John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 72
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This functional equation has eight dimensional variables (and one dimensionless variable, x)in five dimensions (m, kg, s, J, K). We thus obtain three more dimensionless groups to go with x, specificallyqw ρghfb= fn x,,(9.46)hloGhfg ρfIn fact, the situation is even a bit simpler than this, since argumentsrelated to the pressure gradient show that the quality and the densityratio can be combined into a single group, called the convection number :Co ≡1−xx0.8 ρgρf0.5(9.47)The other dimensionless group in eqn. (9.46) is called the boiling number :Bo ≡qwGhfg(9.48)Forced convection boiling in tubes§9.7501Table 9.4 Fluid-dependent parameter F in the Kandlikar correlation for copper tubing.
Additional values are given in [9.47].FluidWaterPropaneR-12R-22R-32F1.02.151.502.201.20FluidR-124R-125R-134aR-152aR-410aF1.901.101.631.101.72so thathfb= fn (Bo, Co)hlo(9.49)When the convection number is large (Co 1), as for low quality,nucleate boiling dominates. In this range, hfb /hlo rises with increasing Boand is approximately independent of Co. When the convection numberis smaller, as at higher quality, the effect of the boiling number declinesand hfb /hlo increases with decreasing Co.Correlations having the general form of eqn.
(9.49) were developedby Schrock and Grossman [9.48], Shah [9.49], and Gungor and Winterton [9.50]. Kandlikar [9.45, 9.47, 9.51] refined this approach further,obtaining good accuracy and better capturing the parametric trends. Hismethod is to calculate hfb /hlo from each of the following two correlationsand to choose the larger value:hfb −0.20.70.80.6683Co=(1−x)f+1058BoFohlo nbdhfb = (1 − x)0.8 1.136 Co−0.9 fo + 667.2 Bo0.7 Fhlo cbd(9.50a)(9.50b)where “nbd” means “nucleate boiling dominant” and “cbd” means “convective boiling dominant”.In these equations, the orientation factor, fo , is set to unity for vertical tubes4 and F is a fluid-dependent parameter whose value is given4The value for horizontal tubes is given in eqn.
(9.52).502Heat transfer in boiling and other phase-change configurations§9.7in Table 9.4. The parameter F arises here for the same reason that fluiddependent parameters appear in nucleate boiling correlations: surfacetension, contact angles, and other fluid-dependent variables influencenucleation and bubble growth. The values in Table 9.4 are for commercial grades of copper tubing. For stainless steel tubing, Kandlikar recommends F = 1 for all fluids.
Equations (9.50) are applicable for the saturated boiling regimes (C through F ) with quality in the range 0 < x ≤ 0.8.For subcooled conditions, see Problem 9.21.Example 9.90.6 kg/s of saturated H2 O at Tb = 207◦ C flows in a 5 cm diameter vertical tube heated at a rate of 184,000 W/m2 . Find the wall temperatureat a point where the quality x is 20%.Solution. Data for water are taken from Tables A.3–A.5.
We firstcompute hlo .G=andRelo =ṁ0.6= 305.6 kg/m2 s=Apipe0.001964GD(305.6)(0.05)== 1.178 × 105µf1.297 × 10−4From eqns. (7.42) and (7.43):1f =!"2 = 0.017361.82 log10 (1.178 × 105 ) − 1.64 (0.01736/8) 1.178 × 105 − 1000 (0.892)3!"= 236.3NuD =1 + 12.7 0.01736/8 (0.892)2/3 − 1Hence,kf0.6590236.3 = 3, 115 W/m2 K0.05DNext, we find the parameters for eqns. (9.50). From Table 9.4, F = 1for water, and for a vertical tube, fo = 1. Also,hlo =Co =1−xxBo =NuD =0.8 ρgρf0.5=1 − 0.200.20.8 9.014856.50.5= 0.3110qw184, 000= 3.147 × 10−4=Ghfg(305.6)(1, 913, 000)Forced convection boiling in tubes§9.7Substituting into eqns. (9.50):hfbnbdhfbcbd= (3, 115)(1 − 0.2)0.8 0.6683 (0.3110)−0.2 (1)+ 1058 (3.147 × 10−4 )0.7 (1) = 11, 950 W/m2 K= (3, 115)(1 − 0.2)0.8 1.136 (0.3110)−0.9 (1)+ 667.2 (3.147 × 10−4 )0.7 (1) = 14, 620 W/m2 KSince the second value is larger, we use it: hfb = 14, 620 W/m2 K.Then,Tw = Tb +qw184, 000= 220◦ C= 207 +hfb14, 620The Kandlikar correlation leads to mean deviations of 16% for waterand 19% for the various refrigerants.
The Gungor and Winterton correlation [9.50], which is popular for its simplicity, does not contain fluidspecific coefficients, but it is somewhat less accurate than either the Kandlikar equations or the more complex Steiner and Taborek method [9.45,9.46]. These three approaches, however, are among the best available.Two-phase flow and heat transfer in horizontal tubesThe preceding discussion of flow boiling in tubes is largely restricted tovertical tubes.
