464 Heat transfer in boiling and other phase-change configurations §9.2 Figure 9.4 Enlarged sketch of a typical metal surface. 9.2 Nucleate boiling Inception of boiling Figure 9.4 shows a highly enlarged sketch of a heater surface. Most metal- finishing operations score tiny grooves on the surface, but they also typ- ically involve some chattering or bouncing action, which hammers small holes into the surface. When a surface is wetted, liquid is prevented by surface tension from entering these holes, so small gas or vapor pockets are formed. These little pockets are the sites at which bubble nucleation occurs. To see why vapor pockets serve as nucleation sites, consider Fig. 9.5. Here we see the problem in highly idealized form. Suppose that a spher- ical bubble of pure saturated steam is at equilibrium with an infinite superheated liquid. To determine the size of such a bubble, we impose the conditions of mechanical and thermal equilibrium. The bubble will be in mechanical equilibrium when the pressure dif- ference between the inside and the outside of the bubble is balanced by the forces of surface tension, σ, as indicated in the cutaway sketch in Fig. 9.5. Since thermal equilibrium requires that the temperature must be the same inside and outside the bubble, and since the vapor inside must be saturated at T sup because it is in contact with its liquid, the force balance takes the form R b = 2σ  p sat at T sup  −p ambient (9.1) The p–v diagram in Fig. 9.5 shows the state points of the internal vapor and external liquid for a bubble at equilibrium. Notice that the external liquid is superheated to (T sup −T sat ) K above its boiling point at the ambient pressure; but the vapor inside, being held at just the right elevated pressure by surface tension, is just saturated. §9.2 Nucleate boiling 465 Figure 9.5 The conditions required for simultaneous mechan- ical and thermal equilibrium of a vapor bubble. Physical Digression 9.1 The surface tension of water in contact with its vapor is given with great accuracy by [9.3]: σ water = 235.8  1 − T sat T c  1.256  1 − 0.625  1 − T sat T c  mN m (9.2a) where both T sat and the thermodynamical critical temperature, T c = 647.096 K, are expressed in K. The units of σ are millinewtons (mN) per meter. Table 9.1 gives additional values of σ for several substances. Equation 9.2a is a specialized refinement of a simple, but quite ac- curate and widely-used, semi-empirical equation for correlating surface Table 9.1 Surface tension of various substances from the collection of Jasper [9.4] a and other sources. Temperature σ = a − bT ( ◦ C) Substance Range ( ◦ C) σ (mN/m) a(mN/m)b(mN/m· ◦ C) Acetone 25 to 50 26.26 0.112 Ammonia −70 42.39 −60 40.25 −50 37.91 −40 35.38 Aniline 15 to 90 44.83 0.1085 Benzene 10 30.21 30 27.56 50 24.96 70 22.40 Butyl alcohol 10 to 100 27.18 0.08983 Carbon tetrachloride 15 to 105 29.49 0.1224 Cyclohexanol 20 to 100 35.33 0.0966 Ethyl alcohol 10 to 100 24.05 0.0832 Ethylene glycol 20 to 140 50.21 0.089 Hydrogen −258 2.80 −255 2.29 −253 1.95 Isopropyl alcohol 10 to 100 22.90 0.0789 Mercury 5 to 200 490.60.2049 Methane 90 18.877 100 16.328 115 12.371 Methyl alcohol 10 to 60 24.00 0.0773 Naphthalene 100 to 200 42.84 0.1107 Nicotine −40 to 90 41.07 0.1112 Nitrogen −195 to −183 26.42 0.2265 Octane 10 to 120 23.52 0.09509 Oxygen −202 to −184 −33.72 −0.2561 Pentane 10 to 30 18.25 0.11021 Toluene 10 to 100 30.90 0.1189 Water 10 to 100 75.83 0.1477 Temperature σ = σ o [ 1 −T(K)/T c ] n Substance Range ( ◦ C) σ o (mN/m) T c (K)n Carbon dioxide −56 to 31 75.00 304.26 1.25 CFC-12 (R12) [9.5] −148 to 112 56.52 385.01 1.27 HCFC-22 (R22) [9.5] −158 to 96 61.23 369.32 1.23 HFC-134a (R134a) [9.6] −30 to 101 59.69 374.18 1.266 Propane [9.7] −173 to 96 53.13 369.85 1.242 a The function σ = σ(T) is not really linear, but Jasper was able to linearize it over modest ranges of temperature [e.g., compare the water equation above with eqn. (9.2a)]. 