John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 61
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If§8.4Natural convection in other situationsNuL∗ 10, the b.l.s will be thick, and they suggest correcting theresult toNucorrected =1.4 ln 1 + 1.4 NuL∗(8.37b)These equations are recommended6 for 1 < RaL∗ < 107 .• In general, for inclined plates in the unstable cases, Raithby andHollands [8.13] recommend that the heat flow be computed firstusing the formula for a vertical plate with g cos θ and then usingthe formula for a horizontal plate with g sin θ (i.e., the componentof gravity normal to the plate) and that the larger value of the heatflow be taken.Stable Cases: For the upper side of cold plates and the lower side of hotplates, the flow is generally stable.
The following results assume that theflow is not obstructed at the edges of the plate; a surrounding adiabaticsurface, for example, will lower h [8.24, 8.25].• For θ < 88◦ and 105 RaL 1011 , eqn. (8.27) is still valid for theupper side of cold plates and the lower side of hot plates when gis replaced with g cos θ in the Rayleigh number [8.16].• For downward-facing hot plates and upward-facing cold plates ofwidth L with very slight inclinations, Fujii and Imura give:1/5NuL = 0.58 RaL(8.38)This is valid for 106 < RaL < 109 if 87◦ θ 90◦ and for 109 RaL < 1011 if 89◦ θ 90◦ . RaL is based on g (not g cos θ).Fujii and Imura’s results are for two-dimensional plates—ones inwhich infinite breadth has been approximated by suppression ofend effects.For circular plates of diameter D in the stable horizontal configurations, the data of Kadambi and Drake [8.26] suggest that1/5NuD = 0.82 RaD Pr0.0346(8.39)Raithby and Hollands also suggest using a blending formula for 1 < RaL∗ < 1010 1010 1/10Nublended,L∗ = Nucorrected+ Nuturb(8.37c)in which Nuturb is calculated from eqn.
(8.36) using L∗ . The formula is useful fornumerical progamming, but its effect on h is usually small.423424Natural convection in single-phase fluids and during film condensation§8.4Natural convection with uniform heat fluxWhen qw is specified instead of ∆T ≡ (Tw − T∞ ), ∆T becomes the unknown dependent variable. Because h ≡ qw /∆T , the dependent variableappears in the Nusselt number; however, for natural convection, it alsoappears in the Rayleigh number. Thus, the situation is more complicatedthan in forced convection.Since Nu often varies as Ra1/4 , we may writeNux =qw x1/4∝ Rax ∝ ∆T 1/4 x 3/4∆T kThe relationship between x and ∆T is then∆T = C x 1/5(8.40)where the constant of proportionality C involves qw and the relevantphysical properties. The average of ∆T over a heater of length L is1∆T =LL0C x 1/5 dx =5C6(8.41)We plot ∆T /C against x/L in Fig.
8.9. Here, ∆T and ∆T (x/L = ½) arewithin 4% of each other. This suggests that, if we are interested in averagevalues of ∆T , we can use ∆T evaluated at the midpoint of the plate inboth the Rayleigh number, RaL , and the average Nusselt number, NuL =qw L/k∆T . Churchill and Chu, for example, show that their vertical platecorrelation, eqn. (8.27), represents qw = constant data exceptionally wellin the range RaL > 1 when RaL is based on ∆T at the middle of the plate.This approach eliminates the variation of ∆T with x from the calculation,but the temperature difference at the middle of the plate must still befound by iteration.To avoid iterating, we need to eliminate ∆T from the Rayleigh number.We can do this by introducing a modified Rayleigh number, Ra∗x , definedasgβqw x 4gβ∆T x 3 qw x=(8.42)να∆T kkναFor example, in eqn.
(8.27), we replace RaL with Ra∗L NuL . The result isRa∗x ≡ Rax Nux ≡NuL = 0.68 + 0.67: 1/4Ra∗L1/4NuL0.4921+Pr9/16 4/9Natural convection in other situations§8.4Figure 8.9 The mean value of ∆T ≡ Tw − T∞ during naturalconvection.which may be rearranged as1/4 NuLNuL − 0.68 = ! 0.67 Ra∗L1/41 + (0.492/Pr)9/16"4/9When NuL 5, the term 0.68 may be neglected, with the result 1/50.73 Ra∗LNuL = !"16/451 + (0.492/Pr)9/16(8.43a)(8.43b)Raithby and Hollands [8.13] give the following, somewhat simpler correlations for laminar natural convection from vertical plates with a uniformwall heat flux:1/5PrRa∗x√Nux = 0.630(8.44a)4 + 9 Pr + 10 Pr6NuL =5PrRa∗√L4 + 9 Pr + 10 Pr1/5(8.44b)These equations apply for all Pr and for Nu 5 (equations for lower Nuor Ra∗ are given in [8.13]).425426Natural convection in single-phase fluids and during film condensation§8.4Example 8.5A horizontal circular disk heater of diameter 0.17 m faces downwardin air at 27◦ C.
