Computational Fluid Mechanics and Heat Transfer Third Edition_14 docx

496 Heat transfer in boiling and other phase-change configurations §9.7 Lienhard [9.39] obtained q max ρ g h fg u jet =  0.21 +0.0017ρ f  ρ g   d jet D  1/3  1000ρ g /ρ f We D  A (9.40) where, if we call ρ f /ρ g ≡ 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 full range of the data. The influence of fluid flow on film boiling. Bromley et al. [9.40] showed that the film boiling heat flux during forced flow normal to a cylinder should take the form q = constant  k g ρ g h  fg ∆Tu ∞ D  1/2 (9.42) for u 2 ∞ /(gD) ≥ 4 with h  fg from eqn. (9.29). Their data fixed the constant at 2.70. Witte [9.41] obtained the same relationship for flow over a sphere and recommended a value of 2.98 for the constant. Additional work in the literature deals with forced film boiling on plane surfaces and combined forced and subcooled film boiling in a vari- ety of geometries [9.42]. Although these studies are beyond our present scope, it is worth noting that one may attain very high cooling rates using film boiling with both forced convection and subcooling. 9.7 Forced convection boiling in tubes Flowing fluids undergo boiling or condensation in many of the cases in which we transfer heat to fluids moving through tubes. For example, such phase change occurs in all vapor-compression power cycles and refrigerators. When we use the terms boiler, condenser, steam generator, or evaporator we usually refer to equipment that involves heat transfer within tubes. The prediction of heat transfer coefficients in these systems is often essential to determining U and sizing the equipment. So let us consider the problem of predicting boiling heat transfer to liquids flowing through tubes. Figure 9.18 The development of a two-phase flow in a vertical tube with a uniform wall heat flux (not to scale). 497 498 Heat transfer in boiling and other phase-change configurations §9.7 Relationship between heat transfer and temperature difference Forced convection boiling in a tube or duct is a process that becomes very hard to delineate because it takes so many forms. In addition to the usual system variables that must be considered in pool boiling, the formation of many regimes of boiling requires that we understand several boiling mechanisms and the transitions between them, as well. Collier and Thome’s excellent book, Convective Boiling and Condensa- tion [9.43], provides a comprehensive discussion of the issues involved in forced convection boiling. Figure 9.18 is their representation of the fairly simple case of flow of liquid in a uniform wall heat flux tube in which body forces can be neglected. This situation is representative of a fairly low heat flux at the wall. The vapor fraction, or quality, of the flow increases steadily until the wall “dries out.” Then the wall temperature rises rapidly. With a very high wall heat flux, the pipe could burn out before dryout occurs. Figure 9.19, also provided by Collier, shows how the regimes shown in Fig. 9.18 are distributed in heat flux and in position along the tube. Notice that, at high enough heat fluxes, burnout can be made to occur at any sta- tion 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 pre- dicted using eqn. (9.37) in Section 9.6. But in the subsequent regimes of slug flow and annular flow (D, E, and F) the heat transfer mechanism changes substantially. Nucleation is increasingly suppressed, and vapor- ization takes place mainly at the free surface of the liquid film on the tube wall. Most efforts to model flow boiling differentiate between nucleate- boiling-controlled heat transfer and convective boiling heat transfer. In those regimes where fully developed nucleate boiling occurs (the later parts of C), the heat transfer coefficient is essentially unaffected by the mass flow rate and the flow quality. Locally, conditions are similar to pool boiling. In convective boiling, on the other hand, vaporization occurs away from the wall, with a liquid-phase convection process dominating at the wall. For example, in the annular regions E and F, heat is convected from the wall by the liquid film, and vaporization occurs at the interface of the film with the vapor in the core of the tube. Convective boiling can also dominate at low heat fluxes or high mass flow rates, where wall nucleate is again suppressed. Vaporization then occurs mainly on en- trained bubbles in the core of the tube. In convective boiling, the heat transfer coefficient is essentially independent of the heat flux, but it is §9.7 Forced convection boiling in tubes 499 Figure 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 trends has presented a challenge to researchers for several decades. One early effort by Chen [9.44] used a weighted sum of a nucleate boiling heat trans- fer coefficient and a convective boiling coefficient, where the weighting depended on local flow conditions. This model represents water data to an accuracy of about ±30% [9.45], but it does not work well with most other fluids. Chen’s mechanistic