BOOKCOMP, Inc. — John Wiley & Sons / Page 432 / 2nd Proofs / Heat Transfer Handbook / Bejan 432 FORCED CONVECTION: INTERNAL 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 [432], (38) Lines: 1540 to 1609 ——— 13.15723pt PgVar ——— Normal Page * PgEnds: Eject [432], (38) q max HLW ≤ 0.57 k L 2 ( T max − T 0 ) Pr 4/99 1 + t D opt −67/99 · Be 47/99 (5.91) with Be = ∆PL 2 µα for the range 10 4 ≤ Re D h ≤ 10 6 10 6 ≤ Re L ≤ 10 8 10 11 ≤ Be ≤ 10 16 • Turbulent flow and entrance lengths: X D 10 X T D (5.60) • Turbulent flow friction factor: f 0.046Re −1/5 D 2 × 10 4 ≤ Re D ≤ 10 5 (see Fig. 5.13) (5.68) • Turbulent flow heat transfer: St ·Pr 2/3 f 2 (5.75) for Pr ≥ 0.5 Nu D = hD k = 0.023Re 4/5 D · Pr 1/3 (5.76) for Pr ≥ 0.50 2 × 10 4 ≤ Re D ≤ 10 6 Nu D = 0.023Re 4/5 D · Pr n (5.77) where n = 0.4 for heating the fluid and n = 0.3 for cooling the fluid in the range L D > 60 0.7 ≤ Pr ≤ 120 2500 ≤ Re D ≤ 1.24 × 10 5 Nu D = 0.027Re 4/5 D · Pr 1/3 µ µ 0 0.14 (5.78) in the range BOOKCOMP, Inc. — John Wiley & Sons / Page 433 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF FORCED CONVECTION RELATIONSHIPS 433 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 [433], (39) Lines: 1609 to 1688 ——— 0.55222pt PgVar ——— Normal Page PgEnds: T E X [433], (39) 0.70 ≤ Pr ≥ 16,700 Re D ≥ 10 4 Here µ 0 = µ(T 0 )(T 0 is the wall temperature) µ = µ(T m )(T m is the bulk temperature) Nu D = (f/2)Re D · Pr 1.07 + 900/Re D − 0.63/(1 + 10Pr) + 12.7(f/2) 1/2 (Pr 2/3 − 1) (5.79a) Nu D = (f/2)Re D · Pr 1.07 + 12.7(f/2) 1/2 (Pr 2/3 − 1) (5.79b) where 0.5 ≤ Pr ≤ 10 6 4000 ≤ Re D ≤ 5 × 10 6 and f from Fig. 5.13. Nu D = (f/2)(Re D − 10 3 )Pr 1 + 12.7(f/2) 1/2 (Pr 2/3 − 1) (5.80) where 0.5 ≤ Pr ≤ 10 6 2300 ≤ Re D ≤ 5 × 10 6 and f from Fig. 5.13. Nu D = 0.0214 Re 0.8 D − 100 Pr 0.4 (5.81a) where 0.5 ≤ Pr ≤ 1.510 4 ≤ Re D ≤ 5 × 10 6 Nu D = 0.012 Re 0.87 D − 280 Pr 0.4 (5.81b) where 1.5 ≤ Pr ≤ 500 3 × 10 3 ≤ Re D ≤ 10 6 Nu D = 6.3 + 0.0167Re 0.85 D · Pr 0.93 q 0 = constant (5.82) 4.8 + 0.0156Re 0.85 D · Pr 0.93 T 0 = constant (5.83) where for eqs. (5.82) and (5.83), 0.004 ≤ Pr ≤ 0.110 4 ≤ Re D ≤ 10 6 BOOKCOMP, Inc. — John Wiley & Sons / Page 434 / 2nd Proofs / Heat Transfer Handbook / Bejan 434 FORCED CONVECTION: INTERNAL 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 [434], (40) Lines: 1688 to 1776 ——— -3.56396pt PgVar ——— Normal Page PgEnds: T E X [434], (40) • Total heat transfer rate: q = hA w ∆T lm (5.85) • Isothermal wall: ∆T lm = ∆T in − ∆T out ln(∆T in /∆T out ) (5.86) q =˙mc p ∆T in 1 − e −hA w / ˙mc p (5.87) • Uniform heat flux: ∆T lm = ∆T in = ∆T out (5.89) NOMENCLATURE Roman Letter Symbols A cross-sectional area, m 2 A w wall area, m 2 (a) pressure at point 1, Pa B cross-section shape number, dimensionless Be Bejan number, dimensionless b length, m (b) pressure at point 2, Pa C cross-section shape factor, dimensionless C f,x local skin friction coefficient, dimensionless c p specific heat at constant pressure, J/kg·K D spacing, diameter, m D h hydraulic diameter, m f friction factor, dimensionless G z Graetz number, dimensionless H height, m h heat transfer coefficient, W/m 2 ·K specific bulk enthalpy, J/kg k thermal conductivity, W/m·K k s size of sand grain, m L flow length, m ˙m mass flow rate, kg/s N number of plate surfaces in one elemental channel, dimensionless Nu Nusselt number, dimensionless Nu x local Nusselt number, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 435 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 435 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 [435], (41) Lines: 1776 to 1794 ——— 0.20847pt PgVar ——— Normal Page PgEnds: T E X [435], (41) P pressure, Pa pressure difference, dimensionless ∆P pressure difference, Pa Pr Prandtl number, dimensionless Pr t turbulent Prandtl number, dimensionless p perimeter of cross section, m q heat flux, W/m 2 r radial position, m r h hydraulic radius, m r 0 tube radius, m Ra Rayleigh