466 §9.2 Nucleate boiling 467 tension: σ = σ o  1 − T sat  T c  11/9 (9.2b) We include correlating equations of this form for CO 2 , propane, and some refrigerants at the bottom of Table 9.1. Equations of this general form are discussed in Reference [9.8]. It is easy to see that the equilibrium bubble, whose radius is described by eqn. (9.1), is unstable. If its radius is less than this value, surface tension will overbalance [p sat (T sup ) − p ambient ]. Thus, vapor inside will condense at this higher pressure and the bubble will collapse. If the bubble radius is slightly larger than the equation specifies, liquid at the interface will evaporate and the bubble will begin to grow. Thus, as the heater surface temperature is increased, higher and higher values of [p sat (T sup )−p ambient ] will result and the equilibrium radius, R b , will decrease in accordance with eqn. (9.1). It follows that smaller and smaller vapor pockets will be triggered into active bubble growth as the temperature is increased. As an approximation, we can use eqn. (9.1) to specify the radius of those vapor pockets that become active nucle- ation sites. More accurate estimates can be made using Hsu’s [9.9] bub- ble inception theory, the subsequent work by Rohsenow and others (see, e.g., [9.10]), or the still more recent technical literature. Example 9.1 Estimate the approximate size of active nucleation sites in water at 1 atm on a wall superheated by 8 K and by 16 K. This is roughly in the regime of isolated bubbles indicated in Fig. 9.2. Solution. p sat = 1.203 × 10 5 N/m 2 at 108 ◦ C and 1.769 × 10 5 N/m 2 at 116 ◦ C, and σ is given as 57.36 mN/matT sat = 108 ◦ C and as 55.78 mN/matT sat = 116 ◦ C by eqn. (9.2a). Then, at 108 ◦ C, R b from eqn. (9.1)is R b = 2(57.36 × 10 −3 ) N/m  1.203 × 10 5 −1.013 ×10 5  N/m 2 and similarly for 116 ◦ C, so the radius of active nucleation sites is on the order of R b = 0.0060 mm at T = 108 ◦ Cor0.0015 mm at 116 ◦ C 468 Heat transfer in boiling and other phase-change configurations §9.2 This means that active nucleation sites would be holes with diameters very roughly on the order of magnitude of 0.005 mm or 5µm—at least on the heater represented by Fig. 9.2. That is within the range of roughness of commercially finished surfaces. Region of isolated bubbles The mechanism of heat transfer enhancement in the isolated bubble regime was hotly argued in the years following World War II. A few con- clusions have emerged from that debate, and we shall attempt to identify them. There is little doubt that bubbles act in some way as small pumps that keep replacing liquid heated at the wall with cool liquid. The ques- tion is that of specifying the correct mechanism. Figure 9.6 shows the way bubbles probably act to remove hot liquid from the wall and intro- duce cold liquid to be heated. It is apparent that the number of active nucleation sites generating bubbles will strongly influence q. On the basis of his experiments, Yam- agata showed in 1955 (see, e.g., [9.11]) that q ∝ ∆T a n b (9.3) where ∆T ≡ T w −T sat and n is the site density or number of active sites per square meter. A great deal of subsequent work has been done to fix the constant of proportionality and the constant exponents, a and b. The exponents turn out to be approximately a = 1.2 and b = 1 3 . The problem with eqn. (9.3) is that it introduces what engineers call a nuisance variable. A nuisance variable is one that varies from system to system and cannot easily be evaluated—the site density, n, in this case. Normally, n increases with ∆T in some way, but how? If all sites were identical in size, all sites would be activated simultaneously, and q would be a discontinuous function of ∆T . When the