approach was substantially improved in a more complex version due to Steiner and Taborek [9.46]. Many other investigators have instead pursued correlations built from dimensional analysis and physical reasoning. To proceed with a dimensional analysis, we first note that the liquid and vapor phases may have different velocities. Thus, we avoid intro- 500 Heat transfer in boiling and other phase-change configurations §9.7 ducing a flow speed and instead rely on the the superficial mass flux, G, through the pipe: G ≡ ˙ m A pipe (kg/m 2 s) (9.43) This mass flow per unit area is constant along the duct if the flow is steady. From this, we can define a “liquid only” Reynolds number Re lo ≡ GD µ f (9.44) which would be the Reynolds number if all the flowing mass were in the liquid state. Then we may use Re lo to compute a liquid-only heat transfer cofficient, h lo from Gnielinski’s equation, eqn. (7.43), using liquid properties at T sat . Physical arguments then suggest that the dimensional functional equa- tion for the flow boiling heat transfer coefficient, h fb , should take the following form for saturated flow in vertical tubes: h fb = fn  h lo ,G,x,h fg ,q w ,ρ f ,ρ g ,D  (9.45) It should be noted that other liquid properties, such as viscosity and con- ductivity, are represented indirectly through h lo . This functional equa- tion has eight dimensional variables (and one dimensionless variable, x) in five dimensions (m, kg, s, J, K). We thus obtain three more dimension- less groups to go with x, specifically h fb h lo = fn  x, q w Gh fg , ρ g ρ f  (9.46) In fact, the situation is even a bit simpler than this, since arguments related to the pressure gradient show that the quality and the density ratio can be combined into a single group, called the convection number: Co ≡  1 −x x  0.8  ρ g ρ f  0.5 (9.47) The other dimensionless group in eqn. (9.46) is called the boiling number: Bo ≡ q w Gh fg (9.48) §9.7 Forced convection boiling in tubes 501 Table 9.4 Fluid-dependent parameter F in the Kandlikar cor- relation for copper tubing. Additional values are given in [9.47]. Fluid F Fluid F Water 1.0 R-124 1.90 Propane 2.15 R-125 1.10 R-12 1.50 R-134a 1.63 R-22 2.20 R-152a 1.10 R-32 1.20 R-410a 1.72 so that h fb h lo = fn ( Bo, Co ) (9.49) When the convection number is large (Co  1), as for low quality, nucleate boiling dominates. In this range, h fb /h lo rises with increasing Bo and is approximately independent of Co. When the convection number is smaller, as at higher quality, the effect of the boiling number declines and h fb /h lo increases with decreasing Co. Correlations having the general form of eqn. (9.49) were developed by Schrock and Grossman [9.48], Shah [9.49], and Gungor and Winter- ton [9.50]. Kandlikar [9.45, 9.47, 9.51] refined this approach further, obtaining good accuracy and better capturing the parametric trends. His method is to calculate h fb /h lo from each of the following two correlations and to choose the larger value: h fb h lo     nbd = (1 −x) 0.8  0.6683 Co −0.2 f o +1058 Bo 0.7 F  (9.50a) h fb h lo     cbd = (1 −x) 0.8  1.136 Co −0.9 f o +667.2Bo 0.7 F  (9.50b) where “nbd” means “nucleate boiling dominant” and “cbd” means “con- vective boiling dominant”. In these equations, the orientation factor, f o , is set to unity for ver- tical tubes 4 and F is a fluid-dependent parameter whose value is given 4 The value for horizontal tubes is given in eqn. (9.52). 502 Heat transfer in boiling and other phase-change configurations §9.7 in Table 9.4. The parameter F arises here for the same reason that fluid- dependent parameters appear in nucleate boiling correlations: surface tension, contact angles, and other fluid-dependent variables influence nucleation and bubble growth. The values in Table 9.4 are for commer- cial grades of copper tubing. For stainless steel tubing, Kandlikar recom- mends F = 1 for all fluids. Equations (9.50) are applicable for the satu- rated boiling regimes (C through F) with quality in the range 0 <x≤ 0.8. For subcooled conditions, see Problem 9.21. Example 9.9 0.6kg/s of saturated H 2 OatT b = 207 ◦ C flows ina5cmdiameter ver- tical tube heated at a rate of 184,000 W/m 2 . Find the wall temperature at a point where the quality x is 20%. Solution. Data for water are taken from Tables A.3–A.5. We first compute h lo . G = ˙ m A pipe = 0.6 0.001964 = 305.6kg/m 2 s and Re lo = GD µ f = (305.6)(0.05) 1.297 ×10 −4 = 1.178 ×10 5 From eqns. (7.42) and (7.43): f = 1  1.82 log 10 (1.178 ×10 5 ) −1.64  2 = 0.01736 Nu D = ( 0.01736/8 )  1.178 ×10 5 −1000  (0.892) 1 +12.7  0.01736/8  (0.892) 2/3 −1  = 236.3 Hence, h lo = k f D Nu D = 0.6590 0.05 236.3 = 3, 115 W/m 2 K Next, we find the parameters for eqns. (9.50). From Table 9.4, F = 1 for water, and for a vertical tube, f o = 1. Also, Co =  1 −x x  0.8  ρ g ρ f  0.5 =  1 −0.20 0.2  0.8  9.014 856.5  0.5 = 0.3110 Bo = q w Gh fg = 184, 000 (305.6)(1, 913, 000) = 3.147 ×10 −4 §9.7 Forced convection boiling in tubes 503 Substituting into eqns. (9.50): h fb    nbd = (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/m 