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HEAT TRANSFER
HANDBOOK
Adrian Bejan
J. A. Jones Professor of Mechanical Engineering Department of Mechanical Engineering Duke University Durham, North Carolina
Allan D. Kraus
Department of Mechanical Engineering University of Akron Akron, Ohio
JOHN WILEY & SONS, INC.
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Designations used by companies to distinguish their products are often claimed as trademarks. In all instances where John Wiley & Sons, Inc. is aware of a claim, the product names appear in initial capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration.
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Copyright © 2003 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data: Bejan, Adrian, 1948– Heat transfer handbook / Adrian Bejan, Allan D. Kraus. p. cm. ISBN 0-471-39015-1 (cloth : alk. paper)
- Heat—Transmission—Handbooks, manuals, etc. I. Kraus, Allan D. II. Title.
TJ250 .B35 2003 621.402'2—dc21 2002028857
Printed in the United States of America
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PREFACE
Heat transfer has emerged as a central discipline in contemporary engineering sci- ence. The research activity of a few decades ago—the material reviewed in the first handbooks—has distilled itself into textbook concepts and results. Heat transfer has become not only a self-standing discipline in the current literature and engineering curricula, but also an indispensable discipline at the interface with other pivotal and older disciplines. For example, fluid mechanics today is capable of describing the transport of heat and other contaminants because of the great progress made in mod- ern convective heat transfer. Thermodynamics today is able to teach modeling, sim- ulation, and optimization of “realistic” energy systems because of the great progress made in heat transfer. Ducts, extended surfaces, heat exchangers, and other features that may be contemplated by the practitioner are now documented in the heat transfer literature. To bring this body of results to the fingertips of the reader is one of the objectives of this new handbook. The more important objective, however, is to inform the reader on what has been happening in the field more recently. In brief, heat transfer marches forward through new ideas, applications, and emerging technologies. The vigor of heat transfer has always come from its usefulness. For example, the challenges of energy self-sufficiency and aerospace travel, which moved the field in the 1970s, are still with us; in fact, they are making a strong comeback. Another example is the miniaturization revolution, which continues unabated. The small-scale channels of the 1980s do not look so small anymore. Even before “small scale” became the fashion, we in heat transfer had “compact” heat exchangers. The direction for the future is clear. The importance of optimizing the architecture of a flow system to make it fit into a finite volume with purpose has always been recognized in heat transfer. It has been and continues to be the driving force. Space comes at a premium. Better and better shapes of extended surfaces are evolving into networks, bushes, and trees of fins. The many surfaces designed for heat transfer augmentation are accomplishing the same thing: They are increasing the heat transfer rate density , the size of the heat transfer enterprise that is packed into a given volume. The smallest features are becoming smaller, but this is only half of the story. The other is the march toward greater complexity. More and more small-scale features must be connected and assembled into a device whose specified size is always macro- scopic. Small-scale technologies demand the optimization of increasingly complex heat-flow architectures. A highly distinguished group of colleagues who are world authorities on the frontiers of heat transfer today have contributed to this new handbook. Their chapters provide a bird’s-eye view of the state of the field, highlighting both the foundations ix
x PREFACE
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and, especially, the edifices that rest on them. Because space comes at a premium, we have allocated more pages to those chapters dedicated to current applications. The latest important references are acknowledged; the classical topics are presented more briefly. One feature of the handbook is that the main results and correlations are summa- rized at the ends of chapters. This feature was chosen to provide quick access and to help the flow of heat transfer knowledge from research to computer-aided design. It is our hope that researchers and practitioners of heat transfer will find this new handbook inspiring and useful. Adrian Bejan acknowledges with gratitude the support received from Professor Kristina Johnson, Dean of the Pratt School of Engineering, and Professor Kenneth Hall, Chairman of the Department of Mechanical Engineering and Materials Science, Duke University. Allan Kraus acknowledges the assistance of his wife, who has helped in the proofreading stage of production. Both authors acknowledge the assistance of our editor at John Wiley, Bob Argen- tieri, our production editor, Milagros Torres, and our fantastic copy editor, known only to us as Barbara from Pennsylvania.
