-vortex cells when Sw ~ 300, as fully developed swirl is established. The evolution of this flow structure can be mapped into three regimes, as shown schematically in Fig. 18, where the influence of Sw on the friction factor is graphed. Sw essentially incorporates the flow rate and tape geometry (y and g / d ) effects, and the phenomenological transitions in the influence of twisted-tape inserts on the laminar flow behavior can be described as follows. 1. Very low Re or large y regime (low Sw): viscous effects dominate the balance between convective inertia and viscous forces; tube blockage and a longer effective helical flow path due to the tape insert produce higher wall shear stress. 2. Moderate Re and y regime (medium Sw): described by the onset and growth of swirl flow that is superimposed over the axial flow; a competing balance among convective inertia, viscous, and tape-helical curvature-induced forces establishes swirl generation. 3. Very small y and/or large Re regime (high Sw): tape-twist-induced swirl is completely established, which grows into two dissimilar counterrotating helical vortices and produces much higher frictional loss.

103

I

t

i

i

r t I

q

I

I

I

i

I I

1

I

l

f

I

I

I

Fully-Developed Swirl Flows (double helical vortices)

..a O

l_

_

N o Swirl Regime r

>0

,-

102 -

9 o

!

D

-(

...."

y--

(5/d) = 0

e~

(8/d)

101 10

[

I

I

l

1

[Jl

I

1

ill

t

101

101

l

i

i

i iJl

101

Sw FIG. 18. Flow regime map for the development of twisted-tape-generated swirl flows and their influence on laminar circular-tube friction factors [34].

212

RAJ M. MANGLIK AND ARTHUR E. BERGLES

It may be noted that the tape thickness primarily alters the partitioning of the circular tube into two parallel helically twisting flow channels that can have a nominally semicircular (8 ~ 0) or circular segment (8 > 0) cross section. This secondary circulation also produces greater "thermal mixing," which results in higher heat transfer coefficients in fully developed laminar flows. Corresponding to the swirl evolution described earlier, typical effects on the Nusselt number are depicted phenomenologically in Fig. 19. Fully developed laminar convection in a circular tube with uniform wall temperature (UWT) is considered in this map, which shows that higher Nusslet numbers are obtained due to tube partitioning, blockage, and flow acceleration (y = e~, S w ~ 1, no-swirl regime) and helical swirl generation (y << ~ , S w >> 1) by a twisted-tape insert. Again, as noted earlier, the scaling of tape-induced swirl by the parameter S w and the consequent correlation with laminar flow f a n d Nu are discussed in the next subsection.

1000

1

I

I

I

1lll

I

t

I

I

~ 1Ill

I

1

1

I

I

III,L

Fully-Developed Swirl Flows 100

Z Pr

10

y =oo Circular Tube

1

/

101

i

J

I J i ]Jll 10 2

i

i

J i iJiJl

J 10 3

I i ] ijli 104

Sw FIG. 19. Flow regime map for the influence of twisted-tape-generated swirl flows on Nu in fully developed laminar convection in a circular tube with UWT.

SWIRL FLOW HEAT TRANSFER

213

B. THERMAL-HYDRAULICDESIGN CORRELATIONS A large number of correlations have been proposed in the literature for predicting friction factors and Nusselt numbers in circular-tube flows with twisted-tape inserts. These include equations derived from empirical data, semiempirical theoretical considerations, and numerical solutions. However, many of them are simply curve fits that best describe the investigator's own data set. In the case of numerical solutions, considerable idealizations have been made in computational modeling, e.g., zero thickness tapes, constant property flows, and negligible entrance effects, which require empirical verification of the various approximations. Compilations of these correlations for f and Nu are given in the next subsections, along with an assessment of their generalized efficacy for predicting the frictional loss and heat transfer performance, and design recommendations. 1. Friction Factor

Most of the available correlations for predicting Fanning friction factors in circular-tube flows with twisted-tape inserts are presented in Table 11.5 Equations for laminar and turbulent flows have been grouped separate and, except where indicated otherwise, all have been modified in terms of parameters based on empty-tube dimensions. Such representation, as per the recommendations of Marner et al. [107], has the advantage of allowing direct comparison with smooth tube flows and the estimation of the relative increase in pressure drop. From the listing, it can be observed that the correlations differ both in form and the manner in which parametric effects of twisted-tape inserts are described. A comparison of the predictions for the frictional loss in laminar flows with a y = 2.5, ( g / d ) = 0.05 twisted-tape insert by some of the correlations in Table II is graphed in Fig. 20. Results for flows in circular and semicircular (y = c~, g = 0) tubes are also included for reference. It is evident that a rather wide scatter band represents the hydrodynamic performance of this typically severe twist-ratio tape. Almost all results are significantly different from each other, except those obtained by Donevski and Kulesza [19] and DuPlessis and Kr6ger [61]. The Shah and London [93] predictions agree with these for Re < 200, but under-predict substantially at higher Re. Watanabe et al. [81] have an incorrect representation of the low Reynolds number asymptotic behavior; their results are less than those for a semicircular duct for Re < 60, which is physically not possible. However, Lecjaks et al. 5Whilethis is not purported to be a complete listing, it is certainly representative of the larger body of the literature on twisted-tape inserts.

RAI M . MANGLIK A N D ARTHUR E.BERGLES

TABLE I1 C H R O N O L ~ C ~LISTING C A L OF CORRELATIO FOR NS FANYINF GR I C T I OFACTOR N I N L A M I N AARN D T U R B L ~ L EFLOWS KT I N C I R C U L ATUBES R W I T H FLLL-LENGTH TWISTED-TA PE INSERTS Laminar Jaws 1 . Date and Singham [59]

2 . Shah and London [93]

where

3. Donevski and Kulesza [I 91

R e < Re,, ( f R e )=

2-n

1

(3a)

.rr - 4(6jd

where n

=

I

-

(0 6092/y0 3,

4. Watanabe er al. [ B l ] Re < Re,,

where

R~< r - ( C ~ I C ~'"-0'1 ~I

+ 2 ;(2S/d

j

I

(4b)

SWIRL FLOW HEAT TRANSFER

215

TABLE II cont&ued 5. Lecjaks et al. [104] 15
[47.825 + (57.25/y)] 19.95Rea

(fRe) =

(5a)

a = (1 - 0.839 + 0.454/y)

(5b)

( f R e ) = A~[15.767 - 0.14706(8/d)][1 + ( R e / 7 0 y l 3 ) l 5 ] 1/3

(6a)

A~ = ('rr/64)[(Pe/d) 2 /(Ae/d2)31

(6b)

P~ = [2d - 28 + (avd/G)], A~ = {[2H2(G - 1)/'rr] - 8d}, G = [1 + ('rr/2y)2] 1/2

(6c)

6. DuPlessis and Kr6ger [61 ]b

where

7. Manglik and Bergles [20] +2-2(a/d)] (fRe)s = { 15.767 [ ax av- 4(8/d)

2}

[1 nt- 10-6Sw255] 1/6

(7a)

where Sw = (Res/x/y) = (Re/x/y){av/[~r - 4(8/d)1}[1 + ('rr/2y)2] 1/2

(7b)

( f R e ) = (fRe)s{'rr/[~ - 4(8/d)]}[1 + ('rr/2y) 2]

(7c)

Turbulent flows

1. Ibragimov et al. [49]

0091 [ 143 1 f = ReO.25 1 +

y4 j

"rr - (48/d)

(8)

~r - O 8 / d )

2. Gambill and Bundy [50]

] 0.046 ['n + 2 - (28/d)] 1.8 ,n" f = ~ L -rr - (48/d) J av - (48/d)

+

oool081 r [~r + 2 -

0.0525 yl.31 [_ Re J

"rr

(28/d).

- (48/a)

]0.3,{

1.2 (9)

'n"

;n" - ( 4 8 / a )

3. Smithberg and Landis [45] j~ =0.116 x / ~ + 0.0249 A(~c) [1125 In (Rehx/~h) - 3170] + 0.046Reh-~ y Reh

(10)

(Because of the implicit relationship and extensive modifications required for expressing parameters in terms of empty tube dimensions, this equation has been retained in its hydraulicdiameter-based form) continues

216

RAJ M. MANGLIK AND ARTHUR E. BERGLES

TABLE II continued 4. Seymour [46] f

yx/'-R~ ~ - (4~/d)

+~

'tr - (4~/d

+ 0.00875 ~ _ (4g/d)

(11)

5. Lopina and Bergles [52] 0.1265 [ rr ]1.8[ ).] (l~ww) rr -rr + -2-(4~/d)(2~/d 1.2 , f = yO.4O6Re~ V - - ~ / d )

(12a)

where

n = 0.35(dh/d) for diabatic conditions and n = 0 for isothermal conditions

(12b)

6. Donevski and Kulesza [19] Re _> Re~, 2 f = ReO.25 1 + y---i~.61] "rr - ~ g / d )

(13a)

"rr - (4~i/d)

where 34000]

Recr'2 = [3300 + (2y)l.5 j

I~ + 2-~ (2~/d )]

(13b)

7. Watanabe et al. [81] Re > Recr

I

ar 3.65 0.046 [ f = ~ l + l + ( 2 y / a r ) 2 ~ - (4~/d)

1"8[

+2-(2~/d "rr - (4~/d)

:]

1.2 Tb o.1

(14)

where Recr is given by Eq. (4b) 8. Manglik and Bergles [21 ] f=

ReO.25 1 + y--55~.291 a'r-(43/d)

a'r-(4~/d)

(15)

aMigai [105] has also presented an implicit, semianalytical correlation for isothermal f However, this equation is quite long and awkward and, hence, is not listed here. Likewise, Donevski and Kulesza [101] have reported a semianalytical, implicit equation for f, which is not included here. In this case, there is good agreement between the empirically derived correlation given here and the theoretical expression. bThe effective flow cross-section area A~ employed by DuPlessis and Kr6ger [61] is that suggested by Nazmeev and Nikolaev [106].

SWIRL FLOW HEAT TRANSFER

I I Illlll

I

1

I

I

217

I I II I

I

I

I

I

q

Shah and London [93]

%

I I I~

..........

Donevski and Kulesza [19]

........

Watanabe et al. [81]

%

Lecjaks et al. [104] .....

DuPlessis and Kroger [61] y = 2.5, ~/d = 0.05

f = (16/Re) 0.1

\

,~-,.

"-,,

[smooth tube] ~

"',>.-,, ~

".:,~.

f = (42.23/Re) [y - 0% 5 -

0.01 101

0]

102

103

104

Re FIG. 20. Comparison of isothermal Fanning friction factor correlations listed in Table II for laminar swirl flows [20].

[104] predict consistently higher than all others. This variability in the friction factor predictions, as seen from the respective equations in Table II, is perhaps due to the different ways in which the swirl flow effects due to the tape twist ratio have been incorporated. Date and Singham [59] and Shah and London [93] use (Re/y)p along with a fourth-order polynomial in y. Donevski and Kulesza [19] use two rather complex groupings: (al + a2/y 2) and the product of Re p/y~ and (1 + b/y~176 Watanabe et al. [81] correlate the swirl flow with the effective curvature at the tube wall [Kc, Eq. (4d), Table II], Lecjaks et al. [104] employ (b/y) and Re p+q/y, and DuPlessis and Kr6ger [61] use the effective flow cross-section area A~ suggested by Nazmeev and Nikolaev [106], along with (Re/cyl3). In order to develop a generalized scaling of twisted-tape-induced swirl flows in the laminar regime, Manglik and Bergles [20] have suggested that it can essentially be represented by the interaction among viscous, convective

218

RAJ M. MANGLIKAND ARTHURE. BERGLES

inertia, and helical-curvature-induced forces. The following force balance 6 describes their interplay:

helical curvature force (p Vz / H ) convective inertia (p Vz / d ) x . viscous force (Ix Vs/d 2) viscous force (Ix Vs/d 2)

(16)

Here the helical curvature force promotes the so-called "centrifugal force" effects, and the twisted-tape 180 ~ pitch H provides the appropriate length scale; the smaller the half-twist pitch H, the more severe the helical curvature and vice versa. The reference velocity Vs, which is common to the scaling of all three forces, is given by the maximum swirl velocity due to the helical flow. This velocity and the helically extended flow length Ls in the tapepartitioned tube is given by

Vs - Va[1 + ('rr/Zy)2] 1/2

(17)

Ls -- L[1 + (Tr/2y)2] 1/2,

(18)

where Va and L are the axial velocity and length, respectively. This scaling or forced-vortex flow representation, and its resolution depicted in Fig. 2, is similar in principle to that suggested by Smithberg and Landis [45], Gutstein et al. [108], and Migay [109] and that inherent in the effective flow crosssection area proposed by Nazmeev and Nikolaev [106]. Thus, the dimensionless grouping of the force balance in Eq. (16) yields the swirl parameter Sw as follows7:

Sw = (Res/x/~), Res = (pVsd/Ix).

(19)

This parameter essentially accounts for the tape thickness, its twist ratio, and the increased helically twisting flow velocity. The swirl-flow Fanning friction factor, based on the increased flow path Ls and maximum swirl velocity Vs, which directly affect the wall shear stresses, can be expressed as

f~ - (APd/Zp VZLs) - f (L/Ls)( Vo/ Vs).

(20)

The second expression on the right-hand side of Eq. (20) describesfs in terms of "empty" tube variables. Thus, based on this scaling of the swirl flow

6It may be noted that by replacing the helical curvature force by a radial centrifugal force on the axial flow, this force balance yields the Dean number for laminar flows in curved tubes; if the buoyancy force is introduced, then the Grashof number is obtained for mixed convection in straight circular tubes [11]. 7This actually yields (Re~/y), but its square root is taken simply because the square of Reynolds number becomes too large.

SWIRL FLOW HEAT TRANSFER

219

behavior by the parameter Sw, the correlation listed as Eq. (7a) in Table II was proposed, which can be rephrased as follows"

( f R e ) s - (fRe)s,y=~(1

+

10-6Sw255) 1/6.

Here, 0 c Re)s,y=o~ is described by the terms in curly brackets in Eq. (7a), (Table II), and this limit represents the frictional loss in a circular tube with a straight tape insert. This correlation was shown to predict experimental data available at that time [22, 29, 90] for 0 _< Sw < 2000, 3.0 <_ y _< ec, to within -t-10%. Its validity and generalized applicability have since been further established [34] by a much larger experimental database, as shown in Fig. 21. Data here represent a very wide range of flow conditions (43 < Re < 2720) and tape geometry (1.5 <_ y < oc, 0.119 < g/d < 0.019). Figure 22 provides the comparison between various turbulent flow correlations listed in Table II for a twisted tape with y - 2.5 and ( g / d ) - 0.05. In the turbulent flow regime, friction factors for y - oc and g - 0 tape inserts can be predicted by hydraulic-diameter-based equations for smooth circular tubes [11,21]. This is represented in Fig. 22 by the two simpler equations given by Blasius [113] and McAdams [114] that are used commonly in the

10

[

I

I

I

Manglik

I

I

I

[

I

l

!

i

1

and Bergles [20] correlation,

!

I

[

I

I

1

I

I

I

I

I_

E q . (7) _

o~ ~

-

8

_

II 9

(D

1

A

A%4~

a~

_

_

(D

Experimental

data

-_

(8/d) -

9

N a i r [48], y - 3.3 a n d 4.8,

/x

Lokanath

[]

Ishikawa and Kamiya [96], y = 1.5, ( g / d ) = 0 . 1 1 9 Agarwal and R a j a R a o [111], y - 2.41 - 4.84, (g/d) = AI-Fahed et al. [112], y - 3 . 6 - 7.1, ( g / d ) = 0 . 0 3 6

O V

0 . 0 0 6 a n d 0.03

_

[110], y - 3 . 4 - 5.2, (~5/d) = 0.02

_

0.04

-

0.1 101

102

103

104

Sw

FIG. 21. Comparison of experimental isothermal Fanning friction factor data with the predictions of the Manglik and Bergles [20] correlation for laminar flow.

220

RAJ M. MANGLIKAND ARTHUR E. BERGLES

I

I

1

I I I I I

y=2.5 ~/d = 0.05

I

i

......... ..........

-

8=0

y = o o

~

'

I

I I I i I

1

t

I

I

t111_

Ibragimov et al. [49] Gambill and Bundy [50] Seymour [46] Lopina and Bergles [52] Donevskiand Kulesza [19] Watanabe et al. [81]

.....

0.1

r

~

.

~ " - ~-'~ ~ . ~ . . . .

..... 9...

0.01 --

-.-.,

- Blasius [11 --

...,

McAdams [114]

0.001 10 3

~

i

i

i

i illl

I

smooth tube

1

10 4

I

I 11111

i 10 5

i

i

I i ill 10 6

Re FIG. 22. Comparison of isothermal Fanning friction factor correlations listed in Table II for turbulent swirl flows [21]. literature. Once again a rather wide performance envelope is seen, where the results of Seymour [46] and Ibragimov et al. [49], respectively, describe its upper and lower ends. There is fair agreement between the Lopina and Bergles [52] and the Donevski and Kulesza [19] correlations, but all others differ significantly. In most instances, the reciprocal of the twist ratio to some power (1/yP) or the wall curvature Kc has been used to correlate swirl flow behavior. Fluctuating velocities and flow instabilities generally characterize the transition to turbulent flow. While the secondary circulation produced by twisted-tape inserts may tend to suppress and delay their onset, scaling of the well-mixed, fully developed turbulent flow field by the swirl parameter S w breaks down [20]. In this case, as proposed previously [21], ( f / f y = ~ ) has been found to correlate with the reciprocal of the tape twist as ( f / f y = ~ ) --[1 + (2.752/y129)].

SWIRL FLOW HEAT TRANSFER

221

This correlation, also listed as Eq. (15) in Table II, is seen to describe within +5%, in Fig. 23, most of the available data in the literature. Its extended validity has been verified further by other experimental investigations [36, 86] as well. Furthermore, given the smooth transition from laminar to turbulent flows observed i n f - Re data (Figs. 12 and 13), by asymptotically matching Eqs. (7) and (15), the friction factor can be predicted by the following single continuous equation [21]: O'lt 9 f = [f/lO -~fll~

(21)

Here, fl and ft are given by Eqs. (7) and (15), respectively, in Table II, and describe all the available data for Re > 0 very well (within + 10% [21, 86]). In addition to its precision, this correlation is a very useful design tool, as it obviates the need to identify the flow regime. For most practical applications the predictions for diabatic friction factors are needed rather than isothermal values, which tend to over- or underpredict the frictional loss depending on the fluid heating or cooling conditions [22, 34, 52, 81]. Lopina and Bergles [52] have shown this to be the case in turbulent water flows, where the power law-type bulk-to-wall viscosity ratio factor is needed to account for liquid heating/cooling effects; the

2.5

I

_

2.0-

I

i

i

I

~

I

f

I

I

l

I

[

Experimental data 9 Manglik and Bergles [90] o Armstrong and Bergles [83] /x Bolla et al. [100] Smithberg and Landis [45] o Saha et al. [29]

l

I

T

I

./..~-'~

(f/fy

= oo)= 1 + (2.752@ 1"29)

-

~ 4

. . . ~ . ' . . . . . ~ ' ~ " ~\

i. ~ " " " " 1.0 ~ ' ~ " ~ - ~ ; " ' I i 0.0 0.1

I

t

l

/

~

...

..-'""

-

_

-5%

_ _

--_

Donevski and Kulesza [19] _

"~ I

I

~. <~;/<.~ "..-'" ~' ~/'" ~ ' ~/ .... ' '.../' ." '"~~. ..........

.

--

I

~.' ~.' ~-" ' " . /,%~"" .~"~'"_ ,-'""

_

_

I

/, .%;.....

_

1.5

I

] -1 Watanabe et al. [81] ./"--~ \ +5% ,-.:/ / ] ~ . . . ; : : ' ~ _. ~/,/~ ,?,~~~,,,"-_"

-

~_~, IJ

I

~ Lopina and Bergles [52] I

I

l

l

t

0.2

I

I

0.3

J

I

I

l

I

0.4

I

I

I

I

0.5

(I/y) FIG. 23. Comparison of Manglik and Bergles [21] correlation for turbulent swirl flow friction factor [Eq. (15), Table II] with experimental data and other predictions [21 ].

