BOOKCOMP, Inc. — John Wiley & Sons / Page 422 / 2nd Proofs / Heat Transfer Handbook / Bejan 422 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 [422], (28) Lines: 1050 to 1053 ——— 8.03699pt PgVar ——— Normal Page * PgEnds: Eject [422], (28) 0.05 0.04 0.04 0.000,01 0.000,05 0.0001 0.0002 0.0004 0.0006 0.001 0.002 0.004 0.008 0.01 0.015 0.02 0.006 0.0008 Relative roughness, /Dk s 0.01 0.02 0.03 0.04 0.05 0.1 4f 0.000,005 0.000,001 Re / D =UDv 10 3 10 4 10 5 10 6 10 7 10 8 Laminar flow, f= 16 Re D Smooth pipes (the Karman- Nikuradse relation) Surface condition k S (mm) Riveted steel Concrete Wood stave Cast iron Galvanized iron Asphalted cast iron Commercial steel or Wrought iron Drawn tubing 0.9–9 0.3–3 0.18–0.9 0.26 0.15 0.12 0.05 0.0015 Figure 5.13 Friction factor for duct flow. (From Bejan, 1995; drawn after Moody, 1944.) Figure 5.13 also documents the effect of wall roughness. It is found experimentally that the performance of commercial surfaces that do not feel rough to the touch departs from the performance of well-polished surfaces. This effect is due to the very small thickness acquired by the laminar sublayer in many applications [e.g., because Uy VSL /ν is of order 10 2 (Bejan, 1995), where y VSL is the thickness of the viscous sublayer]. In water flow through a pipe, with U ≈ 10m/s and ν ≈ 0.01cm 2 /s, y VSL is approximately 0.01 mm. Consequently, even slight imperfections of the surface may interfere with the natural formation of the laminar shear flow contact spots. If the surface irregularities are taller than y VSL , they alone rule the friction process. Nikuradse (1933) measured the effect of surface roughness on the friction factor by coating the inside surface of pipes with sand of a measured grain size glued as tightly as possible to the wall. If k s is the grain size in Nikuradse’s sand roughness, the friction factor fully rough limit is the constant BOOKCOMP, Inc. — John Wiley & Sons / Page 423 / 2nd Proofs / Heat Transfer Handbook / Bejan TURBULENT DUCT FLOW 423 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 [423], (29) Lines: 1053 to 1100 ——— 0.71622pt PgVar ——— Normal Page PgEnds: T E X [423], (29) f 1.74 ln D k s + 2.28 −2 (5.70) The fully rough limit is that regime where the roughness size exceeds the order of magnitude of what would have been the laminar sublayer in time-averaged turbulent flow over a smooth surface, k + s = k s (τ 0 /ρ) 1/2 ν ≥ 10 (5.71) The roughness effect described by Nikuradse is illustrated by the upper curves in Fig. 5.13. 5.6.3 Heat Transfer in Fully Developed Flow There are several empirical relationships for calculating the time-averaged coefficient for heat transfer between the duct surface and the fully developed flow, h = q /(T 0 − T m ). The analytical form of these relationships is based on exploiting the analogy between momentum and heat transfer by eddy rotation. One of the earliest examples is due to Prandtl in 1910 (Prandtl, 1969; Schlichting, 1960), St = f/2 Pr t + ( ¯u 1 /U)(Pr −Pr t ) (5.72) where St, Pr, and Pr t are the Stanton number, Prandtl number, and turbulent Prandtl number, St = h ρc p U Pr = ν α Pr t = M H (5.73) Equation (5.72) holds for Pr ≥ 0.5, and if Pr t is assumed to be 1, the factor ¯u 1 /U is provided by the empirical correlation (Hoffmann, 