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9. Heat transfer in boiling and other phase-change configurations For a charm of powerful trouble, like a Hell-broth boil and bubble.. . . . . .Cool it with a baboon’s blood, then the charm is firm and good. Macbeth, Wm. Shakespeare “A watched pot never boils”—the water in a teakettle takes a long time to get hot enough to boil because natural convection initially warms it rather slowly. Once boiling begins, the water is heated the rest of the way to the saturation point very quickly. Boiling is of interest to us because it is remarkably effective in carrying heat from a heater into a liquid. The heater in question might be a red-hot horseshoe quenched in a bucket or the core of a nuclear reactor with coolant flowing through it. Our aim is to learn enough about the boiling process to design systems that use boiling for cooling. We begin by considering pool boiling—the boiling that occurs when a stationary heater transfers heat to an otherwise stationary liquid. 9.1 Nukiyama’s experiment and the pool boiling curve Hysteresis in the q vs. ∆T relation for pool boiling In 1934, Nukiyama [9.1] did the experiment described in Fig. 9.1. He boiled saturated water on a horizontal wire that functioned both as an electric resistance heater and as a resistance thermometer. By calibrating 457 458 Heat transfer in boiling and other phase-change configurations Figure 9.1 §9.1 Nukiyama’s boiling hysteresis loop. the resistance of a Nichrome wire as a function of temperature before the experiment, he was able to obtain both the heat flux and the temperature using the observed current and voltage. He found that, as he increased the power input to the wire, the heat flux rose sharply but the temperature of the wire increased relatively little. Suddenly, at a particular high value of the heat flux, the wire abruptly melted. Nukiyama then obtained a platinum wire and tried again. This time the wire reached the same §9.1 Nukiyama’s experiment and the pool boiling curve limiting heat flux, but then it turned almost white-hot without melting. As he reduced the power input to the white-hot wire, the temperature dropped in a continuous way, as shown in Fig. 9.1, until the heat flux was far below the value where the first temperature jump occurred. Then the temperature dropped abruptly to the original q vs. ∆T = (Twire − Tsat ) curve, as shown. Nukiyama suspected that the hysteresis would not occur if ∆T could be specified as the independent controlled variable. He conjectured that such an experiment would result in the connecting line shown between the points where the temperatures jumped. In 1937, Drew and Mueller [9.2] succeeded in making ∆T the independent variable by boiling organic liquids outside a tube. Steam was allowed to condense inside the tube at an elevated pressure. The steam saturation temperature—and hence the tube-wall temperature—was varied by controlling the steam pressure. This permitted them to obtain a few scattered data that seemed to bear out Nukiyama’s conjecture. Measurements of this kind are inherently hard to make accurately. For the next forty years, the relatively few nucleate boiling data that people obtained were usually—and sometimes imaginatively—interpreted as verifying Nukiyama’s suggestion that this part of the boiling curve is continuous. Figure 9.2 is a completed boiling curve for saturated water at atmospheric pressure on a particular flat horizontal heater. It displays the behavior shown in Fig. 9.1, but it has been rotated to place the independent variable, ∆T , on the abscissa. (We represent Nukiyama’s connecting region as two unconnected extensions of the neighboring regions for reasons that we explain subsequently.) Modes of pool boiling The boiling curve in Fig. 9.2 has been divided into five regimes of behavior. These regimes, and the transitions that divide them, are discussed next. Natural convection. Water that is not in contact with its own vapor does not boil at the so-called normal boiling point,1 Tsat . Instead, it continues to rise in temperature until bubbles finally to begin to form. On conventional machined metal surfaces, this occurs when the surface is a few degrees above Tsat . Below the bubble inception point, heat is removed by natural convection, and it can be predicted by the methods laid out in 1 This notion might be new to some readers. It is explained in Section 9.2. 459 460 Heat transfer in boiling and other phase-change configurations §9.1 Figure 9.2 Typical boiling curve and regimes of boiling for an unspecified heater surface. Chapter 8. Nucleate boiling. The nucleate boiling regime embraces the two distinct regimes that lie between bubble inception and Nukiyama’s first transition point: 1. The region of isolated bubbles. In this range, bubbles rise from isolated nucleation sites, more or less as they are sketched in Fig. 9.1. As q and ∆T increase, more and more sites are activated. Figure 9.3a is a photograph of this regime as it appears on a horizontal plate. 2. The region of slugs and columns. When the active sites become very numerous, the bubbles start to merge into one another, and an entirely different kind of vapor escape path comes into play. Vapor formed at the surface merges immediately into jets that feed into large overhead bubbles or “slugs” of vapor. This process is shown as it occurs on a horizontal cylinder in Fig. 9.3b. 461 Figure 9.3 d. Film boiling of acetone on a 22 gage wire at earth-normal gravity. The true width of this image is 3.48 cm. b. Two views of transitional boiling in acetone on a 0.32 cm diam. tube. Typical photographs of boiling in the four regimes identified in Fig. 9.2. c. Two views of the regime of slugs and columns. 3.75 cm length of 0.164 cm diam. wire in benzene at earth-normal gravity. q=0.35×106 W/m2 3.45 cm length of 0.0322 cm diam. wire in methanol at 10 earth-normal gravities. q=1.04×106 W/m2 a. Isolated bubble regime—water. 