BOOKCOMP, Inc. — John Wiley & Sons / Page 522 / 2nd Proofs / Heat Transfer Handbook / Bejan 522 FORCED CONVECTION: EXTERNAL FLOWS 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 [522], (84) Lines: 3700 to 3757 ——— 5.66856pt PgVar ——— Short Page PgEnds: T E X [522], (84) T thermal turbulent w width xx-coordinate direction yy-coordinate direction zz-coordinate direction Greek Letter Subscripts ∞ infinity ∆p pressure loss δ thickness of boundary layer, dimensionless Superscripts m exponent N exponent q exponent + normalized variable ∗ normalized variable first derivative second derivative third derivative Other ∂ partial derivative ∇ vector operator REFERENCES Anderson, A. M. (1994). Decoupling of Convective and Conductive Heat Transfer Using the Adiabatic Heat Transfer Coefficient, ASME J. Electron. Packag., 116, 310–316. Arvisu, D. E., and Moffat, R. J. (1982). The Use of Superposition in Calculating Cooling Requirements for Circuit Board Mounted Electronic Components, Proc. 32nd Elect. Comp. Cont., San Diego, CA. Ashiwake, N., Nakayama, W., and Daikoku, T. (1983). Convective Heat Transfer from LSI Packages in an Air-Cooled Wiring Card Array, ASME-HTD-28, ASME, New York, pp. 35–42. Bejan, A., and Sciubba, E. (1992). The Optimal Spacing of Parallel Plates Cooled by Forced Convection, Int. J. Heat Mass Transfer, 35, 3259–3264. Blasius, H. (1908). Grenzschichten in Fl ¨ ussigkeiten mit kleiner Reibung, Z. Math. Phys., 56, 1. Cebeci, T., and Bradshaw, P. (1984). Physical and Computational Aspects of Convective Heat Transfer, Springer-Verlag, New York. Churchill, S. W., and Bernstein, M.(1977). A Correlating Equation for Forced Convection from Gases and Liquids to a Circular Cylinder in Cross Flow, J. Heat Transfer, 99, 300–306. BOOKCOMP, Inc. — John Wiley & Sons / Page 523 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 523 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 [523], (85) Lines: 3757 to 3792 ——— 9.0pt PgVar ——— Short Page PgEnds: T E X [523], (85) Davalath, J., and Bayazitoglu, Y. (1987). Forced Convection Cooling across Rectangular Blocks, J. Heat Transfer, 109, 321–328. DeJong, N. C., Zhang, L. W., Jacobi, A. M., Balachandar, A. M., and Tafti, D. K. (1998). A Complementary Experimental and Numerical Study of the Flow and Heat Transfer in Offset Strip-Fin Heat Exchangers, J. Heat Transfer, 120, 690–698. Gebhart, B. (1971). Convective Heat Transfer, McGraw-Hill, New York, Chap. 7. Gebhart, B. (1980). Convective Heat Transfer, Spring Course Notes, Department of Mechanical Engineering, State University of New York, Buffalo, NY. Gorski, M. A., and Plumb, O. A. (1990). Conjugate Heat Transfer from a Finite Strip Heat Source in a Plane Wall, ASME-HTD-129, AIAA-ASME Thermophysics and Heat Transfer Conference, Seattle, WA, ASME, New York, pp. 47–53. Gorski, M. A., and Plumb, O. A. (1992). Conjugate Heat Transfer from an Isolated Heat Source in a Plane Wall, in Fundamentals of Forced Convection Heat Transfer, ASME-HTD-210, M. A. Ebadian and P. H. Oosthuizen, eds., ASME, New York, pp. 99–105. Incropera, F. P. (1999). Liquid Cooling of Electronic Devices by Single-Phase Convection, Wiley, New York. Incropera, F. P., and DeWitt, D. P. (1996). Fundamentals of Heat and Mass Transfer, 4th ed., Wiley, New York. Joshi, H. M., and Webb, R. L. (1987). Heat Transfer and Friction in the Offset Strip-Fin Heat Exchanger, Int. J. Heat Mass Transfer, 30, 69–83. Kays, W. M., and Crawford, M. (1993). Convective Heat and Mass Transfer, McGraw Hill, New York. Ledezma, G., Morega, A. M., and Bejan, A. (1996). Optimal Spacing between Pin Fins with Impinging Flow, J. Heat Transfer, 118, 570–577. Lehman, G. L., and Wirtz, R. A. (1985). The Effect of Variations in Stream-wise Spacing and Length on Convection from Surface Mounted Rectangular Components, in Heat Transfer in Electronic Equipment, ASME-HTD-48, ASME, New York, pp. 39–47. Martin, H. (1977). Heat and Mass Transfer between Impinging Gas Jets and Solid Surfaces, in Advances in Heat Transfer, Vol. 13, J. P. Hartnett and T. F. Irvine, Jr., eds., Academic Press, New York. Matsushima, H., and Yanagida, T. (1993). Heat Transfer from LSI Packages with Longitudinal Fins in Free Air Stream, in Advances in Electronic Packaging, ASME-EEP-4-2, ASME, New York, pp. 793–800. Moffat, R. J., andOrtega, A. (1988). Direct Air Cooling of Electronic Components, in