BOOKCOMP, Inc. — John Wiley & Sons / Page 391 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 391 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 [391], (131) Lines: 4994 to 5038 ——— 0.0pt PgVar ——— Custom Page (7.0pt) PgEnds: T E X [391], (131) Song, S., Moran, K. P., Augi, R., and Lee, S. (1993a). Experimental Study and Modeling of Thermal Contact Resistance across Bolted Joints, AIAA-93-0844, 31st Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 11–14. Song, S., Yovanovich, M. M., and Goodman, F. O. (1993b). Thermal Gap Conductance of Conforming Rough Surfaces in Contact, J. Heat Transfer, 115, 533–540. Song, S., Lee, S., and Au, V. (1994). Closed-Form Equation for Thermal Constriction/Spread- ing Resistances with Variable Resistance Boundary Condition, Proc. 1994 IEPS Confer- ence, Atlanta, GA, pp. 111–121. Sridhar, M. R. (1994). Elastoplastic Contact Models for Sphere-Flat and Conforming Rough Surface Applications, Ph.D. dissertation, University of Waterloo, Waterloo, Ontario, Canada. Sridhar, M. R., and Yovanovich, M. M. (1994). Review of Elastic and Plastic Contact Con- ductance Models: Comparison with Experiment, J. Thermophys. Heat Transfer, 8(4), 633– 640. Sridhar, M. R., and Yovanovich, M. M. (1996a). Thermal Contact Conductance of Tool Steel and Comparison with Model, Int. J. Heat Mass Transfer, 39(4), 831–839. Sridhar, M. R., and Yovanovich, M. M. (1996b). Empirical Methods to Predict Vickers Micro- hardness, Wear, 193, pp. 91–98. Sridhar, M. R., and Yovanovich, M. M. (1996c). Elastoplastic Contact Model for Isotropic Conforming Rough Surfaces and Comparison with Experments, J. Heat Transfer, 118(1), 3–9. Sridhar, M. R., and Yovanovich, M. M. (1996d). Elastoplastic Constriction Resistance Model for Sphere-Flat Contacts, J. Heat Transfer, 118(1), 202–205. Stevanovi ´ c, M., Yovanovich, M. M., and Culham, J. R. (2002). Modeling Thermal Contact Re- sistance between Elastic Hemisphere and Elastic Layer on Bonded Elastic Substrate, Proc. 8th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, San Diego, CA, May 29–June 1. Stratton, J. A. (1941). Electromagnetic Theory, McGraw-Hill, New York, pp. 207–211. Strong, A. B., Schneider, G. E., and Yovanovich, M. M. (1974). Thermal Constriction Resis- tance of a Disc with Arbitrary Heat Flux: Finite Difference Solutions in Oblate Spheroidal Coordinates, AIAA-74-690, AIAA/ASME 1974 Thermophysics and Heat Transfer Confer- ence, Boston, July 15–17. Tabor, D. (1951). The Hardness of Metals, Oxford University Press, London. Teagan, W. P., and Springer, G. S. (1968). Heat-Transfer and Density Distribution Measure- ments between Parallel Plates in the Transition Regime, Phys. of Fluids, 11(3), 497–506. Thomas, L. B. (1967). Rarefied Gas Dynamics, Academic Press, New York. Thomas, T. R. (1982). Rough Surfaces, Longman Group, London. Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed., McGraw-Hll, New York. Turyk, P. J., and Yovanovich, M. M. (1984). Transient Constriction Resistance for Elemental Flux Channels Heated by Uniform Heat Sources, ASME-84-HT-52, ASME, New York. Veziroglu, T. N. (1967). Correlation of Thermal Contact Conductance Experimental Results, presented at AIAA Thermophysics Specialist Conference, New Orleans, LA. Veziroglu, T. N., and Chandra, S. (1969). Thermal Conductance of Two-Dimensional Con- strictions, Prog. Astronaut. Aeronaut., 21, 617–620. Wachman, H. Y. (1962). The Thermal Accommodation Coefficient: A Critical Survey, ARS J., 32, 2–12. BOOKCOMP, Inc. — John Wiley & Sons / Page 392 / 2nd Proofs / Heat Transfer Handbook / Bejan 392 THERMAL SPREADING AND CONTACT RESISTANCES 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 [392], (132) Lines: 5038 to 5078 ——— 0.0pt PgVar ——— Custom Page (1.0pt) PgEnds: T E X [392], (132) Walowit, J. A., and Anno, J. N. (1975). Modern Developments in Lubrication Mechanics, Applied Science Publishers, Barking, Essex, England. Wesley, D. A., andYovanovich, M.M. (1986). A New Gaseous Gap Conductance Relationship, Nucl. Technol., 72, Jan., 70–74. Whitehouse, D. J., and Archard, J. F. (1970). The Properties of Random Surfaces of Signifi- cance in Their Contact, Proc. R. Soc. London, A316, 97–121. Wiedmann, M. L., and Trumpler, P. R. (1946). Thermal Accommodation Coefficients, Trans. ASME, 68, 57–64. Yovanovich, M. M. (1967). Thermal Contact Resistance across Elastically Deformed Spheres, J. Spacecr. Rockets, 4, 119–122. Yovanovich, M. M. (1969). Overall Constriction Resistance between Contacting Rough, Wavy Surfaces, Int. J. Heat Mass Transfer, 12, 1517–1520. Yovanovich, M. M. (1971). Thermal Constriction Resistance between Contacting Metallic Paraboloids: Application to Instrument Bearings, in Progress in Astronautics and Aeronau- tics: Heat Transfer and Spacecraft Control, Vol. 24, J. W. Lucas, ed., AIAA, New York, pp. 337–358. Yovanovich, M. M. (1972). Effect of Foils on Joint Resistance: Evidence of Optimum Foil Thickness, AIAA-72-283. AIAA 7th Thermophysics Conference, San Antonio, TX, Apr. Yovanovich, M. M. (1976a). General Thermal Constriction Resistance Parameter for Annular Contacts on Circular Flux Tubes, AIAA J., 14(6), 822–824. Yovanovich, M. M. (1976b). General Expressions for Constriction Resistances of Arbitrary Flux Distributions, in Progress in Astronautics and Aeronautics: Radiative Transfer and Thermal Control, Vol. 49, AIAA, New York, pp. 381–396. Yovanovich, M. M. (1976c). Thermal Constriction Resistance of Contacts on a Half-Space: Integral Formulation, in Progress in Astronautics and Aeronautics: Radiative Transfer and Thermal Control, Vol. 49, AIAA, New York, pp. 397–418. Yovanovich, M. M. (1978). Simplified Explicit Elastoconstriction Resistance for Ball/Race Contacts, AIAA-78-84. 16th Aerospace Sciences Meeting, Huntsville, AL, Jan. 16–18. Yovanovich, M. M. (1982). Thermal Contact Correlations, in Progress in Astronautics and Aeronautics: Spacecraft Radiative Transfer and Temperature Control, Vol. 83, T. E. Horton, ed., AIAA, New York, pp. 83–95. Yovanovich, M. M. (1986). Recent Developments in Thermal Contact, Gap and Joint Conduc- tance Theories and Experiments, Proc. 8th International Heat Transfer Conference, San Francisco, Vol. 1, pp. 35–45. Yovanovich, M. M. (1991). Theory and Applications of Constriction and Spreading Resistance Concepts for Microelectronic Thermal Management, in Cooling Techniques for Computers, W. Aung, ed., Hemisphere Publishing, New York, pp. 277–332. Yovanovich, M. M. (1997). Transient Spreading Resistance of Arbitrary Isoflux Contact Areas: Development of a Universal Time Function, AIAA-97-2458, AIAA 32nd Thermophysics Conference, Atlanta, GA, June 23–25. Yovanovich, M. M., and Antonetti, V. W. (1988). Application of Thermal Contact Resistance Theory to Electronic Packages, in Advances in Thermal Modeling of Electronic Compo- nents and Systems, Vol. 1, A. Bar-Cohen and A. D. Kraus, eds., Hemisphere Publishing, New York, pp. 79–128. Yovanovich, M. M., and Burde, S. S. (1977). Centroidal and Area Average Resistances of Nonsymmetric, Singly Connected Contacts, AIAA J., 15(10), 1523–1525. BOOKCOMP, Inc. — John Wiley & Sons / Page 393 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 393 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 [393], (133) Lines: 5078 to 5099 ——— * 168.371pt PgVar ——— Custom Page (1.0pt) * PgEnds: PageBreak [393], (133) Yovanovich, M. M., and Kitscha, W. W. (1974). Modeling the Effect of Air and Oil upon the Thermal Resistance of a Sphere-Flat Contact, in Progress in Astronautics and Aeronautics: Thermophysics