1937) ¯u 1 U 1.5Re −1/8 D h · Pr −1/6 (5.74) Better agreement with measurements is registered by Colburn’s (1933) empirical correlation, St · Pr 2/3 f 2 (5.75) Equation (5.75) is analytically the same as the one derived purely theoretically for boundary layer flow (Bejan, 1995). Equation (5.75) holds for Pr ≥ 0.5 and is to be used in conjunction with Fig. 5.13, which supplies the value of the friction factor f .It applies to ducts of various cross-sectional shapes, with wall surfaces having various degrees of roughness. In such cases D is replaced by D h . In the special case of a pipe BOOKCOMP, Inc. — John Wiley & Sons / Page 424 / 2nd Proofs / Heat Transfer Handbook / Bejan 424 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 [424], (30) Lines: 1100 to 1147 ——— 1.92621pt PgVar ——— Normal Page * PgEnds: Eject [424], (30) with smooth internal surface, eqs. (5.75) and (5.68) can be combined to derive the Nusselt number relationship Nu D = hD k = 0.023Re 4/5 D · Pr 1/3 (5.76) which holds in the range 2 × 10 4 ≤ Re D ≤ 10 6 . A popular version of this is a correlation due to Dittus and Boelter (1930), Nu D = 0.023Re 4/5 D · Pr n (5.77) which was developed for 0.7 ≤ Pr ≤ 120, 2500 ≤ Re D ≤ 1.24 × 10 5 , and L/D > 60. The Prandtl number exponent is n = 0.4 when the fluid is being heated (T 0 >T m ), and n = 0.3 when the fluid is being cooled (T 0 <T m ). All of the physical properties needed for the calculation of Nu D , Re D , and Pr are to be evaluated at the bulk temperature T m . The maximum deviation between experimental data and values predicted using eq. (5.77) is on the order of 40%. For applications in which influence of temperature on properties is significant, the Sieder and Tate (1936) modification of eq. (5.76) is recommended: Nu D = 0.027Re 4/5 D · Pr 1/3 µ µ 0 0.14 (5.78) This correlation is valid for 0.7 ≤ Pr ≤ 16,700 and Re D > 10 4 . The effect of temperature-dependent properties is taken into account by evaluating all the prop- erties (except µ 0 ) at the mean temperature of the stream, T m . The viscosity µ 0 is evaluated at the wall temperature µ 0 = µ(T 0 ). Equations (5.76)–(5.78) can be used for ducts with constant temperature and constant heat flux. More accurate correlations of this type were developed by Petukhov and Kirilov (1958) and Petukhov and Popov (1963); respectively: 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) and Nu D = (f/2)Re D · Pr 1.07 + 12.7(f/2) 1/2 Pr 2/3 − 1 (5.79b) for which f is supplied by Fig. 5.13. Additional information is provided by Petukhov (1970). Equation (5.79a) is accurate within 5% in the range 4000 ≤ Re D ≤ 5 × 10 6 and 0.5 ≤ Pr ≤ 10 6 . Equation (5.79b) is an abbreviated version of eq. (5.79a) and was modified by Gnielinski (1976): Nu D = (f/2)(Re D − 10 3 )Pr 1 + 12.7(f/2) 1/2 Pr 2/3 − 1 (5.80) BOOKCOMP, Inc. — John Wiley & Sons / Page 425 / 2nd Proofs / Heat Transfer Handbook / Bejan TOTAL HEAT TRANSFER RATE 425 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 [425], (31) Lines: 1147 to 1220 ——— 1.55403pt PgVar ——— Normal Page PgEnds: T E X [425], (31) which is accurate within ±10% in the range 0.5 ≤ Pr ≤ 10 6 and 2300 ≤ Re D ≤ 5 ×10 6 . The Gnielinski correlation of eq. (5.80) can be used in both constant-q and constant-T 0 applications. Two simpler alternatives to