Adrian Bejan Allan D. Kraus
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CONTRIBUTORS
A. Aziz, Department of Mechanical Engineering, Gonzaga University, Spokane, WA 99258- Avram Bar-Cohen, Department of Mechanical Engineering, University of Min- nesota, Minneapolis, MN 55455- Current address: Glenn L. Martin Institute of Technology, A. James Clark School of Engineering, Department of Mechanical Engineering, 2181 Glenn L. Martin Hall, College Park, MD 20742- Adrian Bejan, J. A. Jones Professor of Mechanical Engineering, Department of Me- chanical Engineering and Materials Science, Duke University, Durham, NC 27708- 0300 Robert F. Boehm, University of Nevada–Las Vegas, Las Vegas, NV 89154- J. C. Chato, Department of Mechanical and Industrial Engineering, University of Illinois–Urbana-Champaign, Urbana, IL 61801 C. Haris Doumanidis, Department of Mechanical Engineering, Tufts University, Medford, MA 02150 R. T Jacobsen, Idaho National Engineering and Environmental Laboratory, Idaho Falls, ID 83415- Yogesh Jaluria, Mechanical and Aerospace Engineering Department, Rutgers Uni- versity, New Brunswick, NJ 08901- Yogendra Joshi, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332- M. A. Kedzierski, Building and Fire Research Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899 Allan D. Kraus, University of Akron, Akron, OH 44325- José L. Lage, Laboratory of Porous Materials Applications, Mechanical Engineer- ing Department, Southern Methodist University, Dallas, TX 75275- E. W. Lemmon, Physical and Chemical Properties Division, National Institute of Standards and Technology, Boulder, CO 80395- R. M. Manglik, Thermal-Fluids and Thermal Processing Laboratory, Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, 598 Rhodes Hall, P.O. Box 210072, Cincinnati, OH 45221- xiii
xiv CONTRIBUTORS
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E. E. Marotta, Senior Engineer/Scientist, Thermal Technologies Group, IBM Cor- poration, Poughkeepsie, NY 12801 Michael F. Modest, Professor of Mechanical and Nuclear Engineering, College of Engineering, Pennsylvania State University, University Park, PA 16802- Wataru Nakayama, Therm Tech International, Kanagawa, Japan 255- Pamela M. Norris, Associate Professor, Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22903 Jay M. Ochterbeck, College of Engineering and Science, Department of Mechani- cal Engineering, Clemson University, Clemson, SC 29634- S. G. Penoncello, Center for Applied Thermodynamic Studies, College of Engineer- ing, University of Idaho, Moscow, ID 83844- Ranga Pitchumani, Department of Mechanical Engineering, University of Con- necticut, Storrs, CT 06269- Ravi S. Prasher, Intel Corporation, Chandler, AZ 85225 T. J. Rabas, Consultant, Downers Grove, IL 60516 Z. Shan, Center for Applied Thermodynamic Studies, College of Engineering, Uni- versity of Idaho, Moscow, ID 83844- Andrew N. Smith, Department of Mechanical Engineering, United States Naval Academy, Annapolis, MD 21402- Richard N. Smith, Department of Mechanical Engineering, Aeronautical Engineer- ing and Mechanics, Rensselaer Polytechnic Institute, Troy, NY 12180- John R. Thome, Laboratory of Heat and Mass Transfer, Faculty of Engineering Sci- ence, Swiss Federal Institute of Technology Lausanne, CH-1015 Lausanne, Switzer- land Abhay A. Watwe, Intel Corporation, Chandler, AZ 85225 N. T. Wright, Department of Mechanical Engineering, University of Maryland, Baltimore, MD 21250 M. M. Yovanovich, Distinguished Professor Emeritus, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
15. Porous Media 1131 Adrian Bejan 16. Heat Pipes 1181 Jay M. Ochterbeck 17. Heat Transfer in Manufacturing and Materials Processing 1231 Richard N. Smith, C. Haris Doumanidis, and Ranga Pitchumani 18. Microscale Heat Transfer 1309 Andrew N. Smithand Pamela M. Norris 19. Direct Contact Heat Transfer 1359 Robert F. Boehm
