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. Smith and Pamela M. Norris 19. Direct Contact Heat Transfer 1359 Robert F. Boehm Author Index 1401 Subject Index 1427 BOOKCOMP, Inc. — John Wiley & Sons / Page 1 / 2nd Proofs / Heat Transfer Handbook / Bejan 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [First Page] [1], (1) Lines: 0 to 95 ——— 11.25609pt PgVar ——— Normal Page * PgEnds: PageBreak [1], (1) 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 BOOKCOMP, Inc. — John Wiley & Sons / Page 2 / 2nd Proofs / Heat Transfer Handbook / Bejan 2 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [2], (2) Lines: 95 to 139 ——— 6.28009pt PgVar ——— Normal Page PgEnds: T E X [2], (2) 1.1 HEAT TRANSFER FUNDAMENTALS 1.1.1 Introduction Practitioners of the thermal arts and sciences generally deal with four basic thermal transport modes: conduction, convection, phase change, and radiation. The process by which heat diffuses through a solid or a stationary fluid is termed heat conduction. Situations in which heat transfer from a wetted surface is assisted by the motion of the fluid give rise to heat convection, and when the fluid undergoes a liquid–solid or liquid–vapor state transformation at or very near the wetted surface, attention is focused on this phase-change heat transfer. The exchange of heat between surfaces, or between a surface and a surrounding fluid, by long-wavelength electromagnetic radiation is termed thermal heat radiation. It is our intent in this section to describe briefly these modes of heat transfer, with emphasis on an important parameter known as the thermal resistance to heat transfer. Simple examples are given for illustration; detailed descriptions of the same topics are presented in specialized chapters. 1.1.2 Conduction Heat Transfer One-Dimensional Conduction Thermal diffusion through solids is governed by Fourier’s law, which in one-dimensional form is expressible as q =−kA dT dx (W) (1.1) where q is the heat current, k the thermal conductivity of the medium, A the cross- sectional area for heat flow, and dT /dx the temperature gradient, which, because it is negative, requires insertion of the minus sign in eq. (1.1) to assure a positive heat flow q. The temperature difference resulting from the steady-state diffusion of heat is thus related to the thermal conductivity of the material, the cross-sectional area A, and the path length L (Fig. 1.1), according to (T 1 − T 2 ) cd = q L kA (K) (1.2) The form of eq. (1.2), where k and A are presumed constant, suggests that in a way that is analogous to Ohm’s law governing electrical current flow through a resistance, it is possible to define a conduction thermal resistance as R cd ≡ T 1 − T 2 q = L kA (K/W) (1.3) One-Dimensional Conduction with Internal Heat Generation Situations in which a solid experiences internal heat generation, such as that produced by the flow of an electric current, give rise to more complex governing equations and require BOOKCOMP, Inc. — John Wiley & Sons / Page 3 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FUNDAMENTALS 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [3], (3) Lines: 139 to 173 ——— -2.4009pt PgVar ——— Normal Page PgEnds: T E X [3], (3) 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 = T o + q g L 2 2k x L − x L 2 where T o 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 T max = T o + q g L 2 8k (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 T max − T o = q L 2 8kLWδ = q L 8kA (1.5) 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. BOOKCOMP, Inc. — John Wiley & Sons / Page 4 / 2nd Proofs / Heat Transfer Handbook / Bejan 4 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [4], (4) Lines: 173 to 196 ——— -1.48895pt PgVar ——— Long Page PgEnds: T E X [4], (4) x 1 x 2 T max T o 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 R sp = 1 −1.410 +0.344 3 + 0.043 5 + 0.034 7 4ka (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 R sp . 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 BOOKCOMP, Inc. — John Wiley & Sons / Page 5 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FUNDAMENTALS 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [5], (5) Lines: 196 to 214 ——— 7.93205pt PgVar ——— Long Page * PgEnds: Eject [5], (5) Figure 1.3 Thermal resistance for a circular discrete heat source on a two-layer conducting medium. (From Yovanovich and Antonetti, 1988.) across the interface, but this resistance is generally negligible in engineering design. However, when dealing with real interfaces, the asperities present on each of the sur- faces (Fig. 1.4) limit actual contact between the two solids to a very small fraction of the apparent interface area. The flow of heat across the gap between two solids in nominal contact is by solid conduction in areas of actual contact