Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 63
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The corrseponding minimum global thermalresistance between the array and the coolant is given within 20% by(D/L)0.31TD − T∞0.45q/kH1.57Re0.69L · Pr(5.55)The global resistance refers to the entire volume occupied by the array (HL2 ) andis the ratio between the fin-coolant temperature difference (TD − T ) and the total heatBOOKCOMP, Inc. — John Wiley & Sons / Page 418 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 876 to 919———0.32347pt PgVar———Long PagePgEnds: TEX[418], (24)419TURBULENT DUCT FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445current q generated by the H L2 volume.
In the square arrangement of Fig. 5.11c thetotal number of fins is L2 /(S + D)2 . Application of the same geometric optimizationmethod to ducts with turbulent flow is reviewed in Section 5.8.5.6 TURBULENT DUCT FLOW5.6.1 Time-Averaged EquationsThe analysis of turbulent duct flow and heat transfer is traditionally presented in termsof time-averaged quantities, which are denoted by a bar superscript. For example, thelongitudinal velocity is decomposed as u(r, t) = ū(r) + u (r, t), where ū is the timeaveraged velocity and u is the fluctuation, or the time-dependent difference betweenu and ū. In the cylindrical coordinates (r, x) of the round tube shown in Fig. 5.12, thetime-averaged equations for the conservation of mass, momentum, and energy are∂ ū 1 ∂+(r v̄) = 0∂xr ∂r∂ ū∂ T̄∂ ū1 dP̄1 ∂ū+ v̄=−+r(ν + M )∂x∂rρ dxr ∂r∂r∂ T̄∂ T̄1 ∂∂ T̄ū+ v̄=r(α + H )∂x∂rr ∂r∂r(5.56)Lines: 919 to 952(5.57)0.79907pt PgVar———(5.58)These equations have been simplified based on the observation that the duct is a slender flow region; the absence of second derivatives in the longitudinal direction maybe noted.
Note also that dP̄ /dx, which means that P̄ (r, x) P̄ (x). The momentumeddy diffusivity M and the thermal eddy diffusivity H are defined by− ρu v = ρM∂ ū∂rand− ρcp v T = ρcp H∂ T̄∂r(5.59)where u , v , and T are the fluctuating parts of the longitudinal velocity, radial velocity, and local temperature. The eddy diffusivities augment significantly the transportFigure 5.12 Distribution of apparent shear stress in fully developed turbulent flow.BOOKCOMP, Inc. — John Wiley & Sons / Page 419 / 2nd Proofs / Heat Transfer Handbook / Bejan[419], (25)———Long PagePgEnds: TEX[419], (25)420123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSeffect that would occur in the presence of molecular diffusion alone, that is, based onν for momentum and α for thermal diffusion.5.6.2Fully Developed FlowThe entrance region for the development of the longitudinal velocity profile and thetemperature profile is about 10 times the tube diameter,XTX 10 DD(5.60)This criterion is particularly valid for fluids with Pr values of order 1 (air and water).For ducts with other cross sections, D is the smaller dimension of the cross section.The lengths X and XT are considerably shorter than their laminar-flow counterpartswhen ReD ≥ 2000.
Downstream of x = (X,XT ) the flow is fully developed, andv̄ = 0 and ∂ ū/∂x = 0. The left side of eq. (5.57) is zero. For the quantity in brackets,the apparent (or total) shear stress notation is employed:τapp∂ ū= ρ(ν + M )∂y(5.61)[420], (26)Lines: 952 to 1002———3.76613pt PgVar———where y is measured away from the wall, y = r0 −r, as indicated in Fig.
5.12. The twoNormal Pagecontributions to τapp , ρv∂ ū/∂y and ρM ∂ ū/∂y, are the molecular shear stress and the * PgEnds: Ejecteddy shear stress, respectively. Note that τapp = 0 at y = 0. The momentum equation,eq. (5.57), reduces to[420], (26)1 ∂dP̄(5.62)=(rτapp )dxr ∂rwhere both sides of the equation equal a constant. By integrating eq. (5.62) from thewall to the distance y in the fluid, and by using the force balance of eq.
(5.12), onecan show that τapp decreases linearly from τ0 at the wall to zero on the centerline,yτapp = τ0 1 −r0Sufficiently close to the wall, where y r0 , the apparent shear stress is nearlyconstant, τapp 0. The mixing-length analysis that produced the law of the wallfor the turbulent boundary layer applies near the tube wall. Measurements confirmthat the time-averaged velocity profile fits the law of the wall,u+ = 2.5 ln y + + 5.5(5.63)where 2.5 and 5.5 are curve-fitting constants, andu+ =ūu∗y+ =u∗ yνBOOKCOMP, Inc. — John Wiley & Sons / Page 420 / 2nd Proofs / Heat Transfer Handbook / Bejanu∗ =τ0ρ1/2(5.64)TURBULENT DUCT FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445421The group u∗ is known as the friction velocity. The major drawback of the τappapproximation is that the velocity profile deduced from it, eq. (5.63), has a finiteslope at the centerline. An empirical profile that has zero slope at the centerline andmatches eq.
