John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 68
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This provides a short-circuit forcooling the wall. Then, when the bubblecaves in, cold liquid is brought to the wall.Figure 9.6 Heat removal by bubble action during boiling. Darkregions denote locally superheated liquid.where all properties, unless otherwise noted, are for liquid at Tsat . Theconstant Csf is an empirical correction for typical surface conditions.Table 9.2 includes a set of values of Csf for common surfaces (takenfrom [9.12]) as well as the Prandtl number exponent, s. A more extensivecompilation of these constants was published by Pioro in 1999 [9.13].We noted, initially, that there are two nucleate boiling regimes, andthe Yamagata equation (9.3) applies only to the first of them. Rohsenow’sequation is frankly empirical and does not depend on the rational analysis of either nucleate boiling process.
It turns out that it representsq(∆T ) in both regimes, but it is not terribly accurate in either one. Figure 9.7 shows Rohsenow’s original comparison of eqn. (9.4) with data forwater over a large range of conditions. It shows typical errors in heatflux of 100% and typical errors in ∆T of about 25%.Thus, our ability to predict the nucleate pool boiling heat flux is poor.Our ability to predict ∆T is better because, with q ∝ ∆T 3 , a large errorin q gives a much smaller error in ∆T . It appears that any substantialimprovement in this situation will have to wait until someone has managed to deal realistically with the nuisance variable, n.
Current researchefforts are dealing with this matter, and we can simply hope that suchwork will eventually produce a method for achieving reliable heat transfer design relationships for nucleate boiling.470Heat transfer in boiling and other phase-change configurations§9.2Table 9.2 Selected values of the surface correction factor foruse with eqn. (9.4) [9.12]Surface–Fluid CombinationWater–nickelWater–platinumWater–copperWater–brassCCl4 –copperBenzene–chromiumn-Pentane–chromiumEthyl alcohol–chromiumIsopropyl alcohol–copper35% K2 CO3 –copper50% K2 CO3 –coppern-Butyl alcohol–copperCsfs0.0060.0130.0130.0060.0130.0100.0150.00270.00250.00540.00270.00301.01.01.01.01.71.71.71.71.71.71.71.7It is indeed fortunate that we do not often have to calculate q, given∆T , in the nucleate boiling regime.
More often, the major problem isto avoid exceeding qmax . We turn our attention in the next section topredicting this limit.Example 9.2What is Csf for the heater surface in Fig. 9.2?Solution. From eqn. (9.4) we obtainµcp3q3C=sf∆T 3h2fg Pr32 g ρf − ρgσwhere, since the liquid is water, we take s to be 1.0. Then, for water atTsat = 100◦ C: cp = 4.22 kJ/kg·K, Pr = 1.75, (ρf − ρg ) = 958 kg/m3 ,σ = 0.0589 N/m or kg/s2 , hfg = 2257 kJ/kg, µ = 0.000282 kg/m·s.Nucleate boiling§9.2471Figure 9.7 Illustration ofRohsenow’s [9.12] correlation applied todata for water boiling on0.61 mm diameter platinum wire.Thus,kWqC 3 = 3.10 × 10−7 2 3∆T 3 sfm KAt q = 800 kW/m2 , we read ∆T = 22 K from Fig.
9.2. This givesCsf =3.10 × 10−7 (22)38001/3= 0.016This value compares favorably with Csf for a platinum or copper surface under water.472Heat transfer in boiling and other phase-change configurations9.3§9.3Peak pool boiling heat fluxTransitional boiling regime and Taylor instabilityIt will help us to understand the peak heat flux if we first consider theprocess that connects the peak and the minimum heat fluxes. Duringhigh heat flux transitional boiling, a large amount of vapor is gluttedabout the heater.
It wants to buoy upward, but it has no clearly definedescape route. The jets that carry vapor away from the heater in the region of slugs and columns are unstable and cannot serve that function inthis regime. Therefore, vapor buoys up in big slugs—then liquid falls in,touches the surface briefly, and a new slug begins to form.
Figure 9.3cshows part of this process.The high and low heat flux transitional boiling regimes are differentin character. The low heat flux region does not look like Fig. 9.2c but is almost indistinguishable from the film boiling shown in Fig. 9.2d. However,both processes display a common conceptual key: In both, the heater isalmost completely blanketed with vapor.
In both, we must contend withthe unstable configuration of a liquid on top of a vapor.Figure 9.8 shows two commonplace examples of such behavior. Ineither an inverted honey jar or the water condensing from a cold waterpipe, we have seen how a heavy fluid falls into a light one (water or honey,in this case, collapses into air). The heavy phase falls down at one nodeof a wave and the light fluid rises into the other node.The collapse process is called Taylor instability after G. I.