sites have a typical distribution of sizes, n (and hence q) can increase very strongly with ∆T. It is a lucky fact that for a large class of factory-finished materials, n varies approximately as ∆T 5or6 ,soq varies roughly as ∆T 3 . This has made it possible for various authors to correlate q approximately for a large variety of materials. One of the first and most useful correlations for nucleate boiling was that of Rohsenow [9.12] in 1952. It is c p ( T w −T sat ) h fg Pr s = C sf  q µh fg  σ g  ρ f −ρ g   0.33 (9.4) §9.2 Nucleate boiling 469 A bubble growing and departing in saturated liquid. The bubble grows, absorbing heat from the superheated liquid on its periphery. As it leaves, it entrains cold liquid onto the plate which then warms up until nucleation occurs and the cycle repeats. A bubble growing in subcooled liquid. When the bubble protrudes into cold liquid, steam can condense on the top while evaporation continues on the bottom. This provides a short-circuit for cooling the wall. Then, when the bubble caves in, cold liquid is brought to the wall. Figure 9.6 Heat removal by bubble action during boiling. Dark regions denote locally superheated liquid. where all properties, unless otherwise noted, are for liquid at T sat . The constant C sf is an empirical correction for typical surface conditions. Table 9.2 includes a set of values of C sf for common surfaces (taken from [9.12]) as well as the Prandtl number exponent, s. A more extensive compilation of these constants was published by Pioro in 1999 [9.13]. We noted, initially, that there are two nucleate boiling regimes, and the Yamagata equation (9.3) applies only to the first of them. Rohsenow’s equation is frankly empirical and does not depend on the rational anal- ysis of either nucleate boiling process. It turns out that it represents q(∆T) in both regimes, but it is not terribly accurate in either one. Fig- ure 9.7 shows Rohsenow’s original comparison of eqn. (9.4) with data for water over a large range of conditions. It shows typical errors in heat flux 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 error in q gives a much smaller error in ∆T . It appears that any substantial improvement in this situation will have to wait until someone has man- aged to deal realistically with the nuisance variable, n. Current research efforts are dealing with this matter, and we can simply hope that such work will eventually produce a method for achieving reliable heat trans- fer design relationships for nucleate boiling. 470 Heat transfer in boiling and other phase-change configurations §9.2 Table 9.2 Selected values of the surface correction factor for use with eqn. (9.4)[9.12] Surface–Fluid Combination C sf s Water–nickel 0.006 1.0 Water–platinum 0.013 1.0 Water–copper 0.013 1.0 Water–brass 0.006 1.0 CCl 4 –copper 0.013 1.7 Benzene–chromium 0.010 1.7 n-Pentane–chromium 0.015 1.7 Ethyl alcohol–chromium 0.0027 1.7 Isopropyl alcohol–copper 0.0025 1.7 35% K 2 CO 3 –copper 0.0054 1.7 50% K 2 CO 3 –copper 0.0027 1.7 n-Butyl alcohol–copper 0.0030 1.7 It is indeed fortunate that we do not often have to calculate q, given ∆T , in the nucleate boiling regime. More often, the major problem is to avoid exceeding q max . We turn our attention in the next section to predicting this limit. Example 9.2 What is C sf for the heater surface in Fig. 9.2? Solution. From eqn. (9.4) we obtain q ∆T 3 C 3 sf = µc 3 p h 2 fg Pr 3  g  ρ f −ρ g  σ where, since the liquid is water, we take s to be 1.0. Then, for water at T sat = 100 ◦ C: c p = 4.22 kJ/kg·K, Pr = 1.75, (ρ f − ρ g ) = 958 kg/m 3 , σ = 0.0589 N/morkg/s 2 , h fg = 2257 kJ/kg, µ = 0.000282 kg/m·s. §9.2 Nucleate boiling 471 Figure 9.7 Illustration of Rohsenow’s [9.12] correlation applied to data for water boiling on 0.61 mm diameter platinum wire. Thus, q ∆T 3 C 3 sf = 3.10 × 10 −7 kW m 2 K 3 At q = 800 kW/m 2 , we read ∆T = 22 K from Fig. 9.2. This gives C sf =  3.10 × 10 −7 (22) 3 800  1/3 = 0.016 This value compares favorably with C sf for a platinum or copper sur- face under water. 