222

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

exponent in this case is a function of the hydraulic diameter to tube inside diameter ratio [see Eq. (12b), Table II]. Likewise, Watanabe et al. [81] include a bulk-to-wall temperature ratio correction factor for turbulent gas flows [Eq. (14), Table II]. For laminar flows, however, an extended database is not available at the present time to make a quantitative recommendation. The limited data reported by Marner and Bergles [22] for y = 5.39 clearly show the need for including cross-sectional fluid property variation effects. Based on the theoretical results of Harms et al. [115] for the limiting case of a straight-tape (y - ec, 8 = 0) insert, a first-order correction can be employed as

f diabatic = f adiabatic(lXb/ tXw)m,

(22a)

where, depending on the tube-wall heating and cooling conditions, the exponent m takes on the following values: m(UWT)-

{0.65 0.58

heating cooling

or m ( U H F ) -

{0.61 0.54

heating. cooling

(22b)

For laminar gas flows, the viscosity ratio may be replaced with the corresponding temperature ratio, and the recommendations in Ref. [81] should, for the present, provide a reasonable correction for most engineering applications. For turbulent flows, the recommendations of Lopina and Bergles [52] and Watanabe et al. [81] can be adopted. 2. Nusselt N u m b e r

A listing of most of the correlations for enhanced heat transfer in circular tubes with twisted-tape inserts available in the literature is given in Table III. The equations are, once again, grouped separately for laminar and turbulent flows; in laminar flows, those for the UWT and UHF boundary conditions are also listed separately. Laminar flows appear to have received less attention in the literature. This is despite the fact that relatively higher heat transfer enhancement can be achieved in this case [3, 11, 18,20,21]. The early application needs, when twisted tapes were primarily being considered for gas flows in fire-tube boilers, have partly contributed to this. Thermal processing applications generally involve steam heating, or turbulent fluid flow heating/cooling, of laminar in-tubeflows of viscous liquids, which is more closely represented by the uniform wall temperature (UWT) boundary condition. Of the two early attempts, the DuPlessis and Kr6ger [62] equation is based on their limited numerical solutions, whereas the Manglik and Bergles [94] equation was developed on the basis of the Marner

SWIRL FLOW HEAT TRANSFER

223

TABLE III

CHRONOLOGICALLISTINGOF CORRELATIONSFOR NUSSELTNUMBERIN LAMINARAND TURBULENTFLOWSIN CIRCULARTUBESWITH FULL-LENGTH TWISTED-TAPEINSERTSa Laminar flows Tubes with uniform wall temperature (UWT) condition: 1. DuPlessis and Kr6ger [62] Num = 1.58A~[1 + 0.153(X~)-1"~

+ 6.4 • lO-5(pr.Re~/y)3] ~

(23a)

x [1 + O.O02(Re~/y)l4] 1/7 where Re~ = Re(~rd/P~)

,

(23b)

~L[('rrd2/4)-~d]P~}[ 1 +(O.04Pr.Re~/y)3]l/3

X~ = [

(23c)

4-~2~r.Re ~

and A~, P~ and A~ are given by Eqs. (6b) and (6c), respectively, in Table II 2. Manglik and Bergles [94] Num = 4.631

(~b/0"14 ~ww [0.4935 / Pr ( ~ / 3"475 350/

+[1 + 0.0954Gz~

I 0.2

2"6316

(24a)

where R% = (V~d/v), Vo = (m/pA~), and Ar = [('rrd2/4) - 8d] 3. Manglik and Bergles [20]

Num--4.612(~ww)~176

(24b)

10-9(Sw'Pr~ 0"1

(25)

+ 2.132 • 10-14(Rea 9Ra) 223 where Sw is given by Eq. (7b) in Table II, and Rea is as defined in Eq. (24b). Tubes with uniform heat flux (UHF) condition 1. Hong and Bergles [18] Nuz = 5.17211 + 5.484 x 10-3 Pr~

~

(26)

where Rea is given by Eq. (24b) 2. Watanabe et al. [81] Re < Recr

continues

224

RAJ M. MANGLIK AND ARTHUR E. BERGLES

TABLE III continued

Nuz = C1Re~176176

+ -2 -(48/d) (2~/d)]~

"rr - (4~/d ~r

(27)

where Recr and C1 are given by Eqs. (4b)-(4e) in Table II

Turbulent flows 1. Ibragimov et al. [49] { 5.65 x 104 ['rr -+-2 - (2~/d)] 12 [ "rr ).] 1.2} Nu = Nuy=o~ 1 + ~ yRel. "rr - (48/d) ~r - (48/d

(28a)

where

] o8 k,Prw/

"tr - (48/d)

~ - (4~/d)

(28b)

and it represents the Nusselt number for turbulent flows with a straight tape (y = oo) insert 2. Smithberg and Landis [45] Nuh = + I + [350 Pr ~ x

(25.45 ~ \yReh V/-~]

+

dh/(ydRehfh)]

(29a))

O.023(d/dh)( 1 + 0.0219'~~ ~ - Pr -z/3 Re ~ 4y2~ J

where

dp =

{(Tb/Tw)~ (~b/~w)O.36

for gases for liquids

(29b)

and ~ is given by Eq. (10) in Table II. Also, because of its implicit and complex form, this equation is retained in terms of hydraulic-diameter-based parameters 3. Thorsen and Landis [116] (simplified version) Nuh =

ChRe ~ Pr ~ (Tb/Tw)~ + 0.25(x/-G-r~s/Reh)] CcReO.8 pr~/3 (T6/Tw)O.~O[1 _ 0.25(v/_G_~/Reh)]

for heating for cooling

(30a)

where Grs = 0.5~r2(f3fATw)(Reh/y)2(dh/d)

(30b)

Ch = 0.021(1 + 0.07Kc), Cc -- 0.023(1 + 0.07Kc)

(30c)

and Kc is in [ft -1] given by Eq. (4d) in Table II 4. Lopina and Bergles [52]

continues

SWIRL FLOW HEAT TRANSFER

225

TABLE III continued 0"023Re~

Nu = F /

Pr ~ r" 1 -* 0 "25,: -2]0.4 [~r+2-(2g/d)] 0.2 [ zr ] 0.8 / ' k ~-(4~/~ ] t~-(4~/~ol (31)

r 2n 1/3 + O.193[Prf3ATw(Re/y'Z(~_(~/d))J

Here Fis a tape-fin efficiency parameter, and for loose-to-snug fitting tapes F ,,- 1 (see [102]). Also, Eq. (31) is for heating; for cooling, the second term (centrifugal convection) should be dropped 5. Kidd [99] Nu = 0.024Re ~ Pr~

- 1)]ll(Tb/Tw)~

+ (d/L) ~

x ['rr + 2 -- (2~/d).] ~ [ rr .]0.8 "rr - (4g/d) ] ar - ~g/d)

(32)

6. Dri~ius et al. [117] Nu = 0.025Re T M Pr 043(~b/l.Lw)006[0.5 nt- 8(y/'i'r)2] -0"11

]0.84

"rr - (4g/d) J

(33)

- ~g/d)

7. Watanabe et al. [81] Re >_ Recr Nu=

C2RenprO.4(Tb)~

.'rr2,rr + -- --[ ( -(2g/d)] 4 ~ - ~ J l-n ;rr - (-4911"/d)] n

(34)

where Recr, n, and C2 are given by Eqs. (4b), (4c), and (4f), respectively, in Table II 8. Donevski et al. [118] Re < Recr [ 0.967 ] [ +2-(28/d).]l-n[ Nu = 0.023 +(2y)O.768jRe"Pr~ ~ ar - (4g/d) J

In

;rr -

-rr -(4~la)

(35a)

Re _> Recr

[

]a

[

J[

Nu = 0.023 + 0.0175 ReO.8 Pr 0.4 ~ + 2 - (2g/d)q ~ y

rr - - ~ - ~

"

ax ~ - (4g/d

;l

0.8

(35b)

where n = [0.8 - (0.2943/y~ a = 3.398y ~ Recr= [.1 +(24"687/@~ ~ ~

J [ ~ + 2 ~(2~/d)]

(35c) (35d)

(35e) continues

226

RAJ M. MANGLIK AND ARTHUR E. BERGLES TABLE III continued

9. Manglik and Bergles [21] Nu = 0"023Re~ Pr ~ [1 + ~ - ~ ] ["rr+ 2 - (2~/d)] ~ [-rr "rr )] ~ rr - (4~/d) - (4~/d +

(36a)

dp = (l~b / tXw)nor( Tb / Tw) m

(36b)

where

n=

{ 0.18 liquid heating 0.30 liquid cooling

and m = {0.45

gasheating

(36c)

0.15 gas cooling

aThe correlation due to Gambill and Bundy [51] is not included here. This is based on a "centrifugal convection" term, and more refined versions have been reported in later investigations. The analogy-based, semianalytical correlation reported by Migai [105] is not presented here, as it is a rather long and complex equation. Klaczak [119, 120] recommends the same correlation given by Ibragimov et al. [49]. Nazmeev and Nikolaev [106] suggest using correlations for smooth tube with Re evaluated on the basis of an "effective flow cross-section area." DuPlessis and Kr6ger [61,62] have considered this in their correlations.

and Bergles [22, 53] data for a single twist ratio (y = 5.39) insert. In a more recent study [20], with a larger set of experimental data for three different inserts (y -- 3.0, 4.5, and 6.0, ~ / d - 0.0228) and the UWT boundary conditions, it was shown that Nu = ~(Gz, Re, y, Pr, Ra, tXb/IXw).

(37)

Based on the scaling of twisted-tape induced secondary circulation by the swirl parameter Sw, the correlation given by Eq. (25) in Table III was proposed. The following restatement of Eq. (25) highlights the terms that account for the various convection effects: NUm -- 4"612(P~b/IXw)O'14 , , ,_ 9 [ { ( 1 + 0"0951Gz~ ~ fully developed flow

6.413 x 10-9(Sw 9Pr~ swirl flows

3835

/

9

thermal entrance

+ 2.132 x 10-14(Rea 9Ra) 22

:1~

free convection

The interplay between thermal entrance effects and fully developed, twistedtape-generated swirl flows is shown in Fig. 24. Their respective asymptotic conditions for Ra ~ 0 are represented by S w ~ 0, Gz ~ ec (entrance effects) and S w ~ ec, Gz ~ 0 (swirl-dominated flows). Likewise, in flow conditions where Gr > S w 2, free convection effects dominate and are scaled

SWIRL FLOW HEAT TRANSFER

227

103 Nu m - ~ (Gz, Sw, Pr, Ra, gb/gw) .......

Eq. (25), Ra ~ 0

Sw. Pr 0.391 102

. . .7500 ........

~........

4500

t-~- ........

m

A

< =k

-d-~---

_ljO..O_~.~. . . . .

m

Z

Experimental data Manglik and Bergles [90]

101

_

Sw. Pr 0.391 y-oo, 8-0 Eq. (25)

100 101

102

7125- 7875 o 4275 - 4725 o 2850-3150 v 1425- 1575

103

104

-,m

105

Gz FIG. 24. Effect of tape-generated swirl and tube partitioning on the variation of Num with Gz in laminar flows in circular tubes with uniform wall temperature (UWT) [20].

by the grouping (Rea" Ra). This interaction between swirl flows and free convection is depicted in Fig. 25. This correlation [Eq. (25), Table III] predicts the experimental data of Manglik and Bergles [90] and Marner and Bergles [22,53] within +15% and is perhaps the most generalized equation for the UWT heating and cooling conditions. For laminar flows with uniform heat flux (UHF) wall conditions, encountered in two-fluid exchangers with C * = 1 and simulated by electrically heated tubes, Hong and Bergles [18] and Watanabe et al. [81] have reported Nusslet number correlations (Table III). The former is based on their experimental data for water and ethylene glycol with y = 2.45 and 5.08 tape inserts, and a numerical solution [63] for the fully developed asymptote (y = e~); the latter, however, are based on air data. As such, a fair comparison between the two predictive equations cannot be made. Nevertheless, Nu~ results from extrapolations for air (Pr ~ 1) and water (Pr ~ 5) with y = 2.5

228

RAJ M. MANGLIK AND ARTHUR E. BERGLES

102

!

9

i

I

!

i

i

||

!

!

|

|

i

i

i

i

Experimental data Marner and Bergles [53] 1.0x107 < Ra < 3.0x107 m Manglik and Bergles [90] 2.0x106 < Ra < 4.8x106

o

r

=1. =1.

z

101

2. . . . . . . .

.....-m-'0

~"-"~"~:;'/'~0~

10o 102

. SWIRL FLOW, R a ~ 0 i

103

104

S w - P r 0.391 FIG. 25. Effect of buoyancy (or Ra) on laminar fully developed swirl flow heat transfer in circular tubes with uniform wall temperature (UWT) [20].

are shown graphically in Fig. 26, and there is little agreement between them. More notably, the Watanabe et al. [81] equation does not predict the correct asymptotic behavior for Re ~ 0 and/or y --+ c~, and the Hong and Bergles [18] correlation [Eq. (26), Table 3] is recommended for UHF conditions. Inclusion of the classical correction factor for liquids Nuz, vp

-

NUz, cp(~b/I-l,w) 0"14

(38)

to account for temperature-dependent property variation in the flow cross section due to fluid heating and cooling effects is further recommended. In the case of gas flows, the temperature-ratio factor in the Watanabe et al. [81] equation could be used. Furthermore, experimental [54] and computational [55] studies have reported the influence of free convection on swirl flows in uniformly heated (UHF) tubes. The experimental data of Bandyopadhyay et al. [54] for laminar viscous liquid (73 _< Pr _< 404) flows with seven different tape inserts (3.9 < y _< 8.1) provide a quantification of this effect. By extending the Hong and Bergles [18] correlation ( R a * - 0 ) , the following equation has been suggested for fully developed mixed convection:

SWIRL FLOW HEAT TRANSFER

102 I-|-

* Hong * * and ~ ~ *Bergles ~* I , [18] ........ Watanabe et al. [81] . . . . . . . y - oo, ~i - 0

,

~ ~ ~ , ~ r..i~ 5 ~

Pr = 9

"

1.~.. 1

y : 2.5, (k/d) : 0.05

,.-" 1,

229

/.1 ~'I~''1

1.11

.1 ' I

/

1.

.I

~ ! ~ ' ~ ~ ~1 .t" _ , .1.

1.1 1

-

"**"1" ,

Z

101

-" ........

;-:-, .......... -

...........

.1.1.1

iZ-

9149

.,,""

9

..

..--" t.-'T 101

-

;y:.

9

1 .s

10 o

.-""

t

t

~ t Jtl

[81] J

J

J

]

102

J

~ i Jtl

103

i

l

J

; J JJ

104

Re FIG. 26. Prediction of local Nusselt numbers for laminar swirl flows in a circular tube with a twisted-tape insert and uniform wall heat flux (UHF) condition.

Nuz - [Nu9,/_/8 + 1.17(Ra*~

1/9.

(39)

Again, the viscosity-ratio correction factor of Eq. (38) for liquid heating and cooling effects should perhaps be included in this case as well. As mentioned earlier, a relatively larger n u m b e r of Nusselt n u m b e r correlations have been reported in the literature for turbulent flows (see listing in Table III). E q u a t i o n (36) is perhaps the more general predictive equation a m o n g these for Re >_ 104 and can be restated to highlight swirl flow effects due to twisted-tape inserts as ( N u / N u y = ~ ) - [1 + (0.769/y)]. Its predictions are seen in Fig. 27 to describe within + 10% the majority of data reported in the literature for both gas and liquid flows. F u r t h e r m o r e , even though the transition from laminar to turbulent flows is rather " s m o o t h , " the tape-twist ratio y appears to have a role in describing its effect on N u [21]. F o r design purposes, it is therefore r e c o m m e n d e d that a linear interpolation be employed between laminar and t u r b u l e n t f l o w estimates for Sw > 1400 and Re < 104, respectively. It may be noted that for

230

RAJ M. MANGLIK

2.0

~ -

8

=

z

1.5

~

~

r

I

Experimental

~

~

F

I

ARTHUR

'

~

r

E. B E R G L E S

~

I

~

~

~

~

I

i

~

~

data

-

9

Manglik

-

[]

Armstrong

_

zx

Bolla

-

~

AND

and

Bergles and

[90]

Bergles

[83]

e t al. [ 1 0 0 ]

<>

Smithberg

o

Junkhan

and

Landis

[45]

e t al. [27]

.......................

_'-

! ...................................................... i ......

10%

1.0

_

(Nu/Nuy

0.5

t 0.0

J

i

L

I 0.1

~

i

i

~

= =)=

I

i

1 + (0.769/y)

i

l

0.2

l

I

i

z

J

0.3

j

I 0.4

J

J

L

L 0.5

(l/y) FIG. 27. Comparison of Manglik and Bergles [21] correlation for turbulent swirl flow Nusselt number [Eq. (36), Table III] with experimental data.

some twist ratios, where laminar swirl flows might prevail beyond the Sw > 1400 cutoff (and, hence, higher Nu), this strategy would provide a rather conservative prediction for the transition regime.

III. Two-Phase Flow and Heat Transfer A.

GENERAL

COMMENTS

The review of forced-convection boiling heat transfer can be truncated considerably because of the fairly recent and comprehensive survey by Shatto and Peterson [39]. Their review is focused largely on boiling with net vapor generation, although they have mentioned a few studies of subcooled boiling. They did not really make a distinction between these fundamentally different modes of boiling. The object of the present review is to discuss subcooled boiling in detail, both heat transfer and pressure drop, and to add new references that have appeared since the mid-1990s for all modes of boiling. Let us begin by examining the effects of twisted tapes on heat transfer in a uniformly heated, once-through boiler tube. Because the purpose of this

231

SWIRL FLOW HEAT TRANSFER

system is to generate superheated vapor, the intent of the twisted tape is to reduce the wall temperature. Figure 28 shows the progress of wall temperatures for an empty tube and a tube fitted with a twisted tape. This plot is useful as all regions of boiling, and single-phase flow, can be depicted. The heat flux is uniform; and mass flux, pressure level, and inlet temperature are fixed. This results in a common fluid-temperature distribution. According to the previous discussion, the heat transfer coefficient in the single-phase region (1) will be increased considerably so that there is a substantial reduction in the wall temperature. Again referring to Fig. 28, it is observed that subcooled boiling occupies a rather small region of this particular tube (2). This is followed by bulk boiling (3) and dispersed flow film boiling (4). The liquid is finally evaporated, and the single-phase vapor is heated up (5). Reductions of the wall temperature occur in all boiling regions (see later). In bulk boiling (3), the empty tube dries out (critical heat flux condition) at an intermediate quality, with the wall temperature increasing sharply. Due to droplet cooling in dispersed-flow film boiling (4), the wall temperature decreases before increasing again as the vapor is superheated. After dryout, and extending into the high quality region (5), the fluid is in a nonequilibrium state, i.e., the vapor is superheated and there is more liquid in the form of droplets at saturation temperature.

Flow direction

Subcooled boiling enhanced by displacement of vapor from wall; T w reduced

~.,~

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Bulk boiling h enhanced by secondary flow induced in wall liquid layer; Tw reduced

.

Empty tube wall temperature .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Tsa t

o

II I

I

Fluid temperature

~, ~,? ~

~. I , ~ ~ 6 ~ = ._~,

..., ..r

~

-

.....

(4)

-- "

,~

J ~

Liquid temperature with swirl almost at equilibrium value due to centrifuging of droplets to wall

x=

0

X--

:

.

.............. i2) Twisted-tape wall temperature (3) "

Dryout shifted to higher qualitydue to liquid ......... added to heated wall ....... ...... -'" (5) .... . "

1.0

~= az o= ~ ~-5

o.,-, .~_ ; ~ r~,.Q

I

I D i s t a n c e a l o n g tube

FIG. 28. T h e influence o f t w i s t e d - t a p e - i n d u c e d swirl on the e v o l u t i o n o f fluid b u l k a n d t u b e - w a l l t e m p e r a t u r e s a l o n g the tube length in forced c o n v e c t i o n boiling.