Author Index 1401
Subject Index 1427
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CHAPTER 1
Basic Concepts
ALLAN D. KRAUS University of Akron Akron, Ohio
1.1 Heat transfer fundamentals 1.1.1 Introduction 1.1.2 Conduction heat transfer One-dimensional conduction One-dimensional conduction with internal heat generation 1.1.3 Spreading resistance 1.1.4 Interface–contact resistance 1.1.5 Lumped-capacity heating and cooling 1.1.6 Convective heat transfer Heat transfer coefficient Dimensionless parameters Natural convection Forced convection 1.1.7 Phase-change heat transfer 1.1.8 Finned surfaces 1.1.9 Flow resistance 1.1.10 Radiative heat transfer 1.2 Coordinate systems 1.2.1 Rectangular (Cartesian) coordinate system 1.2.2 Cylindrical coordinate system 1.2.3 Spherical coordinate system 1.2.4 General curvilinear coordinates 1.3 Continuity equation 1.4 Momentum and the momentum theorem 1.5 Conservation of energy 1.6 Dimensional analysis 1.6.1 Friction loss in pipe flow 1.6.2 Summary of dimensionless groups 1.7 Units 1.7.1 SI system (Syst`eme International d’Unit´es) 1.7.2 English engineering system (U.S. customary system) 1.7.3 Conversion factors Nomenclature References
1
HEAT TRANSFER FUNDAMENTALS 3
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T
x
Figure 1.1 Heat transfer by conduction through a slab.
greater care in obtaining the appropriate temperature differences. The axial temper- ature variation in the slim, internally heated conductor shown in Fig. 1.2 is found to equal
T = To + q (^) g
L^2
2 k
[
x L
( (^) x L
) 2 ]
where To is the edge temperature. When the two ends are cooled to an identical temperature, and when the volumetric heat generation rate q (^) g (W/m 3 ) is uniform throughout, the peak temperature is developed at the center of the solid and is given by
Tmax = T (^) o + q (^) g
L^2
8 k
(K) (1.4)
Alternatively, because q (^) g is the volumetric heat generation q (^) g = q/LW δ, the center–edge temperature difference can be expressed as
Tmax − T (^) o = q
L^2
8 kLW δ
= q
L
8 kA
where the cross-sectional area A is the product of the width W and the thickness δ. An examination of eq. (1.5) reveals that the thermal resistance of a conductor with a distributed heat input is only one-fourth that of a structure in which all of the heat is generated at the center.
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x 1 x 2
T max
To
Peak temperature
Edge temperature
x
L
Figure 1.2 Temperature variation in an internally heated conductor.
1.1.3 Spreading Resistance In configurations where there is a discrete heat source on the surface of a conducting medium, provision must be made for the lateral spreading of the heat generated in successive “layers” in the conducting medium below the source. The additional resistance associated with this lateral flow of heat is called the spreading resistance. According to Yovanovich and Antonetti (1988), the spreading resistance for a small heat source on a thick conductor or heat spreader (required to be three to five times thicker than the square root of the heat source area) can be expressed as
Rsp =
1 − 1. 410 + 0. 344 ^3 + 0. 043 ^5 + 0. 034 ^7
4 ka
(K/W) (1.6)
where is the ratio of the heat source area to the substrate area, k the thermal conductivity of the conductor, and a the square root of the area of the heat source. For relatively thin conducting layers on thicker substrates, such as encountered in the cooling of microcircuits, eq. (1.6) cannot provide an acceptable prediction of Rsp. Instead, use can be made of the numerical results plotted in Fig. 1.3 to obtain the requisite value of the spreading resistance.
