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~ 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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188

RAJ M. MANGLIKAND ARTHURE. BERGLES

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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SWIRL FLOW HEAT TRANSFER

189

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.

190

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 (2.36 104 n - 0.18 (heating) n - 0.30 (cooling)

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202

RAJ M. MANGLIK AND ARTHUR E. BERGLES

10 3

,

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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

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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

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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 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)-

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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

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VIEWED F R O M U P S T R E A M TO D O W N S T A E A M

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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~

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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

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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 > 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 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