and Spacecraft Control, Vol. 35, R. G. Hering, ed., AIAA, New York, pp. 293–319. Yovanovich, M. M., and Schneider, G. E. (1977). Thermal Constriction Resistance Due to a Circular Annular Contact, in Progress in Astronautics and Aeronautics, Vol. 56, AIAA, New York, pp. 141–154. Yovanovich, M. M., Burde, S. S., and Thompson, J. C. (1977). Thermal Constriction Resis- tance of Arbitrary Planar Contacts with Constant Flux, in Progress in Astronautics and Aeronautics: Thermophysics of Spacecraft and Outer Planet Entry Probes, Vol. 56, AIAA, New York, pp. 127–139. Yovanovich, M. M., Tien, C. H., and Schneider, G. E. (1980). General Solution of Constriction Resistance within a Compound Disk, in Progress in Astronautics and Aeronautics: Heat Transfer, Thermal Control, and Heat Pipes, Vol. 70, AIAA, New York, pp. 47–62. Yovanovich, M. M., Hegazy, A. A., and DeVaal, J. (1982a). Surface Hardness Distribution Effects upon Contact, Gap and Joint Conductances, AIAA-82-0887, AIAA/ASME 3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference, St. Louis, MO, June 7–11. Yovanovich, M. M., DeVaal, J., and Hegazy, A. A. (1982b). A Statistical Model to Predict Ther- mal Gap Conductance between Conforming Rough Surfaces, AIAA-82-0888, AIAA/ASME 3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference, St. Louis, MO, June 7–11. Yovanovich, M. M., Negus, K. J., and Thompson, J. C. (1984). Transient Temperature Rise of Arbitrary Contacts with Uniform Flux by Surface Element Methods, AIAA-84-0397, AIAA 22nd Aerospace Sciences Meeting, Reno, NV, Jan. 9–12. Yovanovich, M. M., Culham, J. R., and Teerstra, P. (1997). Calculating Interface Resistance, Electron. Cool., 3(2), 24–29. Yovanovich, M. M., Culham, J. R., and Teerstra, P. (1998). Analytical Modeling of Spreading Resistance in Flux Tubes, Half Spaces, and Compound Disks, IEEE Trans. Components Packag. Manuf. Technol., A21(1), 168–176. Yovanovich, M. M., Muzychka, Y. S., and Culham, J. R. (1999). Spreading Resistance of Isoflux Rectangles and Strips on Compound Flux Channels, J. Thermophys. Heat Transfer, 13(4), 495–500. BOOKCOMP, Inc. — John Wiley & Sons / Page 395 / 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] [395], (1) Lines: 0 to 85 ——— -0.68391pt PgVar ——— Normal Page PgEnds: T E X [395], (1) CHAPTER 5 Forced Convection: Internal Flows ADRIAN BEJAN Department of Mechanical Engineering and Materials Science Duke University Durham, North Carolina 5.1 Introduction 5.2 Laminar flow and pressure drop 5.2.1 Flow entrance region 5.2.2 Fully developed flow region 5.2.3 Hydraulic diameter and pressure drop 5.3 Heat transfer in fully developed flow 5.3.1 Mean temperature 5.3.2 Thermally fully developed flow 5.4 Heat transfer in developing flow 5.4.1 Thermal entrance region 5.4.2 Thermally developing Hagen–Poiseuille flow 5.4.3 Thermally and hydraulically developing flow 5.5 Optimal channel sizes for laminar flow 5.6 Turbulent duct flow 5.6.1 Time-averaged equations 5.6.2 Fully developed flow 5.6.3 Heat transfer in fully developed flow 5.7 Total heat transfer rate 5.7.1 Isothermal wall 5.7.2 Wall heated uniformly 5.8 Optimal channel sizes for turbulent flow 5.9 Summary of forced convection relationships Nomenclature References 5.1 INTRODUCTION An internal flow is a flow configuration where the flowing material is surrounded by solid walls. Streams that flow through ducts are primary examples of internal flows. Heat exchangers are conglomerates of internal flows. This class of fluid flow and 395 BOOKCOMP, Inc. — John Wiley & Sons / Page 396 / 2nd Proofs / Heat Transfer Handbook / Bejan 396 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 [396], (2) Lines: 85 to 111 ——— -1.92998pt PgVar ——— Normal Page PgEnds: T E X [396], (2) convection phenomena distinguishes