462 Heat transfer in boiling and other phase-change configurations §9.1 Peak heat flux. Clearly, it is very desirable to be able to operate heat exchange equipment at the upper end of the region of slugs and columns. Here the temperature difference is low while the heat flux is very high. Heat transfer coefficients in this range are enormous. However, it is very dangerous to run equipment near qmax in systems for which q is the independent variable (as in nuclear reactors). If q is raised beyond the upper limit of the nucleate boiling regime, such a system will suffer a sudden and damaging increase of temperature. This transition2 is known by a variety of names: the burnout point (although a complete burning up or melting away does not always accompany it); the peak heat flux (a modest descriptive term); the boiling crisis (a Russian term); the DNB, or departure from nucleate boiling, and the CHF, or critical heat flux (terms more often used in flow boiling); and the first boiling transition (which term ignores previous transitions). We designate the peak heat flux as qmax . Transitional boiling regime. It is a curious fact that the heat flux actually diminishes with ∆T after qmax is reached. In this regime the effectiveness of the vapor escape process becomes worse and worse. Furthermore, the hot surface becomes completely blanketed in vapor and q reaches a minimum heat flux which we call qmin . Figure 9.3c shows two typical instances of transitional boiling just beyond the peak heat flux. Film boiling. Once a stable vapor blanket is established, q again increases with increasing ∆T . The mechanics of the heat removal process during film boiling, and the regular removal of bubbles, has a great deal in common with film condensation, but the heat transfer coefficients are much lower because heat must be conducted through a vapor film instead of through a liquid film. We see an instance of film boiling in Fig. 9.3d. Experiment 9.1 Set an open pan of cold tap water on your stove to boil. Observe the following stages as you watch: • At first nothing appears to happen; then you notice that numerous small, stationary bubbles have formed over the bottom of the pan. 2 We defer a proper physical explanation of the transition to Section 9.3. Nukiyama’s experiment and the pool boiling curve §9.1 These bubbles have nothing to do with boiling—they contain air that was driven out of solution as the temperature rose. • Suddenly the pan will begin to “sing.” There will be a somewhat high-pitched buzzing-humming sound as the first vapor bubbles are triggered. They grow at the heated surface and condense very suddenly when their tops encounter the still-cold water above them. This cavitation collapse is accompanied by a small “ping” or “click,” over and over, as the process is repeated at a fairly high frequency. • As the temperature of the liquid bulk rises, the singing is increasingly muted. You may then look in the pan and see a number of points on the bottom where a feathery blur appears to be affixed. These blurred images are bubble columns emanating scores of bubbles per second. The bubbles in these columns condense completely at some distance above the surface. Notice that the air bubbles are all gradually being swept away. • The “singing” finally gives way to a full rolling boil, accompanied by a gentle burbling sound. Bubbles no longer condense but now reach the surface, where they break. • A full rolling-boil process, in which the liquid bulk is saturated, is a kind of isolated-bubble process, as plotted in Fig. 9.2. No kitchen stove supplies energy fast enough to boil water in the slugs-andcolumns regime. You might, therefore, reflect on the relative intensity of the slugs-and-columns process. Experiment 9.2 Repeat Experiment 9.1 with a glass beaker instead of a kitchen pan. Place a strobe light, blinking about 6 to 10 times per second, behind the beaker with a piece of frosted glass or tissue paper between it and the beaker. You can now see the evolution of bubble columns from the first singing mode up to the rolling boil. You will also be able to see natural convection in the refraction of the light before boiling begins. 463 464 Heat transfer in boiling and other phase-change configurations Figure 9.4 9.2 §9.2 Enlarged sketch of a typical metal surface. Nucleate boiling Inception of boiling Figure 9.4 shows a highly enlarged sketch of a heater surface. Most metalfinishing operations score tiny grooves on the surface, but they also typically 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 spherical 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 difference 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 Tsup because it is in contact with its liquid, the force balance takes the form 2σ Rb =  psat at Tsup − pambient (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 (Tsup − Tsat ) 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. Nucleate boiling §9.2 Figure 9.5 The conditions required for simultaneous mechanical 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]:     Tsat 1.256 Tsat mN σwater = 235.8 1 − (9.2a) 1 − 0.625 1 − Tc Tc m where both Tsat and the thermodynamical critical temperature, Tc = 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 accurate and widely-used, semi-empirical equation for correlating surface 465 Table 9.1 Surface tension of various substances from the collection of Jasper [9.4]a and other sources. Substance Acetone Ammonia Aniline Benzene Butyl alcohol Carbon tetrachloride Cyclohexanol Ethyl alcohol Ethylene glycol Hydrogen Isopropyl alcohol Mercury Methane Methyl alcohol Naphthalene Nicotine Nitrogen Octane Oxygen Pentane Toluene Water Temperature Range (◦ C) 25 to 50 −70 −60 −50 −40 15 to 90 10 30 50 70 10 to 100 15 to 105 20 to 100 10 to 100 20 to 140 −258 −255 −253 10 to 100 5 to 200 90 100 115 10 to 60 100 to 200 −40 to 90 −195 to −183 10 to 120 −202 to −184 10 to 30 10 to 100 10 to 100 σ (mN/m) σ = a − bT (◦ C) a (mN/m) b (mN/m·◦ C) 26.26 0.112 44.83 0.1085 27.18 29.49 35.33 24.05 50.21 0.08983 0.1224 0.0966 0.0832 0.089 42.39 40.25 37.91 35.38 30.21 27.56 24.96 22.40 2.80 2.29 1.95 22.90 490.6 0.0789 0.2049 24.00 42.84 41.07 26.42 23.52 −33.72 18.25 30.90 75.83 0.0773 0.1107 0.1112 0.2265 0.09509 −0.2561 0.11021 0.1189 0.1477 18.877 16.328 12.371 n Substance Carbon dioxide CFC-12 (R12) [9.5] HCFC-22 (R22) [9.5] Temperature Range (◦ C) −56 to 31 σ = σo [1 − T (K)/Tc ] σo (mN/m) 75.00 Tc (K) n 304.26 1.25 −148 to 112 56.52 385.01 1.27 −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