John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 46
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Analysis of that momentum balance [6.2] leads to thefollowing expressions for the boundary thickness and the skin frictioncoefficient as a function of x:0.16δ(x)=1/7xRexCf (x) =0.0271/7Rex(6.100)(6.101)To write these expressions, we assume that the turbulent b.l. begins atx = 0, neglecting the initial laminar region. They are reasonably accuratefor Reynolds numbers ranging from about 106 to 109 .
A more accurate322Laminar and turbulent boundary layers§6.8formula for Cf , valid for all turbulent Rex , was given by White [6.10]:0.455Cf (x) = !"2ln(0.06 Rex )6.8(6.102)Heat transfer in turbulent boundary layersLike the turbulent momentum boundary layer, the turbulent thermalboundary layer is characterized by inner and outer regions. In the inner part of the thermal boundary layer, turbulent mixing is increasinglyweak; there, heat transport is controlled by heat conduction in the sublayer.
Farther from the wall, a logarithmic temperature profile is found,and in the outermost parts of the boundary layer, turbulent mixing is thedominant mode of transport.The boundary layer ends where turbulence dies out and uniform freestream conditions prevail, with the result that the thermal and momentum boundary layer thicknesses are the same. At first, this might seemto suggest that an absence of any Prandtl number effect on turbulentheat transfer, but that is not the case.
The effect of Prandtl number isnow found in the sublayers near the wall, where molecular viscosity andthermal conductivity still control the transport of heat and momentum.The Reynolds-Colburn analogy for turbulent flowThe eddy diffusivity for momentum was introduced by Boussinesq [6.11]in 1877. It was subsequently proposed that Fourier’s law might likewisebe modified for turbulent flow as follows:another constant, which ∂T∂T−q = −kreflects turbulent mixing ∂y∂y≡ ρcp · εhwhere T is the local time-average value of the temperature. Therefore,q = −ρcp (α + εh )∂T∂y(6.103)where εh is called the eddy diffusivity of heat. This immediately suggestsyet another definition:εm(6.104)turbulent Prandtl number, Prt ≡εhHeat transfer in turbulent boundary layers§6.8Equation (6.103) can be written in terms of ν and εm by introducing Prand Prt into it.
Thus,εm ∂Tν+(6.105)q = −ρcpPr Prt ∂yBefore trying to build a form of the Reynolds analogy for turbulentflow, we must note the behavior of Pr and Prt :• Pr is a physical property of the fluid. It is both theoretically andactually near unity for ideal gases, but for liquids it may differ fromunity by orders of magnitude.• Prt is a property of the flow field more than of the fluid. The numerical value of Prt is normally well within a factor of 2 of unity. Itvaries with location in the b.l., but, for nonmetallic fluids, it is oftennear 0.85.The time-average boundary-layer energy equation is similar to thetime-average momentum equation [eqn.
(6.90a)]ν∂εm ∂T∂T∂T∂+vq=ρcp+(6.106)=−ρcp u∂x∂y∂y∂yPr Prt ∂yneglect very near walland in the near wall region the convective terms are again negligible. Thismeans that ∂q/∂y 0 near the wall, so that the heat flux is constant iny and equal to the wall heat flux:εm ∂Tν+(6.107)q = qw = −ρcpPr Prt ∂yWe may integrate this equation as we did eqn.
(6.91), with the result that⎧ ∗ u y⎪⎪Prthermal sublayer⎪⎨νTw − T (y)(6.108)= ∗ qw /(ρcp u∗ ) ⎪⎪⎪⎩ 1 ln u y + A(Pr) thermal log layerκνThe constant A depends upon the Prandtl number. It reflects the thermalresistance of the sublayer near the wall. As was done for the constantB in the velocity profile, experimental data or numerical simulation maybe used to determine A(Pr) [6.12, 6.13].
For Pr ≥ 0.5,A(Pr) = 12.8 Pr0.68 − 7.3(6.109)323324Laminar and turbulent boundary layers§6.8To obtain the Reynolds analogy, we can subtract the dimensionlesslog-law, eqn. (6.99), from its thermal counterpart, eqn. (6.108):u(y)Tw − T (y)−= A(Pr) − Bqw /(ρcp u∗ )u∗(6.110a)In the outer part of the boundary layer, T (y) T∞ and u(y) u∞ , sou∞Tw − T∞− ∗ = A(Pr) − B∗qw /(ρcp u )u(6.110b)We can eliminate the friction velocity in favor of the skin friction coefficient by using the definitions of each:22Cfτwu∗(6.110c)==2u∞2ρu∞Hence,Tw − T∞qw /(ρcp u∞ )2Cf22−2= A(Pr) − BCf(6.110d)Rearrangment of the last equation givesCf 2qw4 =(ρcp u∞ )(Tw − T∞ )1 + [A(Pr) − B] Cf 2(6.110e)The lefthand side is simply the Stanton number, St = h (ρcp u∞ ).
Uponsubstituting B = 5.5 and eqn. (6.109) for A(Pr), we obtain the ReynoldsColburn analogy for turbulent flow:Stx =Cf 24 1 + 12.8 Pr0.68 − 1 Cf 2Pr ≥ 0.5(6.111)This result can be used with eqn. (6.102) for Cf , or with data for Cf ,to calculate the local heat transfer coefficient in a turbulent boundarylayer.
The equation works for either uniform Tw or uniform qw . This isbecause the thin, near-wall part of the boundary layer controls most ofthe thermal resistance and that thin layer is not strongly dependent onupstream history of the flow.Equation (6.111) is valid for smooth walls with a mild or a zero pressure gradient.