itself from the class of an external flow, which is treated in Chapters 6 (forced-convection external) and 7 (natural convection). In an external flow configuration, a solid object is surrounded by the flow. There are two basic questions for the engineer who contemplates using an internal flow configuration. One is the heat transfer rate, or the thermal resistance between the stream and the confining walls. The other is the friction between the stream and the walls. The fluid friction part of the problem is the same as calculation of the pressure drop experienced by the stream over a finite length in the flow direction. The fluid friction question is the more basic, because friction is present as soon as there is flow, that is, even in the absence of heat transfer. This is why we begin this chapter with the calculation of velocity and pressure drop in duct flow. The heat transfer question is supplementary, as the duct flow will convect energy if a temperature difference exits between its inlet and the wall. To calculate the heat transfer rate and the temperature distribution through the flow, one must know the flow, or the velocity distribution. When the variation of temperature over the flow field is sufficiently weak so that the fluid density and viscosity are adequately represented by two constants, calculation of the velocity field and pressure drop is independent of that of the temperature field. This is the case in all the configurations and results reviewed in this chapter. When this approximation is valid, the velocity field is “not coupled” to the temperature field, although, as already noted, correct derivation of the temperature field requires the velocity field as a preliminary result, that is, as an input. The following presentation is based on the method developed in Bejan (1995). Alternative reviews of internal flow convection are available in Shah and London (1978) and Shah and Bhatti (1978) and are recommended. 5.2 LAMINAR FLOW AND PRESSURE DROP 5.2.1 Flow Entrance Region Consider the laminar flow through a two-dimensional duct formed between two parallel plates, as shown in Fig. 5.1. The spacing between the plates is D. The flow velocity in the inlet cross section (x = 0) is uniform (U ). Mass conservation means that U is also the mean velocity at any position x downstream, U = 1 A A udA (5.1) where u is the longitudinal velocity component and A is the duct cross-sectional area in general. Boundary layers grow along the walls until they meet at the distance x ≈ X downstream from the entrance. The length X is called entrance length or flow (hydrodynamic) entrance length, to be distinguished from the thermal entrance length discussed in Section 5.4.1. In the entrance length region the boundary layers coexist with a core in which the velocity is uniform (U c ). Mass conservation and the fact that the fluid slows down in the boundary layers requires that U c >U. BOOKCOMP, Inc. — John Wiley & Sons / Page 397 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR FLOW AND PRESSURE DROP 397 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 [397], (3) Lines: 111 to 153 ——— -3.10686pt PgVar ——— Normal Page * PgEnds: PageBreak [397], (3) Figure 5.1 Developing flow in the entrance region of the duct formed between two parallel plates. (From Bejan, 1995.) The length X divides the duct flow into an entrance region (0 <x≤ X) and a fully developed flow region (x ≥ X). The flow friction and heat transfer characteristics of the entrance region are similar to those of boundary layer flows. The features of the fully developed region require special analysis, as shown in Section 5.2.2. The entrance length X is indicated approximately in Fig. 5.1. This is not a precise dimension, for the same reasons