number, dimensionless Re D Reynolds number based on D, dimensionless Re D h Reynolds numbers based on D h , dimensionless Re L Reynolds number based on L, dimensionless S spacing between cylinders, m St Stanton number, dimensionless t plate thickness, m T temperature, K T in inlet temperature, K T out outlet temperature, K T m mean temperature, K ∆T avg average temperature difference, K ∆T lm log-mean temperature difference, K u longitudinal velocity, m/s u ∗ friction velocity, m/s U mean velocity, m/s v transversal velocity, m/s W width, m x longitudinal position, m x ∗ longitudinal position, dimensionless x + longitudinal position, dimensionless X flow entrance length, m X T thermal entrance length, m y transversal position, m y VSL viscous sublayer thickness, m Greek Letter Symbols α thermal diffusivity, m 2 /s H thermal eddy diffusivity, m 2 /s M momentum eddy diffusivity, m 2 /s θ ∗m bulk temperature, dimensionless µ viscosity, kg/s-m ν kinematic visocity, m 2 /s ρ density, kg/m 3 τ app apparent sheer stress, Pa BOOKCOMP, Inc. — John Wiley & Sons / Page 436 / 2nd Proofs / Heat Transfer Handbook / Bejan 436 FORCED CONVECTION: INTERNAL 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 [436], (42) Lines: 1794 to 1847 ——— 3.39505pt PgVar ——— Short Page PgEnds: T E X [436], (42) τ avg averaged wall shear stress, Pa τ w wall shear stress, Pa φ fully developed temperature profile, dimensionless Subscripts in inlet max maximum opt optimum out outlet 0 wall 0-x averaged longitudinally ∞ free stream Superscripts − time-averaged components fluctuating components + wall coordinates REFERENCES Asako, Y., Nakamura, H., and Faghri, M. (1988). Developing Laminar Flow and Heat Transfer in the Entrance Region of Regular Polygonal Ducts, Int. J. Heat Mass Transfer, 31, 2590– 2593. Bar-Cohen, A.,and Rohsenow, W. M. (1984). Thermally Optimum Spacing of Vertical, Natural Convection Cooled, Parallel Plates, J. Heat Transfer, 106, 116–123. Bejan, A. (1984). Convection Heat Transfer. Wiley, New York, p. 157, prob. 11. Bejan, A. (1993). Heat Transfer, Wiley, New York, Chap. 9. Bejan, A. (1995). Convection Heat Transfer, 2nd ed., Wiley, New York. Bejan, A. (2000). Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge. Bejan, A., and Morega, A. M. (1994). The Optimal Spacing of a Stack of Plates Cooled by Turbulent Forced Convection, Int. J. Heat Mass Transfer, 37, 1045–1048. Bejan, A., and Sciubba, E. (1992). The Optimal Spacing of Parallel Plates Cooled by Forced Convection, Int. J. Heat Mass Transfer, 35, 3259–3264. Bhattacharjee, S., and Grosshandler, W. L. (1988). The Formation of a Wall Jet near a High Temperature Wall under Microgravity Environment, ASME-HTD-96, ASME, New York, pp. 711–716. Campo, A. (1999). Bounds for the Optimal Conditions of Forced Convective Flows Inside Multiple Channels Whose Plates Are Heated by Uniform Flux, Int. Commun. Heat Mass Transfer, 26, 105–114. Campo, A., and Li, G. (1996). Optimum Separation of Asymmetrically Heated Sub-channels Forming a Bundle: Influence of Simultaneous Flow and Temperature, Heat Mass Transfer, 32, 127–132. BOOKCOMP, Inc. — John Wiley & Sons / Page 437 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 437 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 [437], (43) Lines: 1847 to 1887 ——— 5.0pt PgVar ——— Short Page PgEnds: T E X [437], (43) Churchill, S. W., and Ozoe, H. (1973). Correlations for Forced Convection with Uniform Heating in Flow over a Plate and in Developing and Fully Developed Flow in a Tube, J. Heat Transfer, 95, 78–84. Colburn, A. P. (1933). A Method of Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction, Trans. Am. Inst. Chem. Eng., 29, 174–210. Dittus, F. W., and Boelter, L. M. K. (1930). Heat Transfer in Automobile Radiators of the Tubular Type, Univ. Calif. Publ. Eng., 2(13), 443–461; Int. Commun. Heat Mass Transfer, 12(1985), 3–22. Drew, T. B. (1931). Mathematical Attacks on Forced Convection