eq. (5.80) are (Gnielinski, 1976) Nu D = 0.0214 Re 0.8 D − 100 Pr 0.4 (5.81a) valid in the range 0.5 ≤ Pr ≤ 1.510 4 ≤ Re D ≤ 5 × 10 6 and Nu D = 0.012 Re 0.87 D − 280 Pr 0.4 (5.81b) valid in the range 1.5 ≤ Pr ≤ 500 3 × 10 3 ≤ Re D ≤ 10 6 The preceding results refer to gases and liquids, that is, to the range Pr ≥ 0.5. For liquid metals, the most accurate correlations are those of Notter and Sleicher (1972): 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) These are valid for 0.004 ≤ Pr ≤ 0.1 and 10 4 ≤ Re D ≤ 10 6 and are based on both computational and experimental data. All the properties used in eqs. (5.82) and (5.83) are evaluated at the mean temperature T m . The mean temperature varies with the position along the duct. This variation is linear in the case of constant q and exponential when the duct wall is isothermal (see Section 5.7). To simplify the recommended evaluation of the physical properties at the T m temperature, it is convenient to choose as representative mean temperature the average value T m = 1 2 (T in + T out ) (5.84) In this definition, T in and T out are the bulk temperatures of the stream at the duct inlet and outlet, respectively (Fig. 5.14). 5.7 TOTAL HEAT TRANSFER RATE The summarizing conclusion is that in both laminar and turbulent fully developed duct flow the heat transfer coefficient h is independent of longitudinal position. This feature makes it easy to express analytically the total heat transfer rate q (watts) between a stream and duct of length L, q = hA w ∆T lm (5.85) BOOKCOMP, Inc. — John Wiley & Sons / Page 426 / 2nd Proofs / Heat Transfer Handbook / Bejan 426 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 [426], (32) Lines: 1220 to 1245 ——— 0.40805pt PgVar ——— Normal Page * PgEnds: Eject [426], (32) ⌬T in ⌬T in ⌬T out ⌬T out Stream, ( )Tx m Stream, ( )Tx m Wall ()Tx 0 Wall ()Tx 0 Wall T in T in T out T out T 0 TT 00LLxx ()a ()b Figure 5.14 Variation of the mean temperature along the duct: (a) isothermal wall; (b) wall with uniform heat flux. In this expression A w is the total duct surface, A w = pL. The effective tempera- ture difference between the wall and the stream is the log-mean temperature differ- ence T lm . 5.7.1 Isothermal Wall When the wall is isothermal (Fig. 5.14) the log-mean temperature difference is ∆T lm = ∆T in − ∆T out ln(∆T in /∆T out ) (5.86) Equations (5.85) and (5.86) express the relationship among the total heat transfer rate q, the total duct surface conductance hA w , and the outlet temperature of the stream. Alternatively, the same equations can be combined to express the total heat transfer rate in terms of the inlet temperatures, mass flow rate, and duct surface conductance, q =˙mc p ∆T in (1 − e −hA w / ˙mc p ) (5.87) In cases where the heat transfer coefficient varies longitudinally, h(x), the h factor on the right side of eq. (5.87) represents the L-averaged heat transfer coefficient: namely, ¯ h = 1 L L 0 h(x) dx (5.88) BOOKCOMP, Inc. — John Wiley & Sons / Page 427 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF FORCED CONVECTION RELATIONSHIPS 427 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 [427], (33) Lines: 1245 to 1291 ——— 6.74318pt PgVar ——— Normal Page * PgEnds: Eject [427], (33) 5.7.2 Wall Heated Uniformly In the analysis of heat