232

RAJ M. MANGLIK AND ARTHUR E. BERGLES

With twisted tapes in bulk boiling, (region 3 of Fig. 28), the liquid is centrifuged to the wall so that a liquid film is maintained [121]. Dryout is thus delayed until a very high quality. The remaining droplets are again centrifuged to the wall, thereby reducing the temperature excursion. An equilibrium fluid condition is promoted so that the wall temperature quickly settles down and tracks the fluid temperature (4). Due to enhancement of the single-phase vapor, the wall temperature is reduced (5). B. SUBCOOLED FLOW BOILING

Subcooled boiling is of interest when cooling high-power devices. The object is to accommodate high heat fluxes with moderate pressure drop penalty, not to generate vapor for use in an energy-conversion device (e.g., Rankine cycle power plant) or for absorbtion of a high heat load (e.g., vaporcompression-cycle air conditioner). The fixed heat flux boundary condition invariably is imposed. This applies to pressurized water nuclear-fission reactors, nuclear-fusion reactors, and electrical and electronic devices (e.g., high-field electromagnets and liquid-cooled radar tubes). The intent then is to operate entirely in the "preheating" region of Fig. 28 (x < 0). Singlephase flow (1) and subcooled boiling (2) occupy the entire tube. 1. Heat Transfer

Gambill et al. [68] and Feinstein and Lundberg [122] provided a few data points characterizing the subcooled boiling curve for water. The location of the boiling curve was clarified in a more detailed study by Lopina and Bergles [102, 123]. Examining data shown in Fig. 29, they concluded that the fully developed boiling curves are essentially the same for various tape twists as for straight flow. It was also found that the point of incipient boiling and the transition from convection to fully developed boiling could be predicted accurately by methods developed for straight flow. These similarities suggested that the mechanism of heat transfer for swirl flow is similar to that for straight flow, even though the bubble mechanics is considerably different in the two cases. The large radial acceleration induced by twisted tapes causes the bubbles to migrate to the center of the tube, resulting in rapid condensation of the vapor. In axial flow, the bubbles remain near the heated surface, condensing rather slowly as they are swept downstream. The bubble disturbance is apparently comparable in both cases, and somewhat similar boiling curves are obtained. When considering the total picture and reexamining these data, it is evident that fully developed, twisted-tape data are somewhat to the left of straight-flow data. This makes subcooled boiling data more compatible

SWIRL FLOW HEAT TRANSFER

I

i

i

i

I

233

l

Degassed, demineralized water L N i c k e l tubes, I n c o n e l tapes Swirl t u b e s d i -- 4.915 m m E m p t y t u b e d i = 5.029 m m 107 9 8 -

(Tsa t - T b ) - -

310.93 K

Pexit - 344.73 k P a Vin = 2 . 7 4 3 - 7.925 m/s

7 o

y=

v

y = 3.15

2.48

og~',

y = 5.26 t'q

[]

y

9

empty tube

= 9.20

%

/x /x D [] [] c 9 []

106

[ 6

J 7

t 8

i I 9 10 ( T w - Tsa 3

I 20

t 30

i 40

[K]

FIG. 29. Influence o f t w i s t e d - t a p e - g e n e r a t e d swirl flow o n fully d e v e l o p e d boiling h e a t transfer [123].

with bulk-boiling data, i.e., there is an enhancement in both subcooled boiling and bulk boiling, as shown in Fig. 28. No correlation is available, however. Twisted tapes are very effective in elevating the critical heat flux in subcooled boiling. The swirl-induced radial pressure gradient promotes vapor removal from the heated surface, thereby permitting higher heat fluxes before vapor blanketing occurs. In fact, the heat fluxes are so high that it is impossible to use burnout protection systems. This means burnout of the test section at CHF or one data point per test section. Data of Gambill et al. [68] that illustrate this improvement are plotted in Fig. 30. In order to permit clear visualization of the major trends in these data it is necessary to consider pressure and geometry as secondary variables and to designate only the

234

RAJ

125

f

I

I

M.

I

I

I

MANGLIK

I

1

~

AND

I

r

I

ARTHUR

I

I

I

E.

r

~

BERGLES

~

9/

J

I

l

f

I

r

I

I

I

I

Gambill et al. [68] data o

r

_

Axial flow

Vin - 45.1 - 47.6

Pexit - 0.1 - 0.43 MPa

d = 4.572, 7.747 mm 7-54

-

L/d=

100

9 Vortex (swirl) flow Pexit = 0.1 - 0.85 MPa d = 4.597 - 10.21 mm

/

JV /

L/d=8-61

_ -

y = 2.08-2.99 ~

75

22.9-33.5

/

/.

_

/. ~

50

o

J

9

/-

J O

<... ,
25

25

-

~ 0

0 .x~ 9 0 .-0~""

9 8.54- 11.6

0

_

o

~,o,-" o i" o o

9/

3.28- 7.32

_

......

0" "-

o 50

Vin - 7.32 - 11.6 m/s

75

(Tsat-Tb)exit

100

-

125

150

[K]

FIG. 30. Influence of twisted-tape-generated swirl flow on subcooled boiling critical heat flux.

v a r i o u s velocities. C e r t a i n l y , t h e s e v a r i a b l e s c o n t r i b u t e to t h e s c a t t e r in t h e d a t a ; h o w e v e r , t h e y s h o u l d n o t be o f t o o g r e a t significance f o r t h e v a r i a b l e s c o v e r e d [124]. It is e v i d e n t t h a t swirl flow p r o d u c e s a s i g n i f i c a n t i n c r e a s e in t h e critical h e a t flux. D a t a s h o w n in Fig. 30 w e r e r e a s o n a b l y well c o r r e l a t e d by G a m b i l l [125] u s i n g a n a d d i t i v e e x p r e s s i o n ( s i n g l e - p h a s e c o n v e c t i o n p l u s p o o l boiling); h o w e v e r , t h e p r e d i c t i o n m e t h o d w a s less successful w h e n a p p l i e d to e t h y l e n e glycol d a t a o f G a m b i l l a n d B u n d y [51].

SWIRL FLOW HEAT TRANSFER

235

Dri~ius et al. [126] reported CHF data for water in a 1.6-mm-diameter tube with tape twist y ranging from 2 to 10. The CHF was dependent on the mass flux, tape twist, and heated length of the tube, but independent of the subcooling. We must now fast forward about 30 years because the subject was neglected until the 1990s. As part of a resurgence of interest in flow subcooled boiling, studies with twisted-tape inserts have been conducted in France, Italy, Japan, and the United States. Almost all of the more recent work has also been carried out with water. Flow boiling experiments with water were carried out by Inasaka et al. [127] and Nariai et al. [128]. They reported CHF data in subcooled flow boiling for 6-mm inside diameter and 100-mm-length stainless-steel tubes, and zirconia twisted tapes with twist ratios ranging from 2 _< y < ~ (untwisted or straight-tape insert). Because the twisted tapes were fabricated with a ceramic material, a relatively large gap between the tape edge and the tube wall was present (0.2-0.4 mm). Thermal-hydraulic test conditions were as follows: exit pressure ranging from 0.1 to 1.5 MPa, inlet temperature of 40~ and mass flux ranging from 6 to 17 m g / m 2 s. For tightly twisted tapes, CHF was found to increase proportionally to y-0.333. Inasaka and Nariai [129] have given a discussion of the different correlations. One of the most important conclusions reached in the latter study is that twisted-tape CHF becomes the same as the empty-tube CHF at pressures above about 1.0 MPa. This deduction is based on the study's data and a modified correlation (using previous data from the group) for empty-tube flow. The empty-tube correlation has a stronger pressure dependence than that for flows with twisted-tape inserts; hence the deterioration in enhancement as the pressure is increased. We suggest that this pressure effect needs to be critically reevaluated. In any case, it is likely that cooling systems with subcooled boiling will be operated at lower pressures, say in the range of 0.1-0.5 MPa. Gaspari and Cattadori [130] found CHF enhancement relative to empty tubes up to a factor of 2.1 with a twist ratio of y - 1.0 and up to 1.4 with y - 2.0. Cardella et al. [131] have provided additional experimental data. Also, empirical correlations of CHF with twisted-tape inserts for water flows have been proposed elsewhere [125-128]. Tong et al. [36] ran experiments with stainless-steel tubes with inside diameters varying from 2.44 to 6.54 mm and several different twist-ratio tapes (1.9 _< y < oo). The gap between the anodized tape edge and the tube wall was between 0.05 and 0.1 mm. A maximum enhancement in CHF of 1.5 was observed with the tightest twist tape insert at a mass flux of 15,000kg/mZs. An interesting effect was found for gently twisted tapes: due to the thermal-insulating effect between the tape and the wall of the heated tube, CHF values are even lower than those for empty tubes. This has

236

RAJ M. MANGLIK AND ARTHUR E. BERGLES

also been reported by Lee et al. [35], who used thermal imaging to confirm that the CHF location was at the edge of the tape. With the smaller twistratio tapes, Tong et al. [36] found that CHF is inversely proportional to y, tube diameter, inlet temperature, and length-to-diameter ratio, but directly proportional to mass flux and exit pressure. An empirical correlation, covering all six parameters, was proposed (for water only) as follows: q'c~r-- 31.554

(a/ao)~ (ee/ee,o) 0"2787 [(Zh/d)o ] 0.2191 (y/yo)~ 0"0735 [ (Lh/d) J ( Zsub,e ~- Zi ) 11.0410

(40)

(T~at,;- Ti,o)J The exit-pressure effect is presented explicitly in the pressure-ratio term and implicitly in the saturation pressure. This equation shows that CHF is proportional to mass flux and exit pressure, but inversely proportional to twist ratio, tube diameter, inlet temperature (can be related to exit subcooling through a heat balance), and length-to-diameter ratio. These trends are consistent with most experimental data, and the correlation describes these data reasonably well, as shown in Fig. 31. In the only study known for another pure fluid, Weisman et al. [85] reported swirl flow CHF data (y = 6.25) at subcooled and low-quality conditions using Refrigerant 113. These results, plus water data from the literature, were compared to a complex phenomenological model that is an extension of an earlier model that Weisman developed for empty tubes. 2. Pressure Drop Pressure drop is crucial to the application of subcooled boiling, yet there have been few studies of this aspect of the problem. The information is needed to size the pumping system, determine the exit pressure so the CHF can be estimated (see earlier discussion), and assess the thermal-hydraulic stability of the system. The latter can be explained as follows. The cooling systems used in a variety of electronic devices usually consist of an array of parallel channels conveying the coolant through the "heat sink." One or more channels is starved for flow when a minimum in the pressure-drop versus flow-rate curve is reached. Thus, the typical experiment that involves collecting data on a forced-flow system, using a single electrically heated tube, is flawed. It fails to capture the Ledinegg, or pressure-drop versus flowrate instability that can occur with subcooled boiling in channels [132]. The hydrodynamic limiting unstable heat flux is lower than the stable critical heat flux described previously.

237

SWIRL FLOW HEAT TRANSFER

100

I

I

$1

s

s

s

[] Gaspari and Cattadori [130] V Inasaka et al. [127] ," +25% /~ Drizius et al. [126] ,." O Gambill et al. (< 10 bar) [68] ," t O Tong et al. [36] s s

t

s

s

t

80

s

s

60 /

sss

pJ

J

s

-25%.-.

..(~

s

s

. ~'/x []

"

/x

s s

o

40

9 ,~,,i

20

~~sss

0

"SSS

20

40

60

80

100

Experimental CHF [MW/m 2] F l a . 31. C o m p a r i s o n o f swirl-flow C H F predictions with experimental data [36].

The general character of the pressure drop as a function of heat flux and twist ratio is shown in Fig. 32. At low heat fluxes, the flow remains single phase over the whole tube length. Pressure drop decreases with an increase in heat flux due to the decrease in the near-wall fluid viscosity--as discussed in an earlier section. As boiling is initiated, the bubble boundary layer causes an increasing frictional pressure drop. Because the nonequilibrium void fraction is closely related to the heat flux, the acceleration component of the pressure drop also increases. Although the experiments were run with test sections vertical, the gravitation component of the pressure drop was very small for the short tube. Further data, with the parametric trends, are presented in Pabisz and Bergles [133]. Test sections burned out at the last point of each curve. Due to the insulating effect of the tape edge (mentioned earlier), the axial CHF is higher than the twisted-tape CHF at the lower twist ratios. This rates as one of the

238

RAJ M. MANGLIK AND ARTHUR E. BERGLES

i

i

i

i

i

i

~

r

i

i

i

i

!

i

i

1.9

~

i

O A /x

y -- 2.6 y-3.6 y - 5.7

din = 2.44 m m

@

y=9.3

Pe-

[]

y-oo

T~-23~

o

o

9

A

A AA

9 A

o

9

1313

i

i

i

I

10

o

9

9

r

i

I

I

10bar

o

O0

o

9

AA

9

A A AA A A

Z3Z~A

i--113 D

I

I

~jI~IK7

~ ~ ~

W~

,

i

o

Empty tubes (two runs)

~ ~ vv ~

i

Lh/din - 24

000

0

<1

i

i

G = 15,000 Kg/m2s

0

0

i

y-

3

_~v

i

9

1

[

1

20

I

I

I

I

1

30

I

I

I

]

40

I

I

50

q" [MW/m 2] FIG. 32. Subcooled boiling pressure drop [36].

"surprises" of enhanced heat transfer, i.e., the normal situation (empty tube) performs better than the enhanced tube (twisted-tape insert). This is due to the necessity of using relatively thick tapes in small-diameter tubes so that the local heat flux is amplified considerably near the edge of the tape. A correlation of these twisted-tape pressure drop data has been proposed by Pabisz and Bergles [133]. This correlation, compared with data in Fig. 33, is A p -- ( A p ~ ~ +

Aps~)l/~,

(41)

SWIRL FLOW HEAT TRANSFER

0.7

1

i

I

i

t

i

I

~

i

!

I

r

239

i

I

I

I

d= 4.4 mm

Pe = 9 bar G = 4,400 kg/m2s 0.6

AP Correlation

04

I

o

.

~'"-

9

69~ 40oc 40oc 40oc 30oc 25oc lOOC lOOC 9oc 9oc

0.2 o []

0.1

A

-

II' V 0.0

I

0

I

l

I

4

J

I

t

I

I

I

l

I

8

12

q"

[ M W / m 2]

I

J

J

I

i

16

_

i

2o

FIG. 33. Correlation of subcooled boiling pressure drop [133].

where any gravitational c o m p o n e n t has been subtracted f r o m the m e a s u r e d pressure drop. The p r o c e d u r e s for calculating, or correlating, the singlephase and subcooled-boiling c o n t r i b u t i o n s are described elsewhere. [133]. The overall pressure d r o p can also be plotted as pressure d r o p versus flow rate so that a m i n i m u m can be spotted easily for a given A p (see Fig. 34). The heat flux c o r r e s p o n d i n g to this m i n i m u m is the expected C H F .

3. Variations in Circumferential Heat Flux Subcooled boiling o f water has been suggested as the m e a n s o f cooling the plasma-facing walls of nuclear-fusion reactors. The cooling channels are

240

RAJ M. MANGLIK AND ARTHUR E. BERGLES

0"311 '1 ! ' ' '

' ' ' ' '

1,,,,

H i l

, . , . . ]

d = 4.4 ram, L / d = 2 5 Pe = 9 bar, T i = 20~

l--] I ! H' il

I1~ I il Hiil

,,,,,

i. i

.

.

.

tMW/m2]

q"

.

-----

2.5

........

5.0

7.5 .......... 10

!

J q

/

//~

i 1-1 i I ~ ~i ! \

'~'

lli~ Hi

0.0

I

0

CHF for multiple channel system at AP = 0.083 bar

,

~

l

"..

i

i

l

~

I

2000

i

i

..-. ....... "

I

i

I

I

4000

l

l

I

6000

/~..-'"~,)/ -1

./~:.f..7-

....y / " ~f.~-:"

i

-

i

I

8000

I

i

I

i

10000

G [kg/m2s] FIG. 34. Two-phase flow pressure drop showing minimum that leads to hydrodynamic CHF [133].

subject to highly nonuniform heating, as the radiation heat input is only from one side of the wall. It is natural to want to increase the allowable heat flux by elevating the critical heat flux through the use of twisted-tape inserts. Various ways have been devised to simulate the heating through only a section of the tube wall. Cardella et al. [131] obtained subcooled C H F data using tubes split in half, with only one section powered. Limited data indicated that, for fixed inlet conditions, the C H F was as much as 100% higher than that for an empty tube with y -- 1, but the pressure drop was increased 10-12 times. It is not clear how a tape with such a tight twist was fabricated. Because earlier tests indicated that the tube-tape gap affected CHF, adversely, tests were run with no gap. Nariai et al. [134] ran experiments with 6-mm inside diameter, 100-mmlength stainless-steel tubes that had the walls thinned by electrochemical machining. With a 90 ~ hot spot, the most intense nonuniform heating condition, C H F (based on the hot-spot heat flux) was 1.8 times the uniform heating value for comparable flow conditions. These tests were extended by Kinoshita

SWIRL FLOW HEAT TRANSFER

241

et al. [135] to a wide range of heating conditions and several tape twists. The test section is shown in Fig. 35. Loose-fitting zirconia tapes were used; it is noted, however, that the gap seems to have an effect, as mentioned earlier. The uniformly heated tubes were instrumented to give boiling curves. It was found that the onset of nucleate boiling (see earlier discussion) and point of net vapor generation in tubes with a twisted-tape insert were similar to the predictions for empty-tube flow. The C H F with a 90 ~ hot spot agreed with the previous results, but the C H F with the 270 ~ hot spot was essentially the same as for the uniformly heated tube with a comparable twisted tape and flow conditions. It is interesting that subsequent tests by this group [136], also for nonuniform heating, do not indicate a sharp decrease in enhancement of the C H F by the twisted tape as the pressure is increased above 1.0 MPa. Suzuki and Kumagai [137] conducted transient subcooled boiling C H F experiments using a thick-walled copper cylinder (190mm OD and 8 mm ID) with inserted aluminum tapes of various twists. The cylinder was heated up prior to establishing the water flow. Their results raised the question as to whether there is a well-defined C H F in the massive plasma-facing

FIO. 35. Schematic of test section showing details of twisted-tape attachment and electrical heating for nonuniform heat flux [135].

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RAJ M. MANGLIK AND ARTHUR E. BERGLES

components that are heated from one side. The wall heat flux probably just redistributes when the high heat flux at the top of the channel precipitates the CHF condition there. It is unlikely that there is a burnout. Araki et al. [138] reported data for one-sided heating (electron beam) of tubes with twisted-tape inserts. Attention was directed to the boiling curve, as heat fluxes were below the CHF condition. An inverse, conjugate'analysis was used to calculate the inside wall heat flux. A new correlation was proposed for fully developed subcooled boiling. More recently, Boscary et al. [139] reported a study of CHF in a copperalloy tube, circular on the inside (D = 18mm), fitted with a twisted tape (y = 2), and square on the outside (24 mm on a side). Heating was accomplished by heating 100 mm of the 400-mm-long test section with an electron-beam gun. The incident heat flux was obtained from an enthalpy balance on the flowing water, and the channel-wall heat flux was obtained numerically. The maximum incident heat flux of 46.7 M W / m 2, corresponding to a maximum wall CHF of 68.6 M W / m 2, was accommodated with the highest mass flux (16Mg/m2s), maximum exit subcooling of 176~ and maximum outlet pressure of 3.6 MPa. Reasonable agreement was obtained with an empty-tube model, modified for swirl flow; however, the effect of the one-sided heating was not considered. It is interesting that there apparently was no coordination between the Japanese groups (Nariai and Araki) or between the Japanese and French/Italian work (Boscary). 4. Ultimate Limits o f Heat Fluxes

It is appropriate to conclude the discussion of subcooled boiling with a summary of the maximum heat fluxes that have been accommodated with this mode of heat transfer. The expectation, of course, is that twisted tapes will allow the maximum heat flux, and a listing of the results from various investigations is given in Table IV. It is evident that the highest heat fluxes have not been recorded with the twisted tapes. However, the channels used to set the empty-tube "records" were impracticably small (d = 0.3-0.5 mm), and the mass fluxes unrealistically high (up to 90,000 kg/m2s). Practical channels would be more like d = 2-3 mm, and these channels could be fitted with twisted tapes. Mass fluxes would likely be constrained to about 40,000 kg/m2s. Under these conditions, the twisted tapes would perform better than the empty tube. The other, more serious problem is that these single-channel CHFs cannot be realized in the parallel channels that are likely to be used in cooling systems designed for subcooled boiling.

SWIRL FLOW HEAT TRANSFER

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TABLE IV HIGHEST REPORTED CHF (MW/m 2) VALUES FOR VARIOUS CONDITIONSa Coolant Channel Empty tube, uniformly heated

Empty tube, nonuniformly heated Twisted tape, uniformly heated tube Twisted tape, nonuniformly heated tube

Pure water 1241124] 133 [1401 2441141] 313 [140] 3371141] 1171122] 144 [36] 34 [134] 80 [139]

Water-alcohol 163 [140]

366 [140] No enhancement reported Not studied

aNumbers in brackets are cited references.