Problems: A Review, Trans. Am. Inst. Chem. Eng., 26, 26–80. Fowler, A. J., Ledezma, G. A., and Bejan, A. (1997). Optimal Geometric Arrangement of Staggered Plates in Forced Convection, Int. J. Heat Mass Transfer, 40, 1795–1805. Gnielinski, V. (1976). New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow, Int. Chem. Eng., 16, 359–368. Graetz, L. (1883). Uber die W ¨ armeleitungf ¨ ahigkeit von Fl ¨ ussigkeiten (On the Thermal Con- ductivity of Liquids), Part 1, Ann. Phys. Chem., 18, 79–94; Part 2 (1885), Ann. Phys. Chem., 25, 337–357. Hagen, G. (1839). Uber die Bewegung des Wassers in engen zylindrischen R ¨ uhren, Pogg. Ann., 46, 423. Hoffmann, E. (1937). Die W ¨ arme ¨ ubertragung bei der Str ¨ omung im Rohr, Z. Gesamte Kaelte- Ind., 44, 99–107. Hornbeck, R. W. (1965). An All-Numerical Method for Heat Transfer in the Inlet of a Tube, ASME-65-WA/HT-36, ASME, New York. Kays, W.M., and Perkins, H. C. (1973). Forced Convection, Internal Flow in Ducts, in Hand- book of Heat Transfer, W. M. Rohsenow and J. P. Hartnett, eds., McGraw-Hill, New York, Sec. 7. Kim, S. J., and Lee, S. W., eds. (1996). Air Cooling Technology for Electronic Equipment, CRC Press, Boca Raton, FL, Chap. 1. Langhaar, H. L. (1942). Steady Flow in the Transition Length of a Straight Tube, J. Appl. Mech., 9, A55–A58. Ledezma, G., Morega, A. M., and Bejan, A. (1996). Optimal Spacing between Pin Fins with Impinging Flow, J. Heat Transfer, 118, 570–577. L ´ ev ` eque, M. A. (1928). Les lois de la transmission de chaleur par convection, Ann. Mines Mem. Ser., 12, 13, 201–299, 305–362, 381–415. Matos, R. S., Vargas, J. V. C., Laursen, T. A., and Saboya, F. E. M. (2001). Optimization Study and Heat Transfer Comparison of Staggered Circular and Elliptic Tubes in Forced Convection, Int. J. Heat Mass Transfer, 44, 3953–3961. Mereu, S., Sciubba, E., and Bejan, A. (1993). The Optimal Cooling of a Stack of Heat Gen- erating Boards with Fixed Pressure Drop, Flow Rate or Pumping Power, Int. J. Heat Mass Transfer, 36, 3677–3686. Moody, L. F. (1944). Friction Factors for Pipe Flow, Trans. ASME, 66, 671–684. Nikuradse, J. (1933). Str ¨ omungsgesetze in rauhen R ¨ ohren, VDI-Forschungsh., 361, 1–22. Notter, R. H., and Sleicher, C. A. (1972). A Solution to the Turbulent Graetz Problem, III: Fully Developed and Entry Region Heat Transfer Rates, Chem. Eng. Sci., 27, 2073–2093. BOOKCOMP, Inc. — John Wiley & Sons / Page 438 / 2nd Proofs / Heat Transfer Handbook / Bejan 438 FORCED CONVECTION: INTERNAL 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] [438], (44) Lines: 1887 to 1919 ——— 168.04701pt PgVar ——— Normal Page PgEnds: T E X [438], (44) Petrescu, S. (1994). Comments on the Optimal Spacing of Parallel Plates Cooled by Forced Convection, Int. J. Heat Mass Transfer, 37, 1283. Petukhov, B. S. (1970). Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, Adv. Heat Transfer, 6, 503–564. Petukhov, B. S., and Kirilov, V. V. (1958). The Problem of Heat Exchange in the Turbulent Flow of Liquids in Tubes, Teploenergetika, 4(4), 63–68. Petukhov, B. S., and Popov, V. N. (1963). Theoretical Calculation of Heat Exchange in Tur- bulent Flow in Tubes of an Incompressible Fluid with Variable Physical Properties, High Temp., 1(1), 69–83. Poiseuille, J. (1840). Recherches exp ´ erimentales sur le mouvement des liquides dans les tubes de tr ` es petit diam ` etres, Comptes Rendus, 11, 961, 1041. Prandtl, L. (1969). Essentials of Fluid Dynamics, Blackie and Son, London, p. 117. Reichardt, H. (1951). Die Grundlagen des turbulenten W ¨ arme ¨ uberganges, Arch. Gesamte Waermetech., 2, 129–142. Rocha, L. A. O., and Bejan, A. (2001). Geometric Optimization of Periodic Flow and Heat Transfer in a Volume Cooled by Parallel Tubes, J. Heat Transfer, 123, 233–239. Schlichting, H. (1960). Boundary Layer Theory, 4th ed., McGraw-Hill, New York, pp. 169, 489. Shah, R. K., and Bhatti, M. S. (1987). Laminar Convective Heat Transfer inDucts, in Handbook of Single-Phase Convective Heat Transfer, S. Kakac¸, R. K. Shah, and W. Aung, Wiley, New York, Chap. 3. Shah, R. K., and London, A. L. (1978). Laminar Flow Forced Convection in Ducts, Suppl. 