§9.6 Transition boiling and system influences 489 or q min = 18, 990 W/m 2 From Fig. 9.2 we read 20,000 W/m 2 , which is the same, within the accuracy of the graph. 9.6 Transition boiling and system influences Many system features influence the pool boiling behavior we have dis- cussed thus far. These include forced convection, subcooling, gravity, surface roughness and surface chemistry, and the heater configuration, among others. To understand one of the most serious of these—the influ- ence of surface roughness and surface chemistry—we begin by thinking about transition boiling, which is extremely sensitive to both. Surface condition and transition boiling Less is known about transition boiling than about any other mode of boiling. Data are limited, and there is no comprehensive body of theory. The first systematic sets of accurate measurements of transition boiling were reported by Berenson [9.30] in 1960. Figure 9.14 shows two sets of his data. The upper set of curves shows the typical influence of surface chem- istry on transition boiling. It makes it clear that a change in the surface chemistry has little effect on the boiling curve except in the transition boiling region and the low heat flux film boiling region. The oxidation of the surface has the effect of changing the contact angle dramatically— making it far easier for the liquid to wet the surface when it touches it. Transition boiling is more susceptible than any other mode to such a change. The bottom set of curves shows the influence of surface roughness on boiling. In this case, nucleate boiling is far more susceptible to roughness than any other mode of boiling except, perhaps, the very lowest end of the film boiling range. That is because as roughness increases the number of active nucleation sites, the heat transfer rises in accordance with the Yamagata relation, eqn. (9.3). It is important to recognize that neither roughness nor surface chem- istry affects film boiling, because the liquid does not touch the heater. Figure 9.14 Typical data from Berenson’s [9.30] study of the influence of surface condition on the boiling curve. 490 §9.6 Transition boiling and system influences 491 Figure 9.15 The transition boiling regime. The fact that both effects appear to influence the lower film boiling range means that they actually cause film boiling to break down by initiating liquid–solid contact at low heat fluxes. Figure 9.15 shows what an actual boiling curve looks like under the influence of a wetting (or even slightly wetting) contact angle. This figure is based on the work of Witte and Lienhard ([9.32] and [9.33]). On it are identified a nucleate-transition and a film-transition boiling region. These are continuations of nucleate boiling behavior with decreasing liquid– solid contact (as shown in Fig. 9.3c) and of film boiling behavior with increasing liquid–solid contact, respectively. These two regions of transition boiling are often connected by abrupt jumps. However, no one has yet seen how to predict where such jumps take place. Reference [9.33] is a full discussion of the hydrodynamic theory of boiling, which includes an extended discussion of the transition boiling problem and a correlation for the transition-film boiling heat flux by Ramilison and Lienhard [9.34]. 492 Heat transfer in boiling and other phase-change configurations §9.6 Figure 9.14 also indicates fairly accurately the influence of roughness and surface chemistry on q max . It suggests that these influences nor- mally can cause significant variations in q max that are not predicted in the hydrodynamic theory. Ramilison et al. [9.35] correlated these effects for large flat-plate heaters using the rms surface roughness, r in µm, and the receding contact angle for the liquid on the heater material, β r in radians: q max q max Z = 0.0336 ( π −β r ) 3.0 r 0.0125 (9.36) This correlation collapses the data to ±6%. Uncorrected, variations from the predictions of hydrodynamic theory reached 40% as a result of rough- ness and finish. Equivalent results are needed for other geometries. Subcooling A stationary pool will normally not remain below its saturation temper- ature over an extended period of time. When heat is transferred to the pool, the liquid soon becomes saturated—as it does in a teakettle (recall Experiment 9.1). However, before a liquid comes up to temperature, or if a very small rate of forced convection continuously replaces warm liquid with cool liquid, we can justly ask what the effect of a cool liquid bulk might be. Figure 9.16 shows how a typical boiling curve might be changed if T bulk <T sat : We know, for example, that in laminar natural convection, q will increase as (T w − T bulk ) 5/4 or as [(T w − T sat ) + ∆T sub ] 5/4 , where ∆T sub ≡ T sat −T bulk . During nucleate boiling, the influence of subcooling on q is known to be small. The peak and minimum heat fluxes are known to increase linearly with ∆T sub . These increases are quite significant. The film boiling heat flux increases rather strongly, especially at lower heat fluxes. The influence of ∆T sub on transitional boiling is not well documented. Gravity The influence of gravity (or any other such body force) is of concern be- cause boiling processes frequently take place in rotating or accelerating systems. The reduction of gravity has a significant impact on boiling processes aboard space vehicles. Since g appears explicitly in the equa- tions for q max , q min , and q film boiling , we know what its influence is. Both q max and q min increase directly as g 1/4 in finite bodies, and there is an additional gravitational influence through the parameter L  . However, when gravity is small enough to reduce R  below about 0.15, the hydrody- §9.6 Transition boiling and system influences 493 Figure 9.16 The influence of subcooling on the boiling curve. namic transitions deteriorate and eventually vanish altogether. Although Rohsenow’s equation suggests that q is proportional to g 1/2 in the nucle- ate boiling regime, other evidence suggests that the influence of gravity on the nucleate boiling curve is very slight, apart from an indirect effect on the onset of boiling. Forced convection The influence of superposed flow on the pool boiling curve for a given heater (e.g., Fig. 9.2) is generally to improve heat transfer everywhere. But flow is particularly effective in raising q max . Let us look at the influence of flow on the different regimes of boiling. 494 Heat transfer in boiling and other phase-change configurations §9.6 Influences of forced convection on nucleate boiling. Figure 9.17 shows nucleate boiling during the forced convection of water over a flat plate. Bergles and Rohsenow [9.36] offer an empirical strategy for predicting the heat flux during nucleate flow boiling when the net vapor generation is still relatively small. (The photograph in Fig. 9.17 shows how a sub- stantial buildup of vapor can radically alter flow boiling behavior.) They suggest that q = q FC     1 +  q B q FC  1 − q i q B  2 (9.37) where • q FC is the single-phase forced convection heat transfer for the heater, as one might calculate using the methods of Chapters 6 and 7. • q B is the pool boiling heat flux for that liquid and that heater from eqn. (9.4). • q i is the heat flux from the pool boiling curve evaluated at the value of (T w −T sat ) where boiling begins during flow boiling (see Fig. 9.17). An estimate of (T w − T sat ) onset can be made by intersecting the forced convection equation q = h FC (T w − T b ) with the following equation [9.37]: (T w −T sat ) onset =  8σT sat q ρ g h fg k f  1/2 (9.38) Equation (9.37) will provide a first approximation in most boiling con- figurations, but it is restricted to subcooled flows or other situations in which vapor generation is not too great. Peak heat flux in external flows. The peak heat flux on a submerged body is strongly augmented by an external flow around it. Although knowledge of this area is still evolving, we do know from dimensional analysis that q max ρ g h fg u ∞ = fn  We D ,ρ f  ρ g  (9.39) §9.6 Transition boiling and system influences 495 Figure 9.17 Forced convection boiling on an external surface. where the Weber number, We, is We L ≡ ρ g u 2 ∞ L σ = inertia force  L surface force  L and where L is any characteristic length. Kheyrandish and Lienhard [9.38] suggest fairly complex expressions of this form for q max on horizontal cylinders in cross flows. For a cylin- drical liquid jet impinging on a heated disk of diameter D, Sharan and 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 [...]