1.1.4 Interface–Contact Resistance Heat transfer across the interface between two solids is generally accompanied by a measurable temperature difference, which can be ascribed to a contact or inter- face thermal resistance. For perfectly adhering solids, geometrical differences in the crystal structure (lattice mismatch) can impede the flow of phonons and electrons
6 BASIC CONCEPTS
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Figure 1.4 Physical contact between two nonideal surfaces.
where k (^) s is the harmonic mean thermal conductivity for solid 1 and solid 2,
k (^) s =
2 k 1 k 2 k 1 + k 2
(W/m · K)
σ is the effective root mean square (rms) surface roughness,
σ =
σ^21 + σ^22
(μm)
m is the effective absolute surface slope,
m = (m^21 + m^22 )^1 /^2
P is the contact pressure, and H is the microhardness of the softer material, both in N/m^2. In the absence of detailed information, the σ/m ratio can be taken as 5 to 9 μm for relatively smooth surfaces. In eq. (1.7a), hg is given by
hg =
k (^) g Y + M
(1.8b)
where kg is the thermal conductivity of the gap fluid, Y is the distance between the mean planes (Fig. 1.4), given by
Y σ
[
− ln
P
H
)] 0. 547
HEAT TRANSFER FUNDAMENTALS 7
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and M is a gas parameter used to account for rarified gas effects,
M = αβΛ
where α is an accommodation parameter (approximately equal to 1.7 for air and clean metals), Λ is the mean free path of the molecules (equal to approximately 0.06 μm for air at atmospheric pressure and 15°C), and β is a fluid property parameter (equal to approximately 1.7 for air and other diatomic gases). Equations (1.8a) and (1.8b) can be added and, in accordance with eq. (1.7a), the contact resistance becomes
R (^) co =
{[
- 25 k (^) s
m σ
P
H
k (^) g Y + M
]
A (^) a
1.1.5 Lumped-Capacity Heating and Cooling An internally heated solid of relatively high thermal conductivity that is experiencing no external cooling will undergo a constant rise in temperature according to
dT dt
q mc
(K/s) (1.10)
where q is the rate of internal heat generation, m the mass of the solid, and c the specific heat of the solid. Equation (1.10) assumes that all the mass can be represented by a single temperature. This approach is commonly called the lumped-capacity model for transient heating. Expanding on the analogy between thermal and electrical resistances suggested previously, the product of mass and specific heat can be viewed as analogous to electrical capacitance and thus to constitute the thermal capacitance. When this same solid is externally cooled, the temperature rises asymptotically toward the steady-state temperature, which is itself determined by the external resis- tance to heat flow, R. Consequently, the time variation of the temperature of the solid is expressible as
T (t) = T (t = 0 ) + qR( 1 − e−t/Rmc^ ) (K) (1.11)
where the product of the external resistance R and the thermal capacitance mc is seen to constitute the thermal time constant of the system.