2 K h fb    cbd = (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/m 2 K Since the second value is larger, we use it: h fb = 14, 620 W/m 2 K. Then, T w = T b + q w h fb = 207 + 184, 000 14, 620 = 220 ◦ C The Kandlikar correlation leads to mean deviations of 16% for water and 19% for the various refrigerants. The Gungor and Winterton corre- lation [9.50], which is popular for its simplicity, does not contain fluid- specific coefficients, but it is somewhat less accurate than either the Kan- dlikar 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 tubes The preceding discussion of flow boiling in tubes is largely restricted to vertical tubes. Several of the flow regimes in Fig. 9.18 will be altered as shown in Fig. 9.20 if the tube is oriented horizontally. The reason is that, especially at low quality, liquid will tend to flow along the bottom of the pipe and vapor along the top. The patterns shown in Fig. 9.20, by the way, will also be observed during the reverse process—condensation—or during adiabatic two-phase flow. Which flow pattern actually occurs depends on several parameters in a fairly complex way. While many methods have been suggested to predict what flow pattern will result for a given set of conditions in the pipe, one of the best is that developed by Dukler, Taitel, and their co- workers. Their two-phase flow-regime maps are summarized in [9.52] and [9.53]. For the prediction of heat transfer, the most important additional parameter is the Froude number, Fr lo , which characterizes the strength of the flow’s inertia (or momentum) relative to the gravitational forces 504 Heat transfer in boiling and other phase-change configurations §9.7 Figure 9.20 The discernible flow regimes during boiling, condensation, or adiabatic flow from left to right in horizontal tubes. that drive the separation of the liquid and vapor phases: Fr lo ≡ G 2 ρ 2 f gD (9.51) When Fr lo < 0.04, the top of the tube becomes relatively dry and h fb /h lo begins to decline as the Froude number decreases further. Kandlikar found that he could modify his correlation to account for gravitational effects in horizontal tubes by changing the value of f o in eqns. (9.50): f o =    1 for Fr lo ≥ 0.04 ( 25 Fr lo ) 0.3 for Fr lo < 0.04 (9.52) Peak heat flux We have seen that there are two limiting heat fluxes in flow boiling in a tube: dryout and burnout. The latter is the more dangerous of the two since it occurs at higher heat fluxes and gives rise to more catastrophic temperature rises. Collier and Thome provide an extensive discussion of the subject [9.43], as does Hewitt [9.54]. §9.8 Forced convective condensation heat transfer 505 One effective set of empirical formulas was developed by Katto [9.55]. He used dimensional analysis to show that q max Gh fg = fn  ρ g ρ f , σρ f G 2 L , L D  where L is the length of the tube and D its diameter. Since G 2 L  σρ f is a Weber number, we can see that this equation is of the same form as eqn. (9.39). Katto identifies several regimes of flow boiling with both saturated and subcooled liquid entering the pipe. For each of these re- gions, he and Ohne [9.56] later fit a successful correlation of this form to existing data. Pressure gradients in flow boiling Pressure gradients in flow boiling interact with the flow pattern and the void fraction, and they can change the local saturation temperature of the fluid. Gravity, flow acceleration, and friction all contribute to pressure change, and friction can be particularly hard to predict. In particular, the frictional pressure gradient can increase greatly as the flow quality rises from the pure liquid state to the pure vapor state; the change can amount to more than two orders of magnitude at low pressures. Data correlations are usually used to estimate the frictional pressure loss, but they are, at best, accurate to within about ±30%. Whalley [9.57] provides a nice introduction such methods. Certain complex models, designed for use in computer codes, can be used to make more accurate predictions [9.58]. 9.8 Forced convective condensation heat transfer When vapor is blown or forced past a cool wall, it exerts a shear stress on the condensate film. If the direction of forced flow is downward, it will drag the condensate film along, thinning it out and enhancing heat transfer. It is not hard to show (see Problem 9.22) that 4µk(T sat −T w )x gh  fg ρ f (ρ f −ρ g ) = δ 4 + 4 3  τ δ δ 3 (ρ f −ρ g )g  (9.53) where τ δ is the shear stress exerted by the vapor flow on the condensate film. Equation (9.53) is the starting point for any analysis of forced convec- tion condensation on an external surface. Notice that if τ δ is negative—if [...]... 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In those regimes where fully developed nucleate boiling occurs (the later parts of C), the heat transfer coefficient is. negative—if 506 Heat transfer in boiling and other phase-change configurations §9.9 the shear opposes the direction of gravity—then it will have the effect of thickening δ and reducing heat transfer.