1 to Advances in Heat Transfer, Academic Press, New York. Sieder, E. N., and Tate, G. E. (1936). Heat Transfer and Pressure Drop of Liquids in Tubes, Ind. Eng. Chem., 28, 1429–1436. Sparrow, E. M. (1955). Analysis of Laminar Forced Convection Heat Transfer in the Entrance Region of Flat Rectangular Ducts, NACA-TN-3331. Stanescu, G., Fowler, A. J., and Bejan, A. (1996). The Optimal Spacing of Cylinders in Free- Stream Cross-Flow Forced Convection, Int. J. Heat Mass Transfer, 39, 311–317. Stephan, K. (1959). W ¨ arme ¨ ubertragang und Druckabfall beinichtausgebildeter Laminar- str ¨ omung in R ¨ ohren und in ebenen Spalten, Chem. Ing. Tech., 31, 773–778. BOOKCOMP, Inc. — John Wiley & Sons / Page 439 / 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] [439], (1) Lines: 0 to 99 ——— 2.76408pt PgVar ——— Normal Page PgEnds: T E X [439], (1) CHAPTER 6 Forced Convection: External Flows YOGENDRA JOSHI George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia WATARU NAKAYAMA Therm Tech International Kanagawa, Japan 6.1 Introduction 6.2 Morphology of external flow heat transfer 6.3 Analysis of external flow heat transfer 6.4 Heat transfer from single objects in uniform flow 6.4.1 High Reynolds number flow over a wedge 6.4.2 Similarity transformation technique for laminar boundary layer flow 6.4.3 Similarity solutions for the flat plate at uniform temperature 6.4.4 Similarity solutions for a wedge Wedge flow limits 6.4.5 Prandtl number effect 6.4.6 Incompressible flow past a flat plate with viscous dissipation 6.4.7 Integral solutions for a flat plate boundary layer with unheated starting length Arbitrarily varying surface temperature 6.4.8 Two-dimensional nonsimilar flows 6.4.9 Smith–Spalding integral method 6.4.10 Axisymmetric nonsimilar flows 6.4.11 Heat transfer in a turbulent boundary layer Axisymmetric flows Analogy solutions 6.4.12 Algebraic turbulence models 6.4.13 Near-wall region in turbulent flow 6.4.14 Analogy solutions for boundary layer flow Mixed boundary conditions Three-layer model for a “physical situation” Flat plate with an unheated starting length in turbulent flow Arbitrarily varying heat flux Turbulent Prandtl number 6.4.15 Surface roughness effect 6.4.16 Some empirical transport correlations Cylinder in crossflow Flow over an isothermal sphere 439 BOOKCOMP, Inc. — John Wiley & Sons / Page 440 / 2nd Proofs / Heat Transfer Handbook / Bejan 440 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 [440], (2) Lines: 99 to 154 ——— -2.0pt PgVar ——— Normal Page PgEnds: T E X [440], (2) 6.5 Heat transfer from arrays of objects 6.5.1 Crossflow across tube banks 6.5.2 Flat plates Stack of parallel plates Offset strips 6.6 Heat transfer from objects on a substrate 6.6.1 Flush-mounted heat sources 6.6.2 Two-dimensional block array 6.6.3 Isolated blocks 6.6.4 Block arrays 6.6.5 Plate fin heat sinks 6.6.6 Pin fin heat sinks 6.7 Turbulent jets 6.7.1 Thermal transport in jet impingement 6.7.2 Submerged jets Average Nusselt number for single jets Average Nusselt number for an array of jets Free surface jets 6.8 Summary of heat transfer correlations Nomenclature References 6.1 INTRODUCTION This chapter is concerned with the characterization of heat transfer and flow under forced convection, where the fluid movement past a heated object is induced by an ex- ternal agent such as a fan, blower, or pump. The set of governing equations presented in Chapter 1 is nonlinear in general, due to the momentum advection terms, vari- able thermophysical properties (e.g., with temperature) and nonuniform volumetric heat generation. Solution methodologies for the governing equations are based on the nondimensional groups discussed in Section 6.3. Solutions can be obtained through analytical means only for a limited number of cases. Otherwise, experimental or nu- merical solution procedures must be employed. 