Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 41
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The average conduction layer thickness A is anarea-weighted harmonic-mean average of Ax taken over the whole body surface. One canshow that A is related to the thin-layer average Nusselt number by-LA - Nur(4.16)Fully Laminar Nusselt Number Nut. As the next step in the correlation method, the thinlayer solution is corrected to account for thick-layer effects [175,223]. The body is surroundedby a uniform layer of stationary fluid of thickness A, and outside that thickness the fluid temperature is taken to be T=. The heat transfer that would occur across this layer is determinedby a conduction analysis and converted to a Nusselt number, and this Nusselt number is Nut.For example, to determine Nut for the case where the body is a very long horizontal isothermal circular cylinder of diameter D, the relevant heat transfer would then be that by heatconduction across a cylindrical annulus of inner diameter D, inner temperature Tw, outerdiameter D + 2A, and outer temperature T= (assumed constant).
Calculating this heat transfer by standard methods, substituting Eq. 4.16, and converting to a Nusselt number yieldsNue = In (1C1+ C1/Nur) '2uLC1- Pi(4.17)where the reference length L is equated to the diameter D, and Pi is the perimeter of the cylinder. The work of Hassani et al. [129] on conduction across layers of uniform thickness on verylong cylinders of various cross section, has shown, in effect, that Eq. 4.17 applies to triangular,rectangular, and trapezoidal cross sections as well, and by implication to cylinders of almostany convex cross section•As a second example, which also will be extended to a more general case, consider the casewhere the body is an isothermal sphere and, again, T= is uniform. The relevant conductio__nheat transfer in this case is that between two concentric spheres of diameters D and D + 2A,respectively.
Solving this elementary problem in heat conduction, recasting in terms of theNusselt number, substituting from Eq. 4.16, and then recognizing that the 2L/D is the conduction Nusselt number NUCOND,one obtainsNue =NUCOND +Nur(4.18)The work of Hassani and Hollands [128] on conduction across layers of uniform thicknessapplied to 3D bodies of various shape has shown that Eq. 4.18 applies approximately to other3D body shapes as well. (Note that the actual NUCONDfor the body at hand must be used.) Thework of Hassani and Hollands also shows that slightly better results for such nonsphericalbodies would be obtained if Eq. 4.18 is modified as follows:S u e = ((NUcoND) n +(NuT)n)TM(4.19)where n is a parameter best determined by fitting to experimental data.
If data are not available, n = 1.07 can be used as a rough approximation; alternatively, a more accurate value canbe obtained by using a formula proposed by Hassani and Hollands [128].A special case arises when the length of a horizontal cylinder greatly exceeds its diameter.If the cylinder is treated as infinite, Eq. 4.17 would yield the result Nue ~ 0 as Ra ~ 0. Nueshould, however, approach NUcoND as Ra ~ 0, where NUCONDis small but nonzero because thecylinder length is not truly infinite.
As a rough approximation for this case:C1Nue = In (1 + C1/Nur) ' NucoNo] max"C1 -2r~Lpiwhere [x, Y]maxrequires that the maximum of x and y is to be taken.(4.20)4.10CHAPTER FOURTurbulent Nusselt Number Nut. This section presents a model for the Nusselt numberRayleigh number relation applying in the limiting case where the flow is turbulent at all locations on the surface, i.e., in the asymptote Ra ~ oo.Model Description.
The chief characteristic of turbulent heat transfer is that (for a givenPrandtl number and Rayleigh number) the heat transfer at a point on the surface dependsonly on the local surface angle ~ (Fig. 4.5), and is independent of how far the point is from theleading edge. It follows thatNu,.x = Ct(~) R a 1/3(4.21)where Nu,,x is the local Nusselt number and the function C,(~) depends only on the Prandtlnumber. (The appearance of x in this equation is an artifact of the nomenclature; when Nut,xand Rag are replaced by their definitions, the x on each side of the equation cancels out.) Ifone integrates this local Nusselt number over the entire body surface to get the total heattransfer, one obtainsNutwhereC, = -~=Ct R a 1/3Ct(~)A To(4.22)dA(4.23)Equation.for Ct. A number of experiments at different Prandtl numbers (mostly on tiltedplates) have been carried out that permit the function C,(~) to be modeled.