If it delivers 15 W, estimate its average surface temperature.Solution. We have no formula for this situation, so the problemcalls for some judicious guesswork. Following the lead of Churchilland Chu, we replace RaD with Ra∗D /NuD in eqn. (8.39): NuD6/5=qw D∆T k6/5 = 0.82 Ra∗D1/5Pr0.034soqw D k∆T = 1.18 1/6gβqw D 4Pr0.028kνα150.172π (0.085)0.02614= 1.18 1/629.8[15/π (0.085) ]0.174(0.711)0.028300(0.02164)(1.566)(2.203)10−10= 140 KIn the preceding computation, all properties were evaluated at T∞ .Now we must return the calculation, reevaluating all properties exceptβ at 27 + (140/2) = 97◦ C:∆T corrected = 1.18 661(0.17)/0.031049.8[15/π (0.085)2 ]0.174300(0.03104)(3.231)(2.277)10−101/6(0.99)= 142 Kso the surface temperature is 27 + 142 = 169◦ C.That is rather hot.
Obviously, the cooling process is quite ineffective in this case.Some other natural convection problemsThere are many natural convection situations that are beyond the scopeof this book but which arise in practice. Two examples follow.§8.4Natural convection in other situationsNatural convection in enclosures. When a natural convection processoccurs within a confined space, the heated fluid buoys up and then follows the contours of the container, releasing heat and in some way returning to the heater. This recirculation process normally enhances heattransfer beyond that which would occur by conduction through the stationary fluid.
These processes are of importance to energy conservation processes in buildings (as in multiply glazed windows, uninsulatedwalls, and attics), to crystal growth and solidification processes, to hotor cold liquid storage systems, and to countless other configurations.Survey articles on natural convection in enclosures have been written byYang [8.27], Raithby and Hollands [8.13], and Catton [8.28].Combined natural and forced convection. When forced convection along,say, a vertical wall occurs at a relatively low velocity but at a relativelyhigh heating rate, the resulting density changes can give rise to a superimposed natural convection process. We saw in footnote 2 on page 4021/2that GrL plays the role of of a natural convection Reynolds number, itfollows that we can estimate of the relative importance of natural andforced convection can be gained by considering the ratiostrength of natural convection flowGrL2 = strength of forced convection flowReL(8.45)where ReL is for the forced convection along the wall.
If this ratio is smallcompared to one, the flow is essentially that due to forced convection,whereasif it is large compared to one, we have natural convection. WhenGrL Re2L is on the order of one, we have a mixed convection process.It should be clear that the relative orientation of the forced flow andthe natural convection flow matters. For example, compare cool air flowing downward past a hot wall to cool air flowing upward along a hot wall.The former situation is called opposing flow and the latter is called assisting flow.
Opposing flow may lead to boundary layer separation anddegraded heat transfer.Churchill [8.29] has provided an extensive discussion of both the conditions that give rise to mixed convection and the prediction of heat transfer for it. Review articles on the subject have been written by Chen andArmaly [8.30] and by Aung [8.31].427428Natural convection in single-phase fluids and during film condensation8.5§8.5Film condensationDimensional analysis and experimental dataThe dimensional functional equation for h (or h) during film condensation is7 h or h = fn cp , ρf , hfg , g ρf − ρg , k, µ, (Tsat − Tw ) , L or xwhere hfg is the latent heat of vaporization.
It does not appear in thedifferential equations (8.4) and (6.40); however, it is used in the calculation of δ [which enters in the b.c.’s (8.5)]. The film thickness, δ, dependsheavily on the latent heat and slightly on the sensible heat, cp ∆T , whichthe film must absorb to condense. Notice, too, that g(ρf −ρg ) is includedas a product, because gravity only enters the problem as it acts upon thedensity difference [cf. eqn. (8.4)].The problem is therefore expressed nine variables in J, kg, m, s, and◦ C (where we once more avoid resolving J into N · m since heat is notbeing converted into work in this situation).
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