Taylor, whofirst predicted it. The so-called Taylor wavelength, λd , is the length ofthe wave that grows fastest and therefore predominates during the collapse of an infinite plane horizontal interface. It can be predicted usingdimensional analysis. The dimensional functional equation for λd is λd = fn σ , g ρf − ρg(9.5)since the wave is formed as a result of the balancing forces of surfacetension against inertia and gravity. There are three variables involving mand kg/s2 , so we look for just one dimensionless group:λd2 g ρf − ρgσ= constantThis relationship was derived analytically by Bellman and Pennington [9.14]for one-dimensional waves and by Sernas [9.15] for the two-dimensionalPeak pool boiling heat flux§9.3473a.
Taylor instability in the surface of the honeyin an inverted honey jarb. Taylor instability in the interface of the water condensing onthe underside of a small cold water pipe.Figure 9.8 Two examples of Taylor instabilities that one mightcommonly experience.waves that actually occur in a plane horizontal interface. The resultswereλd2 g ρf − ρgσ5=√2π √3 for one-dimensional waves2π 6 for two-dimensional waves(9.6)474Heat transfer in boiling and other phase-change configurations§9.3Experiment 9.3Hang a metal rod in the horizontal position by threads at both ends.The rod should be about 30 cm in length and perhaps 1 to 2 cm in diameter.
Pour motor oil or glycerin in a narrow cake pan and lift the pan upunder the rod until it is submerged. Then lower the pan and watch theliquid drain into it. Take note of the wave action on the underside of therod. The same thing can be done in an even more satisfactory way byrunning cold water through a horizontal copper tube above a beaker ofboiling water. The condensing liquid will also come off in a Taylor wavesuch as is shown in Fig. 9.8. In either case, the waves will approximateλd1 (the length of a one-dimensional wave, since they are arrayed on aline), but the wavelength will be influenced by the curvature of the rod.Throughout the transitional boiling regime, vapor rises into liquid onthe nodes of Taylor waves, and at qmax this rising vapor forms into jets.These jets arrange themselves on a staggered square grid, as shown inFig.
9.9. The basic spacing of the grid is λd2 (the two-dimensional Taylorwavelength). Since√λd2 = 2 λd1(9.7)[recall eqn. (9.6)], the spacing of the most basic module of jets is actuallyλd1 , as shown in Fig. 9.9.Next we must consider how the jets become unstable at the peak, tobring about burnout.Helmholtz instability of vapor jetsFigure 9.10 shows a commonplace example of what is called Helmholtzinstability. This is the phenomenon that causes the vapor jets to cave inwhen the vapor velocity in them reaches a critical value.
Any flag in abreeze will constantly be in a state of collapse as the result of relativelyhigh pressures where the velocity is low and relatively low pressureswhere the velocity is high, as is indicated in the top view.This same instability is shown as it occurs in a vapor jet wall inFig. 9.11. This situation differs from the flag in one important particular. There is surface tension in the jet walls, which tends to balance theflow-induced pressure forces that bring about collapse. Thus, while theflag is unstable in any breeze, the vapor velocity in the jet must reach alimiting value, ug , before the jet becomes unstable.a. Plan view of bubbles rising from surfaceb.
Waveform underneath the bubbles shown in a.Figure 9.9 The array of vapor jets as seen on an infinite horizontal heater surface.475476Heat transfer in boiling and other phase-change configurations§9.3Figure 9.10 The flapping of a flag due to Helmholtz instability.Lamb [9.16] gives the following relation between the vapor flow ug ,shown in Fig. 9.11, and the wavelength of a disturbance in the jet wall,λH :2ug =2π σρg λH(9.8)[This result, like eqn. (9.6), can be predicted within a constant usingdimensional analysis. See Problem 9.19.] A real liquid–vapor interfacewill usually be irregular, and therefore it can be viewed as containing allpossible sinusoidal wavelengths superposed on one another.
One problem we face is that of guessing whether or not one of those wavelengthsPeak pool boiling heat flux§9.3Figure 9.11 Helmholtz instability of vapor jets.will be better developed than the others and therefore more liable tocollapse.Example 9.3Saturated water at 1 atm flows down the periphery of the inside of a10 cm I.D. vertical tube. Steam flows upward in the center. The wall ofthe pipe has circumferential corrugations in it, with a 4 cm wavelengthin the axial direction.
Neglect problems raised by curvature and thefinite thickness of the liquid, and estimate the steam velocity requiredto destabilize the liquid flow over these corrugations, assuming thatthe liquid moves slowly.Solution. The flow will be Helmholtz-stable until the steam velocityreaches the value given by eqn. (9.8):2ug =2π (0.0589)0.577(0.04 m)Thus, the maximum stable steam velocity would be ug = 4 m/s.Beyond that, the liquid will form whitecaps and be blown backupward.477478Heat transfer in boiling and other phase-change configurations§9.3Example 9.4Capillary forces hold mercury in place between two parallel steel plateswith a lid across the top. The plates are slowly pulled apart until themercury interface collapses.
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