Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 72
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(6.113a) into eqs. (6.111) provides 51= 1+ν y+(6.114a)y+=−1ν5(6.114b)orWith eq. (6.114b) put into eq. (6.112),1y + − 5 ∂T +1=+Pr5PrT∂y +(6.115)[477], (39)Then integration across the buffer region givesT + − Ts+ =ory+5dy +1/Pr + (y + − 5)/5PrTPr y +T + − Ts+ = 5PrT ln 1 +−1PrT5This may be applied across the entire buffer region:Pr 30−1Tb+ − Ts+ = 5PrT ln 1 +PrT5(6.116)———0.88054pt PgVar(6.117) ∂u+τ=τ0ν ∂y +(6.119)q /ν ∂T + =q0Pr ∂y +(6.120)Similar distributions of shear stress and heat transfer rate in the outer region can beassumed so that(6.121)Because τ and q are constant across the inner two layers, τb = τ0 and q = q0 .Henceq τ= τ0q0BOOKCOMP, Inc.
— John Wiley & Sons / Page 477 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal Page* PgEnds: Eject[477], (39)(6.118)For the outer region (y + > 30), where ν and H ν,q τ= τbqbLines: 1821 to 1886(6.122)478123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSWith eqs. (6.121) and (6.122) in eq. (6.120), ∂u+ ∂T +=ν ∂y +νPrT ∂y +∂u+∂T +=PrT∂y +∂y +and(6.123)Integrating across the outer layer yields+T1+ − Tb+ = PrT (u+1 − ub )(6.124)where T1+ and u+1 are in the free stream andu+b = 5 + 5 ln 6so thatT1+−Tb+= PrTu+1[478], (40)− 5(1 + ln 6)(6.125)Lines: 1886 to 1969Addition of the temperature profiles across the three layers givesT1+ = 5Pr + 5PrT ln———PrT + 5Pr+ PrT u+1 −56PrT(6.126)and becauseT+ =ρcp (T0 − T̄ ) ∗ρcp (T0 − T̄ )v =q0q0τ0ρ1/2=Pr · RexNuxCf21/2Rex (Cf /2)1/2PrT +PrTPrT + 5Pr+5+5ln(u − 5)Pr6PrTPr 1[478], (40)(6.128)This yields the final expression with St = Nux /Rex · Pr:Stx =(Cf/2)1/2 {5Pr(Cf /2)1/2+ 5PrT ln[(PrT + 5Pr)/6PrT ] − 5PrT } + PrT(6.129)ForCf =0.058Re0.20xeq.
(6.129) can be written asNux = 0.029Re0.8x Gwhere the parameter G isBOOKCOMP, Inc. — John Wiley & Sons / Page 478 / 2nd Proofs / Heat Transfer Handbook / Bejan———Long Page* PgEnds: Eject(6.127)the Nusselt number becomesNux =0.84541pt PgVar(6.130)HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445G=1/2 {5Pr(0.029/Re.20x )Pr+ 5 ln[(1 + 5Pr)/6] − 5} + 1479(6.131)As pointed out by Oosthuizen and Naylor (1999), eq.
(6.131) is based on PrT = 1but does apply for various Pr.Flat Plate with an Unheated Starting Length in Turbulent Flow The integral forms of the momentum and energy equations can also be developed for turbulentflow by integrating the mean flow equations across the boundary layer from the surface to a location H outside the boundary layer. The integral momentum equationcan be written asddxHū(ū − U )dy = −0τ0dδ2= −U 2= −(v ∗ )2ρdx(6.132)where the momentum thickness δ2 isδ2 =0HūU[479], (41)Lines: 1969 to 2027ū1−dyU(6.133)———3.1273pt PgVar———Assume that both Pr and PrT are equal to unity and that the mean profiles for theLong Pagemean velocity and temperature in the velocity (δ) and thermal (δT ) boundary layers * PgEnds: Ejectare given by y 1/7ū=UδandT0 − T̄=t0 − T ∞yδT1/7(6.134)Then from the integral momentum equation y 1/7τ=1−τ0δ(6.135)and because the total heat flux at any point in the flow isν + = α + H = y 9/7 y 6/7Cf δτ/ρ= 7νRex 1 −∂ ū/∂y2 xδδ(6.136)the total heat flux at any point in the flow isq = −ρcp (α + H )∂ T̄∂y(6.137)For the assumed profiles this can be written asCfq =ρcp U (T0 − T∞ )2BOOKCOMP, Inc.
— John Wiley & Sons / Page 479 / 2nd Proofs / Heat Transfer Handbook / Bejan1− y 9/7 y 6/7δδ(6.138)[479], (41)480123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSThe wall heat flux can be determined from eq. (6.138). This expression is substituted on the right-hand side of the integral energy equation. The ū and T̄ profiles andthe δ as a function of x relations are substituted on the left side and δT is solved foras a function of x. The final result for the Stanton number St isSt =Cf21− x 9/10 −1/90x(6.139)With the same Prandtl number correction as in the case of the fully heated plate, x 9/10 −1/901−St · Pr0.40 = 0.0287Re−0.2xx(6.140)Just as for laminar flow, the result of eq.