exchangers (Bejan, 1993), it is found that the applicability of eqs. (5.85) and (5.86) is considerably more general than what is suggested by Fig. 5.14. For example, when the heat transfer rate q is distributed uniformly along the duct, the temperature difference ∆T does not vary with the longitudinal position. This case is illustrated in Fig. 5.14, where it was again assumed that A, p, h, and c p are independent of x. Geometrically, it is evident that the effective value ∆T lm is the same as the constant ∆T recorded all along the duct, ∆T lm = ∆T in = ∆T out (5.89) Equation (5.89) is a special case of eq. (5.86): namely, the limit ∆T in /∆T out → 1. 5.8 OPTIMAL CHANNEL SIZES FOR TURBULENT FLOW The optimization of packing of channels into a fixed volume, which in Section 5.5 was outlined for laminar duct flow, can also be pursued when the flow is turbulent (Bejan and Morega, 1994). With reference to the notation defined in Fig. 5.10, where the dimension perpendicular to the figure is W , the analysis consists of intersecting the two asymptotes of the design: a few wide spaces with turbulent boundary layers and many narrow spaces with fully developed turbulent flow. The plate thickness (t) is not negligible with respect to the spacing D. The optimal spacing and maximal global conductance of the HWL package are D opt /L (1 + t/D opt ) 4/11 = 0.071Pr −5/11 · Be −1/11 (5.90) and 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) where Be = ∆PL 2 /µα. These results are valid in the range 10 4 ≤ Re D h ≤ 10 6 and 10 6 ≤ Re L ≤ 10 8 , which can be shown to correspond to the pressure drop number range 10 11 ≤ Be ≤ 10 16 . 5.9 SUMMARY OF FORCED CONVECTION RELATIONSHIPS • Laminar flow entrance length: X/D Re D ≈ 10 −2 (5.2) BOOKCOMP, Inc. — John Wiley & Sons / Page 428 / 2nd Proofs / Heat Transfer Handbook / Bejan 428 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 [428], (34) Lines: 1291 to 1348 ——— 10.81242pt PgVar ——— Short Page * PgEnds: Eject [428], (34) • Skin friction coefficient definition: C f,x = τ w 1 2 ρU 2 (5.3) • Laminar fully developed (Hagen–Poiseuille) flow between parallel plates with spacing D: u(y) = 3 2 U 1 − y D/2 2 (5.7) with U = D 2 12µ − dP dx (5.8) • Laminar fully developed (Hagen–Poiseuille) flow in a tube with diameter D: u = 2U 1 − r r 0 2 (5.9) with U = r 2 0 8µ − dP dx (5.10) • Hydraulic radius and diameter: r h = A p hydraulic radius (5.14) D h = 4A p hydraulic diameter (5.15) • Friction factor: f = τ w 1 2 ρU 2 see Tables 5.1–5.3 (5.11) 24 Re Dh D h = 2D parallel plates (D = spacing) (5.17) 16 Re Dh D h = D round tube (D = diameter) (5.18) BOOKCOMP, Inc. — John Wiley & Sons / Page 429 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF FORCED CONVECTION RELATIONSHIPS 429 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 [429], (35) Lines: 1348 to 1415 ——— 5.55528pt PgVar ——— Short Page * PgEnds: Eject [429], (35) • Pressure drop: ∆P = f 4L D h 1 2 ρU 2 (5.16) • Nusselt number (see Tables 5.1 through 5.3): Nu = hD k = D ∂T /∂r r=r 0 T 0 − T m (5.24) • Laminar thermal entrance length: X T ≈ 10 −2 Pr · D h · Re D h (5.26) • Thermally developing Hagen–Poiseuille flow (Pr =∞): • Round tube, isothermal wall: Nu x = 1.077x −1/3 ∗ − 0.70 x ∗ ≤ 0.01 3.657 + 6.874(10 3 x ∗ ) −0.488 e −57.2x ∗ x ∗ > 0.01 (5.32) Nu 0−x = 1.615x −1/3 ∗ − 0.70 x ∗ ≤ 0.005 1.615x −1/3 ∗ − 0.20 0.005 <x ∗ < 0.03 3.657 + 0.0499/x ∗ x ∗ > 0.03 (5.33) • Round tube, uniform heat flux: Nu x = 3.302x −1/3 ∗ − 1.00 x ∗ ≤ 0.00005 1.302x −1/3 ∗ − 0.50 0.00005 <x ∗ ≤ 0.0015 4.364 + 8.68(10 3 x ∗ ) −0.506 e −41x ∗ x ∗ > 0.001 (5.34) Nu 0−x = 1.953x −1/3 ∗ x ∗ ≤ 0.03 4.364 + 0.0722/x ∗ x ∗ > 0.03 (5.35) • Parallel plates, isothermal surfaces: Nu 0−x = 1.233x −1/3 ∗ + 0.40 x ∗ ≤ 0.001 7.541 + 6.874(10 3 x ∗ ) −0.488 e −245x ∗ x ∗ > 0.001 (5.37) Nu 0−x = 1.849x −1/3 ∗ x ∗ ≤ 0.0005 1.849x −1/3 ∗ + 0.60 0.0005 <x ∗ ≤ 0.006 7.541 + 0.0235/x ∗ x ∗ > 0.006 (5.38) BOOKCOMP, Inc. — John Wiley & Sons / Page 430 / 2nd Proofs / Heat Transfer Handbook / Bejan 430 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 [430], (36) Lines: 1415 to 1478 ——— 10.42749pt PgVar ——— Short Page * PgEnds: Eject [430], (36) • Parallel plates, uniform heat flux: Nu x = 1.490x −1/3 ∗ x ∗ ≤ 0.0002 1.490x −1/3 ∗ − 0.40 0.0002 <x ∗ ≤ 0.001 8.235 + 8.68(10 3 x ∗ ) −0.506 e −164x ∗ x ∗ > 0.001 (5.39) Nu 0−x = 2.236x −1/3 ∗ x ∗ ≤ 0.001 2.236x −1/3 ∗ + 0.90 0.001 <x ∗ ≤ 0.01 8.235 + 0.0364/x ∗ x ∗ ≥ 0.01 (5.40) • Thermally and hydraulically developing flow: • Round tube, isothermal wall: Nu x = 7.55 + 0.024x −1.14 ∗ 0.0179Pr 0.17 x −0.64 ∗ − 0.14 1 + 0.0358Pr 0.17 x −0.64 ∗ 2 (5.41) Nu 0−x = 7.55 + 0.024x −1.14 ∗ 1 + 0.0358Pr 0.17 x −0.64 ∗ (5.42) ∆P 1 2 ρU 2 = 13.74(x + ) 1/2 + 1.25 + 64x + − 13.74(x + ) 1/2 1 + 0.00021(x + ) −2 (5.43) x + = x/D Re D (5.44) • Round tube, uniform heat flux: Nu x 4.364 1 + (Gz/29.6) 2 1/6 = 1 + Gz/19.04 1 + (Pr/0.0207) 2/3 1/2 1 + (Gz/29.6) 2 1/3 3/2 1/3 (5.46) • Optimal channel sizes: • Laminar flow, parallel plates: D opt L 2.7Be −1/4 Be = ∆PL 2 µα (5.47) q max HLW 0.60 k L 2 (T max − T 0 )Be 1/2 (5.48) BOOKCOMP, Inc. — John Wiley & Sons / Page 431 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF FORCED CONVECTION RELATIONSHIPS 431 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 [431], (37) Lines: 1478 to 1540 ——— 2.59857pt PgVar ——— Short Page PgEnds: T E X [431], (37) • Staggered plates: D opt L 5.4Pr −1/4 Re L L b −1/2 (5.50) for the range Pr = 0.72 10 2 ≤ Re L ≤ 10 4 0.5 ≤ Nb L ≤ 1.3 • Bundle of cylinders in cross flow: S opt D 1.59 (H/D) 0.52 ˜ P 0.13 · Pr 0.24 ˜ P = ∆PD 2 µν (5.51) for the range 0.72 ≤ Pr ≤ 50 10 4 ≤ ˜ P ≤ 10 8 25 ≤ H D ≤ 200 S opt D 1.70 (H/D) 0.52 Re 0.26 D · Pr 0.24 (5.52) T D − T ∞ qD/kLW 4.5 Re 0.90 D · Pr 0.64 (5.53) with Re D = U ∞ D ν 140 ≤ Re D ≤ 14,000 • Array of pin fins with impinging flow: S opt L 0.81Pr −0.25 · Re −0.32 L (5.54) for the range 0.06 ≤ D L ≤ 0.14 0.28 ≤ H L ≤ 0.56 0.72 ≤ Pr ≤ 7 10 ≤ Re D ≤ 700 90 ≤ Re L ≤ 6000 • Turbulent duct flow: D opt /L 1 + t/D opt 4/11 = 0.071Pr −5/11 · Be −1/11 (5.90) . Proofs / Heat Transfer Handbook / Bejan TOTAL HEAT TRANSFER RATE 425 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 [425],. analytically the total heat transfer rate q (watts) between a stream and duct of length L, q = hA w ∆T lm (5.85) BOOKCOMP, Inc. — John Wiley & Sons / Page 426 / 2nd Proofs / Heat Transfer Handbook /. upper curves in Fig. 5.13. 5.6.3 Heat Transfer in Fully Developed Flow There are several empirical relationships for calculating the time-averaged coefficient for heat transfer between the duct surface