John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 44
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Is it safe? What isthe average temperature of the plate?Solution. In accordance with eqn. (6.71),∆Tmax = ∆Tx=L =qLqL/k=1/2k Nux=L0.453 Rex Pr1/3or if we evaluate properties at (85 + 15)/2 = 50◦ C, for the moment,∆Tmax =420(0.6)/0.0278= 91.5◦ C"1/2!−51/3(0.709)0.453 0.6(1.8)/1.794 × 10This will give Twmax = 15 + 91.5 = 106.5◦ C. This is very close to105◦ C. If 105◦ C is at all conservative, q = 420 W/m2 should be safe—particularly since it only occurs over a very small distance at the endof the plate.The Reynolds analogy§6.6311From eqn.
(6.72) we find that∆T =0.453∆Tmax = 61.0◦ C0.6795soTw = 15 + 61.0 = 76.0◦ C6.6The Reynolds analogyThe analogy between heat and momentum transfer can now be generalized to provide a very useful result. We begin by recalling eqn. (6.25),which is restricted to a flat surface with no pressure gradient: 1Cfuyud=−δ(6.25)−1 ddxδ20 u∞ u∞and by rewriting eqns. (6.47) and (6.51), we obtain for the constant walltemperature case:1 qwuT − T∞ydφδd=(6.74)dxδtρcp u∞ (Tw − T∞ )0 u∞ Tw − T∞But the similarity of temperature and flow boundary layers to one another[see, e.g., eqns. (6.29) and (6.50)], suggests the following approximation,which becomes exact only when Pr = 1:uT − T∞δtδ= 1−Tw − T∞u∞Substituting this result in eqn.
(6.74) and comparing it to eqn. (6.25), weget 1Cfqwuuyd=−δ=−−1 d−dxδ2ρcp u∞ (Tw − T∞ )φ20 u∞ u∞(6.75)Finally, we substitute eqn. (6.55) to eliminate φ from eqn. (6.75). Theresult is one instance of the Reynolds-Colburn analogy:8CfhPr2/3 =ρcp u∞28(6.76)Reynolds [6.6] developed the analogy in 1874. Colburn made important use of it inthis century. The form given is for flat plates with 0.6 ≤ Pr ≤ 50.
The Prandtl numberfactor is usually a little different for other flows or other ranges of Pr.312Laminar and turbulent boundary layers§6.6For use in Reynolds’ analogy, Cf must be a pure skin friction coefficient.The profile drag that results from the variation of pressure around thebody is unrelated to heat transfer. The analogy does not apply whenprofile drag is included in Cf .The dimensionless group h/ρcp u∞ is called the Stanton number. Itis defined as follows:St, Stanton number ≡Nuxh=ρcp u∞Rex PrThe physical significance of the Stanton number isSt =h∆Tactual heat flux to the fluid=ρcp u∞ ∆Theat flux capacity of the fluid flow(6.77)The group St Pr2/3 was dealt with by the chemical engineer Colburn, whogave it a special symbol:j ≡ Colburn j-factor = St Pr2/3 =NuxRex Pr1/3(6.78)Example 6.7Does the equation for the Nusselt number on an isothermal flat surface in laminar flow satisfy the Reynolds analogy?Solution.
If we rewrite eqn. (6.58), we obtainNux0.3322/3= 31/3 = St PrRexRex Pr(6.79)But comparison with eqn. (6.33) reveals that the left-hand side ofeqn. (6.79) is precisely Cf /2, so the analogy is satisfied perfectly. Likewise, from eqns. (6.68) and (6.34), we getCf0.664NuL2/3= 3=1/3 ≡ St Pr2ReLReL Pr(6.80)The Reynolds-Colburn analogy can be used directly to infer heat transfer data from measurements of the shear stress, or vice versa. It can alsobe extended to turbulent flow, which is much harder to predict analytically. We shall undertake that problem in Sect.