472 Heat transfer in boiling and other phase-change configurations §9.3 9.3 Peak pool boiling heat flux Transitional boiling regime and Taylor instability It will help us to understand the peak heat flux if we first consider the process that connects the peak and the minimum heat fluxes. During high heat flux transitional boiling, a large amount of vapor is glutted about the heater. It wants to buoy upward, but it has no clearly defined escape route. The jets that carry vapor away from the heater in the re- gion of slugs and columns are unstable and cannot serve that function in this regime. Therefore, vapor buoys up in big slugs—then liquid falls in, touches the surface briefly, and a new slug begins to form. Figure 9.3c shows part of this process. The high and low heat flux transitional boiling regimes are different in character. The low heat flux region does not look like Fig. 9.2c but is al- most indistinguishable from the film boiling shown in Fig. 9.2d. However, both processes display a common conceptual key: In both, the heater is almost completely blanketed with vapor. In both, we must contend with the unstable configuration of a liquid on top of a vapor. Figure 9.8 shows two commonplace examples of such behavior. In either an inverted honey jar or the water condensing from a cold water pipe, 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 node of a wave and the light fluid rises into the other node. The collapse process is called Taylor instability after G. I. Taylor, who first predicted it. The so-called Taylor wavelength, λ d , is the length of the wave that grows fastest and therefore predominates during the col- lapse of an infinite plane horizontal interface. It can be predicted using dimensional analysis. The dimensional functional equation for λ d is λ d = fn  σ,g  ρ f −ρ g   (9.5) since the wave is formed as a result of the balancing forces of surface tension against inertia and gravity. There are three variables involving m and kg/s 2 , so we look for just one dimensionless group: λ d  g  ρ f −ρ g  σ = constant This relationship was derived analytically by Bellman and Pennington [9.14] for one-dimensional waves and by Sernas [9.15] for the two-dimensional §9.3 Peak pool boiling heat flux 473 a. Taylor instability in the surface of the honey in an inverted honey jar b. Taylor instability in the interface of the water condensing on the underside of a small cold water pipe. Figure 9.8 Two examples of Taylor instabilities that one might commonly experience. waves that actually occur in a plane horizontal interface. The results were λ d  g  ρ f −ρ g  σ =  2π √ 3 for one-dimensional waves 2π √ 6 for two-dimensional waves (9.6) [...]... shown that collapse would begin at 1 3 times this value, or at 1 .2 cm.) Prediction of qmax General expression for qmax The heat flux must be balanced by the latent heat carried away in the jets when the liquid is saturated Thus, we can write immediately qmax = ρg hfg ug Aj Ah (9.9) where Aj is the cross-sectional area of a jet and Ah is the heater area that supplies each jet For any heater configuration,... first temptation might Film boiling §9.4 487 be simply to add a radiation heat transfer coefficient, hrad to hboiling as obtained from eqn (9 .28 ) or (9.30), where hrad 4 4 εσ Tw − Tsat qrad = = Tw − Tsat Tw − Tsat and where ε is a surface radiation property of the heater called the emittance (see Section 10. 