BOOKCOMP, Inc. — John Wiley & Sons / Page 644 / 2nd Proofs / Heat Transfer Handbook / Bejan 644 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [644], (10) Lines: 356 to 370 ——— -0.45691pt PgVar ——— Normal Page PgEnds: T E X [644], (10) TABLE 9.1 Advancing Contact Angles Liquid Surface β (deg) Water Copper 86 Brass 84 Benzene Copper 25 Brass 23 Ethanol Copper 14–19 Brass 14–18 Methanol Copper 25 Brass 22 n-Propanol Copper 13 Brass 8 Source: Shakir and Thome (1986). 9.3.3 Size Range of Active Nucleation Sites Above, a single uniform temperature was assumed for the wall and liquid. A more practical case is when there is a temperature gradient in the form of a thermal bound- ary layer in the liquid adjacent to the wall, such as illustrated in Fig. 9.5 for a conical nucleation site, where a vapor nucleus of radius r nuc sits at the cavity mouth. The bulk liquid temperature is T ∞ , the wall temperature is T w (where T w ≥ T ∞ ) and a linear temperature gradient is assumed in the thermal boundary layer of thickness δ.Ifα nc is the natural convection heat transfer coefficient and λ L is the thermal conductivity of the liquid, the boundary layer thickness is approximately δ = λ L α nc (9.15) Hsu (1962) postulated that a nucleus sitting in such a temperature gradient activates if the superheat at the top of the vapor nucleus is greater than that required for its equilibrium [i.e., eq. (9.10)], including the distortion of the temperature profile by the bubble nucleus itself. Nucleation occurs if the local liquid temperature profile intersects the equilibrium nucleation curve. The first site to activate is at the tangency between the nucleation superheat curve and the liquid temperature profile line. Hsu assumed that distortion put the location of this temperature at a distance 2r nuc from the surface, while Han and Griffith (1965) put the distance at 1.5r nuc based on potential flow theory. If the liquid pool is at the saturation temperature (i.e., if T ∞ = T sat and 1.5r nuc is assumed for displacement of the isotherms), the cavity size satisfying the condition of tangency is r nuc = δ/2, which is approximately 50 µm for water at its normal boiling point. Much larger superheats or heat fluxes are typically necessary, however, to initiate boiling on a heated surface. This discrepancy results from the fact that a range of cav- ity sizes exists on a real surface, and large cavities that respect the criterion above may BOOKCOMP, Inc. — John Wiley & Sons / Page 645 / 2nd Proofs / Heat Transfer Handbook / Bejan BOILING NUCLEATION 645 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [645], (11) Lines: 370 to 374 ——— 4.097pt PgVar ——— Normal Page PgEnds: T E X [645], (11) Figure 9.5 Nucleation in a temperature gradient. not be available. In this case, T w must be increased until the liquid temperature pro- file intersects the equilibrium bubble curve corresponding to a smaller cavity. Thus, there is a maximum nucleation radius r max and a minimum nucleation radius r min that will activate among those available, where r max is the largest cavity size available that meets the nucleation criterion, and similarly, r min is the smallest cavity size that meets the criterion nearer the surface. In reality, no such cavity may be available, or only one at the tangency point, or finally, numerous cavities in the size range from r min to r max , and hence this introduces the concept of a size range of active nucleation sites. Actual prediction of the nucleation superheat for a surface is complicated by our lack of knowledge of the size and shape of the cavities actually available. Consequently, it is more typical to observe the superheat at which nucleation occurs experimentally and then to back-calculate the effective nucleation radius using eq. (9.10). 9.3.4 Nucleation Site Density As the heat flux at the surface is increased, more and more nucleation sites activate and the question arises as to how many sites are active per unit area. The nucleation site density can be determined using several approaches. First, active boiling sites can be counted with heat flux or wall superheat as the independent variable, typically using photographs or videos of the boiling process. Second, the nucleation site den- sity can be inferred from the measured heat flux by postulating some heat transfer mechanisms. There is rarely agreement between these two approaches. Third, the BOOKCOMP, Inc. — John Wiley & Sons / Page 646 / 2nd Proofs / Heat Transfer Handbook / Bejan 646 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [646], (12) Lines: 374 to 393 ——— -2.0pt PgVar ——— Short Page PgEnds: T E X [646], (12) size and distribution of cavities on the surface can be determined using a scanning electron microscope. However, this is tedious and the resulting distribution of cavi- ties only describes the actual surface investigated. As a consequence, prediction of the nucleation site density is only an approximate estimate. 