Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 64

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One of the earliest examplesis due to Prandtl in 1910 (Prandtl, 1969; Schlichting, 1960),St =f/2Pr t + (ū1 /U )(Pr − Pr t )(5.72)where St, Pr, and Pr t are the Stanton number, Prandtl number, and turbulent Prandtlnumber,St =hρcp UPr =ναPr t =MH(5.73)Equation (5.72) holds for Pr ≥ 0.5, and if Pr t is assumed to be 1, the factor ū1 /U isprovided by the empirical correlation (Hoffmann, 1937)ū1−1/8 1.5ReDh · Pr −1/6U(5.74)Better agreement with measurements is registered by Colburn’s (1933) empiricalcorrelation,St · Pr 2/3 f2(5.75)Equation (5.75) is analytically the same as the one derived purely theoretically forboundary layer flow (Bejan, 1995).

Equation (5.75) holds for Pr ≥ 0.5 and is to beused in conjunction with Fig. 5.13, which supplies the value of the friction factor f . Itapplies to ducts of various cross-sectional shapes, with wall surfaces having variousdegrees of roughness. In such cases D is replaced by Dh . In the special case of a pipeBOOKCOMP, Inc. — John Wiley & Sons / Page 423 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1053 to 1100———0.71622pt PgVar———Normal PagePgEnds: TEX[423], (29)424123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSwith smooth internal surface, eqs. (5.75) and (5.68) can be combined to derive theNusselt number relationshipNuD =hD4/5= 0.023ReD · Pr 1/3k(5.76)which holds in the range 2 × 104 ≤ ReD ≤ 106 . A popular version of this is acorrelation due to Dittus and Boelter (1930),NuD = 0.023ReD · Pr n4/5(5.77)which was developed for 0.7 ≤ Pr ≤ 120, 2500 ≤ ReD ≤ 1.24 × 105 , andL/D > 60.

The Prandtl number exponent is n = 0.4 when the fluid is being heated(T0 > Tm ), and n = 0.3 when the fluid is being cooled (T0 < Tm ). All of the physicalproperties needed for the calculation of NuD , ReD , and Pr are to be evaluated at thebulk temperature Tm . The maximum deviation between experimental data and valuespredicted using eq. (5.77) is on the order of 40%.For applications in which influence of temperature on properties is significant, theSieder and Tate (1936) modification of eq. (5.76) is recommended: 0.14µ4/51/3NuD = 0.027ReD · Pr(5.78)µ0This correlation is valid for 0.7 ≤ Pr ≤ 16,700 and ReD > 104 .

The effect oftemperature-dependent properties is taken into account by evaluating all the properties (except µ0 ) at the mean temperature of the stream, Tm . The viscosity µ0 isevaluated at the wall temperature µ0 = µ(T0 ). Equations (5.76)–(5.78) can be usedfor 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:NuD =(f/2)ReD · Pr1.07 + 900/ReD − 0.63/(1 + 10Pr) + 12.7(f/2)1/2 (Pr 2/3 − 1)(5.79a)andNuD =(f/2)ReD · Pr1.07 + 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 ≤ ReD ≤ 5 × 106and 0.5 ≤ Pr ≤ 106 . Equation (5.79b) is an abbreviated version of eq. (5.79a) andwas modified by Gnielinski (1976):NuD =(f/2)(ReD − 103 )Pr1 + 12.7(f/2)1/2 Pr 2/3 − 1BOOKCOMP, Inc. — John Wiley & Sons / Page 424 / 2nd Proofs / Heat Transfer Handbook / Bejan(5.80)[424], (30)Lines: 1100 to 1147———1.92621pt PgVar———Normal Page* PgEnds: Eject[424], (30)TOTAL HEAT TRANSFER RATE123456789101112131415161718192021222324252627282930313233343536373839404142434445425which is accurate within ±10% in the range 0.5 ≤ Pr ≤ 106 and 2300 ≤ ReD ≤5 × 106 . The Gnielinski correlation of eq.

(5.80) can be used in both constant-q andconstant-T0 applications. Two simpler alternatives to eq. (5.80) are (Gnielinski, 1976) 0.4NuD = 0.0214 Re0.8(5.81a)D − 100 Prvalid in the range0.5 ≤ Pr ≤ 1.5104 ≤ ReD ≤ 5 × 106and 0.4NuD = 0.012 Re0.87D − 280 Pr(5.81b)[425], (31)valid in the range1.5 ≤ Pr ≤ 5003 × 103 ≤ ReD ≤ 106The preceding results refer to gases and liquids, that is, to the range Pr ≥ 0.5. Forliquid metals, the most accurate correlations are those of Notter and Sleicher (1972):0.93q0 = constant(5.82)6.3 + 0.0167Re0.85D · PrNuD =0.850.934.8 + 0.0156ReD · PrT0 = constant(5.83)These are valid for 0.004 ≤ Pr ≤ 0.1 and 104 ≤ ReD ≤ 106 and are basedon both computational and experimental data.