1) Unfortunately, such addition is not correct, because the additional radiative heat transfer will... “flood.” Kutateladze did the dimensional analysis of qmax based on the flooding mechanism and obtained the following relationship, which, lacking a characteristic length and being of the same form as eqn (9.11), is really valid only for an infinite horizontal plate: 1 /2 qmax = C ρg hfg 3 Readers are reminded that √ n x ≡ x 1/n 4 g ρf − ρ g σ 480 Heat transfer in boiling and other phase-change configurations... accordance with Section 4.3, we note that no significant conversion from work to heat is occurring so that J must be retained as a separate unit There are thus two pi-groups The first group can 481 4 82 Heat transfer in boiling and other phase-change configurations §9.3 arbitrarily be multiplied by 24 /π to give qmax qmax Π1 = = 1 /2 qmaxz (π /24 ) ρg hfg 4 σ g(ρf − ρg ) (9.13) Notice that the factor of 24 /π... conservativism of the period prevented the idea from gaining acceptance for another decade Peak pool boiling heat flux §9.3 Example 9.5 Predict the peak heat flux for Fig 9 .2 Solution We use eqn (9.11) to evaluate qmax for water at 100 ◦ C on an infinite flat plate: 1 /2 qmax = 0.149 ρg hfg 4 g(ρf − ρg )σ = 0.149(0.597)1 /2 (2, 25 7, 000) 9.8(958 .2 − 0.6)(0.0589) 4 = 1 .26 0 × 106 W/m2 = 1 .26 0 MW/m2 Figure 9 .2 shows... (9 .28 ) where vapor and liquid properties should be evaluated at Tsat + ∆T /2 and at Tsat , respectively The latent heat correction in this case is similar in form to that for film condensation, but with different constants in it Sadasivan and Lienhard [9 .25 ] have shown it to be hfg = hfg 1 + 0.968 − 0.163 Prg Jag (9 .29 ) for Prg ≥ 0.6, where Jag = cpg (Tw − Tsat ) hfg Dhir and Lienhard [9 .26 ] did the same... forces that bring about collapse Thus, while the flag is unstable in any breeze, the vapor velocity in the jet must reach a limiting value, ug , before the jet becomes unstable a Plan view of bubbles rising from surface b Waveform underneath the bubbles shown in a Figure 9.9 The array of vapor jets as seen on an infinite horizontal heater surface 475 476 Heat transfer in boiling and other phase-change configurations... intricate, brings together an enormous range of heat transfer data for cylinders, within 20 % It is worth noting that radiation is seldom important when the heater temperature is less than 300◦ C The use of the analogy between film condensation and film boiling is somewhat questionable during film boiling on a vertical surface In this case, the liquid–vapor interface becomes Helmholtz-unstable at a short... equation should be used directly 488 Heat transfer in boiling and other phase-change configurations 9.5 §9.5 Minimum heat flux Zuber [9.17] also provided a prediction of the minimum heat flux, qmin , along with his prediction of qmax He assumed that as Tw − Tsat is reduced in the film boiling regime, the rate of vapor generation eventually becomes too small to sustain the Taylor wave action that characterizes... than this, there is a small integral number of jets on a plate which may be larger or smaller in area than 16/π per jet When this is the case, the actual qmax may be larger or smaller than that predicted by eqn (9.11) (see Problem 9.13) The form of the preceding prediction is usually credited to Kutateladze [9 .21 ] and Zuber [9.17] Kutateladze (then working in Leningrad and later director of the Heat . 24 .00 0.0773 Naphthalene 100 to 20 0 42. 84 0. 1107 Nicotine −40 to 90 41.07 0.11 12 Nitrogen −195 to −183 26 . 42 0 .22 65 Octane 10 to 120 23 . 52 0.09509 Oxygen 20 2 to −184 −33. 72 −0 .25 61 Pentane 10. 30 .21 30 27 .56 50 24 .96 70 22 .40 Butyl alcohol 10 to 100 27 .18 0.08983 Carbon tetrachloride 15 to 105 29 .49 0. 122 4 Cyclohexanol 20 to 100 35.33 0.0966 Ethyl alcohol 10 to 100 24 .05 0.08 32 Ethylene. eventually produce a method for achieving reliable heat trans- fer design relationships for nucleate boiling. 470 Heat transfer in boiling and other phase-change configurations §9 .2 Table 9 .2 Selected