Advances in Thermal Modeling of Electronic Components and Systems, Vol. 1, A. Bar-Cohen and A. D. Kraus, eds., Hemisphere Publishing, New York. Morega, A. M., Bejan, A., and Lee, S. W. (1995). Free Stream Cooling of Parallel Plates, Int. J. Heat Mass Transfer, 38(3), 519–531. Nakayama, W., and Park, S H. (1996). Conjugate Heat Transfer from a Single Surface- Mounted Block to Forced Convective Air Flow in a Channel, J. Heat Transfer, 118, 301– 309. Nakayama, W., Matsushima, H., and Goel, P. (1988). Forced Convective Heat Transfer from Arrays of Finned Packages, in Cooling Technology for Electronic Equipment, W. Aung, ed., Hemisphere Publishing, New York, pp. 195–210. BOOKCOMP, Inc. — John Wiley & Sons / Page 524 / 2nd Proofs / Heat Transfer Handbook / Bejan 524 FORCED CONVECTION: EXTERNAL FLOWS 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 [Last Page] [524], (86) Lines: 3792 to 3820 ——— 161.04701pt PgVar ——— Normal Page PgEnds: T E X [524], (86) Oosthuizen, P. H., and Naylor, D. (1999). Convective Heat Transfer Analysis, WCB/McGraw- Hill, New York, Chap. 6. Ortega, A. (1996). Conjugate Heat Transfer in Forced Air Cooling of Electronic Components, in Air Cooling Technology for Electronic Equipment, S. J. Kim and S. W. Lee, eds., CRC Press, Boca Raton, FL, pp. 103–171. Ortega, A., Wirth, U., and Kim, S. J. (1994). Conjugate Forced Convection from a Discrete Heat Source on a Plane Conducting Surface: A Benchmark Experiment, in Heat Transfer in Electronic Systems, ASME-HTD-292, ASME, New York, pp. 25–36. Pohlhausen, E. (1921). Der W ¨ armeaustausch zwishen festen K ¨ orpern und Fl ¨ ussigkeiten mit kleiner Reibung und kleiner W ¨ armeleitung, Z. Angew. Math. Mech., 1, 115–121. Roeller, P. T., and Webb, B. (1992). A Composite Correlation for Heat Transfer from Isolated Two- and Three Dimensional Protrusions in Channels, Int. J. Heat Mass Transfer, 35(4), 987–990. Roeller, P. T., Stevens, J., and Webb, B. (1991). Heat Transfer and Turbulent Flow Characteris- tics of Isolated Three-Dimensional Protrusions in Channels, J. Heat Transfer, 113, 597–603. Smith, A. G., and Spalding, D. B. (1958). Heat Transfer in a Laminar Boundary Layer with Constant Fluid Properties and Constant Wall Temperature, J. R. Aeronaut. Soc., 62, 60–64. Whitaker, S. (1972). Forced Convection Heat Transfer Correlations for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres and for Flow in Packed Beds and Tube Bundles, AIChE J., 18, 361–371. Wirtz, R. A., and Dykshoorn, P. (1984). Heat Transfer from Arrays of Flat Packs in Channel Flow, Proc. 4th International Electronics Packaging Society Conferences, New York, pp. 318–326. Wirtz, R. A., Sohal, R., and Wang, H. (1997). Thermal Performance of Pin-Fin Fan-Sink Assemblies, J. Electron. Packag., 119, 26–31. Womac, D. J., Ramadhyani, S., and Incropera, F. P. (1993). Correlating Equations for Impinge- ment Cooling of Small Heat Sources with Single Circular Liquid Jets, J. Heat Transfer, 115, 106–115. Zhukauskas, A. (1972). Heat Transfer from Tubes in Cross Flow, in Advances in Heat Transfer, J. P. Hartnett and T. F. Irvine, eds., Vol. 8, Academic Press, New York. Zhukauskas, A. (1987). Convective Heat Transfer in Cross Flow, in Convective Heat Transfer, S. Kakac¸, R. K. Shah, and W. Aung, eds., Wiley, New York. BOOKCOMP, Inc. — John Wiley & Sons / Page 525 / 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] [525], (1) Lines: 0 to 83 ——— -0.65392pt PgVar ——— Normal Page PgEnds: T E X [525], (1) CHAPTER 7 Natural Convection YOGESH JALURIA Mechanical and Aerospace Engineering Department Rutgers University New Brunswick, New Jersey 7.1 Introduction 7.2 Basic mechanisms and governing equations 7.2.1 Governing equations 7.2.2 Common approximations 7.2.3 Dimensionless parameters 7.3 Laminar natural convection flow over flat surfaces 7.3.1 Vertical surfaces 7.3.2 Inclined and horizontal surfaces 7.4 External laminar natural convection flow in other circumstances 7.4.1 Horizontal cylinder and sphere 7.4.2 Vertical cylinder 7.4.3 Transients 7.4.4 Plumes, wakes, and other free boundary flows 7.5 Internal natural convection 7.5.1 Rectangular enclosures 7.5.2 Other configurations 7.6 Turbulent flow 7.6.1 Transition from laminar flow to turbulent flow 7.6.2 Turbulence 7.7 Empirical correlations 7.7.1 Vertical flat surfaces 7.7.2 Inclined and horizontal flat surfaces 7.7.3 