C. BULK BOILING

1. General C o m m e n t s

It was mentioned earlier that this review deferred to that of Shatto and Peterson [39] for evaluation of the bulk boiling behavior of twisted-tape inserts in circular tubes. They showed that twisted tapes increase the heat transfer coefficient throughout the entire quality region, increase the CHF (dryout quality), and increase the heat transfer coefficient in postdryout (dispersed-flow film boiling). These are, respectively, regions 3, 4, and 5 of Fig. 28. However, there are dozens of studies not cited in [39] that should be commented on. The majority of these papers are listed in the bibliographic survey of Bergles et al. [6], which appeared about the same time as Shatto and Peterson [39]. Suffice it to say, twisted tapes have improved convective vaporization of water (low-pressure and high-pressure), refrigerants (R-11, R-113, R-12, and R-22), a cryogenic fluid (liquid nitrogen), and liquid metals (cesium, mercury, potassium, rubidium, and sodium). Full-length tapes, with various tube-tape gaps, and intermittent tapes have been tested. Twisted tapes have also been shown to be effective when they are inserted in annuli and rod clusters (either continuously or as spacers). Very tight twists (y < 1) can be produced by winding a strip on a core rod. Much of this work appeared during the "golden era" of research on twophase flow and heat transfer, the 1960s, and is described elsewhere [142]. One of the old studies is worth discussing. Matzner et al. [143] studied the effect of full-length and short twisted-tape inserts on bulk-boiling CHF in 10-mm inside diameter tubes with light water at 7 MPa. The application was

244

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

nuclear reactors, particularly a potential replacement of pressurized heavy water cooling with boiling light water cooling in the CANDU reactor. The full-length tapes increased CHF by 30-50%. Instabilities, observed in this study with full-length tapes, were speculated to be a form of parallel-channel instability--the tightly fitting tape dividing the tube into two parallel channels. Additional inlet throttling stopped the instability. Kakaq [144] summarized studies that reported the effect of enhanced surfaces (spring-insert roughness and porous metallic matrix coating) on instability parameters for R-12 flowing in horizontal and vertical tubes. Although twisted-tape inserts were not considered, it is likely that these would give similar results, i.e., enhanced surfaces decrease system stability. There are several cautions associated with the use of twisted tapes. The degradation of CHF in subcooled boiling with tapes of high twist ratio in tubes of small diameter has been mentioned in the preceding section; this is expected to hold for bulk boiling as well. Another consideration is that the tape essentially constitutes a large unheated wall. The liquid captured by the tape is lost to the wall. At very high quality where the total liquid flow is small, this effect produces a decrease in the CHF. The question of fouling/corrosion is of concern, especially with impure water. An appreciable tube-tape gap is often allowed to avoid collection of impurities that might cause corrosion. The gap can fill up with deposits, however, effectively "cementing" the tape in the tube. In the as-new condition, the thermal performance will not be so effective due to the leakage flow through the gap. Then, too, there is the possibility of fretting corrosion if the loose tape vibrates and abrades the tube wall. In any case, the tape must have a slightly loose fit in order to be inserted into the tube; this raises the issue of securing the tape in the tube [39]. 2. Recent Studies

Relatively little new information has been contributed to this area of enhanced heat transfer. Lee et al. [35] obtained CHF data with vaporizing R-113. They confirmed that CHF can be decreased with tight-fitting tapes of a large twist ratio. They did not indicate any problems with decomposition of R-113, when the film-boiling condition was reached, as is observed in pool-boiling CHF. Kedzierski and Kim [145] presented extensive data for convective boiling of refrigerants and refrigerant mixtures in a horizontal tube with a twistedtape insert. Fluid heating was used, with the test section divided into 20 segments connected by U-bends and T-connections. This yielded sectional-

SWIRL FLOW HEAT TRANSFER

245

average, almost local, heat transfer coefficients that were correlated in a single expression consisting of a product of dimensionless properties. More recently, Doeffler et al. [146] and Groeneveld et al. [147] addressed the important problem of fluid-to-fluid modeling and twisted-tape enhancement. With appropriate scaling, it is possible to run experiments with lowpressure refrigerant instead of high-pressure water, the actual nuclear reactor coolant. Refrigerant tests require less expensive test sections, slightly lower mass fluxes, and much lower pressures, temperatures, and power to reach CHF as compared to water. It was demonstrated that the ratio of the CHF for the tube with a twisted tape to that for the empty tube is virtually identical to that for water, if the conditions for R-134a are chosen properly. D. CONDENSATION HEAT TRANSFER

Condensation is somewhat easier to describe. The sequence of events is basically the reverse of Fig. 28; entering from the right, the flow is desuperheated and there is then condensation (x = 1 to x - 0) and subcooling of the liquid. Fluid cooling is the only possibility so it is difficult to get local data. A heat balance can be run on the coolant, from the flow rate and numerous thermocouples, to get the local heat flux. Knowing the heat transfer coefficient on the coolant side and the local temperature of the test fluid, the condensation heat transfer coefficient can be calculated. For a substantial temperature gradient, the coolant flow rate must be low; however, this means a low heat transfer coefficient. The net result is considerable uncertainty in the condensation coefficient. Alternatively, the tube can be sectioned, and from the coolant flow rate and thermocouples at the beginning and end of each section, the sectionalaverage heat transfer coefficient can be obtained. The heat transfer coefficient of the coolant need not be known if the temperature of the condensing tube wall is obtained, but this is a difficult procedure. Often, there is only one section, and only an average condensing heat transfer coefficient is obtained. A stepwise calculation of condensing behavior is desirable, which requires local, or quasi-local, data. Just as in boiling, the coefficients can be strongly affected by stratification in horizontal tubes at low flow rates. As in the case of boiling, phase-change coefficients may be large, so when deciding to use enhancement, due attention must be given to the controlling thermal resistance. The twisted-tape insert will increase the heat transfer coefficient, but an assessment as to whether it is necessary should be made, especially considering the increased pressure drop. This means a reduced saturation temperature and a reduced temperature difference in the heat exchanger.

246

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

1. Steam Condensation

The limited literature is best discussed in two groups: steam and refrigerant vapor. Carey and Shah [148] discussed the earlier literature in both areas. In-tube condensation of steam occurs in power plant reheaters, air-cooled condensers for power plants, and distillation facilities for the production of potable water. Royal and Bergles [149] appear to have conducted the only study of enhancement of in-tube condensation of steam by twisted-tape inserts. The 4.8-m test condenser was divided into four annular sections that were instrumented for cooling water temperature rise and inner tube wall temperatures. The flow rates of both streams were measured, and the pressure of the steam was monitored at numerous axial intervals. Steam pressures were up to 0.6 MPa and the maximum mass flux was 440 kg/mZs. Snug-fitting tapes of y = 3.3 and 7.0 were found to increase the average heat transfer coefficients (slightly superheated steam inlet and slightly subcooled water outlet) by up to 30% above the empty-tube values on a nominal area basis. The coefficient was higher for the tighter tape twist. For design purposes, curve fits were presented for each pressure in the form: havg C . Gn [149, 150]. An empty-tube model for the heat transfer coefficient, modified for twisted-tape effects, agreed reasonably well with the data [69,149]. The tubes with twisted-tape inserts result in rather large pressure drops (two to four times the empty-tube value at constant mass flux and pressure) due to the increased wetted perimeter, the spiraling effects of a longer flow path, and secondary flow. Performance evaluation criteria (see Section V) were applied to thermal-hydraulic data [70, 149]. In a work related somewhat to steam condensation, Helmer and Iqbal [151] studied the impact of a twisted tape ( y - - 3) on dehumidification of moist air. =

2. Refrigerant Condensation

Luu and Bergles [152] ran extensive tests of the condensation of R-113 vapor, including tubes with twisted-tape inserts. The apparatus mentioned in the previous section was modified for this purpose. Tapes of both y = 2.8 and 4.6 yielded overall average heat transfer coefficients approximately 30% above those of a smooth tube on a nominal area basis. However, the pressure drop was as much as 3.5 times higher. Curve fits and a modification of an empty-tube correlation were provided [71, 152]. Azer and Said [153] and Said and Azer [154] reported a comparable study of R-113 with tapes of similar twist that yielded comparable heat transfer coefficients and pressure drops. More recently, Kedzierski and Kim [145]

SWIRL FLOW HEAT TRANSFER

247

studied in-tube, twisted-tape condensation of a wide range of refrigerants (see earlier discussion for boiling). An equation is proposed that uses the swirl parameter Sw discussed in the early part of this review. Ninety-five percent of the sectional-average coefficients are correlated to within 20%.

IV. Modified Tapes and Compound Techniques A. MODIFIED TWISTED-TAPE INSERTS

A few studies have been published sporadically in the literature that describe tests of different modifications of twisted-tape inserts. The drive behind this work has been to obtain heat transfer gains due to tape,induced swirl, but reduce the associated flow friction losses. The concept of initially generating swirl and then allowing it to decay was applied in perhaps one of the earliest modified usage of twisted-tape inserts. To achieve this, short tapes can be placed in the tube inlet. Their length and fluid flow rates would determine the extent of swirl generated, and its subsequent postinsert sustenance and decay in the remainder of the tube. The intent here is to achieve the swirl promoted enhanced heat transfer with a relatively lower pressure drop. The experimental work of Klepper [25] and the theoretical analysis of Migay and Golubev [73] first considered such usage in single-phase, turbulent gas/air flows [11]. With short tapes of different lengths and of twist ratios in the range 2.38 <_ y < 8.05, Klepper [25] reported 35 to 85% higher Nusselt numbers in comparison with smooth tubes for turbulent flows (2 x 104 < Reh < 3.8 x 105) of nitrogen. Considering constant pumping power and the performance of full-length tapes, this amounts to up to 7% enhancement in heat transfer. The theoretical predictions in a later study by Algifri and Bhardwaj [74] are in good agreement with these experimental results. In a study [87] of laminar viscous liquid flows (Re < 1150, 205 < Pr < 518), up to 22% improvement in heat transfer over full-length tapes, for the same pumping power, has been found. This was achieved by a y - 2.5 tape that, from the inlet, covered one-fourth of the tube length. An interesting variation of the short tape insert application is to consider a continuously varying twist pitch. By tightening the tape twist gradually, Zozulya and Shkuratov [75] found the heat transfer performance to be comparable to that with constant-twist tape, but with up to a 25% reduction in the pressure drop penalty. Another extension of this concept is the periodic generation and decay of swirl by interspacing short twisted-tape elements along the length of the tube. Burfoot and Rice [28] considered this in an experimental study with

248

RAJ M. M A N G L I K AND A R T H U R E. BERGLES

turbulent water flows and found no significant improvement over full-length tapes. Data of Saha et al. [76] for water flows (5000 < Re < 4.3 x 104) in electrically heated tubes further corroborate this finding. In the case of laminar flows (Re < 2300), some enhancement has been reported [24,29, 76, 110] in experiments with viscous Newtonian as well as non-Newtonian fluids. Theoretical predictions of isothermal friction factors and Nusselt numbers, using finite-difference techniques and an idealized model, have also been reported [155]. In all of these studies by the group at IIT-Mumbai (Bombay), India, short twisted-tape elements of one- to three-pitch turns, with each succeeding element connected by a thin central rod of a fixed length, were interspaced throughout the length of the tube. The better thermal-hydraulic performance was obtained with inserts that had threepitch tape elements. However, two experimental studies by an investigator from the same group [87, 88] showed no appreciable improvement over the performance of full-length tapes. A major shortcoming of this body of work is that the respective hydrodynamic lengths for the development of swirl and its decay have not been established. Conceptually, the twisted-tape length should be sufficient to generate fully developed swirl flow, and the subsequent empty-tube length should correspond to its characteristic decay length. Entrance effects may also have to be factored into this scaling. Monheit [26] and Jayaraj et al. [77] have considered perforated full-length tapes in laminar, single-phase liquid flows; in the former experiments, tapes with slit edges have also been tested. The intent here was to achieve enhanced heat transfer with a relatively smaller pressure drop penalty. Contrary to this expectation, the frictional loss was significantly higher with no improvement in the heat transfer. In another modified tape usage, Penney [156] has reported up to 95% higher heat transfer coefficients in heating of corn syrup with a loose-fitting tape that was rotated mechanically at 100 rpm. Such a usage would perhaps have to be treated as a compound technique. There do not appear to be any studies of a modified or an interrupted twisted tape in subcooled boiling. In the case of interrupted or interspaced tapes, the intent would be to shift the boiling curve to the left and elevate CHF, while reducing the pressure drop. The elements would have to be spaced reasonably close together because of the rather rapid decay of the swirl after an element. The short tapes utilized by Matzner et al. [143] were 51 mm long, y = 5, inserted at the inlet and 2.4m from the inlet of 3.7- and 4.9-m vertical tubes. It was not stated how the tape segments were "fitted tightly" into the tube. The mass flux ranged from 1200 to 9800kg/m2s. These inserts increased CHF by 0-35%, depending on the mass flux and quality at the second insert. With the considerable scatter, data are difficult to interpret. Mayinger et al. [157] reported significant increases in CHF for both sub-

SWIRL FLOW HEAT TRANSFER

249

cooled and bulk boiling when twisted tapes were used to generate vortex motion at the inlet of short tubes. Twisted-tape inserts have been used as throttling devices in the inlets of boiler tubes to reduce or eliminate flow instabilities [158]. Due to the enhanced heat transfer downstream of the tape, this type of throttling would be preferred over a simple orifice plate. It is noted that inlet swirl generated by twisted-tape inserts is really no different from swirl generation by spiral ramps and tangential slots [159] or tangential injection (see Fig. 1b and Ref. [160]). B. COMPOUND ENHANCEMENT TECHNIQUES

Compound techniques are a relatively new area of enhancement that hold promise for practical applications, as heat transfer coefficients can usually be increased above each of the techniques acting alone. Some examples involving single-phase flow and twisted tapes are rough tube wall with twisted-tape inserts [30,31,89], internally finned tubes with twisted-tape inserts [32, 161,162], and rotating tubes with twisted-tape inserts [33,163]. As noted earlier, a rotating twisted tape was also studied [156]. Muralidhar Rao and Sastri [163] reported the compound enhancement of rotating tubes (along a parallel axis) with twisted-tape inserts. The rotation, and resulting enhancement, is an inherent condition, e.g., the rotor windings of large turbogenerators. Figure 36 shows that as the rotational speed (Ja) increases, the Nusselt numbers for laminar flow of air are elevated considerably above those predicted by a correlation for a stationary tube with a twisted-tape insert. This correlation was confirmed by these investigators with their own data, who also found that its predictions were above data for a rotating empty tube. The compound enhancement is thus above either of the techniques applied separately. Compound enhancement with twisted tapes has also been applied to boiling situations [164, 165]. There do not appear to be any examples of twisted-tape compound enhancement with convective condensation. Boyd et al. [164] reported experiments with top heating of circular channels with integral spiral fins and a twisted-tape insert. Subcooled boiling was included among the flow modes. Although the fins alone exhibited the greatest enhancement, it was speculated that if the tape twist had been reduced, the compound configuration would give the greatest enhancement. Compound enhancement has been sought using a twisted tape together with a volatile liquid added to the coolant. Pabisz and Bergles [165] studied the effect of a 1-pentanol additive in water on CHF. A 3-4% by weight mixture was used, as that approximate percentage had been found to give

250

RAJ M. MANGLIK AND ARTHUR E. BERGLES

r

i

i

r

l

I

l

Ja (Y : 5.0) 30

20

<> 0 /x

25

[]

50

o

100

o [] o o

Z

121

10 9 O

8

J

~

Stationary tube with twisted-tape insert

7

J

1

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5

6

7

8

9 10 3

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2

Re FIG. 36. Effect of tube rotational speed (Ja) on Nu in laminar flow heat transfer in a circular tube with a twisted-tape insert [163].

superior performance in previous studies of pool and flow boiling. Contrary to expectations, the C H F was reduced below that with the twisted-tape, pure fluid value (d = 4.4 mm, P = 9 bar, G = 4400kg/m2s, and ATsub = 70-86~ This rates as one of the "surprises" of enhanced heat transfer, and it is speculated as being due to many small bubbles restricting flow of the cold core liquid to the wall.

V. Performance Evaluation Criteria A.

SINGLE-PHASE F L O W

Numerous, and sometimes conflicting, factors enter into the ultimate decision to use an enhancement technique: heat duty increase or area reduction that can be obtained, initial cost, pumping power or operating cost, maintenance cost (especially cleaning), safety, and reliability, among others. These factors are difficult to quantify, and a generally acceptable criterion

251

SWIRL FLOW HEAT TRANSFER

may not exist. It is possible, however, to suggest some performance criteria for preliminary design guidance. Pumping power has been mentioned earlier as a constraint for twisted-tape inserts, with single-phase flow, in numerous studies [3, 8, 11,25, 29, 39, 67]. It is, therefore, useful to derive the conditions for this figure of merit as an example. Consider the basic geometry and the pumping power fixed, with the objective of creasing the heat transfer. The following ratio is then of interest: R3 --

qa

= d , L , N , P , Ti, A T

= --. Nuo

(42)

qo

With the pumping power (neglecting entrance and exit losses) given by P - N V A c 4 f ( L / d )P V2/2,

(43)

the fixed pumping power constraint requires that (fRe3)~ = ( f Re3)o-

(44)

The calculation best proceeds by picking Rea and calculating (or reading from a graph and data set) the value of Nua and fa for the twisted tape case. Next, for the empty tube, using its conventionalfcorrelation, Reo is obtained from Eq. (44) and Nuo is then calculated from the appropriate smooth-tube correlation. The desired ratio of Eq. (42) is thus obtained; R3 > 1 would indicate enhanced performance for fixed geometry and pumping power. A typical set of results for laminar flows in a uniform wall temperature (UWT) circular tube with twisted-tape inserts is graphed in Fig. 37 [91]. As seen from Fig. 37 for typical tube lengths ( L i d = 200) used in shell-and-tube heat exchangers, and depending on the flow rates, type of fluid, thermal condition, and twist ratio, up to 5.5 times higher heat transfer rates can be sustained for the same pumping power. The computation of a wide range of performance evaluation criteria (PEC or Rs) is discussed in Bergles et al. [166]. The extension of these PEC to include thermal resistances other than the tubeside has been presented by Bergles et al. [167], and analysis of the broad range of possible PEC (256!) is given by Nelson and Bergles [168]. An application of PEC to compound enhancement, rough tube plus twisted tape, is given by Zimparov [89,169]. B. TwO-PHASE FLOW The application of PEC to boiling or condensing is more complicated, as it should involve the pressure gradient in the enhanced tube. However, it is usual to ignore the pressure gradient and consider a constant pressure level.

252

RAJ M. MANGLIK AND ARTHUR E. BERGLES

6[

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|

(8/d) .......

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(L/d)

= 0.04,

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+

t

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= 200, Ra ~ 0, (UWT)

y=2

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Res FIG. 37. Enhanced heat transfer performance of twisted-tape inserts for typical laminar flow conditions in a circular tube at (UWT) for fixed pumping power and geometry [91].

Gambill et al. [68] related the CHF in subcooled boiling to pumping power for twisted-tape and empty-tube flow. It was found that CHFs for swirl flow are approximately twice those for straight flow at the same pumping power. As discussed elsewhere [39], Matzner et al. [143] developed an efficiency index based on the increased exit quality, or critical power, that could be achieved with the inserts. The "efficiency" was generally higher for both fulllength and segmented tapes--in accordance with the CHF results. A further comparison was made by plotting pumping power versus critical power, where it was demonstrated that generally there was no penalty involved by using twisted-tape inserts; i.e., the pressure drop increase was accompanied by a commensurate critical power increase. This can be of benefit in a heavy water-moderated, light water-cooled nuclear reactor due to the greater removal of liquid light water from the core.

SWIRL FLOW HEAT TRANSFER

253

Royal and Bergles [70] have devised two performance evaluation criteria for enhanced condensation. The first relates to the size reduction made possible by replacing the plain tubes with enhanced tubes of similar nominal diameter. It assumes constant pressure level. The second criterion serves as a measure of the pressure drop consequences of an enhanced surface or insert. In terms of these criteria, the tubes with twisted tapes were found to be less favorable than internally finned tubes. Megerlin et al. [170] used a presentation for enhanced tubes, including tubes with twisted-tape inserts, that required the dissipated power at CHF divided by the pumping power. However, as mentioned earlier, the pressure drops with twisted tapes can be quite large; this, in turn, affects the LMTD in two-fluid heat exchangers. Webb [171] developed the only known analysis of PEC for two-phase flow that considers this effect. He considered both work-producing (e.g., Rankine power cycle) and work-absorbing (e.g., vapor-compression refrigeration) systems. Procedures are also presented for computing generalized PECs.