9.4 BUBBLE DYNAMICS The growth of vapor bubbles can be a particularly complex physical phenomenon. For example, bubble growth and departure are influenced by the orientation of the surface, the thickness and temperature profile in the thermal boundary layer, the proximity of neighboring bubbles, transient thermal diffusion within the wall to the adjacent liquid, wake effects of the previous bubble, bubble shape during growth, and so on. Bubble dynamics play a key role in the development of any analytical model purporting to predict nucleate pool boiling heat transfer coefficients. The simplest case to analyze is that of a single spherical bubble growing within an infinite, uniformly superheated liquid remote from a wall. This is presented below. A comprehensive treatment of bubble growth theory can be found in van Stralen and Cole (1979). 9.4.1 Bubble Growth A spherical bubble growing in a uniformly superheated liquid is the simplest geom- etry to analyze. The pressure and temperature inside the bubble are p G and T G ; the bubble radius is R and is a function of time t from the initiation of growth (growth rate is dR/dt). The pressure and temperature in the liquid are p ∞ and T ∞ . Other effects influencing the bubble are ignored, such as the static head of the liquid, and the center point of the bubble is assumed to be immobile. At inception, the superheat is sufficient for nucleation. Then, as the bubble grows, the pressure inside the bubble decreases and with it, T sat at the bubble interface. Enthalpy stored in the superheated liquid adja- cent to the interface is converted into latent heat at the bubble interface and hence the interfacial temperature falls, creating a thermal diffusion shell around the bubble. Mo- mentum is imparted to the surrounding liquid as the bubble grows and heat diffuses from the superheated bulk to the interface at a rate equal to the rate at which latent heat is liberated at the interface. In addition, the equilibrium vapor pressure curve is assumed to describe this dynamic process, and it is assumed that the vapor pressure in the bubble corresponds to the saturation pressure at the vapor temperature [i.e., that p G = p sat (T G )]. Bubble growth under these conditions is controlled by two factors: 1. Inertia. The initial growth of a bubble is very fast, limited only by the momen- tum available to displace the surrounding liquid from its path. That is, inertia must be imparted to the liquid to accelerate it away in front of the growing bubble. 2. Heat diffusion. As the bubble grows in size, the effect of inertia becomes negli- gible, and growth continues by virtue of diffusion of heat from the superheated BOOKCOMP, Inc. — John Wiley & Sons / Page 647 / 2nd Proofs / Heat Transfer Handbook / Bejan BUBBLE DYNAMICS 647 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [647], (13) Lines: 393 to 443 ——— 0.32526pt PgVar ——— Short Page * PgEnds: Eject [647], (13) liquid to the interface, although at a much slower growth rate than during the inertia-controlled stage of growth. For inertia-controlled bubble growth during the initial stage of bubble growth, Rayleigh (1917) modeled incompressible radially symmetric flow of the liquid sur- rounding a bubble for a spherical bubble with a differential element of radius r and thickness dr for a bubble of radius R. The Rayleigh equation is R d 2 R dt 2 + 3 2 dR dt 2 = 1 ρ L p G − p ∞ − 2σ R (9.16) where the vapor pressure in the bubble is p G and that at the interface is p L . Since 2σ/R p G − p ∞ , the term 2σ/R can be ignored. Utilizing a linearized version of the Clapeyron equation for the small pressure differences involved, the Rayleigh equation reduces to R d 2 R dt 2 + 3 2 dR dt 2 = ρ G ρ L T ∞ − T sat (p ∞ ) T sat (p ∞ ) h LG (9.17) Finally, integrating from the initial condition of R = 0att = 0, the Rayleigh bubble growth equation for inertia-controlled growth is obtained: R(t) = 2 3 T ∞ − T sat (p ∞ ) T sat (p ∞ ) h LG ρ G ρ L 1/2 t (9.18) For heat diffusion–controlled growth, Plesset and Zwick (1954) derived the following bubble growth equation for relatively large superheats: R(t) = Ja 12a L t π (9.19) where a L is the thermal diffusivity of the liquid: a L = λ L ρ L c pL (9.20) and the Jakob number is Ja = ρ L c pL (T ∞ − T sat ) ρ G h LG (9.21) Thus, for heat diffusion–controlled