Cylinders and spheres 7.7.4 Enclosures 7.8 Summary Nomenclature References 7.1 INTRODUCTION The convective mode of heat transfer involves fluid flow along with conduction, or diffusion, and is generally divided into two basic processes. If the motion of the 525 BOOKCOMP, Inc. — John Wiley & Sons / Page 526 / 2nd Proofs / Heat Transfer Handbook / Bejan 526 NATURAL CONVECTION 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 [526], (2) Lines: 83 to 95 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [526], (2) fluid arises from an external agent, for instance, a fan, a blower, the wind, or the motion of the heated object itself, which imparts the pressure to drive the flow, the process is termed forced convection. If, on the other hand, no such externally induced flow exists and the flow arises “naturally” from the effect of a density difference, resulting from a temperature or concentration difference in a body force field such as gravity, the process is termed natural convection. The density difference gives rise to buoyancy forces due to which the flow is generated. A heated body cooling in ambient air generates such a flow in the region surrounding it. The buoyant flow arising from heat or material rejection to the atmosphere, heating and cooling of rooms and buildings, recirculating flow driven by temperature and salinity differences in oceans, and flows generated by fires are other examples of natural convection. There has been growing interest in buoyancy-induced flows and the associated heat and mass transfer over the past three decades, because of the importance of these flows in many different areas, such as cooling of electronic equipment, pollution, materials processing, energy systems, and safety in thermal processes. Several books, reviews, and conference proceedings may be consulted for detailed presentations on this subject. See, for instance, the books by Turner (1973), Jaluria (1980), Kakac¸et al. (1985), and Gebhart et al. (1988). The main difference between natural and forced convection lies in the mechanism by which flow is generated. In forced convection, externally imposed flow is generally known, whereas in natural convection it results from an interaction of the density difference with the gravitational (or some other body force) field and is therefore inevitably linked with and dependent on the temperature and/or concentration fields. Thus, the motion that arises is not known at the onset and has to be determined from a consideration of the heat and mass transfer process which are coupled with fluid flow mechanisms. Also, velocities and the pressure differences in natural convection are usually much smaller than those in forced convection. The preceding differences between natural and forced convection make the ana- lytical and experimental study of processes involving natural convection much more complicated than those involving forced convection. Special techniques and methods have therefore been devised to study the former, with a view to providing information on the flow and on the heat and mass transfer rates. To understand the physical nature of natural convection transport, let us consider the heat transfer from a heated vertical surface placed in an extensive quiescent medium at a uniform temperature, as shown in Fig. 7.1. If the plate surface tem- perature T w is greater than the ambient temperature T ∞ , the fluid adjacent to the vertical surface gets heated, becomes lighter (assuming that it expands on heating), and rises. Fluid from the neighboring areas moves in, due to the generated pressure differences, to take the place of this rising fluid. Most fluids expand on heating, result- ing in a decrease in density as the temperature increases, a notable exception being water between 0 and 4°C. If the vertical surface is initially at temperature T ∞ , and then, at a given instant, heat is turned on, say through an electric current, the flow undergoes a transient before the flow shown is achieved. It is the analysis and study of this time-dependent as well as steady flow that yields the desired information on the heat transfer rates, flow and temperature fields, and other relevant process variables. BOOKCOMP, Inc. — John Wiley & Sons / Page 527 / 2nd Proofs / Heat Transfer Handbook / Bejan INTRODUCTION 527 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 [527], (3) Lines: 95 to 105 ——— * 19.097pt PgVar ——— Normal Page * PgEnds: Eject [527], (3) T w T w x x Flow Flow Velocity boundary layer Entrainment Entrainment T ϱϱ , T ϱϱ , TT w > ϱ TT w < ϱ y y Body force (gravity field) ()a ()b Figure 7.1 Natural convection flow over a vertical surface, together with the coordinate system. The flow adjacent to a cooled surface is downward, as shown in Fig. 7.1b, provided that the fluid density decreases with an increase in temperature. Heat transfer from the vertical surface may be expressed in terms of the commonly used Newton’s law of cooling, which gives the relationship between the heat transfer rate q and the temperature difference between the surface and the ambient as q = ¯ hA(T w − T ∞ ) (7.1) where ¯ h is the average convective heat transfer coefficient and A is the total areaof the vertical surface. The coefficient ¯ h depends on the flow configuration, fluid properties, dimensions of the heated surface, and generally also on the temperature difference, because of which the dependence of q on T w −T ∞ is not linear. Since the fluid motion becomes zero at the surface due to the no-slip condition, which is generally assumed to apply, the heat transfer from the heated surface to the fluid in its immediate vicinity is by conduction. It is therefore given by Fourier’s law as BOOKCOMP, Inc. — John Wiley & Sons / Page 528 / 2nd Proofs / Heat Transfer Handbook / Bejan 528 NATURAL CONVECTION 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 [528], (4) Lines: 105 to 137 ——— 1.76207pt PgVar ——— Normal Page PgEnds: T E X [528], (4) q =−kA ∂T ∂y 0 (7.2) Here the temperature gradient is evaluated at the surface, y = 0, in the fluid and k is the thermal conductivity of the fluid. From this equation it is obvious that the natural convection flow largely affects the temperature gradient at the surface, since the remaining parameters remain essentially unaltered. The analysis is therefore directed at determining this gradient, which in turn depends on the nature and characteristics of the flow, temperature field, and fluid properties. The heat transfer coefficient ¯ h represents an integrated value for the heat transfer rate over the entire surface, since, in general, the local value h x would vary with the vertical distance from the leading edge, x = 0, of the vertical surface. The local heat transfer coefficient h x is defined by the equation q = h x (T w − T ∞ ) (7.3) where q is the rate of heat transferper unit area perunit time at alocation x,where the surface temperature difference is T w − T ∞ , which may itself be a function of x. The average heat transfer coefficient ¯ h is obtained from eq. (7.3) through integration over the entire surface area. Both ¯ h and h x are generally given in terms of a nondimensional parameter called the Nusselt number Nu. Again, an overall (or average) value Nu, and a local value Nu x , may be defined as Nu = ¯ hL k Nu x = h x x k (7.4) where L is the height of the vertical surface and thus represents a characteristic dimension. The fluid far from the vertical surface is stationary, since an extensive medium is considered. The fluid next to the surface is also stationary, due to the no-slip condition. Therefore, flow exists in a layer adjacent to the surface, with zero vertical velocity on either side, as shown in Fig. 7.2. A small normal velocity component does exist at the edge of this layer, due to entrainment into the flow. The temperature varies from T w to T ∞ . Therefore, the maximum vertical velocity occurs at some distance away from the surface. Its exact location and magnitude have to be determined through analysis or experimentation. The flow near the bottom or leading edge of the surface is laminar, as indicated by a well-ordered and well-layered flow, with no significant disturbance. However, as the flow proceeds vertically upward or downstream, the flow gets more and more disorderly and disturbed, because of flow instability, eventually becoming chaotic and random, a condition termed turbulent flow. The region between the laminar and turbulent flow regimes is termed the transition region. Its location and extent depend on several variables, such as the temperature of the surface, the fluid, and the nature and magnitude of external disturbances in the vicinity of the flow. Most of the processes