Several of the flow regimes in Fig. 9.18 will be alteredas shown in Fig. 9.20 if the tube is oriented horizontally. The reason isthat, especially at low quality, liquid will tend to flow along the bottom ofthe pipe and vapor along the top. The patterns shown in Fig. 9.20, by theway, will also be observed during the reverse process—condensation—orduring adiabatic two-phase flow.Which flow pattern actually occurs depends on several parametersin a fairly complex way.
While many methods have been suggested topredict what flow pattern will result for a given set of conditions in thepipe, one of the best is that developed by Dukler, Taitel, and their coworkers. Their two-phase flow-regime maps are summarized in [9.52]and [9.53].For the prediction of heat transfer, the most important additionalparameter is the Froude number, Frlo , which characterizes the strengthof the flow’s inertia (or momentum) relative to the gravitational forces503504Heat transfer in boiling and other phase-change configurations§9.7Figure 9.20 The discernible flowregimes during boiling, condensation, oradiabatic flow from left to right inhorizontal tubes.that drive the separation of the liquid and vapor phases:Frlo ≡G2ρf2 gD(9.51)When Frlo < 0.04, the top of the tube becomes relatively dry and hfb /hlobegins to decline as the Froude number decreases further.Kandlikar found that he could modify his correlation to account forgravitational effects in horizontal tubes by changing the value of fo ineqns.
(9.50):fo =⎧⎨1⎩(25 Frlo )for Frlo ≥ 0.040.3for Frlo < 0.04(9.52)Peak heat fluxWe have seen that there are two limiting heat fluxes in flow boiling in atube: dryout and burnout. The latter is the more dangerous of the twosince it occurs at higher heat fluxes and gives rise to more catastrophictemperature rises. Collier and Thome provide an extensive discussion ofthe subject [9.43], as does Hewitt [9.54].§9.8Forced convective condensation heat transferOne effective set of empirical formulas was developed by Katto [9.55].He used dimensional analysis to show thatρg σ ρf Lqmax,= fn,Ghfgρf G2 L Dwhere L is the length of the tube and D its diameter.
Since G2 L σ ρfis a Weber number, we can see that this equation is of the same formas eqn. (9.39). Katto identifies several regimes of flow boiling with bothsaturated and subcooled liquid entering the pipe. For each of these regions, he and Ohne [9.56] later fit a successful correlation of this form toexisting data.Pressure gradients in flow boilingPressure gradients in flow boiling interact with the flow pattern and thevoid fraction, and they can change the local saturation temperature of thefluid. Gravity, flow acceleration, and friction all contribute to pressurechange, and friction can be particularly hard to predict.
In particular, thefrictional pressure gradient can increase greatly as the flow quality risesfrom the pure liquid state to the pure vapor state; the change can amountto more than two orders of magnitude at low pressures. Data correlationsare usually used to estimate the frictional pressure loss, but they are,at best, accurate to within about ±30%. Whalley [9.57] provides a niceintroduction such methods.
Certain complex models, designed for usein computer codes, can be used to make more accurate predictions [9.58].9.8Forced convective condensation heat transferWhen vapor is blown or forced past a cool wall, it exerts a shear stresson the condensate film. If the direction of forced flow is downward, itwill drag the condensate film along, thinning it out and enhancing heattransfer. It is not hard to show (see Problem 9.22) thatτδ δ344µk(Tsat − Tw )x4=δ +(9.53)ghfg ρf (ρf − ρg )3 (ρf − ρg )gwhere τδ is the shear stress exerted by the vapor flow on the condensatefilm.Equation (9.53) is the starting point for any analysis of forced convection condensation on an external surface.
Notice that if τδ is negative—if505506Heat transfer in boiling and other phase-change configurations§9.9the shear opposes the direction of gravity—then it will have the effect ofthickening δ and reducing heat transfer. Indeed, if for any value of δ,τδ = −3g(ρf − ρg )4δ,(9.54)the shear stress will have the effect of halting the flow of condensatecompletely for a moment until δ grows to a larger value.Heat transfer solutions based on eqn. (9.53) are complex because theyrequire that one solve the boundary layer problem in the vapor in orderto evaluate τδ ; and this solution must be matched with the velocity atthe outside surface of the condensate film. Collier and Thome [9.43,§10.5] discuss such solutions in some detail.
One explicit result has beenobtained in this way for condensation on the outside of a horizontalcylinder by Shekriladze and Gomelauri [9.59]:⎧⎡1/2 ⎤⎫1/2⎨ ρ u∞ D⎬ghfg µf Df⎣1 + 1 + 1.69⎦NuD = 0.64(9.55)⎩ µf⎭u2∞ kf (Tsat − Tw )where u∞ is the free stream velocity and NuD is based on the liquidconductivity. Equation (9.55) is valid up to ReD ≡ ρf u∞ D µf = 106 .Notice, too, that under appropriate flow conditions (large values of u∞ ,for example), gravity becomes unimportant and3NuD → 0.64 2ReD(9.56)The prediction of heat transfer during forced convective condensation in tubes becomes a different problem for each of the many possibleflow regimes. The reader is referred to [9.43, §10.5] or [9.60] for details.9.9Dropwise condensationAn automobile windshield normally is covered with droplets during alight rainfall.
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