that the thickness of a boundary layer (δ) is known only as an order-of-magnitude length. The scale of X can be determined from the scale of δ, which according to the Blasius solution is δ ∼ 5x Ux ν −1/2 The transition from entrance flow to fully developed flow occurs at x ∼ X and δ ∼ D/2, and therefore it can be concluded that X/D Re D ≈ 10 −2 (5.2) where Re D = UD/ν. The heat transfer literature also recommends the more precise value 0.04 in place of the 10 −2 in eq. (5.2) (Schlichting, 1960), although as shown in Fig. 5.2 and 5.3, the transition from entrance flow to fully developed flow is smooth. The friction between fluid and walls is measured as the local shear stress at the wall surface, τ x (x) = µ ∂u ∂y y=0 or the dimensionless local skin friction coefficient C f,x = τ w 1 2 ρU 2 (5.3) BOOKCOMP, Inc. — John Wiley & Sons / Page 398 / 2nd Proofs / Heat Transfer Handbook / Bejan 398 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 [398], (4) Lines: 153 to 160 ——— -3.02599pt PgVar ——— Normal Page * PgEnds: Eject [398], (4) 10 Ϫ4 10 Ϫ3 10 Ϫ2 10 10 2 12 xD/ Re D C fx D, Re Figure 5.2 Local skin-friction coefficient in the entrance region of a parallel-plate duct. (From Bejan, 1995.) 10 Ϫ3 10 Ϫ2 10 Ϫ1 10 100 16 x DRe D C fx D, Re () ReC fxD0Ϫ Figure 5.3 Local and average skin friction coefficients in the entrance region of a round tube. (From Bejan, 1995.) Figure 5.2 shows a replotting (Bejan, 1995) of the integral solution (Sparrow, 1955) for C f,x in the entrance region of a parallel-plate duct. The dashed-line asymptote indicates the C f,x estimate based on the Blasius solution for the laminar boundary layer between a flat wall and a uniform free stream (U ). If numerical factors of order 1 are neglected, the boundary layer asymptote is BOOKCOMP, Inc. — John Wiley & Sons / Page 399 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR FLOW AND PRESSURE DROP 399 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 [399], (5) Lines: 160 to 223 ——— 0.3832pt PgVar ——— Normal Page * PgEnds: Eject [399], (5) C f,x ≈ Ux ν −1/2 or C f,x · Re D ≈ x D ·Re D −1/2 The solid-line asymptote, C f,x · Re D = 12, represents the skin friction solution for the fully developed flow region (Section 5.2.2). The local skin friction coefficient in the entrance region of a round tube is indicated by the lower curve in Fig. 5.3. This is a replotting (Bejan, 1995) of the solution reported by Langhaar (1942). The upper curve is for the averaged skin friction co- efficient, C f 0−x = 1 x x 0 C f,ξ (ξ)dξ (5.4a) or C f 0−x = ¯ τ 1 2 ρU 2 (5.4b) where ¯ τ = 1 x x 0 τ w (ξ)dξ The horizontal asymptote serves both curves, C f,x = 16 = C f 0−x and represents the solution for fully developed skin friction in a round tube as shown subsequently in eq. (5.18). 5.2.2 Fully Developed Flow Region The key feature of the flow in the region downstream of x ∼ X is that the transverse velocity component (v = 0 in Fig. 5.1) is negligible. In view of the equation for mass conservation, ∂u ∂x + ∂v ∂y = 0 the vanishing of v is equivalent to ∂u/∂x = 0, that is, a velocity distribution that does not change or does not develop further from one x to the next. This is why this flow region is called fully developed. It is considered as defined by BOOKCOMP, Inc. — John Wiley & Sons / Page 400 / 2nd Proofs / Heat Transfer Handbook / Bejan 400 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 [400], (6) Lines: 223 to 271 ——— 0.44623pt PgVar ——— Normal Page * PgEnds: Eject [400], (6) v = 0or ∂u ∂x = 0 (5.5) This feature is a consequence of the geometric constraint that downstream of x ∼ X, the boundary layer thickness δ cannot continue to grow. In this region, the length scale for changes in the transverse direction is the constant D, not the freely