VI. Concluding Remarks A. SUMMARY An extended review of heat transfer enhancement in tubular heat-exchange devices by means of twisted-tape inserts has been presented. Both single-phase and two-phase (flow boiling and condensation) have been considered, and the salient features of twisted-tape applications and their thermal-hydraulic performance are as follows. 1. A twisted-tape insert is a very simple and effective device (a thin, helically twisted metallic or nonmetallic tape) that can be used to generate swirl flows and promote heat transfer enhancement in a tube. There is, of course, an added pressure drop penalty. 2. The literature on twisted-tape inserts dates back to 1896, when perhaps its first usage in fire-tube boilers was reported [12]. Today, they find applications in waste-heat recovery systems, process heat exchangers, boilers and condensers, domestic hot water heaters, refrigerant evaporators, and many other heat exchangers. 3. Phenomenologically, the enhancement in heat transfer can be ascribed to the partitioning and blockage of the flow cross section by the tape insert, reduced hydraulic diameter, effectively longer fluid flow path (or residence time), tape-twist induced swirl or secondary circulation, and, in some instances (particularly with very tight-fitting or mechanically attached metallic tapes), tape fin effects.

254

RAJ M. MANGLIK AND ARTHUR E. BERGLES

4. The tape-generated swirl is characterized by a helical vortex, which grows and breaks up into a two-vortex cell structure with a decreasing tape-twist ratio (y) and/or an increasing axial flow Reynolds number. In laminar single-phase flows, this behavior can be scaled by the swirl parameter (Sw), which effectively correlates fully developed swirl flow friction factors and Nusslet numbers for the UWT condition. Also, in short tubes, thermal entrance effects, represented by Gz or (L/d), and tube partition may be the dominant effect, and, in flows where Gr > Sw 2, buoyancy or mixed-convection effects may be present in both UWT and UHF tube-wall conditions. In the turbulent flow regime, however, ( 1 / ~ ) provides the correlating scale for both f and Nu. 5. Experimental data, as well as predictive correlations for f and Nu in single-phase flows, are available for all the flow regimes and heating/cooling conditions, and their evaluation with suitable recommendations has been outlined in this review for the design and performance rating of tubular heat exchangers with twisted-tape inserts. 6. A detailed review of the use of twisted tapes in subcooled boiling is presented, with the bulk boiling behavior having been described by others [39]. Twisted tapes are especially effective in elevating the CHF for single tubes. Many studies have defined the CHF for water, and a comprehensive correlation is available. Pressure drop is emphasized, not just for design of cooling systems, but also because it defines a hydrodynamic CHF for multiple, parallel channels that is lower than the single-tube value. Variations in circumferential heat flux, particularly the contemporary problem of one-sided heating, are treated. Several recent studies of bulk boiling are discussed. 7. The early work on condensation still stands and provides design information for the use of twisted tapes, particularly in steam condensers and refrigerant condensers (where there are some new data). The application of twisted tapes to this service--and also convective vaporization--is hampered by the large pressure drop penalty. 8. A few modified and compound usages of twisted tapes have been considered in the literature, primarily in single-phase flows. The more promising modification is to use a short tape insert in the tube inlet (to generate swirl and then allow it to decay, thereby reducing the pressure drop). An extension of this strategy is to periodically or intermittently produce swirl by employing short, interspaced twisted-tape elements. Short tape inserts have also been used effectively in both subcooled and bulk boiling. Compound usage of twisted tapes is a promising new area of enhancement, which has received rather sparse and sporadic attention. This technique is particularly attractive where one form of en-

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255

hancement preexists. In rotating coolant channels of electrical machines, for example, the heat transfer improvement can be more than what could be achieved with either tube rotation or the twisted tape acting alone. There are at least four examples of compound techniques. 9. Finally, several performance evaluation criteria can be employed for the optimized usage of twisted tapes in different applications. For example, in single-phase flows, one figure of merit is to consider fixed pumping power to evaluate the increased heat load that can be sustained over an empty smooth tube case by using a twisted tape. The rigorous methodology for two-phase flows is somewhat complex because of the pressure gradient in the tube and often entails a complete "thermal system" rather than a "unit heat exchanger" approach. Several investigations have compared twisted tapes, other enhancement devices, and empty tubes on the basis of constant pumping power but assuming constant pressure in the tube. B. RECOMMENDATIONSFOR FUTURE WORK While the performance of twisted-tape inserts has been studied for over a century and there is a considerably large body of literature, there are several unresolved questions and challenging new applications. 1. In single-phase Newtonian fluid flows, where the use of twisted tapes in conventional heat exchange systems seems to be quite well established for both laminar and turbulent regimes, the entrance region behavior and nonuniform heating effects require further attention. In particular, establishing the entry length for fully developed swirl flow (as well as the swirl decay length) is essential for the optimized usage of short inlet inserts and interspaced inserts. Furthermore, most chemical, food, pharmaceutical, biochemical, and other such process fluid media have a non-Newtonian rheology, where significant benefits from the enhanced performance with twisted tapes can be realized, and very limited data and design information are currently available. 8 Another exciting and uncharted application is the miniaturization of twisted tapes for microchannel systems. 2. Fully developed subcooled boiling heat transfer should be studied to clarify the amount of enhancement (shift of the boiling curve to the left). The pressure effect on C H F - - a p p a r e n t lack of enhancement with 8Here we would especially like to acknowledge the fundamental work of the late Professor Thomas F. Irvine, Jr., on non-Newtonian flow and heat transfer in smooth channels, which provide the basic scientific framework for investigating the thermal-hydraulic behavior with twisted-tape inserts.

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water above 1.0 barmshould be clarified. Much work is still needed on nonuniform heating, particularly when the channel is in a massive block that is heated from one side. Attention should be directed to the conjugate problem that defines the onset of the critical condition and to define the outer wall heat flux that leads to overall CHF and burnout in such channels. Pressure-drop data for such channels do not appear to exist. Finally, the application of performance evaluation criteria to two-phase systems should be accelerated. 3. Fouling of twisted tapes has received too little attention. It is known that particulate fouling is reduced with twisted tapes [172]. For example, twisted tapes minimized settling, plugging, and erosion in nitrogen-graphite suspensions [173], and twisted tapes are proposed to reduce fouling and enhance heat transfer in gas-solids suspensions with fixed-size particles [174]. In effect, these are compound techniques. 4. The future of twisted tapes may lie with compound enhancement. Ways should be sought to define and exploit the conditions under which several enhancement techniques can increase the heat transfer performance above that for each technique operating alone. Some really high heat transfer coefficients are waiting to be exhibited.

Nomenclature A Ac Ac, o cp C*

CHF d

dh do f

fh

is g G Gr

heat transfer surface area flow cross section of free-flow area of the tube (= "rrd2/4 - ~d) empty tube cross-section area (-- "rrd2/4) specific heat of the fluid ratio of the minimum to maximum heat capacity rates in a two-fluid heat exchanger critical heat flux inside diameter of a circular tube hydraulic diameter (= 4Ac/Pw) tube outside diameter empty tube Fanning friction factor Fanning friction factor based on actual mean-flow axial velocity and hydraulic diameter swirl-flow Fanning friction factor, Eq. (20) gravitational acceleration mass flux (= th/Ac) Grashof number (= gp2d3~ATw/p~2)

Gz h H

Ja k

Kc L

Ls rh

N Nu Num Nuz P

Pw

Ap Pr

q

q"

Graetz number (= &cp/kL) heat transfer coefficient pitch for a 180~ tape twist dimensionless tube rotation speed (Fig. 36) fluid thermal conductivity effective helical wall curvature of the tape-partitioned tube, Eq. (4d) axial length of tube or tape effective swirl flow length, Eq. (18) mass flow rate number of tubes Nusselt number (= hd/k) length-averaged mean Nusselt number axially local Nusselt number pressure; pumping power, Eq. (43) wetted perimeter pressure drop Prandtl number (= ~cp/k) heat transfer rate heat flux (= q/A)

SWIRL FLOW HEAT TRANSFER

q~r Ra Re

Rea Reh Res

Sw

Tb Tw AT UHF UWT

critical heat flux Rayleigh number (= G r - P r ) Reynolds number based on tube inside diameter and "empty" tube flow (= God/Ix) Reynolds number based on the actual mean flow axial velocity (= Gd/Ix) Reynolds number based on the hydraulic diameter Reynolds number based on the swirl velocity (= p Vsd/Ix) dimensionless swirl parameter (= Res/v/y), Eq. (19) bulk fluid temperature tube wall temperature temperature difference uniform or constant axial wall heat flux condition on the tube wall uniform or constant wall temperature condition on the tube wall

Va Vo Vs Vt w x y z a 13

Ix v p

257

mean axial flow velocity (= &/pAc) empty tube flow velocity (= &/pAc, o) swirl velocity, Eq. (17) tangential component of swirl velocity, Fig. 2 width of twisted tape local axial length of tube; vapor quality in liquid-vapor system, Fig. 28 dimensionless twist ratio (= H/d) dimensionless axial length (= ar/4Gz) helix angle for tape twist, Fig. 2b coefficient of isobaric thermal expansion thickness of twisted-tape insert dynamic viscosity of fluid kinematic viscosity fluid density tape thickness correction parameter, Eq. (2b)

Subscripts a avg b cp e

fo h i & 0 S

sat sb sub vp w

pertaining to the augmented or enhanced performance average at bulk fluid temperature with constant fluid properties at the exit of the tube fluid only or single-phase condition, Eq. (41) based on the hydraulic diameter at tube inlet inner surface of tube; tube inlet pertaining to the smooth empty tube performance; reference condition in Eq. (40) pertaining to swirl flow conditions at saturation temperature for subcooled boiling condition, Eq. (41) for subcooled fluid with variable property (viscosity) effects in flow cross section at tube-wall temperature

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103. DuPlessis, J. P. (1982). "Laminar Flow and Heat Transfer in a Smooth Tube with a Twisted-Tape Insert." Ph.D. Dissertation, University of Stellenbosch, South Africa. 104. Lecjaks, Z., Macha6, I., and Sir, J. (1984). Druckverlust bei der str6mung einer fltissigkeit durch ein rohr mit schraubeneinbauten. Chem. Engin. Process. 18, 67-72 (in German); English translation republished as Pressure loss in fluids flowing in pipes equipped with helical screws. Int. Chem. Engin. 27, 205-209, 1987. 105. Migai, V. K. (1966). Friction and heat transfer in twisted flow inside a tube. Izuvestiia Akademii Nauk S S S R Energetika I Transp. 5, 143-151. [in Russian] 106. Nazmeev, V. G., and Nikolaev, N. A. (1980). Correlation of experimental data on heat transfer in tubes with twisted tape turbulence promoters. Therm Engin. 27(3), 151-152. 107. Marner, W. J., Bergles, A. E., and Chenoweth, J. M. (1983). On the presentation of performance data for enhanced tubes used in shell-and-tube heat exchangers. J. Heat Transfer 105, 358-365. 108. Gutstein, M. U., Converse, G. L., and Peterson, J. R. (1970). "Theoretical Analysis and Measurement of Single-Phase Pressure Losses and Heat Transfer for Helical Flow in a Tube." Technical Note D-6097, National Aeronautics and Space Administration, Washington, DC. " 109. Migay, V. K. (1990). Hydraulic drag and heat transfer in tubes with internal twisted-tape swirl generators: Case of single-phase incompressible flow. Heat Transfer Soviet Res. 22(6), 824-830. 110. Lokanath, M. S. (1992). "A Study of the Effect of Partial and Full Length Twisted Tape Inserts on Laminar Flow Heat Transfer in Tubes." Ph.D. Thesis, University of Poona, India. 111. Agarwal, S. K., and Raja Rao, M. (1996). Heat transfer augmentation for the flow of a viscous liquid in circular tubes using twisted-tape inserts. Int. J. Heat Mass Transfer 39, 3547-3557. 112. A1-Fahed, S., Chamra, L. M., and Chakroun, W. (1999). Pressure drop and heat transfer comparison for both microfin tube and twisted-tape inserts in laminar flow. Exp. Therm. Fluid Sci. 18, 323-333. 113. Blasius, H. (1913). Das ~ihnlichkeitsgesetz bei reibungsvorg~ingen in fltissigkeiten. Forsch. Arb. Ing.- Wes., No. 131, Berlin. 114. McAdams, W. H. (1954). "Heat Transmission," 3ra Ed. McGraw-Hill, New York. 115. Harms, T. M., Jog, M. A., and Manglik, R. M. (1998). Effects of temperature-dependent viscosity variations and boundary conditions on fully developed laminar forced convection in a semicircular duct. J. Heat Transfer 120, 600-605. 116. Thorsen, R. S., and Landis, F. (1968). Friction and heat transfer characteristics in turbulent swirl flow subjected to large transverse temperature gradients. J. Heat Transfer 90, 87-98. 117. Dri~ius, M.-R. M., Shkema, R. K., and Shlan6iauskas, A. A. (1980). Heat transfer in a twisted stream of water in a tube. Int. Chem. Engin. 20, 486-489. 118. Donevski, B., Plocek, M., Kulesza, J., and Sasic, M. (1990). Analysis of tubeside laminar and turbulent flow heat transfer with twisted tape inserts. In "Heat Transfer Enhancement and Energy Conservation" (S.-J. Deng et al., eds.), pp. 175-185. Hemisphere, New York. 119. Klaczak, A. (1971). Analysis of the operation of exchanger tubes with internal axial helical screw generator of swirl. Rev. Gene. Therm. 10(112), 341-351. 120. Klaczak, A. (1973). Heat transfer in tubes with spiral and helical turbulators. J. Heat Transfer 95, 557-558. 121. Fryer, P. J., and Whalley, P. B. (1982). The effect of swirl on the liquid distribution in annular two-phase flow. Int. J. Multiphase Flow 8, 285-289. 122. Feinstein, L., and Lundberg, R. E. (1963). "Study Of Advanced Techniques For Cooling Very High Power Microwave Tubes." Report No. RADC-TDR-63-242, Rome Air Development Center.

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123. Lopina, R. F., and Bergles, A. E. (1973). Subcooled boiling of water in tape generated swirl flow. J. Heat Transfer 95, 281-283. 124. Vandervort, C. L., Bergles, A. E., and Jensen, M. K. (1994). An experimental study of critical heat flux in high heat flux subcooled boiling. Int. J. Heat Mass Transfer 37(Suppl. 1), 161-173. 125. GambiU, W. R. (1963). Generalized prediction of burnout heat flux for flowing, subcooled, wetting liquids. Chem. Engin. Progr. Symp. Ser. 59(41), 71-87. 126. Dri~ius, M.-R., Shkema, R. K., and Shlan~iauskas, A. A. (1978). Boiling crisis in swirled flow of water in pipes. Heat Transfer Soviet Res. 10(4), 1-7. 127. Inasaka, F., Nariai, H., Fujisaki, W., and Ishiguro, H. (1991). Critical heat flux of subcooled flow boiling in tubes with internal twisted tape. In "Proceedings of the ASME/JSME Thermal Engineering Joint Conference," Vol. 2, pp. 65-70. ASME, New York. 128. Nariai, H., Inasaka, F., Fujisaka, W., and Ishiguro, H. (1991). Critical heat flux of subcooled flow boiling in tubes with internal twisted tapes. In "Proceedings of the ANS Winter Meeting," pp. 38-46. ANS, New York. 129. Inasaka, F., and Nariai, H. (1992). Evaluation of subcooled critical heat flux correlations for tubes with and without internal twisted tapes. In "Proceedings of the 5th International Topical Meeting on Reactor Thermal Hydraulics," Vol. 4, pp. 919-928. 130. Gaspari, G. P., and Cattadori, G. (1994). Subcooled flow boiling in tubes with and without turbulence promoters. Exp. Therm. Fluid Sci. 8, 28-34. 131. Cardella, A., Celata, G. P., Dell'Orco, G., Gaspari, G. P., Cattadori, G., and Mariani, A. (1992). Thermal hydraulic experiments for the NET divertor. In "Proceedings of the 17th Symposium on Fusion Technology," Vol. 1, pp. 206-210. Rome, Italy. 132. Tong, L. S., and Tang, Y. S. (1997). "Boiling Heat Transfer and Two Phase Flow," pp. 457-480. Taylor & Francis, Washington, DC. 133. Pabisz, R. A., Jr., and Bergles, A. E. (1997). Using presure drop to predict the critical heat flux in multiple tube, subcooed boiling systems. In "Experimental Heat Transfer, Fluid Mechanics and Thermodynamics 1997" (M. Giot et al., eds.) pp. 851- 858. Edizioni ETS, Pisa, Italy. 134. Nariai, H., Inasaka, F., Ishikawa, A., and Kinoshita, H. (1994). Effect of internal twisted tape on critical heat flux of subcooled flow boiling under nonuniform heating condition. Transactions of the JSME 60, 4215-4221. 135. Kinoshita, K., Yoshida, T., Nariai, H., and Inasaka, F. (1996). Study on the mechanism of critical heat flux enhancement for subcooled flow boiling in a tube with internal twisted tape under nonuniform heating conditions. Heat Transfer Jpn. Res. 25, 293-307. 136. Nariai, H., Inasaka, F., Ishikawa, A., and Kinoshita, H. (1993-95). Critical heat flux of subcooled flow boiling in tube with internal twisted tape under non-uniform heating: Experimental results and discussion of mechanism. In "Proceedings of 30th Japan Heat Transfer Symposium," pp. 490-492. Heat Transfer Society of Japan, Tokyo, Japan. 137. Suzuki, S., and Kumugai, S. (1996). Transient behavior of subcooled forced convective boiling with twisted tape inserts. Heat Transfer Jpn. Res. 25, 178-191. 138. Araki, M., Ogawa, M., Kunugi, T., Satoh, K., and Suzuki, S. (1996). Experiments on heat transfer of smooth and swirl tubes under one-sided heating conditions. Int. J. Heat Mass Transfer 39, 3045-3055. 139. Boscary, J., Fabre, J., and Schlosser, J. (1999). Critical heat flux of water subcooled flow in one-side heated swirl tubes. Int. J. Heat Mass Transfer 42, 287-301. 140. Ornatskii, A. P., and Vinyarskii, L. S. (1965). Critical heat transfer in the forced motion of unheated water-alcohol mixtures in tubes of diameter 0.5 mm. High Temp. 3, 881-882. 141. Ornatskii, A. P., and Vinyarskii, L. S. (1965). Heat transfer crisis in a forced flow of underheated water in small-bore tubes. High Temp. 3, 400-405.

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265

142. Bergles, A. E., (1969). Survey and evaluation of techniques to augment convective heat and mass transfer. In "Progress in Heat and Mass Transfer," Vol. I, pp. 331--424. Pergamon Press, Oxford. 143. Matzner, B., Casterline, J. E., Moek, E. O., and Wikhammer, G. A. (1965). Critical heat flux in long tubes at 1000 psi. A S M E Paper No. 65-WA/HT-30, ASME, New York. 144. Kaka~, S. (1999). The effect of augmented surfaces on two-phase flow instabilities. In "Heat Transfer Enhancement of Heat Exchangers" (S. Kakag et al., eds.), pp. 447-465. Kluwer Academic, Dordrecht, The Netherlands. 145. Kedzierski, M. A., and Kim, M. S. (1997). "Convective Boiling and Condensation Heat Transfer with a Twisted-Tape Insert for R12, R22, R152a, R134a, R290, R32/R134a, R32/R152a, R290/R134a. R134a/R600a." Report NISTIR 5905, National Institute of Standards and Technology, Gaithersburg, MD. 146. Doeffler, S., Groeneveld, D. C., and Schenk, J. R. (1996). Experimental study of the effects of flow inserts on heat transfer and critical heat flux. In "Proceedings of the 4th International Conference on Nuclear Engineerng (ICONE-4)," Vol. 1, Part A, pp. 41-49. New Orleans, LA. 147. Groeneveld, D. C., Doeffler, S., Tain, R. M., Hammouda, N., and Cheng, S. C. (1997). Fluid-to-fluid modelling of the critical heat flux and post-dryout heat transfer. In "Experimental Heat Transfer, Fluid Mechanics and Thermodynamics 1997" (M. Giot et al., eds.), Vol. 2, pp. 859-866. Edizioni ETS, Pisa, Italy. 148. Carey, V. P., and Shah, R. K. (1988). Design of compact and enhanced heat exchangers for liquid-vapor phase-change applications. In "Two-Phase Flow Heat Exchangers," pp. 909968. Kluwer Academic, Dordrecht, The Netherlands. 149. Royal, J. H., and Bergles, A. E. (1975). "Augmentation of Horizontal In-Tube Condensation of Steam." Heat Transfer Laboratory Report No. HTL-9, Iowa State University, Ames, IA. 150. Royal, J. H., and Bergles, A. E. (1976). Experimental study of the augmentation of horizontal in-tube condensation. A S H R A E Trans. 82, (Part 1), 919-931. 151. Helmer, W. A., and Iqbal, I. (1980). Laminar heat transfer in a circular tube with twisted tapes during condensation. A S H R A E Trans. 86, (Part 2), 662-674. 152. Luu, M., and Bergles, A. E. (1979). "Augmentation of Horizontal In-Tube Condensation of R-113." Heat Transfer Laboratory Report No. HTL-23, Iowa State University, Ames, IA. 153. Azer, N. Z., and Said, S. A. (1981). "Augmentation of Condensation Heat Transfer by Internally Finned Tubes and Twisted Tapes." Final Technical Report, Department of Mechanical Engineering, Kansas State University. 154. Said, S. A., and Azer, N. Z. (1983). Heat transfer and pressure drop during condensation inside horizontal tubes with twisted-tape inserts. A S H R A E Trans. 89, (Part 1), 96-113. 155. Date, A. W., and Saha, S. K. (1990). Numerical prediction of laminar flow and heat transfer characteristics in a tube fitted with regularly spaced twisted-tape elements. Int. J. Heat Fluid Flow 11(4), 346-354. 156. Penney, W. R. (1965). The spiralator: Initial tests and correlations. AIChE Preprint No. 16, 8th National Heat Transfer Conference, Los Angeles, CA. 157. Mayinger, F., Schad, O., and Weiss, E. (1966). "Investigations into The Critical Heat Flux In Boiling." Mannesmann Augsburg Niirnberg Final Report No. 09.03.01, MAN, Niirnberg, Germany. 158. Berenson, P. J. (1965). An experimental investigation of flow stability in multitube forcedconvection vaporizers. ASME Paper No. 65-HT-61, ASME, New York. 159. Gambill, W. R., and Greene, N. D. (1958). A preliminary study of boiling burnout heat fluxes for water in vortex flow. Chem. Engin. Prog. 54(10), 68-76.