bubble growth, the radius R increases with time as t 1/2 , while it grows linearly with time during the initial inertia-controlled stage of growth. Mikic et al. (1970) combined the Rayleigh and Plesset–Zwick equations to arrive at an asymptotic bubble growth equation valid for the entire bubble growth period: BOOKCOMP, Inc. — John Wiley & Sons / Page 648 / 2nd Proofs / Heat Transfer Handbook / Bejan 648 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [648], (14) Lines: 443 to 476 ——— 0.33908pt PgVar ——— Normal Page PgEnds: T E X [648], (14) R + = 2 3 (t + + 1) 3/2 − (t + ) 3/2 − 1 (9.22) where R + = RA B 2 (9.23) t + = tA 2 B 2 (9.24) A = 2 [ T ∞ − T sat (p ∞ ) ] h LG ρ G 3ρ L T sat (p ∞ ) 1/2 (9.25) B = 12a L π 1/2 Ja (9.26) This expression reduces to eqs. (9.18) and (9.19) at the two extremes of t + , respec- tively. Bubble growth at heated walls differs significantly from these ideal conditions since growth occurs in a thermal boundary layer that may bethicker or thinnerthanthe bubble itself. The velocity field in the liquid created by the growing bubble is affected by the wall and with it the inertia force imposed on the liquid, which may change the bubble shape from spherical to hemispherical or to some other more complex shape. The hydrodynamic wake of a departing bubble may disturb the velocity field of the next bubble or that of adjacent bubbles. Furthermore, rapidly growing bubbles trap a thin evaporating liquid microlayer on the heated surface. For example, Fig. 9.6 illustrates microlayer evaporation underneath a growing bubble and macrolayer evaporation from the thermal boundary layer to the bubble as proposed in a model by van Stralen (1966). Figure 9.6 Bubble growth model of van Stralen (1966). BOOKCOMP, Inc. — John Wiley & Sons / Page 649 / 2nd Proofs / Heat Transfer Handbook / Bejan BUBBLE DYNAMICS 649 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [649], (15) Lines: 476 to 507 ——— 0.64607pt PgVar ——— Normal Page * PgEnds: Eject [649], (15) 9.4.2 Bubble Departure Bubble departure is another fundamental process of importance in nucleate boiling. The diameter at which a bubble departs from the surface during its growth is con- trolled by buoyancy and inertia forces (each attempting to detach the bubble from the surface) and surface tension and hydrodynamic drag forces (both resisting its departure). Furthermore, the shape of the bubble may deviate significantly from the idealized spherical shape. While slowly growing bubbles tend to remain spherical, rapidly growing bubbles tend to be hemispherical. Numerous other shapes are ob- served using high-speed movie cameras or videos. The simplest case to analyze is that of a large, slowly growing bubble on a flat surface facing upward, for which the hydrodynamic and inertia forces are negligible. Departure occurs when the buoyancy force trying to lift the bubble off overcomes the surface tension force trying to hold it on. The surface tension force also depends on the contact angle β (i.e., a contact angle approaching 90° increases the surface tension force and hence the bubble departure diameter). Fritz (1935) proposed the first bubble departure equation that equated these two forces. The Fritz equation, utilizing the contact angle β (i.e., β = π/2 = 90° for a right angle) and surface tension σ,gives the bubble departure diameter as d oF = 0.0208β σ g(ρ L − ρ G ) 1/2 (9.27) The contact angle relative to the surface (through the liquid) is input in degrees. The Fritz equation has been extended empirically to pressures ranging from 0.1 to 19.8 MPa with the following correction to d o F : d o = 0.0012 ρ L − ρ G ρ G 0.9 d oF (9.28) More complex bubble departure models include the inertia, buoyancy, drag, and surface tension forces. The surface tension may also act to pinch off the bubble as it begins to depart from the surface. As an illustration of one of these theories, that of Keshock and Siegel (1964) is presented here. Assuming a spherical bubble, their force balance of the static and dynamic forces acting on a departing bubble is F b + F p = F i + F σ + F D (9.29) where F b and F p are the buoyancy and the excess pressure forces, respectively, acting to lift the bubble off the surface and F i ,F σ , and F D are the inertia, surface tension, and liquid drag forces resisting bubble departure. The buoyancy force is F b = πd 3 o 6 (ρ L − ρ G )g (9.30) The excess pressure on the dry area where the bubble is attached to the wall is BOOKCOMP, Inc. — John Wiley & Sons / Page 650 / 2nd Proofs / Heat Transfer Handbook / Bejan 650 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [650], (16) Lines: 507 to 559 ——— 1.60626pt PgVar ——— Normal Page * PgEnds: Eject [650], (16) ∆p = 2σ sin β d b + σ r b 2σ sin β d b (9.31) The first term comes from the Laplace equation (9.1) applied to the bubble base diameter, including the contact angle β, while the second term accounts for the effect of curvature at the base of the bubble. Keshock and Siegel assumed that the second term is negligible compared to the first, such that the excess pressure force acting on the base area of diameter d b is F p = πd b 2 σ sin β (9.32) The inertia force imparted on the surrounding liquid by the growing bubble is F i = d dt µ (9.33) where m is the mass of the liquid displaced by the bubble and u is the interfacial velocity. Assuming that the liquid affected is 11/16 of the bubble volume, the inertia force is F i = d dt 11 16 ρ L 4π [ R(t) ] 3 3 dR(t) dt d=d o (9.34) The interfacial velocity dR(t)/dt may be determined with the Plesset and Zwick bubble growth model presented earlier. The surface tension force acting on the dry perimeter at the base of the bubble of diameter d b is F σ = πd b σ sin β (9.35) Assuming a spherical bubble rising freely in a liquid at a velocity equal to that of bubble growth at the moment of departure, the hydrodynamic drag force resisting bubble departure is F D = 1 2 ρ L C D π [ R(t) ] 2 dR(t) dt 2 (9.36a) or F D = π 4 C D Re bub µ L [R(t)] dR(t) dt (9.36b) where the drag coefficient is C D and the bubble Reynolds number is Re bub = ρ L [ 2R(t) ][ dR(t)/dt ] µ L (9.37) BOOKCOMP, Inc. — John Wiley & Sons / Page 651 / 2nd Proofs / Heat Transfer Handbook / Bejan POOL BOILING HEAT TRANSFER 651 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [651], (17) Lines: 559 to 599 ——— 4.75613pt PgVar ——— Normal Page PgEnds: T E X [651], (17) The drag coefficient is related to the bubble Reynolds number as C D = a Re bub (9.38) (where Keshock and Siegel use a = 45), so the drag force is F D = π 4 aµ L [ R(t) ] dR(t) dt (9.39) In the model above, the drag force is typically negligible, while the inertia force becomes significant only at large ∆T . The excess pressure and surface tension forces are difficult to calculate for a real case since d b is unknown (d b would typically be larger than the cavity mouth since the liquid film trapped under a growing bubble partially dries out). 9.4.3 Bubble Departure Frequency The bubble departure frequency is f = 1 t g + t w (9.40) where t g is the bubble growth time and t w is the waiting time between the departure of one bubble and the initiation of growth of the next. Bubble departure frequencies range from as low as 1 Hz at very small superheats to over 100 Hz at high superheats. The bubble growth time t g can be obtained by calculating the bubble departure diam- eter [e.g., using eq. (9.28)] and solving for time t in the Plesset–Zwick bubble growth equation presented earlier. After a bubble departs, the length of pause before the next bubble begins to grow depends on the rate at which the surface and adjacent liquid are reheated by transient heat conduction from the wall. Once the bubble departure diameter and frequency are known, the volumetric vapor flow rate from a single boiling site may be determined. Combining this with the bubble nucleation site density, the volumetric vapor flow rate per unit area from the heated surface can be estimated, and hence latent heat transport from the surface can also be calculated. Thus, the latent heat flux can be determined, and subtracting this from the total heat flux measured, the sensible heat flux leaving the surface in the form of superheated liquid is obtained. The effects of sequential bubbles and neighboring bubbles on one another and their competition for the heat stored within the thermal boundary layer, however, make an accurate prediction of the vapor flow rate from the surface very difficult to obtain. 9.5 POOL BOILING HEAT TRANSFER Methods for predicting heat transfer coefficients in the various pool boiling regimes are described in this section. Most of the attention is placed on nucleate boiling, but BOOKCOMP, Inc. — John Wiley & Sons / Page 652 / 2nd Proofs / Heat Transfer Handbook / Bejan 652 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [652], (18) Lines: 599 to 616 ——— 0.897pt PgVar ——— Normal Page PgEnds: T E X [652], (18) heat transfer in the film boiling and transition boiling regimes are also addressed. Methods are also provided to predict departure from nucleate boiling (referred to as DNB or critical heat flux) and the minimum heat flux of film boiling. 