(6.140) can be generalized to an arbitrarilyvarying wall temperature through superposition (Kays and Crawford, 1980).[480], (42)Arbitrarily Varying Heat Flux Based on eq. (6.134), which is Prandtl’s 17 powerlaw for thermal boundary layers on smooth surfaces with uniform specified surfaceheat flux, a relationship that is nearly identical to eq. (6.140) for uniform surfacetemperature is recommended:Lines: 2027 to 2074St · Pr0.40 = 0.03Re−0.2x(6.141)For a surface with an arbitrarily specified heat flux distribution and an unheatedsection, the temperature rise can be determined from x0 =x(6.142)Γ(x0 , x)q (x0 ) dx0T0 (x) − T∞ =x0 =0whereΓ(x0 ,x) =9 x 9/10 −8/9Pr−0.60 · Re−0.800x101 8 1−xΓ 9 Γ 9 (0.0287k)(6.143)Turbulent Prandtl Number An assumption of PrT = 1 has been made in theforegoing analyses.
The turbulent Prandtl number is determined experimentally usingPrT =u v (∂ T̄ /∂y)v T (∂ ū/∂y)(6.144)Kays and Crawford (1993) have pointed out that, in general, it is difficult to measureall four quantities accurately. With regard to the mean velocity data, mean temperaturedata in the logarithmic region show straight-line behavior when plotted in the wallcoordinates. The slope of this line, relative to that of the velocity profile, providesPrT . Data for air and water reveal a PrT range from 0.7 to 0.9.
No noticeable effectBOOKCOMP, Inc. — John Wiley & Sons / Page 480 / 2nd Proofs / Heat Transfer Handbook / Bejan———1.5713pt PgVar———Normal PagePgEnds: TEX[480], (42)HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445481due to surface roughness is noted, but at very low Pr, a Prandtl number effect on PrTis observed. This is believed to be due to the higher thermal conductivity based onexisting analyses. At higher Pr, there is no effect on PrT . The PrT values are foundto be constant in the “law of the wall” region but are higher in the sublayer, and thewall value of PrT approaches 1.09 regardless of Pr.6.4.15Surface Roughness EffectIf the size of the roughness elements is represented by a mean length scale ks , aroughness Reynolds number may be defined as Rek = ks (v ∗ /ν).
Three regimes ofroughness may then be prescribed:SmoothTransitional roughRek < 55 < Rek < 70Fully roughRek > 70[481], (43)Under the fully rough regime, the friction coefficient becomes independent ofviscosity and Rek . The role of the roughness is to destabilize the sublayer. For Rek >70, the sublayer disappears and the shear stress is transmitted to the wall by pressuredrag on the roughness elements. The mixing length near a rough surface is given as = κ(y+δyo ), which yields a finite eddy diffusivity at y = 0.
In the wall coordinates,based on experimental data,*δy0+ =δy0 v ∗= 0.031Rekν(6.145)The law of the wall for a fully rough surface isdu+1=++dyκ(y + δy0+ )(6.146)Because no sublayer exists for fully rough surface, eq. (6.146) may be integrated from0 to y + to provideu+ =11 32.6ln y + + lnκκ Rek(6.147)The friction coefficient is calculated by evaluating u+∞ while including the additiveeffect of the outer wake. The latter results in a displacement by 2.3. The result isu+∞ =11 848= ln1/2(Cf /2)κks(6.148)The heat transfer down to the plane of the roughness elements is by eddy conductivity, but the final transfer to the surface is by molecular conduction through thealmost stagnant fluid in the roughness cavities. The law of the wall is written asBOOKCOMP, Inc.
— John Wiley & Sons / Page 481 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2074 to 2114———3.49532pt PgVar———Normal PagePgEnds: Eject[481], (43)482123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWST+−δT0+=y+0dy +H /ν(6.149)where the second term on the left is the temperature difference across which heat istransferred by conduction.
WithH1 κ +y + δy0+==νPrT νPrT(6.150a)thenT + = δT0+ +32.6y +PrTlnκRek(6.150b)[482], (44)A local heat transfer coefficient can be defined based on δT0 :δT0+ =ρcp v ∗1=hkStk(6.151)———2.71744pt PgVarBased on experimental data for roughness from spheres,Stk = C ·Re−0.20k· Pr−0.40(6.152)with C ≈ 0.8 and PrT ≈ 0.9, the law of the wall can be rewritten asT+ =1PrT32.6y ++lnStkκRek(6.153)and using a procedure similar to that used for a smooth surface,St =6.4.16Cf /2PrT + (Cf /2)1/2 /Stk(6.154)Some Empirical Transport CorrelationsCylinder in Crossflow The analytical solutions described in Sections 6.4.9 and6.4.10 provide local convection coefficients from the front stagnation point (usingsimilarity theory) to the separation point of a cylinder using the Smith–Spaldingmethod.
At the front stagnation point, the free stream is brought to rest, with anaccompanying rise in pressure. The initial development of the boundary layer alongthe cylinder following this point is under favorable pressure gradient conditions;that is, dp/dx < 0. However, the pressure reaches a minimum at some value ofx, depending on the Reynolds number. Farther downstream, the pressure gradientis adverse (i.e., dp/dx > 0), until the point of boundary layer separation wherethe surface shear stress becomes zero. This results in the formation of a wake.
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