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160. Weeds, J., and Dhir, V. K. (1983). Experimental study of the enhancement of critical heat flux using tangential flow injection. In "Fundamentals of Heat Transfer in Fusion Energy Systems," HTD-Vol. 24, pp. 11-18. ASME, New York. 161. Van Rooyen, R. S., and Kr6ger, D. G. (1978). Laminar flow heat transfer in internally finned tubes with twisted-tape inserts. In "Heat Transfer 1978," Vol. 2, pp. 577-581, Hemisphere, Washington, DC. 162. Zhou, Q. T., Ye, S. Y., Ou-Yang, Y. H., and Gu, N. Z. (1990). Combined enhancement of tube-side heat transfer in cast-iron air preheater. In "Heat Transfer 1990," Vol. 4, pp. 39-44. Hemisphere, New York. 163. Muralidhar Rao, M., and Sastri, V. M. K. (1995). Experimental investigation of fluid flow and heat transfer in a rotating tube with twisted-tape inserts. Heat Transfer Engin. 16(2), 19-28. 164. Boyd, R. D., Smith, A., and Turknett, J. C. (1995), Two-dimensional wall temperature measurements and heat transfer enhancement for top-heated horizontal channels with flow boiling. Exp. Therm. Fluid Sr 11, 372-386. 165. Pabisz, R. A., Jr., and Bergles, A. E. (1996). "Enhancement of Critical Heat Flux in Subcooled Flow Boiling Using Alcohol Additives and Twisted-Tape Inserts." Heat Transfer Laboratory Report No. HTL-25, Rensselaer Polytechnic Institute, Troy, NY. 166. Bergles, A. E., Blumenkrantz, A. R., and Taborek, J. (1974). Performance evaluation criteria for enhanced heat transfer. In "Heat Transfer 1974, Proceedings of the 5th International Heat Transfer Conference," Vol. II, pp. 234-238. The Japan Society of Mechanical Engineers, Tokyo, Japan. 167. Bergles, A. E., Junkhan, G. H., and Bunn, R. L. (1974). Extended performance evaluation criteria for enhanced heat transfer surfaces. Lett. Heat Mass Transfer 1, 113-119. 168. Nelson, R. M., and Bergles, A. E. (1986). Performance evaluation for tubeside heat transfer enhancement of a flooded evaporator water chiller. A S H R A E Trans. 92, (Part 1B), 739-755. 169. Zimparov, V. (2000). Enhancement of heat transfer by a combination of a single start spirally corrugated tubes with a twisted tape. In "Thermal Sciences 2000, Proceedings of the ASME-ZSITS International Thermal Science Seminar," pp. 395-401. ZSITS, Ljubljana, Slovenia. 170. Megerlin, F. E., Murphy, R. W., and Bergles, A. E. (1974). Augmentation of heat transfer in tubes by use of mesh and brush inserts. J. Heat Transfer 96, 145-151. 171. Webb, R. L. (1988). Performance evaluation criteria for enhanced tube geometries used in two-phase heat exchangers. In "Heat Transfer Equipment Design" (R. K. Shah et al., eds.), pp. 697-704. Hemisphere, New York. 172. Somerscales, E. F. C., and Bergles, A. E. (1997). Enhancement of heat transfer and fouling mitigation. Adv. Heat Transfer 30, 197-253. 173. Anonymous (1960). "Gas-suspension Task III Final Report". Babcock and Wilcox Reports 1201 and 1207. 174. Schmidt, W. A. (1933). "Heat Interchanging Apparatus." U. S. Patent No. 1,916,337.

AUTHOR INDEX

Numerals in parentheses following the page numbers refer to reference numbers citied in the text ASTM B388-96, 121(34) ASTM B389-81,121(29) ASTM B478-85, 121(30) ASTM B753-86, 121(31) ASTM E1137, 126(59) ASTM PEN 28-012093-40, 118(4), 124(4) ASTM Vol. 14.03 E1-95, 120(16) ASTM Vol. 14.03 E77-92, 120(17) Astrov, D. A., 123(42) Astrov, D. N., 123(48) Attal-Tretout, B., 160(258) Auld, B. A., 164(291) Azer, N. Z., 246(153; 154) Aziz, R. A., 123(36)

A Actis, A., 133(94) Adriaans, M. J., 135(117) Agarwal, K. N., 194(67), 251(67) Agarwal, S. K., 219(111) Agatsuma, K., 135(121; 123) Akai, M., 196(84) Alaruri, S., 140(149; 150) Alessandretti, G. C., 160(264) A1-Fahed, S., 219(112) Algifri, A. H., 195(74), 247(74) Alifanov, O. M., 18(3; 4) Allison, S. W., 138(139; 141), 140(146) Alnot, P., 160(268) Ambrosini, D., 164(293) American Institute of Physics, 165(312) Amtmann, K., 158(252) Anderson, T. J., 160(265) Anderssen, R. S., 53(56), 106(56) Andresen, P., 162(280) Andrews, M. J., 184(2) Annen, K. D., 157(243) Anno, M., 190(16), 192(16) Anonymous, 256(173) ANSI/ASTM B388-96, 136(126) Antcliff, R. R., 160(261; 269) Araki, M., 242(138) Aranda, P., 134(108) Arenz, R. W., 134(99) Armstrong, R. D., 196(83), 202(83), 203(83), 221(83), 230(83) Armstrong, R. L., 158(255) Artyukhin, E. A., 18(4) AS 2190, 120(22) Asanuma, T., 157(245) Asawaroengchai, C., 158(247) Aselage, T. L., 135(117) Assmann, C., 133(97) Astill, A. G., 161(270) ASTM B106-96, 121(33) ASTM B223-97, 121(32)

Bak, J., 155(216) Ballico, M., 133(95), 139(95), 140(95), 163(95) Bandyopadhyay, P. S., 193(54), 228(54) Barat, R. B., 157(238) Barlow, R. S., 158(254) Barow, R., 147(199) Bar-Ziv, E., 157(238) Baughn, J. W., 140(158) Bazan, C., 134(102) Beakley, W. R., 130(82) Beck, J. V., 26(13), 27(13) Becker, J. A., 130(80) Beckermann, C., 196(80) Beerman, H. P., 136(128) Beliansky, L. B., 123(42) Bell, J., 138(133) Ben-Amotz, D., 158(250) Benjaminson, A., 135(122) Benne, M. E., 138(134) Bensler, H. P., 192(38), 194(38) Bentley, R. E., 157(237), 165(305; 306; 307) Berenson, P. J., 249(158) Berg, G. P., 134(104) 267

268

AUTHOR INDEX

Bergles, A. E., I84(3). 185(3; 6: 7). 186(3; 6: 7). 188(3; 11; 201, 190(3; 7; 11: 15: 18; 20; 21: 22; 23), 191(6; 7; 11; 27; 30). 192(20: 21; 34; 36; 52), 193(11; 18; 20; 21; 22: 23: 53). 19q3; 18: 64; 65). 195(69; 70; 71 ; 72), 196(7; 27: 34; 83), 197(18; 90; 921, 198(18; 22; 53; 6 4 94). 199(11; 90), 200(22; 52; 53; 641, 201(90), 202(11; 52: 83; 90), 203(11; 21: 27: 65; 83: 90), 204(20; 21: 34; 65; 90). 205(11; 65). 206(34; 102), 209(102), 21 l(34). 21 3(107), 216(21: 521, 218(11), 219(21; 22; 34: 90), 220(2l; 52). 221(21: 22; 34; 36; 52: 83; 901, 222(3; 1 1; 18: 21; 22; 52; 94). 223(18; 20; 94): 224(52j. 226(20; 21; 22; 53). 227(18; 20; 22; 53: 90), 228(18; 20; 53; 90), 229(18; 21), 230(27; 83; 90). 232(133; 123). 233(123), 234(124). 235(36), 236(36). 237(36; 133), 238(36; 133), 239(133), 240(133), 243(6: 36; 124; 142). 24H69; 70; 71: 149; 150; 152). 247(11). 249(30 165). 251(3: 11: 166; 167; 168). 253(70 170). 256(172) Berry, K. H., 123(37) Berry, R. J., 128(63; 64; 65) Beshears, D. L.. 138(139), 140(146; 147: 148: 155) Besley, L. M., 123(38) Bestwick, T. D.. 162(284) Beushausen. V.. 158(257) Bhalla, A. S.. 158(253) Bhardwaj, R. K., 195(74). 247(74) Bhatnagar, R. K., 185(9) Bird. C., 137(132) Bizzak. D.. 140(152) Blackwell, B., 26(13), 27(13) Blalock, T. V., 133(93). 134(93) Blasius, H., 219(113) Blumenkrantz. A. R.,251(166) Bockle. S.. 157(240), 162(240) Bolcs, A,, 142(168) Bolla, G., 202(100), 221(100), 230(100) Bonczyk, P. A., 155(2I9) Bonnier. G., 134(109) Bonsett, T., 140(150) Borella, H. M., 140(147) Boscary. J., 242(139), 243(139) Bosch, W., 134(109) Bouchardy, P., 160(258) Boudouard, 0.. 127(61)

Boutard. D , 134(110) Boutard-Gablllet. D , 134(108) Boj d, R D . 249(164) Bradley. L C , 138(143) Brandt. B. L , 130(78), 165(78) Bratfalean, R , 161(272) Breton, Y , 155(218) Brew R , 190(16). 192(16) Brewington. A , 140(149. 150) Brewster. M Q 27(23) Brixy, H , 133(94) Brown, M S , 161(271) Brown. R , 147(196) BS 593, 120(12) BS 692. I20(13) BS 1041. 119(5), 120(5) BS 1041 Part 3. 126(58) BS 1041 Part 7. 136(130) BS 1365, 120(14) BS EN 60751. 126(60) BS I S 0 4795, 120(15) Bugos, A . 140(155) Bundy. R D 192(50, 51), 194(68). 215(50), ZlO(50). 226(5 11, 232(68), 233(68). 234(51, 68). 237(68). 252(68j Bunn. R L . 251(167) Burfoot. D 191(28). 195(28), 247(28) Burlbaw, E J . 158(255) Busch, P A 135(114, 1 15)

.

.

.

.

.

Cadars, P 195(78) Callendar, H L , 125(55, 56, 57) Callls, J 138(133) Camc~.C , 141(161). 142(162) Campbell, R P , 142(166) Capps, G J . 140(146. 147. 155) Cardella, A , 235(13 l), 240(131) Carey, V P , 246( 148) Carley. J S . 123(36) Carlson. D J , 155(226) Carne\ale, E H 163(287.288) Caroll, B F . 138(136) Carter, C D , 157(239) Casterline, J E , 243(143), 248(143), 252(143) Cates. M R , 138(139). 140(146. 147. 155) Cattadori, G., 235(130, 131). 237(130), 240(131) Cattafesta, L N 138(135)

.

.

.

AUTHOR INDEX Celata, G. P., 235(131), 240(131) Cetas, T. C., 135(111; 120) Chakroun, W., 219(112) Chamra, L. M., 219(112) Chan, C., 162(278) Chen, C.-K., 27(15) Chen, J., 147(199) Chen, P. H., 151(209), 153(209), 154(209) Chen, Z. R., 155(215) Chenevier, M., 155(218; 221) Cheng, S. C., 245(147) Chenoweth, J. M., 213(107) Cheskis, S., 155(221) Chetwynd, J. H., 145(181) Chew, J. W., 142(169) Chiang, H. G., 151(209), 153(209), 154(209) Childs, P. R. N., 151(208), 165(208; 303) ChingPrado, E., 158(253) Chua, S. J., 158(248) Chung, O. Y., 27(16) Chyu, M., 140(152) Claggett, T. J., 124(53) Clark, C. F., 134(99) Clausen, S., 155(216) Clement, J. R., 130(76) Cohen, E. R., 143(171) Colantonio, A., 147(202; 203) Colburn, A. P., 192(41) Colwell, J. H., 133(96) Connolly, J. J., 129(72) Converse, G. L., 218(108) Corana, A., 26( 11), 33(11) Cordero, J., 158(253) Corwin, R. R., 144(172) Counterman, W. S., 196(82) Crafton, B. T., 138(137) Crites, R. C., 138(134) Cubertafon, J. C., 155(218) Cui, J. B., 158(252) Curry, B. P., 27(23) Cutler, A. D., 160(261; 269) Cyr, M., 140(146) Czysz, P., 140(144; 145)

D Daily, J. W., 162(278) Darabi, J., 186(10) Dasmahapatra, J. K., 190(24), 193(24), 248(24)

269

Date, A. W., 191(29), 193(55), 194(59; 60), 195(29; 76), 214(59), 217(59), 219(29), 221(29), 228(55), 248(29; 76; 155), 251(29) Davidson, D. M., 196(79) Day, P. K., 135(117) Daykin, C. I., 129(73) Debarber, P. A., 161(271) Dec, J. E., 162(274) Dedikov, Y. A., 123(42) De Giorgio, G., 202(100), 221 (100), 230(100) de Groot, M. J., 133(94), 134(109) Dell-Orco, G., 235(131), 240(131) Deming, C., 128(66) Dessiatoun, S. V., 186(10) Dever, M., 140(155) DeWitt, D. P., 144(174; 175; 176) Dhal, S. K., 196(88), 248(88) Dhanak, V. R., 155(222) Dhir, V. K., 249(160) Dibble, R. W., 158(254) Dils, R. R., 147(186) DIN 1715, 121(28) Diskin, G. S., 160(261) Dittman, R. H., 111(1) Dixon, W. P., 140(144; 145) Dobbs, G. M., 160(265; 267) Dodrill, B. C., 132(88) Doeffler, S., 245(146; 147) Doi, J., 157(245) Donevski, B., 190(19), 192(19), 206(101), 208(101), 213(19), 214(19), 216(19; 101), 217(19), 220(19), 221(19), 225(118) Donnelly, V. M., 155(217) Donovan, J. F., 138(134) Drake, M. C., 158(247) Drapcho, D. L., 158(247) Drizius, M.-R. M., 224(117), 235(126), 237(126) Drung, D., 133(97) Dubbledam, J., 133(94) Duncan, R. V., 135(117) Dunn, W. L., 27(17) DuPlessis, J. P., 194(61; 62), 210(103), 213(61), 215(61), 216(61), 217(61), 222(62), 223(62), 226(61; 62) Durieux, M., 123(39; 40; 41), 135(118) Dutta, A., 196(87; 88), 247(87), 248(87; 88) Dyer, F., 140(155)

270

AUTHOR INDEX

E East, R. A., 141(160) Eaton, J. K., 140(159), 142(159) Eckbreth, A. C., 160(259; 265; 267) Edge, A. C., 140(153) Edsinger, R. E., 123(45) Edwards, G. J., 158(246) Eid, A., 29(45), 30(45) Elder, F., 133(94) Elliott, G. S., 157(239) Erf, R. K., 164(299) Erturk, H., 28(38) Ervin, J., 140(152) Etemad, S. Gh., 193(58) Evans, S. I., 192(42) Ewan, B. C. R., 147(189) Ewart, P., 161(272) Ezekoye, O. A., 28(38), 30(49; 51)

Fabre, J., 242(139), 243(139) Fairchild, C. O., 136(127) Farina, D. J., 140(159), 142(159) Farmer, A. J. D., 157(231; 232) Farrell, P. V., 164(294) Farrow, R. L., 160(262) Fausett, L., 26(8) Feinstein, L., 232(122), 243(122) Feist, J. P., 140(151) Fellmuth, B., 123(43), 134(109) Feng, Z. C., 158(248) Ferrier, 26(10), 33(10) Filliben, J. J., 130(81) Fletcher, L. S., 184(2) Fogle, W. E., 133(96) Fonger, W. H., 138(142) Forbes, A., 134(109), 163(289) Forkey, J. N., 157(234) Fran9a, F., 28(41), 29(46), 30(46; 48; 49; 51), 31(53; 54) Frank, R., 132(86) Friedman, J. N., 55(55) Fryer, P. J., 232(121) Fujisaka, W., 235(128) Fujisaki, W., 235(127), 237(127) Fujita, Y., 196(86), 200(86), 202(86), 221(86) Furukawa, G. T., 112(3), 123(3), 124(3), 128(3)

G Gaitonde, U. N., 191(29), 193(54), 195(29; 76), 219(29), 221(29), 228(54), 248(29; 76), 251(29) Galleano, R., 133(94) Gallery, J., 138(133) Gallop, J. C., 133(94; 98) Gambill, W. R., 190(17), 192(50; 51), 194(68), 215(50), 220(50), 226(51), 232(68), 233(68), 234(51; 68; 125), 235(125), 237(68), 249(159), 252(68) Gaspari, G. P., 235(130; 131), 237(130), 240(131) Gaydon, A. G., 155(225) Gershanik, A. P., 123(48) Gicquel, A., 155(218) Gillies, G. T., 138(141) Glikman, M. S., 123(48) Glumac, N., 157(239) Gnielinski, V., 203(98) Goffe, 26(10), 33(10) Goldschmidt, V. W., 196(80) Goldstein, L., 28(41) Goldstein, R. J., 151(209), 153(209), 154(209) Golubev, L. K., 195(73), 247(73) Gomez, A., 158(25I) Goodfellow Catalogue, 121(27) Goto, K., 156(229) Gouterman, M., 138(133) Grattan, K. T. V., 147(190; 191) Green, C. B., 130(80) Greene, N. D., 249(159) Greenhalgh, D. A., 160(263) Greenwood, J. R., 165(303) Greywall, D. S., 135(114; 115) Grill, L., 155(222) Groeneveld, D. C., 245(146; 147) Grohmann, K., 123(43) Grunefeld, G., 158(257) Gu, N. Z., 249(162) Guildner, L. A., 123(45) Guille, N., 138(137) Gupta, J., 154(213) Gutstein, M. U., 218(108)

H Hacker, J. M., 140(159), 142(159) Haddad, G. N., 157(231)

AUTHOR INDEX Hahn, J. W., 161(273) Hall, J. A., 119(7) Hall, R. J., 155(219) Hammouda, N., 245(147) Hamner, M., 138(138) Hansen, G. P., 162(285) Hansen, P. C., 10(1), 16(1), 20(5), 22(5) Hanson, R. K., 162(276; 277) Hanuza, J., 158(249) Hao, L., 133(98) Harasgama, S. P., 142(168) Harms, T. M., 222(115) Hart, S. R., 130(85) Hartl, J. C., 147(187) Harutunian, V., 28(39; 40) Hashemian, H. M., 129(69) Hassouni, K., 155(218) Hauge, R. H., 162(285) Hay, J. L., 142(163) Head, D. I., 134(109), 135(116) Hechtfischer, D., 134(109) Helmer, W. A., 246(151) Hendricks, T. J., 27(12) Hentschel, W., 162(280) Heping, T., 27(32) Herlin, N., 160(268) Hermier, Y., 134(109) Hernicz, R. S., 144(176) Herring, G. C., 161(271) Herzfeld, C. H., 165(308) Heyes, A. L., 140(151) Hippensteele, S. A., 142(162) Hofeldt, D. L., 164(294) Hoffman, D., 157(241) Hoffmann, A., 134(109) Hoffs, A., 142(168) Holden, M. S., 138(136) Hollingsworth, D. K., 142(163) Hoist, G. C., 147(205), 149(205), 150(205; 207) Hong, S. W., 190(18), 193(18), 194(18; 63), 197(18; 92), 198(18; 63), 222(18), 223(18), 227(18; 63), 228(18), 229(18) Honma, H., 154(210) Horton, J. F., 157(242) Howell, J. R., 27(12), 28(7; 38; 39; 40; 43), 29(46), 30(46; 48; 49; 50; 51), 31(52; 53; 54) Hsu, P.-T., 27(15) Huang, C. H., 27(14) Hubner, J. P., 138(136) Hughes, I. G., 161(272)