All the properties used in eqs. (5.82)and (5.83) are evaluated at the mean temperature Tm . The mean temperature varieswith 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 simplifythe recommended evaluation of the physical properties at the Tm temperature, it isconvenient to choose as representative mean temperature the average valueTm = 21 (Tin + Tout )(5.84)In this definition, Tin and Tout are the bulk temperatures of the stream at the duct inletand outlet, respectively (Fig. 5.14).5.7 TOTAL HEAT TRANSFER RATEThe summarizing conclusion is that in both laminar and turbulent fully developedduct flow the heat transfer coefficient h is independent of longitudinal position.

Thisfeature makes it easy to express analytically the total heat transfer rate q (watts)between a stream and duct of length L,q = hAw ∆TlmBOOKCOMP, Inc. — John Wiley & Sons / Page 425 / 2nd Proofs / Heat Transfer Handbook / Bejan(5.85)Lines: 1147 to 1220———1.55403pt PgVar———Normal PagePgEnds: TEX[425], (31)426123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSTT⌬TinWallT0 (x)WallT0⌬ToutTout⌬ToutTout⌬TinStream, Tm (x)Tin0Stream, Tm (x)TinLx0(a)Lx(b)Figure 5.14 Variation of the mean temperature along the duct: (a) isothermal wall; (b) wallwith uniform heat flux.[426], (32)Lines: 1220 to 1245———0.40805pt PgVar———Normal PageIn this expression Aw is the total duct surface, Aw = pL.

The effective tempera- * PgEnds: Ejectture difference between the wall and the stream is the log-mean temperature difference Tlm .[426], (32)5.7.1Isothermal WallWhen the wall is isothermal (Fig. 5.14) the log-mean temperature difference is∆Tlm =∆Tin − ∆Toutln(∆Tin /∆Tout )(5.86)Equations (5.85) and (5.86) express the relationship among the total heat transfer rateq, the total duct surface conductance hAw , and the outlet temperature of the stream.Alternatively, the same equations can be combined to express the total heat transferrate in terms of the inlet temperatures, mass flow rate, and duct surface conductance,q = ṁcp ∆Tin (1 − e−hAw /ṁcp )(5.87)In cases where the heat transfer coefficient varies longitudinally, h(x), the h factor onthe right side of eq.

(5.87) represents the L-averaged heat transfer coefficient: namely,h̄ =1L0BOOKCOMP, Inc. — John Wiley & Sons / Page 426 / 2nd Proofs / Heat Transfer Handbook / BejanLh(x) dx(5.88)SUMMARY OF FORCED CONVECTION RELATIONSHIPS1234567891011121314151617181920212223242526272829303132333435363738394041424344454275.7.2 Wall Heated UniformlyIn the analysis of heat exchangers (Bejan, 1993), it is found that the applicability ofeqs. (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 theduct, 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 cpare independent of x.

Geometrically, it is evident that the effective value ∆Tlm is thesame as the constant ∆T recorded all along the duct,∆Tlm = ∆Tin = ∆Tout(5.89)Equation (5.89) is a special case of eq. (5.86): namely, the limit ∆Tin /∆Tout → 1.[427], (33)5.8OPTIMAL CHANNEL SIZES FOR TURBULENT FLOWLines: 1245 to 1291The 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 turbulent6.74318ptPgVar(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 intersectingNormal Pagethe two asymptotes of the design: a few wide spaces with turbulent boundary layersand many narrow spaces with fully developed turbulent flow. The plate thickness (t) * PgEnds: Ejectis not negligible with respect to the spacing D.

The optimal spacing and maximalglobal conductance of the HWL package are[427], (33)Dopt /L= 0.071Pr −5/11 · Be−1/11(1 + t/Dopt )4/11(5.90)qmaxt −67/99k· Be47/99≤ 0.57 2 (Tmax − T0 )Pr 4/99 1 +HLWLDopt(5.91)andwhere Be = ∆PL2 /µα. These results are valid in the range 104 ≤ ReDh ≤ 106 and106 ≤ ReL ≤ 108 , which can be shown to correspond to the pressure drop numberrange 1011 ≤ Be ≤ 1016 .5.9SUMMARY OF FORCED CONVECTION RELATIONSHIPS• Laminar flow entrance length:X/D≈ 10−2ReDBOOKCOMP, Inc.

