John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 71
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However, before a liquid comes up to temperature, or ifa very small rate of forced convection continuously replaces warm liquidwith cool liquid, we can justly ask what the effect of a cool liquid bulkmight be.Figure 9.16 shows how a typical boiling curve might be changed ifTbulk < Tsat : We know, for example, that in laminar natural convection,q will increase as (Tw − Tbulk )5/4 or as [(Tw − Tsat ) + ∆Tsub ]5/4 , where∆Tsub ≡ Tsat − Tbulk . During nucleate boiling, the influence of subcoolingon q is known to be small. The peak and minimum heat fluxes are knownto increase linearly with ∆Tsub . These increases are quite significant.The film boiling heat flux increases rather strongly, especially at lowerheat fluxes.
The influence of ∆Tsub on transitional boiling is not welldocumented.GravityThe influence of gravity (or any other such body force) is of concern because boiling processes frequently take place in rotating or acceleratingsystems. The reduction of gravity has a significant impact on boilingprocesses aboard space vehicles.
Since g appears explicitly in the equations for qmax , qmin , and qfilm boiling , we know what its influence is. Bothqmax and qmin increase directly as g 1/4 in finite bodies, and there is anadditional gravitational influence through the parameter L . However,when gravity is small enough to reduce R below about 0.15, the hydrody-Transition boiling and system influences§9.6Figure 9.16 The influence of subcooling on the boiling curve.namic transitions deteriorate and eventually vanish altogether. AlthoughRohsenow’s equation suggests that q is proportional to g 1/2 in the nucleate boiling regime, other evidence suggests that the influence of gravityon the nucleate boiling curve is very slight, apart from an indirect effecton the onset of boiling.Forced convectionThe influence of superposed flow on the pool boiling curve for a givenheater (e.g., Fig.
9.2) is generally to improve heat transfer everywhere. Butflow is particularly effective in raising qmax . Let us look at the influenceof flow on the different regimes of boiling.493494Heat transfer in boiling and other phase-change configurations§9.6Influences of forced convection on nucleate boiling. Figure 9.17 showsnucleate boiling during the forced convection of water over a flat plate.Bergles and Rohsenow [9.36] offer an empirical strategy for predictingthe heat flux during nucleate flow boiling when the net vapor generationis still relatively small.
(The photograph in Fig. 9.17 shows how a substantial buildup of vapor can radically alter flow boiling behavior.) Theysuggest that>2??qiqB@1−q = qFC 1 +qFCqB(9.37)where• qFC is the single-phase forced convection heat transfer for the heater,as one might calculate using the methods of Chapters 6 and 7.• qB is the pool boiling heat flux for that liquid and that heater fromeqn. (9.4).• qi is the heat flux from the pool boiling curve evaluated at the valueof (Tw −Tsat ) where boiling begins during flow boiling (see Fig. 9.17).An estimate of (Tw − Tsat )onset can be made by intersecting theforced convection equation q = hFC (Tw − Tb ) with the followingequation [9.37]:(Tw − Tsat )onset =8σ Tsat qρg hfg kf1/2(9.38)Equation (9.37) will provide a first approximation in most boiling configurations, but it is restricted to subcooled flows or other situations inwhich vapor generation is not too great.Peak heat flux in external flows.
The peak heat flux on a submergedbody is strongly augmented by an external flow around it. Althoughknowledge of this area is still evolving, we do know from dimensionalanalysis that qmax= fn WeD , ρf ρgρg hfg u∞(9.39)§9.6Transition boiling and system influencesFigure 9.17 Forced convection boiling on an external surface.where the Weber number, We, isρg u2∞ Linertia force L=WeL ≡σsurface force Land where L is any characteristic length.Kheyrandish and Lienhard [9.38] suggest fairly complex expressionsof this form for qmax on horizontal cylinders in cross flows.
For a cylindrical liquid jet impinging on a heated disk of diameter D, Sharan and495496Heat transfer in boiling and other phase-change configurations§9.7Lienhard [9.39] obtained qmax= 0.21 + 0.0017ρf ρgρg hfg ujetdjetD1/3 1000ρg /ρfAWeD(9.40)where, if we call ρf /ρg ≡ r ,A = 0.486 + 0.06052 ln r − 0.0378 (ln r )2 + 0.00362 (ln r )3(9.41)This correlation represents all the existing data within ±20% over the fullrange of the data.The influence of fluid flow on film boiling. Bromley et al. [9.40] showedthat the film boiling heat flux during forced flow normal to a cylindershould take the formq = constantkg ρg hfg ∆T u∞1/2D(9.42)for u2∞ /(gD) ≥ 4 with hfg from eqn. (9.29).