encountered in nature are generally turbulent. However, flows in many BOOKCOMP, Inc. — John Wiley & Sons / Page 529 / 2nd Proofs / Heat Transfer Handbook / Bejan BASIC MECHANISMS AND GOVERNING EQUATIONS 529 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 [529], (5) Lines: 137 to 155 ——— 0.69102pt PgVar ——— Normal Page PgEnds: T E X [529], (5) Figure 7.2 Velocity and temperature distributions in natural convection flow over a vertical surface. industrial applications, such as those in electronic systems, are often in the laminar or transition regime. A determination of the regime of the flow and its effect on the flow parameters and heat transfer rates is therefore important. Natural convection flow may also arise in enclosed regions. This flow, which is generally termed internal natural convection, is different in many ways from the ex- ternal natural convection considered in the preceding discussion on a vertical heated surface immersed in an extensive, quiescent, isothermal medium. Buoyancy-induced flows in rooms, transport in complete or partial enclosures containing electronic equipment, flows in enclosed water bodies, and flows in the liquid melts of solidify- ing materials are examples of internal natural convection. In this chapter we discuss both external and internal natural convection for a variety of flow configurations and circumstances. 7.2 BASIC MECHANISMS AND GOVERNING EQUATIONS 7.2.1 Governing Equations The governing equations for a convective heat transfer process are obtained by con- siderations of mass and energy conservation and of the balance between the rate of momentum change and applied forces. These equations may be written, for constant viscosity µ and zero bulk viscosity, as (Gebhart et al., 1988) Dρ Dt = ∂ρ ∂t + V ·∇ρ =−ρ∇·V (7.5) BOOKCOMP, Inc. — John Wiley & Sons / Page 530 / 2nd Proofs / Heat Transfer Handbook / Bejan 530 NATURAL CONVECTION 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 [530], (6) Lines: 155 to 188 ——— 5.08006pt PgVar ——— Long Page * PgEnds: Eject [530], (6) ρ DV Dt = ρ ∂V ∂t + V ·∇V = F −∇p +µ∇ 2 V + µ 3 ∇(∇·V) (7.6) ρc p DT Dt = ρc p ∂T ∂t + V ·∇T =∇·(k∇T)+q + βT Dp Dt + µΦ v (7.7) where V is the velocity vector, T the local temperature, t the time, F the body force per unit volume, c p the specific heat at constant pressure, p the static pressure, ρ the fluid density, β the coefficient of thermal expansion of the fluid, Φ v the viscous dissipation (which is the irreversible part of the energy transfer due to viscous forces), and q the energy generation per unit volume. The coefficient of thermal expansion β =−(1/ρ)(∂ρ/∂T ) p , where the subscript p denotes constant pressure. For a perfect gas, β = 1/T , where T is the absolute temperature. The total, or particle, derivative D/Dt may be expressed in terms of local derivative as ∂/∂t + V ·∇. As mentioned earlier, in natural convection flows, the basic driving force arises from the temperature (or concentration) field. The temperature variation causes a difference in density, which then results in a buoyancy force due to the presence of the body force field. For a gravitational field, the body force F = ρg, where g is the gravitational acceleration. Therefore, it is the variation of ρ with temperature that gives rise to the flow. The temperature field is linked with the flow, and all the preceding conservation equations are coupled through variation in the density ρ. Therefore, these equations have to be solved simultaneously to determine the velocity, pressure, and temperature distributions in space and in time. Due to this complexity in the analysis of the flow, several simplifying assumptions and approximations are generally made to solve natural convection flows. In the momentum equation, the local static pressure p may be broken down into two terms: one, p a , due to the hydrostatic pressure, and other other, p d , the dynamic pressure due to the motion of the fluid (i.e., p = p a + p d ). The former pressure component, coupled with the body force acting on the fluid, constitutes the buoyancy force that is driving mechanism for the flow. If ρ ∞ is the density of the fluid in the ambient medium, we may write the buoyancy term as F −∇p = (ρg −∇p a ) −∇p d = (ρg −ρ ∞ g) −∇p d = (ρ −ρ ∞ )g −∇p d (7.8) If g is downward and the x direction is upward (i.e., g =−ig, where i is