growing δ, and the mass conservation equation requires that U/L ≈ v/D, where L is the flow dimension in the downstream direction. The v scale is then v ≈ UD /L and this scale vanishes as L increases, that is, as the flow reaches sufficiently far into the duct. Another consequence of the full development of the flow is that the pressure is essentially uniform in each constant-x cross section (∂P /∂y = 0). This feature is derived by substituting v = 0 into the momentum (Navier–Stokes) equation for the y direction. With reference to Fig. 5.1, the pressure distribution is P(x), and the momentum equation for the flow direction x becomes dP dx = µ d 2 u dy 2 (5.6) Both sides of this equation must equal the same constant, because at most, the left side is a function of x and the right side a function of y. That constant is the pressure drop per unit length, ∆P L =− dP dx The pressure drop and the flow distribution u(y) are obtained by solving eq. (5.6) subject to u = 0 at the walls (y ± D/2), where y = 0 represents the center plane of the parallel plate duct: u(y) = 3 2 U 1 − y D/2 2 (5.7) with U = D 2 12µ − dP dx (5.8) In general, for a duct of arbitrary cross section, eq. (5.6) is replaced by dP dx = µ ∇ 2 u = constant where the Laplacian operator ∇ 2 accounts only for curvatures in the cross section, ∇ 2 = ∂ 2 ∂y 2 + ∂ 2 ∂z 2 BOOKCOMP, Inc. — John Wiley & Sons / Page 401 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR FLOW AND PRESSURE DROP 401 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 [401], (7) Lines: 271 to 331 ——— -0.74576pt PgVar ——— Normal Page * PgEnds: Eject [401], (7) that is, ∂ 2 /∂x 2 = 0. The boundary conditions are u = 0 on the perimeter of the cross section. For example, the solution for fully developed laminar flow in a round tube of radius r 0 is u = 2U 1 − r r 0 2 (5.9) with U = r 2 0 8µ − dP dx (5.10) This solution was first reported by Hagen (1839) and Poiseuille (1840), which is why the fully developed laminar flow regime is also called Hagen–Poiseuille flow or Poiseuille flow. 5.2.3 Hydraulic Diameter and Pressure Drop Equations (5.8) and (5.10) show that in fully developed laminar flow the mean veloc- ity U (or the mass flow rate ˙m = ρAU) is proportional to the longitudinal pressure gradient P/L. In general, and especially in turbulent flow, the relationship between ˙m and ∆P is nonlinear. Fluid friction results for fully developed flow in ducts are reported as friction factors: f = τ w 1 2 ρU 2 (5.11) where τ w is the shear stress at the wall. Equation (5.11) is the same as eq. (5.3), with the observation that in fully developed flow, τ w and f are x-independent. The shear stress τ w is proportional to ∆P/L. This proportionality follows from the longitudinal force balance on a flow control volume of cross section A and length L, A ∆P = τ w pL (5.12) where p is the perimeter of the cross section. Equation (5.12) is general and is independent of the flow regime. Combined with eq. (5.11), it yields the pressure drop relationship ∆P = f pL A 1 2 ρU 2 (5.13) where A/p represents the transversal length scale of the duct: r h = A p hydraulic radius (5.14) D h = 4A p hydraulic diameter (5.15) . on Compound Flux Channels, J. Thermophys. Heat Transfer, 13(4), 495–500. BOOKCOMP, Inc. — John Wiley & Sons / Page 395 / 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. Time-averaged equations 5.6.2 Fully developed flow 5.6.3 Heat transfer in fully developed flow 5.7 Total heat transfer rate 5.7.1 Isothermal wall 5.7.2 Wall heated uniformly 5.8 Optimal channel sizes for. with Experments, J. Heat Transfer, 118(1), 3–9. Sridhar, M. R., and Yovanovich, M. M. (1996d). Elastoplastic Constriction Resistance Model for Sphere-Flat Contacts, J. Heat Transfer, 118(1), 202–205. Stevanovi ´ c,