Observation hasalso revealed that there are two patterns of turbulent flow: detached and attached. Attachedflow, where the flow sticks to the body contour, is best exemplified by the flow on a verticalplate, and the C,(~), = C, (90 °) applying in this situation is denoted CY. Detached flow, whereturbulent eddies rise away from the heated surface, is best exemplified by the flow on a horizontal upward-facing (heated) plate, and the C,(~) = C,(0 °) applying in this situation isdenoted C v. The first step in establishing the C,(~) function has been to model how C,v and CYdepend on the Prandtl number; the equations for these quantities (justified later) that havebeen found to best fit currently available data areand0.13 Pr °'22CV= (1 + 0.61 Pr°81)°42(4.24)C7 = 0"14( 11++0"01070.0Pr1Pr)(4.25)Ctv and CtUare tabulated in Table 4.1.For other angles, we can make use of the fact that the attached flow on an inclined platewould look just like the corresponding flow on the vertical plate if the gravity on the verticalplate were changed from g to g sin ~: in other words, the flow is driven by the component ofthe buoyancy force that is directed along the surface.
Since (from Eq. 4.22) Nux Rag^1/3, it follows that, in attached flow,Ct = Ctv sin 1'3 ~(4.26)In detached flow, the mixing of the boundary layer fluid and the ambient fluid is driven bythe component of gravity normal to the surface. Since this mixing dominates the heatexchange, the heat transfer from a tilted upward-facing plate can be obtained from the equation for a horizontal plate by replacing g by g cos ~. For the flow to remain detached, however,g cos ~ must remain positive.
Since Nux R~tx...1/3, it follows directly that, in detached flow,C, = C,V[cos ~, "JOll/3max(4.27)Equation 4.26 applies for attached flow and Eq. 4.27 for detached flow. Experience hasindicated that the flow pattern actually observed at a given location on the body will be theone that maximizes the local heat transfer. From this it follows thatNATURAL CONVECTIONG---- [CU[ cOS ~' "~'.l~]l'3max,CV Sin1/3 ~]max4.11(4.28)Equation 4.28 for G is substituted into Eq. 4.23 to obtain G.
This requires an integration overthe surface. The values of C, in the equations provided in this chapter were obtained in this way.For geometries not covered in this chapter, it may be more convenient to use the followingapproximate equation for C, [128], which applies for 0.7 Pr < 2000:-Ah-C, = 0.0972 - (0.0157 + 0.462C v) ~+ (0.615C v - 0.0548 - 6 x 10-6 Pr) LI@A(4.29)Ah is the area of any horizontal downward-facing heated parts (or upward-facing horizontalcooled parts) of the body's surface.The recommended equation for C,(~) is provisional because of the lack of data and the disagreement among different sets of measurements. Since the recommended Ct(~) affects manyof the correlations in this chapter, the user of these correlations should understand the experimental foundation for Eq. 4.24 for C v and Eq. 4.25 for C,v.
The following paragraphs providethe necessary background.Discussion of Cv Equation. Equation 4.24 for C,v has been forced to pass through 0.103for gases (Pr = 0.71), and through the value of 0.064 measured by Lloyd et al. [187] for Pr =2000. There is no experimental confirmation of Eq. 4.24 for Pr < 0.7. The data for gases (Pr --0.7) are tightly clustered around 0.103, provided AT/T << 1. For water, the measurements ofFujii and Fujii [99] yield C v = 0.13, compared to 0.11 by Vliet and Liu [275]; the equationpasses near the lower of these values.
The equation agrees with values obtained from the dataof Fujii and Fujii [99] for oils (Pr = 20-200) if all properties are evaluated at a reference temperature of Te - 0.75 Tw + 0.25 T~. For Pr = 2000, the value obtained from the data of Moranand Lloyd [197] falls about 15 percent below the benchmark value of 0.064 obtained by Lloydet al. [187] using the same technique.The form of Eq. 4.24 is surprising since, based on Eq. 4.9a and b, one would expect C v toincrease monotonically with increasing Pr and approach the Pr ~ oo asymptote from below. Itis therefore not clear why the high-quality data of Lloyd et al. [187] at Pr -- 2000 give a Ctvvalue less than that for water and air. The apparent corroboration by Fujii and Fujii [99] foroils is also highly uncertain because of the extreme sensitivity to the reference temperatureused.
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