271

Hurley, T. L., 147(194) Huston, W. D., 121(35)

Ibragimov, M. M., 192(49), 202(49), 203(49), 215(49), 220(49), 224(49), 226(49) Ichimaru, H., 156(229) IEC-751, 129(70) Iinuma, K., 157(245) Inasaka, F., 235(127; 128; 129), 237(127), 240(134), 241(135; 136), 243(134) Incropera, F., 27(18) Inoue, A., 192(35), 236(35), 244(35) Iqbal, I., 246(151) Ircon Inc., 146(185) Ireland, P. T., 140(157), 142(165; 169) Ishigami, S., 135(121; 123) Ishiguro, H., 235(127; 128), 237(127) Ishikawa, A., 240(134), 241(136), 243(134) Ishikawa, T., 200(96), 204(96), 205(96), 219(96) ISO 651, 120(18) ISO 653, 120(19) ISO 654, 120(20) ISO 1770, 120(21) Iwashita, K., 191(32), 249(32)

J Jarrett, O., 160(261; 269) Jayaraj, D., 195(77), 248(77) Jensen, M. K., 185(6; 7; 8), 186(6; 7; 8), 190(7; 8), 191(6; 7; 8), 192(36; 37; 38), 193(8), 194(8; 38), 196(7), 221(36), 234(124), 235(36), 236(36), 237(36), 238(36), 243(6; 36; 124), 251(8) Ji, H. F., 138(136) Jinde, Z., 128(66) Jinroug, S., 128(66) Jochemsen, R., 134(109) Jog, M. A., 222(115) Johnson, J. B., 133(90) Joklik, R. G., 156(228) Jones, M. R., 27(23; 24), 29(47) Jones, T. V., 129(71), 140(157), 142(165) Joshi, S. D., 190(23), 193(23) Joule, J. P., 185(5) Ju, J. J., 161(273)

272

AUTHOR INDEX

Junkhan, G. H., 190(15), 191(27), 196(27), 203(27), 230(27), 251 (167)

K Kachanov, A., 155(221) Kaka~;, S., 244(144) Kamiya, T., 200(96), 204(96), 205(96), 219(96) Kamper, R. A., 133(92) Kaplan, H., 145(184), 150(184) Karpetis, A. N., 158(251) Katiyar, R. S., 158(253) Kato, S., 154(211) Kazenwadel, J., 157(240), 162(240) Kedzierski, M. A., 244(145), 246(145) Kelble, C. A., 138(138) Keller, J. O., 162(274) Kemnal, R. L., 190(14) Kemp, R. C., 123(38) Kemp, W. R. C., 123(38) Kempkens, H., 160(266) Kerlin, T. W., 124(51), 165(304) Khalil, G., 138(133) Khuriyakub, B. T., 164(290) Kidd, G. J., Jr., 202(99), 203(99), 225(99) Kido, A., 162(275) Kim, K., 142(162) Kim, M. S., 244(145), 246(145) King, W. J., 192(41) Kinosada, T., 162(283) Kinoshita, H., 240(134), 241 (136), 243(134) Kinoshita, K., 241 (135) Kirov, N. Y., 190(13), 192(13) Klaczak, A., 226(119; 120) Klemme, B. J., 135(117) Klepper, O. H., 190(25), 195(25), 247(25), 251(25) Kneizys, F. X., 145(181) Koch, A., 162(280) Koch, R., 192(43), 203(43) Kohler, S. T., 142(169) Kojima, S., 158(249) Kosugi, S., 154(210) Kourous, H. E., 147(193) Krause, J. K., 132(88) Krauss, R. H., 140(153) Kreith, F., 192(44) Krishnan, S., 162(285) Kroesen, G. M. W., 162(284)

Kr6ger, D. G., 194(61; 62), 213(61), 215(61), 216(61), 217(61), 222(62), 223(62), 226(61; 62), 249(161) Krohn, D. A., 147(192) Kubota, S., 162(275) Kudo, K., 27(26), 28(44), 29(45), 30(45) Kulesza, J., 190(19), 192(19), 206(101), 208(101), 213(19), 214(19), 216(19; 101), 217(19), 220(19), 221(19), 225(118) Kumugai, S., 241 (137) Kunugi, T., 242(138) Kunzelmann, T., 157(240), 162(240) Kuroda, A., 27(26), 29(45), 30(45) Kusama, H., 140(154) Kychakoff, G., 162(276)

L Lachendro, J., 138(137) Ladieu, F., 134(108; 110) Lallemand, M., 27(36) Lan, C., 29(46), 30(46; 50) Landis, F., 192(45), 202(45), 203(45), 206(45), 208(45), 215(45), 218(45), 221(45), 224(45; 116), 230(45) Landl, G., 53(56), 106(56) Laplant, F., 158(250) Laufer, G., 140(153) Laurence, G., 158(250) Lawless, W. N., 134(99; 103) Leaver, V. M., 119(7) Le Chatelier, H., 127(61) Lecjaks, Z., 215(104), 217(104) Lee, R. A., 191(30), 249(30) Lee, S. J., 142(170), 192(35), 236(35), 244(35) Lee, Y. J., 164(290) Lefebvre, M., 160(268) Lehtiniemi, R. K., 147(198; 201) Leipertz, A., 157(241), 160(260) Lempert, W. R., 157(234; 244) Leong, K. H., 27(23) Lewis, W., 140(147; 148) Ley, L., 158(252) Li, H. Y., 27(27; 30; 31) Li, H. Z., 155(215) Li, R. R., 134(104) Li, W. S., 158(248) Liberatore, L. C., 119(6) Lin, J.-D., 27(20) Linhua, L., 27(32)

AUTHOR INDEX Lipinski, L., 132(87) Liptak, B. G., 124(53), 165(302) Liu, L. H., 27(33; 34; 35) Liu, T., 138(135) Lloyd, G. M., 161(272) Lokanath, M. S., 219(110), 248(110) London, A. L., 197(93), 200(93), 213(93), 214(93), 217(93) Long, C. A., 165(303) Longwell, J. P., 157(238) Lopez, A. M., 196(86), 200(86), 202(86), 221(86) Lopina, R. F., 192(52), 200(52), 202(52), 206(102), 209(102), 216(52), 220(52), 221(52), 222(52), 224(52), 232(102; 123), 233(123) Lou, Y. H., 155(215) Lundberg, R. E., 232(122), 243(122) Luster, S. D., 147(193) Luther, H., 123(43) Luu, M., 195(71; 72), 246(71; 152) Lynnworth, L. C., 163(288)

M Maan, J. C., 134(100; 101; 105) MacArthur, C., 140(152) MacFarlane, J., 133(98) Machac, I., 215(104), 217(104) Maczka, M., 158(249) Madding, R. P., 144(173) Maeno, K., 154(210) Magens, E., 160(260) Magre, P., 160(258) Magunov, A. N., 162(282; 286) Maior, M. M., 134(100; 101; 105) Mallery, C. F., 155(223) Malyshev, M. V., 155(217) Manglik, R. M., 185(7; 9), 186(7), 188(11; 20), 190(7; 11; 20; 21; 23), 191(7; 11), 192(20; 21; 34; 47), 193(11; 20; 21; 23; 57), 194(47; 64; 65; 66), 196(7; 34; 66), 197(11; 90; 91), 198(64; 94), 199(11; 90), 200(64), 201 (90), 202(11; 90), 203(11; 21; 65; 90), 204(20; 21; 34; 65; 90), 205(11; 65), 206(20; 34; 47), 208(66), 209(47), 210(20; 66), 211(34), 215(20), 216(21), 217(20), 218(11), 219(20; 21; 34; 90), 220(20; 21), 221(21; 34; 90), 222(11; 20; 21; 94; 115), 223(20; 94),

273

226(20; 21), 227(20; 90), 228(20; 90), 229(21), 230(21; 90), 247(11), 251(11; 91), 252(91) Mangum, B. W., 112(3), 123(3), 124(3), 128(3; 68), 130(81) Maramraju, S., 192(34), 196(34), 204(34), 206(34), 211(34), 219(34), 221(34) Marchesi, M., 26(11), 33(11) Margolis, D., 192(44) Margrave, J. L., 162(285) Mariani, A., 235(131), 240(131) Marner, W. J., 190(22), 193(22; 53), 198(22; 53), 200(22; 53), 213(107), 219(22), 221(22), 222(22), 226(22; 53), 227(22; 53), 228(53) Martin, D. M., 155(224) Martini, C., 26(11), 33(11) Maruyama, N., 154(211) Masilamani, J. G., 195(77), 248(77) Masri, A. R., 158(254) Mast, D. B., 134(104) Matacotta, F. C., 123(39) Matsumura, M., 28(42; 43), 31(52) Mattern, P. L., 160(262) Matthews, L. K., 27(18) Matyjasik, S., 134(102) Matzner, B., 243(143), 248(143), 252(143) Mayinger, F., 248(157) McAdams, W. H., 219(114) McClatchey, R. A., 145(178; 179; 180; 181) McClean, I. P., 138(140) McConville, G. T., 123(36; 39; 46) McCormick, N. J., 27(25) McFarland, D., 140(149) McGee, T. D., 136(129), 165(129) McLachlan, B., 138(133) McPheeters, E., 140(150) Medvecz, P. J., 155(224) Megahed, M., 162(281) Megerlin, F. E., 253(170) Meier, U. E., 162(279) Meijer, G. C. M., 121(26), 162(280) Meininger, M., 142(167) Mengfic, M. P., 27(21) Menkel, S., 133(97) Metcalfe, E., 155(214) Metz, P. D., 128(62) Migai, V. K., 216(105), 226(105) Migay, V. K., 195(73), 218(109), 247(73) Mikic, B. B., 191(30), 249(30)

274

AUTHOR INDEX

Miles, R. B., 157(233; 234; 244) Mitev, C., 158(256) Miyamoto, N., 162(275) Mochizuki, T., 128(67) Moek, E. O., 243(143), 248(143), 252(143) Moffat, R. J., 140(159), 142(159; 164) Mohandas, P., 134(109), 135(116) Moiseeva, N. P., 128(68) Molezzi, M. J., 142(166) Molinar, G., 123(50) Molnar, S. B., 134(101) Monheit, M., 191(26), 195(26), 248(26) Morales, J. C., 26(9), 28(39; 40; 43), 29(6), 31(52; 53; 54) Mori, Y., 196(81; 84), 203(81), 213(81), 214(81), 216(81), 217(81), 220(81), 221(81), 222(81), 223(81), 225(81), 227(81 ), 228(81), 229(81) Morizov, V. A., 20(6) Morris, M. J., 138(134) Morrison, R., 133(89) Moulin, A. M., 120(25), 121(25) Muhs, J. D., 140(147) Mujumdar, A. S., 193(58) Mulholland, G. P., 27(19) Muller, T., 158(257) Munch, K. U., 157(241) Muralidhar Rao, M., 249(163), 250(163) Murawski, C., 140(152) Murphy, A. B., 157(232) Mutton, J. E., 137(132)

Nieuwenhuys, G. J., 134(109) Nikolaev, N. A., 216(106), 217(106), 218(106), 226(106) Nirala, A. K., 164(295; 296; 297; 298) Nirmalan, V., 191(27), 196(27), 203(27), 230(27) Noel, B. W., 140(146; 147; 148) Nomsfelov, E. V., 192(49), 202(49), 203(49), 215(49), 220(49), 224(49), 226(49) Northam, G. B., 160(261; 269) Nyguist, H., 133(91)

O O'Connor, L., 120(23), 121(23) Oehrlein, G. S., 162(284) Ogawa, H., 162(275) Ogawa, M., 242(138) Oguma, M., 27(26), 28(7; 40), 29(45), 30(45; 48), 31(52; 53; 54) Ohadi, M. M., 186(10) Ohsawa, T., 157(245) Ohtake, K., 156(229) Oppenheim, R. M., 130(83) Oppermann, W., 162(280) Ornatskii, A. P., 243(140; 141) Otugen, M. V., 157(236; 243) Ou, S., 142(167) Ou-Yang, Y. H., 249(162) Owens, L. R., 138(138) Ozaki, E., 27(26) Ozisik, M. N., 27(14; 27)

N Nain, V. P. S., 123(36) Nair, R. K., 192(48), 200(48), 219(48) Nakajima, M., 155(220) Namer, I., 157(235) Nariai, H., 235(127; 128; 129), 237(127), 241(135; 136) Nariari, H., 240(134), 243(134) Naruse, I., 156(229) Nazmeev, V. G., 216(106), 217(106), 218(106), 226(106) Nazmeev, Y. G., 193(56) Nelson, R. M., 251 (168) Neumann, R. D., 137(131) Nicholas, J. V., 119(8), 120(8; 11), 126(8), 128(8), 129(8), 165(8) Nichols, K. M., 155(224)

P Pabisz, R. A., Jr., 237(133), 238(133), 239(133), 240(133), 249(165) Pal, A., 138(134) Palmer, A. W., 147(190) Palmieri, A., 190(16), 192(16) Paoletti, D., 164(293) Pari, P., 134(108; 110) Park, C. W., 161(273) Park, H. M., 27(16) Pattee, H. A., 156(230) Pavese, F., 123(44; 49; 50) Peacock, G. R., 147(200) Pealat, M., 160(258; 268) Pearson, G. L., 130(80) Peden, D., 133(98)

AUTHOR INDEX Pedrocchi, E., 202(100), 221 (100), 230(100) Penney, W. R., 248(156), 249(156) Penning, F. C., 134(100; 101; 105) Pepler, S. J., 158(256) Perch-Nielsen, T., 147(195) Perrin, J., 160(268) Petaccia, L., 155(222) Peters, M. F., 136(127) Petersen, K. M., 129(69) Peterson, G. P., 192(39), 193(39), 194(39), 230(39), 243(39), 244(39), 251(39), 252(39), 254(39) Peterson, J. E., 157(242) Peterson, J. R., 218(108) Peterson, R. B., 156(230) Pilavachi, P. A., 184(1) Pitimada, D., 190(16), 192(16) Pitre, L., 134(109) Plocek, M., 225(118) Plumb, H. H., 165(309) Pokhodun, A. I., 128(68) Pollock, D. D., 124(52) Polunin, S. P., 123(42) Popernack, T. G., 138(138) Porter, F. M., 160(263) Poss, H. L., 163(287) Pramila, J., 164(297) Preston-Thomas, H., 112(2) Proceedings of Tempmeko, 165(314) Proceedings of Thermosense, 165(313)

Q Qizheng, Y., 27(32) Quinn, T. J., 135(112), 165(112) Quinnell, E. H., 130(76)

Rahn, L. A., 160(262) Raja Rao, M., 190(24), 191(31), 193(24; 31), 219(111), 248(24), 249(31) Ranganathan, C., 192(47), 194(47), 206(47), 209(47) Rantala, J., 147(198) Rao, K. S., 191(33), 249(33) Ravigururajan, T. S., 191(27), 196(27), 203(27), 230(27) Raynaud, M., 27(37) Reesink, A. L., 133(94)

275

Rice, P., 191(28), 195(28), 247(28) Ridella, S., 26(11), 33(11) Ristein, J., 158(252) Rivir, R., 142(167) Roberts, G. T., 141(160) Roberts, W. L., 161(271) Rodenburgh, A., 144(172) Rogatto, W. D., 145(183) Rogers, 26(10), 33(10) Roland, F., 135(122) Romans, E., 133(98) Rosenblatt, G. M., 158(247) Rothe, E., 162(280) Rotter, M., 134(108; 110) Royal, J. H., 195(69; 70), 246(69; 70; 149; 150), 253(70) Royds, R., 192(40) Rubin, L. G., 130(77; 78; 79), 165(77; 78; 79) Rumyantsev, S. V., 18(4) Runciman, H. M., 148(206), 150(206) Ruperti, N. J., 27(37) Rusby, R. L., 129(74), 134(109), 135(113; 116) Ryu, J. S., 161(273)

Saaski, E. W., 147(187) Sacadura, J. F., 27(37) Sacha, J. P., 147(193) Saenger, K. L., 154(213) Saha, S. K., 191(29), 195(29; 76), 196(87; 88), 219(29), 221(29), 247(87), 248(29; 76; 87; 88; 155), 251(29) Said, S. A., 246(153; 154) Saito, T., 29(45), 30(45) Saitoh, Y., 135(123) Sakurai, H., 133(94) Salehi, M., 186(10) Sallee, N., 140(149) Sample, H. H., 130(78), 165(78) Sano, Y., 191(32), 249(32) Santoni, A., 155(222) Sapoff, M., 130(83; 84) Saraswat, K. C., 164(290) Sarfim, A. F., 157(238) Sarjant, R. J., 192(42) Sasic, M., 225(118) Sastri, V. M. K., 249(163), 250(163) Satoh, K., 242(138) Satoh, M., 135(121; 123)

276

AUTHOR INDEX

Sawada, S., 128(67) Schad, O., 248(157) Schanze, K. S., 138(136) Schefer, R. W., 157(235) Schenk, J. R., 245(146) Schlosser, J., 242(139), 243(139) SchliJter, H., 162(280) Schmidt, G., 123(47) Schmidt, W. A., 256(174) Schmieder, D. E., 147(204) Schooley, J. F., 165(310; 311) Schulz, C., 157(240), 162(240) Schurig, T. A., 133(97) Schuster, G., 129(75), 134(109), 135(119) Schwarz, A., 154(212) Seasholtz, R. G., 157(243) Seeger, T., 160(260) Seetharamu, K. N., 195(77), 248(77) Seger, 136(125) Seinhart, J. S., 130(85) Seitzman, J. M., 162(276; 277) Selby, J. E. A., 145(178; 179; 180; 181) Sergatskov, D. A., 135(117) Seymour, E. V., 192(46), 205(46), 206(46), 207(46), 216(46), 220(46) Shabestari, B. N., 147(193) Shah, R. K., 197(93), 200(93), 213(93), 214(93), 217(93), 246(148) Shakher, C., 164(295; 296; 297; 298) Shatto, D. P., 192(39), 193(39), 194(39), 230(39), 243(39), 244(39), 251(39), 252(39), 254(39) Shen, Z. F., 55(55) Shen, Z. X., 158(248) Shepard, R. L., 133(93; 94), 134(93) Shepherd, R., 137(132) Shilling, R. L., 200(97) Shivkumar, C., 191(31), 193(31), 249(31) Shkema, R. K., 224(117), 235(126), 237(126) Shkuratov, I. Y., 195(75), 247(75) Shlanciauskas, A. A., 224(117), 235(126), 237(126) Sholes, R. R., 140(156) Shome, B., 185(6; 8), 186(6; 8), 190(8), 191(6; 8), 193(8), 194(8), 243(6), 251(8) Sieder, E. N., 199(95) Siemens, W. H., 125(54) Siewert, C. E., 27(28; 29) Simmons, C. M., 138(139) Simonich, J. C., 142(164)