(5.64) as y + → 0 is that of Reichardt (1951):)3(1+r/r0 y + + 5.5u+ = 2.5 ln(5.65)2 1 + 2(r/r0 )2The friction factor is defined by eq. (5.11) and is related to the friction velocity(τ0 /ρ)1/2 :τ0ρ1/2=U 1/2f2(5.66)[421], (27)+An analysis based on a velocity curve fit where, instead of eq. (5.63), u is proportional to (y + )1/7 (Prandtl, 1969) leads tof −1/40.078ReD(5.67)where ReD = UD/ν and D = 2r0 . Equation (5.67) agrees with measurements upto ReD = 8 × 104 . An empirical relation that holds at higher Reynolds numbers insmooth tubes (Fig. 5.13) is−1/5f 0.046ReD2 × 104 < ReD < 106(5.68)An alternative that has wider applicability is obtained by using the law of the wall,eq. (5.63), instead of Prandtl’s 17 power law, u+ ∼ (y + )1/7 . The result is (Prandtl,1969)1f 1/2= 1.737 ln ReD f 1/2 − 0.396(5.69)which agrees with measurements for ReD values up to O(106 ).
The heat transferliterature refers to eq. (5.69) as the Kármán–Nikuradse relation (Kays and Perkins,1973); this relation is displayed as the lowest curve in Fig. 5.13. This figure is knownas the Moody chart (Moody, 1944). The laminar flow line in Fig. 5.13 is for a roundtube. The figure shows that the friction factor in turbulent flow is considerably greaterthan that in laminar flow in the hypothetical case that the laminar regime can exist atsuch large Reynolds numbers.For fully developed flow through ducts with cross sections other than round, theKármán–Nikuradse relation of eq. (5.69) still holds if ReD is replaced by the Reynoldsnumber based on hydraulic diameter, ReDh .
Note that for a duct of noncircular crosssection, the time-averaged τ0 is not uniform around the periphery of the cross section;hence, in the friction factor definition of eq. (5.11), τ0 is the perimeter-averaged wallshear stress.BOOKCOMP, Inc. — John Wiley & Sons / Page 421 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1002 to 1050———0.0161pt PgVar———Normal PagePgEnds: TEX[421], (27)422Surface conditionRiveted steelConcreteWood staveCast ironGalvanized ironAsphalted cast ironCommercial steel orWrought ironDrawn tubingkS (mm)0.9–90.3–30.18–0.90.260.150.120.050.00150.10.040.03Laminar flow,0.02f = 16ReDSmooth pipes(the KarmanNikuradse relation)0.01103104105106ReD = UD/v107[422], (28)Relative roughness, ks /D0.050.040.040.020.0150.010.0080.0060.0040.0020.0010.00080.00060.00040.00020.00010.000,050.054f123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWS0.000,010.80.0000,01000, 05001Figure 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 experimentallythat the performance of commercial surfaces that do not feel rough to the touchdeparts from the performance of well-polished surfaces. This effect is due to the verysmall thickness acquired by the laminar sublayer in many applications [e.g., becauseUyVSL /ν is of order 102 (Bejan, 1995), where yVSL is the thickness of the viscoussublayer].
In water flow through a pipe, with U ≈ 10m/s and ν ≈ 0.01cm2 /s, yVSLis approximately 0.01 mm. Consequently, even slight imperfections of the surfacemay interfere with the natural formation of the laminar shear flow contact spots. Ifthe surface irregularities are taller than yVSL , they alone rule the friction process.Nikuradse (1933) measured the effect of surface roughness on the friction factorby coating the inside surface of pipes with sand of a measured grain size glued astightly as possible to the wall. If ks is the grain size in Nikuradse’s sand roughness,the friction factor fully rough limit is the constantBOOKCOMP, Inc.
— John Wiley & Sons / Page 422 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1050 to 1053———8.03699pt PgVar———Normal Page* PgEnds: Eject[422], (28)TURBULENT DUCT FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445Df 1.74 ln + 2.28ks423−2(5.70)The fully rough limit is that regime where the roughness size exceeds the order ofmagnitude of what would have been the laminar sublayer in time-averaged turbulentflow over a smooth surface,ks+ =ks (τ0 /ρ)1/2≥ 10ν(5.71)The roughness effect described by Nikuradse is illustrated by the upper curves inFig. 5.13.5.6.3[423], (29)Heat Transfer in Fully Developed FlowThere are several empirical relationships for calculating the time-averaged coefficientfor heat transfer between the duct surface and the fully developed flow, h = q /(T0 −Tm ). The analytical form of these relationships is based on exploiting the analogybetween momentum and heat transfer by eddy rotation.
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