... correlation for copper tubing Additional values are given in [9.47] Fluid Water Propane R -1 2 R-22 R -3 2 F 1. 0 2 .15 1. 50 2.20 1. 20 Fluid R -1 2 4 R -1 2 5 R- 13 4a R -1 5 2a R- 41 0a F 1. 90 1. 10 1. 63 1. 10 1. 72 so that hfb = 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 Bo and is approximately independent of Co... nbd = (3, 11 5) (1 − 0.2)0.8 0.66 83 (0 . 31 10 )−0.2 (1) + 10 58 (3. 14 7 × 10 −4 )0.7 (1) = 11 , 950 W/m2 K hfb cbd = (3, 11 5) (1 − 0.2)0.8 1. 13 6 (0 . 31 10 )−0.9 (1) + 667.2 (3. 14 7 × 10 −4 )0.7 (1) = 14 , 620 W/m2 K Since the second value is larger, we use it: hfb = 14 , 620 W/m2 K Then, T w = Tb + qw 18 4, 000 = 220◦ C = 207 + hfb 14 , 620 The Kandlikar correlation leads to mean deviations of 16 % for water and 19 % for... of 18 4,000 W/m2 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 hlo G= and Relo = ˙ m 0.6 = 30 5.6 kg/m2 s = Apipe 0.0 019 64 GD (30 5.6)(0.05) = = 1. 178 × 10 5 µf 1. 297 × 10 −4 From eqns (7.42) and (7. 43) : f = NuD = 1 1.82 log10 (1. 178 × 10 5 ) − 1. 64 2 = 0. 01 736 (0. 01 736 /8) 1. 178 × 10 5 − 10 00 (0.892) = 236 .3 1 + 12 .7... on having a heat pipe with much larger condenser area than evaporator area Thus, the heat fluxes on the condenser are kept relatively low This facilitates such uncomplicated means for the ultimate heat disposal as using a small fan to blow air over the condenser One heat- pipe-based electronics heat sink is shown in Fig 9. 23 The copper block at center is attached to a microprocessor, and the evaporator... evaporator heat loads, for example One way to vary its performance is through the introduction of a non-condensible gas in the pipe This gas will collect at the condenser, limiting the area of the condenser that vapor can reach By varying the amount of gas, the thermal resistance of the heat pipe can be controlled In the absence of active control of the gas, an increase in the heat load at the evaporator... 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 fluiddependent parameters appear in nucleate boiling correlations: surface tension, contact angles, and other fluid-dependent variables influence nucleation and... water can be superheated at low pressure How many degrees of superheat does this suggest that water can sustain at the low pressure of 1 atm? (It turns out that this calculation is accurate within about 10 %.) What would Rb be at this superheat? 9.7 Use Yamagata’s equation, (9 .3) , to determine how nucleation site density increases with ∆T for Berenson’s curves in Fig 9 .14 (That is, find c in the relation... will raise the pressure in the pipe, compressing the noncondensible gas and lowering the thermal resistance of the pipe The result is that the temperature at the evaporator remains essentially constant even as the heat load rises as falls Heat pipes have proven useful in cooling high power-density electronic devices The evaporator is located on a small electronic component 511 512 Chapter 9: Heat transfer. .. the vast majority of technical applications 9 .10 The heat pipe A heat pipe is a device that combines the high efficiencies of boiling and condensation It is aptly named because it literally pipes heat from a hot region to a cold one The operation of a heat pipe is shown in Fig 9.22 The pipe is a tube that can be bent or turned in any way that is convenient The inside of the tube is lined with a layer... 9 .36 ) • Selection of the tube material The tube material must be compatible with the working fluid Gas generation and corrosion are particular considerations Copper tubes are widely used with water, methanol, and acetone, but they cannot be used with ammonia Stainless steel §9 .10 The heat pipe tubes can be used with ammonia and many liquid metals, but are not suitable for long term service with water . tubing. Additional values are given in [9.47]. Fluid F Fluid F Water 1. 0 R -1 2 4 1. 90 Propane 2 .15 R -1 2 5 1. 10 R -1 2 1. 50 R- 13 4a 1. 63 R-22 2.20 R -1 5 2a 1. 10 R -3 2 1. 20 R- 41 0a 1. 72 so that h fb h lo = fn ( Bo,. flows ina5cmdiameter ver- tical tube heated at a rate of 18 4,000 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 impinging on a heated disk of diameter D, Sharan and 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 + 0.0 017 ρ f  ρ g   d jet D  1 /3  10 00ρ g /ρ f We D  A