1.1.6 Convective Heat Transfer
Heat Transfer Coefficient Convective thermal transport from a surface to a fluid in motion can be related to the heat tranfser coefficient h, the surface-to-fluid temperature difference, and the “wetted” surface area S in the form
q = hS(Ts − Tf ) (W) (1.12)
HEAT TRANSFER FUNDAMENTALS 9
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h = C
k L
(Ra) n^ (W/m 2 · K) (1.14)
where n is found to be approximately 0.25 for 10 3 < Ra < 10 9 , representing laminar flow; 0.33 for 10 9 < Ra < 10 12 , the region associated with the transition to turbulent flow; and 0.4 for Ra > 10^12 , when strong turbulent flow prevails. The precise value of the correlating coefficient C depends on fluid, the geometry of the surface, and the Rayleigh number range. Nevertheless, for common plate, cylinder, and sphere configurations, it is found to vary in the relatively narrow range of 0.45 to 0.65 for laminar flow and 0.11 to 0.15 for turbulent flow past the heated surface. Natural convection in vertical channels such as those formed by arrays of lon- gitudinal fins is of major significance. Elenbaas (1942) was the first to document a detailed study of this configuration, and his experimental results for isothermal plates were later confirmed numerically by Bodoia and Osterle (1964). A uniform picture of the thermal transport in such a vertical channel has emerged from these and com- plementary studies. It has been shown that the value of the Nusselt number lies between two extremes that are based on the size of the space between the plates or width of channel. For wide spacing, the plates appear to have little influence on one another, and the Nusselt number in this case achieves its isolated plate limit. On the other hand, for closely spaced plates or for relatively long channels, the fluid attains its fully developed value and the Nusselt number reaches its fully developed limit. Intermediate values of the Nusselt number can be obtained from a correlating method suggested by Churchill and Usagi (1972) for smoothly varying processes, and these values have been verified by a plethora of detailed experimental and numerical studies. Thus, the correlation for the average value of h along isothermal vertical channels spaced z units apart is
h =
k z
El
El 1 /^2
where El is the channel Elenbaas number :
El ≡
ρ^2 βgcp z^4 ∆T μkL and ∆T is the surface temperature minus the bulk fluid temperature, ∆T = Ts − Tf. Natural convection fundamentals and results are covered in more detail in Chapter 7.
Forced Convection For forced flow in long or very narrow parallel-plate chan- nels, the heat transfer coefficient attains an asymptotic value (the fully developed limit), which for symmetrically heated channel surfaces is equal approximately to
h =
4 k d (^) e
(W/m 2 · K) (1.16)
where d (^) e is the hydraulic diameter defined in terms of the flow area A and the wetted perimeter of the surfaces p:
10 BASIC CONCEPTS
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d (^) e ≡
4 A
p
In the inlet zones of such parallel-plate channels and along isolated plates, the heat transfer coefficient varies with the distance from the leading edge. The low-velocity, or laminar flow, average convective heat transfer coefficient along a surface of length L for Re ≤ 3 × 10 5 is found to be
h = 0. 664
k L
Re 1 /^2 · Pr 1 /^3 (W/m 2 · K) (1.17)
where k is the fluid thermal conductivity, L the characteristic dimension of the sur- face, and Re the Reynolds number based on L: namely, V L/ˆ ν. A similar relation applies to a flow in tubes, pipes, annuli or channels, with the equivalent diameter d (^) e serving as the characteristic dimension in both the Nusselt and Reynolds numbers. For laminar flow, Re ≤ 2100,
hd (^) e k
Re · Pr
d (^) e L
μ μw
which is attributed to Sieder and Tate (1936) and where μw is the viscosity of the convective medium at the channel wall temperature. Observe that this relationship shows that the heat transfer coefficient attains its maximum value at the inlet to the channel and decreases as de /L decreases. In higher-velocity turbulent flow along plates, the dependence of the convective heat transfer coefficient on the Reynolds number increases, and in the range Re ≥ 3 × 10 5 ,
h = 0. 036
k L
Re 0.^8 · Pr 1 /^3 (W/m 2 · K) (1.19)
In pipes, tubes, annuli, and channels, turbulent flow occurs at an equivalent diameter-based Reynolds number of 10,000, with the regime bracketed by 2100 ≤ Re ≤ 10,000 usually referred to as the transition region. For the transition region, Hausen (1943) has provided the correlating equation
hd (^) e k
= 0. 116 (Re − 125 )Pr 1 /^3
[
d (^) e L
) 2 / 3 ] (
μ μw
and Sieder and Tate (1936) give for turbulent flow
hde k
= 0 .023Re 0.^8 · Pr 1 /^3
μ μw
Forced convection in internal and external flows is treated in greater detail in Chapters 5 and 6.