6.8.Turbulent boundary layers§6.7Example 6.8How much drag force does the air flow in Example 6.5 exert on theheat transfer surface?Solution. From eqn. (6.80) in Example 6.7, we obtainCf =2 NuLReL Pr1/3From Example 6.5 we obtain NuL , ReL , and Pr1/3 :Cf =2(367.8)= 0.002135(386, 600)(0.707)1/3soτyx = (0.002135)1(0.002135)(1.05)(15)2ρu2∞ =22= 0.2522 kg/m·s2and the force isτyx A = 0.2522(0.5)2 = 0.06305 kg·m/s2 = 0.06305 N= 0.23 oz6.7Turbulent boundary layersTurbulenceBig whirls have little whirls,That feed on their velocity.Little whirls have littler whirls,And so on, to viscosity.This bit of doggerel by the English fluid mechanic, L.
F. Richardson, tellsus a great deal about the nature of turbulence. Turbulence in a fluid canbe viewed as a spectrum of coexisting vortices in which kinetic energyfrom the larger ones is dissipated to successively smaller ones until thevery smallest of these vortices (or “whirls”) are damped out by viscousshear stresses.The next time the weatherman shows a satellite photograph of NorthAmerica on the 10:00 p.m.
news, notice the cloud patterns. There will be313Laminar and turbulent boundary layers314§6.7one or two enormous vortices of continental proportions. These hugevortices, in turn, feed smaller “weather-making” vortices on the order ofhundreds of miles in diameter. These further dissipate into vortices ofcyclone and tornado proportions—sometimes with that level of violencebut more often not. These dissipate into still smaller whirls as they interact with the ground and its various protrusions.
The next time the windblows, stand behind any tree and feel the vortices. In the great plains,where there are not many ground vortex generators (such as trees), youwill see small cyclonic eddies called “dust devils.” The process continuesright on down to millimeter or even micrometer scales. There, momentum exchange is no longer identifiable as turbulence but appears simplyas viscous stretching of the fluid.The same kind of process exists within, say, a turbulent pipe flow athigh Reynolds number.
Such a flow is shown in Fig. 6.17. Turbulencein such a case consists of coexisting vortices which vary in size from asubstantial fraction of the pipe radius down to micrometer dimensions.The spectrum of sizes varies with location in the pipe. The size andintensity of vortices at the wall must clearly approach zero, since thefluid velocity goes to zero at the wall.Figure 6.17 shows the fluctuation of a typical flow variable—namely,velocity—both with location in the pipe and with time. This fluctuationarises because of the turbulent motions that are superposed on the average local flow.
Other flow variables, such as T or ρ, can vary in the samemanner. For any variable we can write a local time-average value as1u≡TTu dt(6.81)0where T is a time that is much longer than the period of typical fluctuations.9 Equation (6.81) is most useful for so-called stationary processes—ones for which u is nearly time-independent.If we substitute u = u + u in eqn.
(6.81), where u is the actual localvelocity and u is the instantaneous magnitude of the fluctuation, weobtain1u=TT1u dt +0 T=u9T0u dt(6.82)=uTake care not to interpret this T as the thermal time constant that we introducedin Chapter 1; we denote time constants are as T .§6.7Turbulent boundary layers315Figure 6.17 Fluctuation of u and other quantities in a turbulent pipe flow.This is consistent with the fact thatu or any other average fluctuation = 0(6.83)since the fluctuations are defined as deviations from the average.We now want to create a measure of the size, or lengthscale, of turbulent vortices.
This might be done experimentally by placing two velocitymeasuring devices very close to one another in a turbulent flow field.When the probes are close, their measurements will be very highly correlated with one one another. Then, suppose that the two velocity probesare moved apart until the measurements first become unrelated to oneanother. That spacing gives an indication of the average size of the turbulent motions.Prandtl invented a slightly different (although related) measure of thelengthscale of turbulence, called the mixing length, .
He saw as anaverage distance that a parcel of fluid moves between interactions. Ithas a physical significance similar to that of the molecular mean freepath. It is harder to devise a clean experimental measure of than of the316Laminar and turbulent boundary layers§6.7correlation lengthscale of turbulence. But we can still use the concept of to examine the notion of a turbulent shear stress.The shear stresses of turbulence arise from the same kind of momentum exchange process that gives rise to the molecular viscosity. Recallthat, in the latter case, a kinetic calculation gave eqn. (6.45) for the laminar shear stress ∂u(6.45)τyx = (constant) ρC∂y =uwhere was the molecular mean free path and u was the velocity difference for a molecule that had travelled a distance in the mean velocitygradient.
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