— John Wiley & Sons / Page 427 / 2nd Proofs / Heat Transfer Handbook / Bejan(5.2)428123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWS• Skin friction coefficient definition:Cf,x =τw1ρU 22(5.3)• Laminar fully developed (Hagen–Poiseuille) flow between parallel plates withspacing D:3y 2u(y) = U 1 −(5.7)2D/2withD2dPU=−12µdx[428], (34)(5.8)• Laminar fully developed (Hagen–Poiseuille) flow in a tube with diameter D: 2 ru = 2U 1 −(5.9)r0withr02dPU=−8µdx(5.10)• Hydraulic radius and diameter:rh =Dh =Aphydraulic radius4Ahydraulic diameterp(5.14)(5.15)• Friction factor: τw1ρU 22 24f =ReDh 16ReDhsee Tables 5.1–5.3(5.11)Dh = 2Dparallel plates (D = spacing)(5.17)Dh = Dround tube (D = diameter)(5.18)BOOKCOMP, Inc. — John Wiley & Sons / Page 428 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1291 to 1348———10.81242pt PgVar———Short Page* PgEnds: Eject[428], (34)SUMMARY OF FORCED CONVECTION RELATIONSHIPS123456789101112131415161718192021222324252627282930313233343536373839404142434445429• Pressure drop:∆P = f4LDh1 2ρU2(5.16)• Nusselt number (see Tables 5.1 through 5.3):∂T /∂r r=r0hDNu ==DkT0 − T m(5.24)• Laminar thermal entrance length:XT ≈ 10−2 Pr · Dh · ReDh• Thermally developing Hagen–Poiseuille flow (Pr = ∞):• Round tube, isothermal wall:−1/31.077x∗− 0.70x∗ ≤ 0.01Nux =3−0.488 −57.2x∗3.657 + 6.874(10 x∗ )ex∗ > 0.01Nu0−x−1/3− 0.70 1.615x∗−1/3= 1.615x∗− 0.203.657 + 0.0499/x∗0.005 < x∗ < 0.03———(5.32)5.55528pt PgVar———Short Page* PgEnds: Eject(5.33)x∗ > 0.03[429], (35)x∗ ≤ 0.000050.00005 < x∗ ≤ 0.0015 (5.34)x∗ > 0.001• Parallel plates, isothermal surfaces:−1/3+ 0.40x∗ ≤ 0.0011.233x∗Nu0−x =3−0.488 −245x∗7.541 + 6.874(10 x∗ )ex∗ > 0.001−1/3x∗ ≤ 0.0005 1.849x∗−1/3Nu0−x = 1.849x∗+ 0.600.0005 < x∗ ≤ 0.0067.541 + 0.0235/x∗x∗ > 0.006[429], (35)Lines: 1348 to 1415x∗ ≤ 0.005• Round tube, uniform heat flux:−1/3− 1.00 3.302x∗−1/3Nux = 1.302x∗− 0.504.364 + 8.68(103 x∗ )−0.506 e−41x∗−1/3x∗ ≤ 0.031.953x∗Nu0−x =4.364 + 0.0722/x∗x∗ > 0.03BOOKCOMP, Inc.

— John Wiley & Sons / Page 429 / 2nd Proofs / Heat Transfer Handbook / Bejan(5.26)(5.35)(5.37)(5.38)430123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWS• Parallel plates, uniform heat flux:−1/3x∗ ≤ 0.0002 1.490x∗−1/3Nux = 1.490x∗− 0.400.0002 < x∗ ≤ 0.001 (5.39)3−0.506 −164x∗8.235 + 8.68(10 x∗ )ex∗ > 0.001−1/3x∗ ≤ 0.001 2.236x∗−1/3Nu0−x = 2.236x∗(5.40)+ 0.900.001 < x∗ ≤ 0.018.235 + 0.0364/x∗x∗ ≥ 0.01• Thermally and hydraulically developing flow:• Round tube, isothermal wall:Nux = 7.55 +Nu0−x = 7.55 +0.024x∗−1.141+0.0179Pr 0.17 x∗−0.64− 0.1420.0358Pr 0.17 x∗−0.64(5.41)Lines: 1415 to 1478———0.024x∗−1.14(5.42)1 + 0.0358Pr 0.17 x∗−0.64∆P1.25 + 64x+ − 13.74(x+ )1/21/2=13.74(x)++11 + 0.00021(x+ )−2ρU 22x+ =[430], (36)x/DReD———Short Page(5.43) * PgEnds: Eject(5.44)• Round tube, uniform heat flux:Nux1/64.364 1 + (Gz/29.6)23/2 1/3Gz/19.04= 1+ 1/21/31 + (Pr/0.0207)2/31 + (Gz/29.6)2(5.46)• Optimal channel sizes:• Laminar flow, parallel plates:Dopt 2.7Be−1/4LBe =∆PL2µαqmaxk 0.60 2 (Tmax − T0 )Be1/2HLWLBOOKCOMP, Inc.

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