The factor 12.8 (Pr0.68 − 1) in the denominator accountsfor the thermal resistance of the sublayer. If the walls are rough, thesublayer will be disrupted and that term must be replaced by one thattakes account of the roughness (see Sect. 7.3).Heat transfer in turbulent boundary layers§6.8Other equations for heat transfer in the turbulent b.l.Although eqn. (6.111) gives an excellent prediction of the local value of hin a turbulent boundary layer, a number of simplified approximations toit have been suggested in the literature. For example, for Prandtl numbersnot too far from unity and Reynolds numbers not too far above transition,the laminar flow Reynolds-Colburn analogy can be usedCfPr−2/3for Pr near 1(6.76)Stx =2The best exponent for the Prandtl number in such an equation actuallydepends upon the Reynolds and Prandtl numbers. For gases, an exponentof −0.4 gives somewhat better results.A more wide-ranging approximation can be obtained after introducing a simplifed expression for Cf .
For example, Schlichting [6.3, Chap. XXI]shows that, for turbulent flow over a smooth flat plate in the low-Re range,Cf 0.05921/5Rex,5 × 105 Rex 107(6.112)With this Reynolds number dependence, Žukauskas and coworkers [6.14,6.15] found thatCf0.7 ≤ Pr ≤ 380(6.113)Pr−0.57 ,Stx =2so that when eqn. (6.112) is used to eliminate Cf0.43Nux = 0.0296 Re0.8x Pr(6.114)Somewhat better agreement with data, for 2 × 105 Rex 5 × 106 , isobtained by adjusting the constant [6.15]:0.43Nux = 0.032 Re0.8x Pr(6.115)The average Nusselt number for uniform Tw is obtained from eqn.(6.114) as follows: L0.0296 Pr0.43 L k L 10.8Rex dxNuL = h =kkL 0 x325326Laminar and turbulent boundary layers§6.8where we ignore the fact that there is a laminar region at the front of theplate.
Thus,0.43NuL = 0.0370 Re0.8L Pr(6.116)This equation may be used for either uniform Tw or uniform qw , and forReL up to about 3 × 107 [6.14, 6.15].A flat heater with a turbulent b.l. on it actually has a laminar b.l. between x = 0 and x = xtrans , as is indicated in Fig. 6.4. The obvious wayto calculate h in this case is to writeL1h=q dxL∆T 0(6.117)Lxtrans1=hlaminar dx +hturbulent dxL0xtranswhere xtrans = (ν/u∞ )Retrans . Thus, we substitute eqns. (6.58) and (6.114)in eqn.
(6.117) and obtain, for 0.6 Pr 50,;0.8−0.097NuL = 0.037 Pr0.43 Re0.8(Retrans )1/2L − Retrans − 17.95 Pr(6.118)If ReL Retrans , this result reduces to eqn. (6.116).Whitaker [6.16] suggested setting Pr−0.097 ≈ 1 and Retrans ≈ 200, 000in eqn. (6.118):0.43NuL = 0.037 PrRe0.8L− 9200µ∞µw1/40.6 ≤ Pr ≤ 380(6.119)This expression has been corrected to account for the variability of liquidviscosity with the factor (µ∞ /µw )1/4 , where µ∞ is the viscosity at the freestream temperature, T∞ , and µw is that at the wall temperature, Tw ; otherphysical properties should be evaluated at T∞ .
If eqn. (6.119) is usedto predict heat transfer to a gaseous flow, the viscosity-ratio correctionterm should not be used and properties should be evaluated at the filmtemperature. This is because the viscosity of a gas rises with temperatureinstead of dropping, and the correction will be incorrect.Finally, it is important to remember that eqns. (6.118) and (6.119)should be used only when ReL is substantially above the transitionalvalue.Heat transfer in turbulent boundary layers§6.8A correlation for laminar, transitional, and turbulent flowA problem with the two preceding relations is that they do not reallydeal with the question of heat transfer in the rather lengthy transitionregion.
Both eqns. (6.118) and (6.119) are based on the assumption thatflow abruptly passes from laminar to turbulent at a critical value of x,and we have noted in the context of Fig. 6.4 that this is not what occurs.The location of the transition depends upon such variables as surfaceroughness and the turbulence, or lack of it, in the stream approachingthe heater.Churchill [6.17] suggests correlating any particular set of data withNux = 0.45 + 0.3387 φ1/2⎧⎨⎪where3/5⎫1/2⎪⎬(φ/2, 600)1+ !"2/5 ⎪⎭1 + (φu /φ)7/2⎪⎩0.04681+Prφ ≡ Rex Pr2/3(6.120a)2/3 −1/2(6.120b)and φu is a number between about 105 and 107 . The actual value of φumust be fit to the particular set of data. In a very “clean” system, φuwill be larger; in a very “noisy” one, it will be smaller.
If the Reynoldsnumber at the end of the turbulent transition region is Reu , an estimateis φu ≈ φ(Rex = Reu ).The equation is for uniform Tw , but it may be used for uniform qwif the constants 0.3387 and 0.0468 are replaced by 0.4637 and 0.02052,respectively.Churchill also gave an expression for the average Nusselt number:NuL = 0.45 + 0.6774 φ1/2⎧⎨⎪⎪⎩3/5⎫1/2⎪⎬(φ/12, 500)1+ !"2/5 ⎪⎭1 + (φum /φ)7/2(6.120c)where φ is defined as in eqn. (6.120b), using ReL in place of Rex , andφum ≈ 1.875 φ(ReL = Reu ).
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