the unit vector in the x direction and g is the magnitude of the gravitational acceleration, as is generally the case for vertical buoyant flows), then F −∇p = (ρ ∞ − ρ)gi −∇p d (7.9) and the buoyancy term appears only in the x-direction momentum equation. There- fore, the resulting governing equations for natural convection are the continuity equa- tion, eq. (7.5), the energy equation, eq. (7.7), and the momentum equation, which becomes ρ DV Dt = (ρ −ρ ∞ )g −∇p d + µ∇ 2 V + µ 3 ∇(∇·V) (7.10) BOOKCOMP, Inc. — John Wiley & Sons / Page 531 / 2nd Proofs / Heat Transfer Handbook / Bejan BASIC MECHANISMS AND GOVERNING EQUATIONS 531 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 [531], (7) Lines: 188 to 216 ——— 3.00002pt PgVar ——— Long Page PgEnds: T E X [531], (7) 7.2.2 Common Approximations The governing equations for natural convection flow are coupled, elliptic, partial differential equations and are therefore of considerable complexity. Another problem in obtaining a solution to these equations lies in the inevitable variation of the density ρ with temperature or concentration. Several approximations are generally made to simplify these equations. Two of the most important among these are the Boussinesq and the boundary layer approximations. The Boussinesq approximations involve two aspects. First, the density variation in the continuity equation is neglected. Thus, the continuity equation, eq. (7.5), be- comes ∇·V = 0. Second, the density difference, which causes the flow, is ap- proximated as a pure temperature or concentration effect (i.e., the effect of pressure on the density is neglected). In fact, the density difference is estimated for thermal buoyancy as ρ ∞ − ρ = ρβ(T −T ∞ ) (7.11) These approximations are employed very extensively for natural convection. An im- portant condition for the validity of these approximations is that β(T − T ∞ ) 1 (Jaluria, 1980). Therefore, the approximations are valid for small temperature dif- ferences if β is essentially unchanged. However, they are not valid near the density maximum of water at 4°C, where β is zero and changes sign as the temperature varies across this value (Gebhart, 1979). Similarly, for large temperature differences en- countered in fire and combustion systems, these approximations are generally not applicable. Another approximation made in the governing equations is the extensively em- ployed boundary layer assumption. The basic concepts involved in using the bound- ary layer approximation in natural convection flows are very similar to those in forced flow. The main difference lies in the fact that the pressure in the region outside the boundary layer is hydrostatic instead of being the externally imposed pressure, as is the case in forced convection. The velocity outside the layer is only the entrainment velocity due to the motion pressure and is not an imposed free stream velocity. How- ever, the basic treatment and analysis are quite similar. It is assumed that the flow and the energy, or mass, transfer, from which it arises, are restricted predominantly to a thin region close to the surface. Several experimental studies have corroborated this assumption. As a consequence, the gradients along the surface are assumed to be much smaller than those normal to it. The main consequences of the boundary layer approximations are that the down- stream diffusion terms in the momentum and energy equations are neglected in com- parison with the normal diffusion terms. The normal momentum balance is neglected since it is found to be of negligible importance compared to the downstream balance. Also, the velocity and thermal boundary layer thicknesses, δ and δ T , respectively, are given by the order-of-magnitude expressions δ L = O 1 Gr 1/4 (7.12) . Offset Strip-Fin Heat Exchangers, J. Heat Transfer, 120, 690–698. Gebhart, B. (1971). Convective Heat Transfer, McGraw-Hill, New York, Chap. 7. Gebhart, B. (1980). Convective Heat Transfer, Spring. Fundamentals of Heat and Mass Transfer, 4th ed., Wiley, New York. Joshi, H. M., and Webb, R. L. (1987). Heat Transfer and Friction in the Offset Strip-Fin Heat Exchanger, Int. J. Heat Mass Transfer, . Cooling of Small Heat Sources with Single Circular Liquid Jets, J. Heat Transfer, 115, 106–115. Zhukauskas, A. (1972). Heat Transfer from Tubes in Cross Flow, in Advances in Heat Transfer, J. P.