6.2 MORPHOLOGY OF EXTERNAL FLOW HEAT TRANSFER Various cases arise from the geometry of a heated object and the constraint imposed on the fluid flow. Figure 6.1 shows the general configuration in which it is assumed that the body is being cooled by the flow. The heated object is an arbitrary shape enclosed in a rectangular envelope. The dimensions of the envelope are L, the length in the streamwise direction, W , the length in the cross-stream direction (the width), and H , the height. Generally, the fluid flow is constrained by the presence of bounding BOOKCOMP, Inc. — John Wiley & Sons / Page 441 / 2nd Proofs / Heat Transfer Handbook / Bejan MORPHOLOGY OF EXTERNAL FLOW HEAT TRANSFER 441 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 [441], (3) Lines: 154 to 167 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [441], (3) Figure 6.1 Heated object in a flow over a bounding surface. surfaces. The bounding surface may be a solid wall or an interface with a fluid of a different kind, for instance, a liquid–vapor interface. The distance s x signifies the extent of the bounding surface in the streamwise direction, and s z is the distance to the bounding surface. When s z H or L and s z s x , the flow around the object is uniform and free of the effect of the bounding surface. Otherwise, the object is within a boundary layer developing on a larger object. In laboratory experiments and many types of industrial equipment, one often finds a situation where the object is placed in a duct. When the duct cross section has dimensions comparable to the object size, the flow has a velocity distribution defined by the duct walls and the object. Hence, the foregoing relations between H, L, s x , and s z can be put into more precise forms involving the velocity and viscosity of the fluid as well. In an extreme case, the object is in contact with the bounding surface; that is, s z = 0. In such cases the flow and temperature fields are generally defined by both the bounding surface and the object. Only in cases where the object dimensions are much smaller than those of the bounding surface is the external flow defined primarily by the bounding surface. Several external flow configurations are illustrated in Fig. 6.2. The symbols used to define the dimensions are conventionally related to the flow direction. For the flat plate in Fig. 6.2a, is the plate length in the streamwise direction, w is the length (width) in the cross-stream direction, and t is the plate thickness. The cylinder in Fig. 6.2b has length and diameter d. For the rectangular block of Fig. 6.2c, is oriented in the streamwise direction, h is the height, and w is the width. Sometimes, these letters can be used as subscripts to a common symbol for the block. The sphere (Fig. 6.2d)is defined by only one dimension, that is, the diameter d. Although an infinite number of configurations can be conceived from the combination of external flow and object geometry, only a limited number of cases have been the subject of theoretical studies as well as practical applications. The most common are two-dimensional objects in uniform flow, which are used in basic research and teaching. . Natural Convection Cooled, Parallel Plates, J. Heat Transfer, 106, 116–123. Bejan, A. (1984). Convection Heat Transfer. Wiley, New York, p. 157, prob. 11. Bejan, A. (1993). Heat Transfer, Wiley, New York, Chap Forced Convection, Int. J. Heat Mass Transfer, 37, 1283. Petukhov, B. S. (1970). Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, Adv. Heat Transfer, 6, 503–564. Petukhov,. International Kanagawa, Japan 6.1 Introduction 6.2 Morphology of external flow heat transfer 6.3 Analysis of external flow heat transfer 6.4 Heat transfer from single objects in uniform flow 6.4.1 High Reynolds