9.5.1 Nucleate Boiling Heat Transfer Mechanisms Before presenting a selection of nucleate pool boiling correlations, the heat transfer mechanisms playing a role in nucleate pool boiling, illustrated in Fig. 9.7, are identi- fied as follows: 1. Bubble agitation. The systematic pumping motion of thegrowing and departing bubbles agitates the liquid, pushing it back and forth across the heater surface, which in effect transforms the otherwise natural convection process into a localized forced convection process. Sensible heat is transported away in the form of superheated liquid and depends on the intensity of the boiling process. ()a ()b ()c Bubble Liquid motion Hot liquid Superheated liquid Heated surface Microlayer Evaporation Figure 9.7 Heat transfer mechanisms in nucleate pool boiling: (a) bubble agitation; (b) vapor–liquid exchange; (c) evaporation. BOOKCOMP, Inc. — John Wiley & Sons / Page 653 / 2nd Proofs / Heat Transfer Handbook / Bejan POOL BOILING HEAT TRANSFER 653 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [653], (19) Lines: 616 to 637 ——— 2.0pt PgVar ——— Normal Page * PgEnds: Eject [653], (19) 2. Vapor–liquid exchange. The wakes of departing bubbles remove the thermal boundary layer from the heated surface, and this creates a cyclic thermal boundary layer stripping process. Sensible heat is transported away in the form of superheated liquid, whose rate of removal is proportional to the thickness of the layer, its mean temperature, the area of the boundary layer removed by a departing bubble, the bubble departure frequency, and the density of active boiling sites. 3. Evaporation. Heat is conducted into the thermal boundary layer and then to the bubble interface, where it is converted to latent heat. Macroevaporation occurs over the top of the bubble while microevaporation occurs underneath the bubble across the thin liquid layer trapped between the bubble and the surface, the latter often referred to as microlayer evaporation. The rate of latent heat transport depends on the volumetric flow of vapor away from the surface per unit area. The mechanisms above compete for the same heat in the liquid and hence overlap with one another, thermally speaking. At low heat fluxes characteristic of the isolated bubble region, natural convection also occurs on inactive areas of the surface, where no bubbles are growing. 9.5.2 Nucleate Pool Boiling Correlations The complexity of the nucleate pool boiling process is such that accurate, reliable analytically based design theories are yet to be available. Unresolved problems are related primarily to predicting such things as boiling nucleation superheats, boiling site densities for a given surface, and thermal interaction between neighboring boil- ing sites. As a consequence, completely empirical methods are used for predicting nucleate pool boiling heat transfer coefficients. In a pool boiling experiment, the wall superheat ∆T is measured versus the heat flux q, and the nucleate boiling heat trans- fer coefficient is obtained from its definition (α nb ≡ q/∆T ). These data may be fit with expressions such as q ∝ ∆T n , α nb ∝ ∆T n ,orα nb ∝ q n , where the exponent n is on the order of 3, 2, or 0.7, respectively. Pool boiling correlations are typically formulated in similar fashion where expressions in the form α nb ∝ q n are the easiest to apply since heat flux is an imposed design variable while the wall temperature in ∆T is unknown and part of the solution. Literally hundreds of pool boiling correla- tions have been proposed; below a representative selection of recommended methods is presented plus the classic Rohsenow method. Bubble Agitation Correlation of Rohsenow Rohsenow (1962) proposed the first widely quoted correlation. He assumed the boiling process is dominated by the bubble agitation mechanism depicted in Fig. 9.7, whose bubble-induced forced- convective heat transfer process could be correlated with the standard single-phase forced-convection correlation relation: Nu = C 1 · Re x · Pr y (9.41) where the Nusselt number for boiling is defined as . postulating some heat transfer mechanisms. There is rarely agreement between these two approaches. Third, the BOOKCOMP, Inc. — John Wiley & Sons / Page 646 / 2nd Proofs / Heat Transfer Handbook /. growth continues by virtue of diffusion of heat from the superheated BOOKCOMP, Inc. — John Wiley & Sons / Page 647 / 2nd Proofs / Heat Transfer Handbook / Bejan BUBBLE DYNAMICS 647 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [647],. a model by van Stralen (1 966) . Figure 9.6 Bubble growth model of van Stralen (1 966) . BOOKCOMP, Inc. — John Wiley & Sons / Page 649 / 2nd Proofs / Heat Transfer Handbook / Bejan BUBBLE DYNAMICS 649 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [649],