Simons, A. J., 138(140) Singh, Y. H., 193(57) Singham, J. R., 194(59), 214(59), 217(59) Sir, J., 215(104), 217(104) Small, J. G., 140(156) Smith, A. P., 161(270), 249(164) Smith, F. G., 145(182) Smith, M. D. W., 137(132) Smith, M. W., 160(261; 269) Smith, S. P., 157(238) Smith, T. F., 55(55) Smithberg, E., 192(45), 202(45), 203(45), 206(45), 208(45), 215(45), 218(45), 221(45), 224(45), 230(45) Soechting, F., 142(167) Somerscales, E. F. C., 185(7), 186(7), 190(7), 191(7), 196(7), 256(172) Sorrensen, J. C., 147(195) Sostman, H. E., 128(62) Soulen, R. J., 133(96) Spagnolo, G. S., 164(293) Spiegel, H., 160(260) St. Clair, Jr., 26(13), 27(13) Starner, S. H., 158(254) Stephenson, R. J., 120(25), 121(25) Steur, P. P. M., 123(39; 40; 41; 44) Stevens, R., 138(140) Stoeckel, F., 155(221) Storm, A., 134(109) Strehlow, P., 134(105; 106; 107; 109) Stricker, W., 162(279) Strouse, G. F., 128(68) Struck, C. W., 138(142) Stufflebeam, J. H., 160(267) Subbotin, V. I., 192(49), 202(49), 203(49), 215(49), 220(49), 224(49), 226(49) Subramaniam, S., 27(21) Sugimoto, H., 135(123) Sukhatme, S. P., 193(54), 228(54) Sullivan, J. P., 138(135; 137) Sun, M., 147(188) Suter, J. J., 135(124) Suzuki, S., 241 (137), 242(138) Swenson, C. A., 135(111; 113) Szmyrka-Grzebyk, A., 132(87)

T Tabbita, M., 142(167) Taborek, J., 251 (166)

AUTHOR INDEX Tain, R. M., 245(147) Taira, T., 196(81), 203(81), 213(81), 214(81), 216(81), 217(81), 220(81), 221(81), 222(81), 223(81), 225(81), 227(81), 228(81), 229(81) Takahashi, M., 192(35), 236(35), 244(35) Tan, H. P., 27(33; 34) Tang, Y. S., 236(132) Taniguchi, H., 28(44) Taran, J. P., 160(258) Tate, G. E., 199(95) Taylor, B. N., 143(171) Taylor, D. J., 160(269) Taylor, J. H., 145(177) Taylor, W. L., 123(36) Tellex, P. A., 160(267) ter Harmsel, H., 135(118) Tezuka, A., 27(24) Thomas, L., 158(256) Thomas, M., 140(149) Thorsen, R. S., 224(116) Thynell, S. T., 155(223) Tikhonov, A. N., 18(2) Tillett, S. B., 130(81) Timnat, Y. M., 156(227) Timoshenko, S. P., 120(24), 121(24) Tobin, K. W., 140(146; 147; 148; 155) Tomita, T., 162(283) Tong, L. S., 236(132) Tong, W., 192(36), 221(36), 235(36), 236(36), 237(36), 238(36), 243(36) Touloukian, Y. S., 144(174; 175; 176) Tsai, J.-H., 27(20; 22) Tukamoto, K., 135(123) Turknett, J. C., 249(164) Turley, W. D., 140(147; 148)

U Uchiyama, F., 135(121; 123) Uchiyama, H., 155(220) Uhlenbusch, J., 160(266) Usman, S., 196(85), 236(85) Usui, H., 191(32), 249(32)

V Valvano, J. W., 165(300) Vandervort, C. L., 234(124), 243(124) van Dijk, H., 135(118)

277

Van Dijk, S. J., 119(7) van Herwaarden, A. W., 121(26) Van Kempen, H., 134(100; 101; 105) van Rijn, C., 135(118) Van Rooyen, R. S., 249(161) Varis, J., 147(201) Varma, H. K., 194(67), 251(67) Vaughan, G., 158(256) Veirs, K. D., 158(247) Verma, S. K., 164(297) Vinyarskii, L. S., 243(140; 141) Violino, P., 160(264) Viskanta, R., 27(18) Voges, H., 162(280) Vuohelainen, R., 147(201) Vysochanskii, Y. M., 134(101)

W Wadley, H. N. G., 164(292) Walker, G. W., 147(204) Wang, J. D., 155(215) Wang, Y., 155(215) Wang, Z., 142(169) Wansbrough, R. W., 194(68), 232(68), 233(68), 234(68), 237(68), 252(68) Wareing, D. P., 158(256) Watanabe, K., 196(81), 203(81), 213(81), 214(81), 216(81), 217(81), 220(81), 221(81), 222(81), 223(81), 225(81), 227(81), 228(81), 229(81) Watson, H. M. L., 137(132) Webb, R. L., 184(4), 190(15), 253(171), 296(4) Weber, S., 123(47) Webster, J. G., 165(301) Weeds, J., 249(160) Ween, S., 120(9) Weisman, J., 196(85), 236(85) Weiss, E., 248(157) Welland, M. E., 120(25), 121(25) Whalley, P. B., 232(121) Whitcomb, H. J., 119(6) White, D. R., 119(8), 120(8), 126(8), 128(8), 129(8), 133(94), 165(8) Whitham, J. M., 190(12), 253(12) Wiegers, S. A. J., 134(100; 101; 105) Wikhammer, G. A., 243(143), 248(143), 252(143) Wilson, M., 140(150) Wise, J. A., 120(10)

278

AUTHOR INDEX

Wolfendale, P. C. F., 129(73) Wolff-Gassmann, D., 162(279) Wolfhard, H. G., 155(225) Wood, S. D., 130(81) Worrall, R. W., 124(53) Wu, W. J., 27(19) Wurzbach, R. N., 147(197)

X Xu, H. Q., 155(215) Xumo, L., 128(66)

Y Yalin, A. P., 157(233) Yamada, Y., 27(24), 196(84) Yamamoto, Y., 156(229) Yamashita, T., 162(283) Yang, J. Y., 196(85), 236(85) Yang, K.-T., 27(15) Yang, W.-J., 28(44)

Yates, H. W., 145(177) Ye, S. Y., 249(162) Yera, K., 197(91), 251(91), 252(91) Yewen, J. D., 129(73) Yoon, J. H., 142(170) Yos, J. M., 163(287) Yoshida, T., 241(135) You, L., 194(66), 196(66), 208(66), 210(66) Yousefian, F., 27(36) Yu, Q. Z., 27(33; 34) Yuta, S., 155(220)

Z Zakharov, A. A., 123(42) Zemansky, M. W., 111(1) Zhang, Z., 147(190; 191) Zhou, Q. T., 249(162) Zimparov, V., 196(89), 249(89), 251(89; 169) Zozulya, N. V., 195(75), 247(75)

SUBJECT INDEX

convergence criteria, 79 different size heaters, 82-85 energy balance, 74-77 example, description, 73-74 grid independence study, 86-87 heating devices, small number, 87-90 iterative procedure, 78 parametric study, 90- 93 relaxation, 79 same number equations/unknowns, 80-82 Critical heat flux enhancement, 194-195 flow boiling experiments, 235-236 limits, 242 variations, 239-242

A Absorption spectroscopy, 155 Acoustic measurement methods, 163-164

B Bimetallic thermometers, 120-121 Boiling bulk, 243-244 subcooled flow heat transfer, 232-236 objective, 232 pressure drop, 236-239 Bullers rings, 135-136

C

D

Capacitance thermometers, 134 Carbon resistors, 129-130 CARS. s e e Coherent anti-stokes Raman scattering CGR. s e e Conjugate gradient regularization CHF. s e e Critical heat flux Coherent anti-stokes Raman scattering, 158-160 Condensation events sequence, 245 refrigerant, 246-247 steam, 246 Conduction -radiation, inverse heat source conduction mechanism, 68 distribution, 70-71 energy balance, 63-64 heat fluxes, 65-68 SVD decomposition, 69-70 temperature, 66-69 Conjugate gradient regularization application, 71-73 aspects, discussion, 24-26 description, 22"24 Convection -radiation, inverse boundary design

Decomposition MTSVD applications, 28- 31 aspects, discussion, 24-26 -CGR,-comparision, 22-24 description, 16-17 -Tikhonov, comparison, 19-21 SVD description, 11-13 heat source placement, 45-47 inverse thermal designs, 69-70 TSVD aspects, discussion, 24-26 -CGR, comparison, 19-21 description, 13-16 heat source placement, 42-45 regularization, 81 -Tikhonov, comparison, 19-21 Degenerative four wave mixing, 160-161 Design. s e e Inverse design; Thermal systems DFWM. s e e Degenerative four wave mixing Diode thermometers, 132-133 Discretization, numerical, 10-11 Doped germanium resistors, 129-130 279

280

SUBJECT INDEX

E Ellipsometry, 162 Emission spectroscopy, 155

Flows gas, 192 swirl (see Swirl flows) turbulent, 192-193 Fog particles, 149-150 Fredholm integral equation, 8-9

G Gas constant, 122 Gas flows, in-tube, 192 Germanium. see Doped germanium resistors

H Heat flux conduction-radiation, inverse heat source, 65-68 critical bulk boiling, 244-245 enhancement, 194-195 flow boiling experiments, 235-236 limits, 242 variations, 239-242 devices, 98-103 uniform wall condition, 197-198 Heat source inverse (see Inverse heat source) placement, radiative system equations, 38-42 specifications, 36- 38 SVD regulation, 45-47 TSVD regulation, 42-45, 48 Heat transfer application, 184-185 classification, 185 process, importance, 184 swirl flow bulk boiling early studies, 243-244 recent studies, 244-245 condensation events sequence, 245

refrigerant, 246-247 steam, 246 performance evaluation single-phase flow, 250--251 two-phase flow, 251-253 pressure drop data, 197-205 structure, 205-212 subcooled flow boiling circumferential heat flux, 239-242 heat flux limits, 242 heat transfer, 232-236 pressure drop, 236-239 thermal-hydraulic design friction factor, 213, 217-222 Nusselt number, 222, 226-230 twisted tape compound enhancement, 249-250 description, 188-190 effects of, 230-232 history, 190-196 modified inserts, 247-249 types, 185-186 Holographic interferometers, 154

I Imaging, thermal, 147-150 Industrial platinum resistance thermometers, 125-129 Infrared thermometry, 143-147 Interferometers function, 153 holographic, 154 speckle shearing, 164 Inverse design boundary, radiation-convection convergence criteria, 79 different size heaters, 82-85 energy balance, 74-77 example, description, 73-74 grid independence study, 86-87 heating devices, small number, 87-90 heat transfer, 77-78 iterative procedure, 78 parametric study, 90-93 relaxation, 79 same number equations/unknowns, 80-82 surface interactions, 76 case for, 104-105

SUBJECT INDEX

constraints, imposing, 103-104 -conventional design, differences associated-problems, physics of, 5 existence, 4 uniqueness, 4-5 energy generation shape, 93-98 further research areas, 105-106 linear systems radiative system, heat source placement equations, 38-42 specifications, 36- 38 SVD regulation, 45-47 TSVD regulation, 42-45, 48 surface radiative exchange determination of geometry by, 31-35 existence, 35- 36 two- and three-dimensional results, 31 uniqueness, 35- 36 literature review, 27-28 rationale for, 2-3 regularization methods conjugate gradient, 22-24 discussion, 24-26 equation, 8- 9 Levenberg-Marquardt, 26 Monte Carlo, 26 MTSVD applications, 28- 31 aspects, discussion, 24-26 -CGR, -comparision, 22-24 description, 16-17 -Tikhonov, comparison, 19-21 numerical discretization, 10-11 SVD, 11-13 Tikhonov aspects, discussion, 24-26 description, 18-21 -TSVD, comparison, 19-21 TSVD aspects, discussion, 24-26 -CGR, comparison, 19-21 description, 13-16 -Tikhonov, comparison, 19-21 state of the art, 28-31 Inverse heat source conduction mechanism, 68 distribution, 70-71 energy balance, 63-64 heat fluxes, 65-68 SVD decomposition, 69-70

281

temperature, 66-69 IPRTS. s e e Industrial platinum resistance thermometers Iron rhodium resistors, 129-130 Isothermal tubes, 199-200, 202 ITS-90, 112-113

Jacobi scheme, 28 L Laminar flows in circular tubes, 208, 210-212 coefficient data, 198-199 friction factor, 213-222 Nusselt numbers, 193-194 smoke-injection techniques, 206-208 Laser-induced fluorescence, 161-162 Levenberg-Marquardt multivariate optimization, 26 LIF. s e e Laser-induced fluorescence Linear systems inverse design radiative system, heat source placement equations, 38-42 specifications, 36- 38 SVD regulation, 45-47 TSVD regulation, 42-45, 48 surface radiative exchange determination of geometry by, 31-35 existence, 35- 36 two- and three-dimensional results, 31 uniqueness, 35- 36 Line reversal, 155-156 Liquid crystals, thermographic, 140-142 Liquid-in-glass thermometers, 118-120 M Manometric thermometry, 121-123 Measurement temperature criteria availability, 118 cost, 118 level of contact, 115 range, 115 sensor protection, 117-118

282

SUBJECT INDEX

Measurement (continued) sensor size, 115, 117 thermal disturbance, 115 transient response, 117 uncertainty, 113-114 methods acoustic, 162 bimetallic thermometers, 120-121 bullers rings, 135-136 capacitance thermometers, 134 carbon resistors, 129-130 CARS, 158-160 DFWM, 160-161 diode thermometers, 132-133 doped germanium resistors, 129-130 ellipsometry, 163-164 infrared thermometry, 143-147 LIF, 161-162 line reversal, 155-156 liquid-in-glass thermometers, 118-120 manometric thermometry, 121-123 noise thermometry, 133-134 peak temperature indicating devices, 137 PRTs, 125-129 pyrometric cones, 135-136 quartz thermometers, 135 Raman scattering, 158 Rayleigh scattering, 156-157 refraction, index of, 150-154 rhodium iron resistors, 129-130 semiconductor devices, 131-132 Speckle, 164 spectroscopy, 155 temperature-sensitive paints, 137-138 thermal imaging, 147-150 thermistors, 130 thermocouples, 123-124 thermographic liquid crystals, 140-142 thermographic phosphors, 138-140 thermoscope bars, 135-136 Modified truncated singular value decomposition applications, 28- 31 aspects, discussion, 24-26 -CGR,-comparision, 22-24 description, 16-17 -Tikhonov, comparison, 19-21 Monte Carlo methods application, 28

aspects, discussion, 27 description, 26 MTSVD. see Modified truncated singular value decomposition

N Noise thermometry, 133-134 Nongray media energy balance, 52-55 problems, 55-63 temperature distribution, 52 walls, 51-52 Nuclear reactors, 190 Nusselt number, 222, 226-230

Peak temperature indicating devices, 137 Phosphors, thermographic, 140-142 Photography, Speckle, 164 Platinum resistance thermometers, 125-129 Pressure drop, 236-239 PRTs. see Platinum resistance thermometers Pyrometric cones, 135-136

Q Quartz thermometers, 135

R Radiation -conduction, in inverse heat source conduction mechanism, 68 distribution, 70-71 energy balance, 63-64 heat fluxes, 65-68 SVD decomposition, 69-70 temperature, 66-69 -convection, inverse boundary design convergence criteria, 79 different size heaters, 82-85 energy balance, 74-77 example, description, 73-74 grid independence study, 86-87 heating devices, small number, 87-90 iterative procedure, 78 parametric study, 90- 93

SUBJECT INDEX

relaxation, 79 same number equations/unknowns, 80-82 effects of fog particles, 149-150 systems heat source placement equations, 38-42 specifications, 36- 38 SVD regulation, 45-47 TSVD regulation, 42-45, 48 surface exchange determination of geometry by, 31- 35 existence, 35- 36 two- and three-dimensional results, 31 uniqueness, 35- 36 Raman scattering, 158-160 Rayleigh scattering, 156-157 READ method, 28 Refraction, index of, 150-154 Resistance temperature detectors, 124, 129-130 Resistors carbon, 129-130 doped germanium, 129-130 rhodium iron, 129-130 Rhodium iron resistors, 129-130 RTDs. s e e Resistance temperature detectors

S Saturation vapor, 123 Scales, temperature, 112-113 Schlieren system, 152 Semiconductor devices, 131-132 Sensors, 115, 117-118 Shadowgraph system, 152 Singular value decomposition applications, 29- 30 description, 11-13 heat source placement, 45-47 inverse thermal designs, 69-70 Smoke-injection techniques, 206-208 Speckle photography, 164 Speckle shearing interferometry, 164 Spectroscopy, 155 Speed of sound, 163-164 SPRTs. s e e Standard platinum resistance thermometers Standard platinum resistance thermometers, 125-129

283

Subcooled flow boiling circumferential heat flux, 239-242 heat flux limits, 242 heat transfer, 232-236 objective, 232 pressure drop, 236-239 Surface exchange systems radiative transfer determination of geometry by, 31- 35 existence, 35- 36 two- and three-dimensional results, 31 uniqueness, 35- 36 SVD. s e e Singular value decomposition Swirl flows modified twisted-tape inserts, 244-247 performance evaluation single-phase flow, 250-251 two-phase flow, 251-253 structure bulk boiling condensed heat transfer, 244-247 early studies, 243-244 recent studies, 244-245 overview, 205-212 subcooled flow boiling circumferential heat flux, 239-242 heat flux limits, 242 heat transfer, 232-236 pressure drop, 236-239 thermal-hydraulic design friction factor, 213, 217-222 twisted tapes compound enhancement, 249-250 description, 188-190 history, 190-196 modified inserts, 247-249 types, 185-186

Temperature distribution, 153 measurement criteria availability, 118 cost, 118 level of contact, 115 range, 115 sensor protection, 117-118 sensor size, 115, 117

284

SUBJECT INDEX

Temperature (continued) thermal disturbance, 115 transient response, 117 uncertainty, 113-114 methods acoustic, 162 bimetallic thermometers, 120-121 bullers rings, 135-136 capacitance thermometers, 134 carbon resistors, 129-130 CARS, 158-160 DFWM, 160-161 diode thermometers, 132-133 doped germanium resistors, 129-130 ellipsometry, 163-164 infrared thermometry, 143-147 LIF, 161-162 line reversal, 155-156 liquid-in-glass thermometers, 118-120 manometric thermometry, 121-123 noise thermometry, 133-134 peak temperature indicating devices, 137 PRTs, 125-129 pyrometric cones, 135-136 quartz thermometers, 135 Raman scattering, 158 Rayleigh scattering, 156-157 refraction, index of, 150-154 rhodium iron resistors, 129-130 semiconductor devices, 131-132 Speckle, 164 spectroscopy, 155 temperature-sensitive paints, 137-138 thermal imaging, 147-150 thermistors, 130 thermocouples, 123-124 thermographic liquid crystals, 140-142 thermographic phosphors, 138-140 thermoscope bars, 135-136 saturation vapor, 123 scales, 111-112 -sensitive paints, 137-138 Thermal systems disturbance, 115 with highly nonlinear characteristics nongray medium energy balance, 52-55 problems, 55-63 temperature distribution, 52

walls, 51-52 problems, techniques for, 50-51 imaging, 147-150 inverse boundary design convergence criteria, 79 different size heaters, 82-85 energy balance, 74-77 example, description, 73-74 grid independence study, 86-87 heating devices, small number, 87-90 heat transfer, 77-78 iterative procedure, 78 parametric study, 90-93 relaxation, 79 same number equations/unknowns, 80-82 surface interactions, 76 inverse design case for, 104-105 constraints, imposing, 103-104 -conventional design, differences associated-problems, physics of, 5 existence, 4 uniqueness, 4-5 energy generation shape, 93-98 further research areas, 105-106 literature review, 27-28 rationale for, 2-3 regularization methods analytical solution, 9-10 conjugate gradient method, 22-24 discussion, 24-26 equation, 8- 9 Levenberg-Marquardt, 26 Monte Carlo, 26 MTSVD, 16-17 numerical discretization, 10-11 SVD, 11-13 Tikhonov, 18-21 TSVD, 13-16 state of the art, 28-31 uniform heat flux devices, 98-103 inverse heat source CGR, 71-73 conduction mechanism, 68 distribution, 70-71 energy balance, 63-64 heat fluxes, 65-68 SVD decomposition, 69-70 temperature, 66-69

SUBJECT INDEX thermal-hydraulic design, 213, 217-222 Thermistors, 130 Thermocouples, 123-124 Thermodynamic scale, 112-113 Thermographic liquid crystals, 140-142 Thermographic phosphors, 138-140 Thermometers bimetallic, 120-121 capacitance, 134 diode, 132-133 infrared, 143-147 liquid-in-glass, 118-120 manometric, 121-123 noise, 133-134 platinum, 125-129 quartz, 135 Thermoscope bars, 135-136 Tikhonov regularization aspects, discussion, 24-26 description, 18-21 -TSVD, comparison, 19-21 Transient response, 117

285

Truncated singular value decomposition aspects, discussion, 24-26 -CGR, comparison, 19-21 description, 13-16 heat source placement, 42-45, 48 regularization, 81 -Tikhonov, comparison, 19-21 TSVD. s e e Truncated singular value decomposition Twisted tapes compound enhancement, 249-250 description, 188-190 history, 190-196 modified inserts, 247-249

U Uncertainty, definition, 113-114 Uniform wall heat flux condition, 197-198

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