and fluid conduction across the “open” spaces. Radiation across the gap can be important in a vacuum environment or when surface temperatures are high. The heat transferred across an interface can be found by adding the effects of solid-to-solid conduction and conduc- tion through the fluid and recognizing that solid-to-solid conduction in the contact zones involves heat flowing sequentially through the two solids. With the total con- tact conductance h co , taken as the sum of solid-to-solid conductance h c and the gap conductance h g , h co = h c + h g (W/m 2 · K) (1.7a) the contact resistance based on the apparent contact area A a may be defined as R co = 1 h co A a (K/W) (1.7b) In eq. (1.7a), h c is given by (Yovanovich and Antonetti, 1988) h c = 1.25k s m σ P H 0.95 (1.8a) BOOKCOMP, Inc. — John Wiley & Sons / Page 6 / 2nd Proofs / Heat Transfer Handbook / Bejan 6 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [6], (6) Lines: 214 to 261 ——— 0.00914pt PgVar ——— Normal Page * PgEnds: Eject [6], (6) 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 = 2k 1 k 2 k 1 + k 2 (W/m · K) σ is the effective root mean square (rms) surface roughness, σ = σ 2 1 + σ 2 2 1/2 (µm) m is the effective absolute surface slope, m = (m 2 1 + m 2 2 ) 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), h g is given by h g = k g Y + M (1.8b) where k g is the thermal conductivity of the gap fluid, Y is the distance between the mean planes (Fig. 1.4), given by Y σ = 1.185 −ln 3.132 P H 0.547 BOOKCOMP, Inc. — John Wiley & Sons / Page 7 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FUNDAMENTALS 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [7], (7) Lines: 261 to 319 ——— 12.20604pt PgVar ——— Normal Page PgEnds: T E X [7], (7) 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 = 1.25k s m σ P H 0.95 + k g Y + M A a −1 (1.9) 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(T s − T f )(W) (1.12) BOOKCOMP, Inc. — John Wiley & Sons / Page 8 / 2nd Proofs / Heat Transfer Handbook / Bejan 8 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [8], (8) Lines: 319 to 371 ——— -3.84573pt PgVar ——— Long Page * PgEnds: Eject [8], (8) The differences between convection to a rapidly moving fluid, a slowly flowing fluid, or a stagnant fluid, as well as variations in the convective heat transfer rate among various fluids, are reflected in the values of h. For a particular geometry and flow regime, h may be found from available empirical correlations and/or theoret- ical relations. Use of eq. (1.12) makes it possible to define the convective thermal resistance as R cv ≡ 1 hS (K/W) (1.13) Dimensionless Parameters Common dimensionless quantities that are used in the correlation of heat transfer data are the Nusselt number Nu, which relates the convective heat transfer coefficient to the conduction in the fluid: Nu ≡ h k/L = hL k The Prandtl number Pr, which is a fluid property parameter: Pr ≡ c p µ k = ν α the Grashof number Gr, which accounts for the bouyancy effect: Gr ≡ ρ 2 βgL 3 ∆T µ 2 and the Reynolds number Re, which relates the momentum in the flow to the viscious dissipation: Re ≡ ρ ˆ VL µ All thermal properties in the foregoing dimensionless groups apply to the fluid at its bulk temperature. Natural Convection In natural convection, fluid motion is induced by density differences resulting from temperature gradients in the fluid. The heat transfer coef- ficient for this regime can be related to the buoyancy and the thermal properties of the fluid through the Rayleigh number Ra, which is the product of the Grashof and Prandtl numbers: Ra ≡ ρ 2 βgc p µk L 3 ∆T = gβ αν L 3 ∆T where the fluid properties ρ, β,c p , µ, and k are evaluated at the fluid bulk temperature. Empirical correlations for the heat transfer coefficient in natural convection boundary layer flow have taken the form BOOKCOMP, Inc. — John Wiley & Sons / Page 9 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FUNDAMENTALS 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [9], (9) Lines: 371 to 411 ——— -1.95479pt PgVar ——— Long Page * PgEnds: Eject [9], (9) 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 576 El + 2.873 El 1/2 1/2 (1.15) where El is the channel Elenbaas number: El ≡ ρ 2 βgc p z 4 ∆T µkL and ∆T is the surface temperature minus the bulk fluid temperature, ∆T = T s −T f . 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 = 4k 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: . Page 2 / 2nd Proofs / Heat Transfer Handbook / Bejan 2 BASIC CONCEPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [2] ,. / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FUNDAMENTALS 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [3],. / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FUNDAMENTALS 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [5],