Their data fixed the constantat 2.70. Witte [9.41] obtained the same relationship for flow over a sphereand recommended a value of 2.98 for the constant.Additional work in the literature deals with forced film boiling onplane surfaces and combined forced and subcooled film boiling in a variety of geometries [9.42]. Although these studies are beyond our presentscope, it is worth noting that one may attain very high cooling rates usingfilm boiling with both forced convection and subcooling.9.7Forced convection boiling in tubesFlowing fluids undergo boiling or condensation in many of the cases inwhich we transfer heat to fluids moving through tubes. For example,such phase change occurs in all vapor-compression power cycles andrefrigerators.
When we use the terms boiler, condenser, steam generator,or evaporator we usually refer to equipment that involves heat transferwithin tubes. The prediction of heat transfer coefficients in these systemsis often essential to determining U and sizing the equipment. So let usconsider the problem of predicting boiling heat transfer to liquids flowingthrough tubes.Figure 9.18 The development of a two-phase flow in a verticaltube with a uniform wall heat flux (not to scale).497498Heat transfer in boiling and other phase-change configurations§9.7Relationship between heat transfer and temperature differenceForced convection boiling in a tube or duct is a process that becomes veryhard to delineate because it takes so many forms.
In addition to the usualsystem variables that must be considered in pool boiling, the formationof many regimes of boiling requires that we understand several boilingmechanisms and the transitions between them, as well.Collier and Thome’s excellent book, Convective Boiling and Condensation [9.43], provides a comprehensive discussion of the issues involvedin forced convection boiling. Figure 9.18 is their representation of thefairly simple case of flow of liquid in a uniform wall heat flux tube inwhich body forces can be neglected. This situation is representative of afairly low heat flux at the wall. The vapor fraction, or quality, of the flowincreases steadily until the wall “dries out.” Then the wall temperaturerises rapidly.
With a very high wall heat flux, the pipe could burn outbefore dryout occurs.Figure 9.19, also provided by Collier, shows how the regimes shown inFig. 9.18 are distributed in heat flux and in position along the tube. Noticethat, at high enough heat fluxes, burnout can be made to occur at any station in the pipe. In the subcooled nucleate boiling regime (B in Fig. 9.18)and the low quality saturated regime (C), the heat transfer can be predicted using eqn.
(9.37) in Section 9.6. But in the subsequent regimesof slug flow and annular flow (D, E, and F ) the heat transfer mechanismchanges substantially. Nucleation is increasingly suppressed, and vaporization takes place mainly at the free surface of the liquid film on thetube wall.Most efforts to model flow boiling differentiate between nucleateboiling-controlled heat transfer and convective boiling heat transfer. Inthose regimes where fully developed nucleate boiling occurs (the laterparts of C), the heat transfer coefficient is essentially unaffected by themass flow rate and the flow quality.
Locally, conditions are similar to poolboiling. In convective boiling, on the other hand, vaporization occursaway from the wall, with a liquid-phase convection process dominatingat the wall. For example, in the annular regions E and F , heat is convectedfrom the wall by the liquid film, and vaporization occurs at the interfaceof the film with the vapor in the core of the tube. Convective boilingcan also dominate at low heat fluxes or high mass flow rates, where wallnucleate is again suppressed. Vaporization then occurs mainly on entrained bubbles in the core of the tube.
In convective boiling, the heattransfer coefficient is essentially independent of the heat flux, but it is§9.7Forced convection boiling in tubesFigure 9.19 The influence of heat flux on two-phase flow behavior.strongly affected by the mass flow rate and quality.Building a model to capture these complicated and competing trendshas presented a challenge to researchers for several decades. One earlyeffort by Chen [9.44] used a weighted sum of a nucleate boiling heat transfer coefficient and a convective boiling coefficient, where the weightingdepended on local flow conditions. This model represents water data toan accuracy of about ±30% [9.45], but it does not work well with mostother fluids. Chen’s mechanistic approach was substantially improvedin a more complex version due to Steiner and Taborek [9.46].
Many otherinvestigators have instead pursued correlations built from dimensionalanalysis and physical reasoning.To proceed with a dimensional analysis, we first note that the liquidand vapor phases may have different velocities. Thus, we avoid intro-499500Heat transfer in boiling and other phase-change configurations§9.7ducing a flow speed and instead rely on the the superficial mass flux, G,through the pipe:G≡ṁApipe(kg/m2 s)(9.43)This mass flow per unit area is constant along the duct if the flow issteady.
From this, we can define a “liquid only” Reynolds numberRelo ≡GDµf(9.44)which would be the Reynolds number if all the flowing mass were inthe liquid state. Then we may use Relo to compute a liquid-only heattransfer cofficient, hlo from Gnielinski’s equation, eqn. (7.43), using liquidproperties at Tsat .Physical arguments then suggest that the dimensional functional equation for the flow boiling heat transfer coefficient, hfb , should take thefollowing form for saturated flow in vertical tubes:(9.45)hfb = fn hlo , G, x, hfg , qw , ρf , ρg , DIt should be noted that other liquid properties, such as viscosity and conductivity, are represented indirectly through hlo .
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