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8 BOILING 8.1 Introduction When a solid surface is immersed in liquid and the solid surface temperature, Tw, exceeds the saturation temperature, Tsat ( p ) , for the liquid at that pressure, vapor bubbles can form, grow, and detach from the solid surface. This phase change process is called boiling, and the energy transport involved is classified as convective heat transfer with phase change. Two types of boiling are commonly distinguished: pool boiling and flow or forced convective boiling. In the former case, the bulk liquid is quiescent while the liquid near the heating surface moves due to free convection and the mixing induced by bubble growth and detachment. In the latter case, bulk liquid motion driven by some external means is superimposed on the motion that also occurs in pool boiling. Vapor bubbles within the liquid phase are the primary visual characteristics that distinguish boiling from evaporation. The influence of bubbles is also the primary cause of differences in the thermodynamic and hydrodynamic analyses of these two phase change modes. Fig. 8.1 compares, in schematic fashion, the temperature field of a quiescent volume of fluid experiencing evaporation with that of one experiencing boiling. As the figure indicates, boiling requires a much larger temperature Figure 8.1 Schematic comparison of evaporation and nucleate boiling temperature profiles. Chapter 8 Boiling 665 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press (a) T < Tsat (b) T = Tsat vapor, Figure 8.2 Effects of liquid temperature: (a) subcooled boiling, (b) saturated boiling ( liquid). difference between the bulk liquid and the heating surface than evaporation does. A second distinction that arises from the figure is that the thermal boundary layer is less precise in the case of boiling; this may be attributed to the mixing action caused by the release of bubbles from the heating surface. In addition to the pool and forced convection classifications, boiling can also be categorized according to the initial temperature of the liquid. Fig. 8.2 shows pool boiling for different initial liquid temperatures. Subcooled boiling occurs if the liquid temperature starts below the saturation temperature. In this case, bubbles formed at the heater surface experience condensation as they rise through the cooler bulk liquid and may collapse before reaching the free surface. Saturated boiling, on the other hand, occurs if the temperature of the liquid equals the saturation temperature. In this case, bubbles form at the heating surface, travel intact through the liquid, and escape at the free surface. Section 8.2 introduces the pool boiling curve and the various regimes (free convection, nucleate boiling, transition boiling, film boiling) of which it is composed. After this introduction, the four regimes and the characteristic points of the curve are then discussed in detail. Section 8.3 discusses nucleate boiling, including nucleation, bubble dynamics and detachment, nucleation site density, numerical simulation of bubble growth and merger, and heat transfer analysis. The critical heat flux is presented in Section 8.4, followed by discussions on transition boiling and minimum heat flux in Section 8.5. Section 8.6 discusses film boiling, including film boiling laminar boundary layer analysis and correlations, direct numerical simulations, and Leidenfrost phenomena. 8.2 Pool Boiling Regimes The classical pool boiling curve is a plot of heat flux, q′′, versus excess temperature, ΔT = Tw – Tsat. As the value of the excess temperature increases, the curve traverses four different regimes: (1) natural or free convection, (2) nucleate boiling, (3) transition boiling, and (4) film boiling. Different experimental 666 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press methods may be used to define the pool boiling curve; constant temperature control and constant heat flux control are the two most commonly cited. A typical boiling curve for saturated pool boiling of water at atmospheric pressure for a temperature-controlled environment is shown in Fig. 8.3. When the excess temperature ΔT is less than 5 °C, no bubbles form. Instead, heat is transferred from the solid surface to the bulk liquid via natural convection. Heat transfer coefficients in this regime can be calculated using the semi-empirical correlations for natural convection. When the excess temperature increases beyond 5 °C, the system enters the nucleate boiling regime – point A on Fig. 8.3. Vapor bubbles are generated at certain preferred locations on the heater surface called nucleation sites; these are often microscopic cavities or cracks on the solid surface. Nucleation occurs repeatedly from the same sites, indicating a causal link between bubble formation and some surface feature as well as the cyclical nature of the bubble-forming process. Small cavities and surface cracks act as sites for bubble generation because: (1) the contact area between the liquid and heating surface increases relative to a perfectly-smooth surface, so liquid trapped in these areas vaporizes first; and (2) the presence of trapped gases in such cracks creates liquid-vapor interfaces, which serve as sites where transfer of energy in the form of latent heat from the liquid to the vapor phase takes place. Once a vapor bubble has been initiated at a nucleation site, under the right conditions the bubble grows to a certain required diameter, detaches from the heating surface, and rises to the liquid free surface. If the excess temperature remains at the low end of the nucleate boiling regime, shown between points A and B of Fig. 8.3, each bubble generated can grow and detach from the surface independently – that is, without interaction between bubbles. As the bubble-generating process occurs at the Figure 8.3 Pool boiling curve for saturated water. Chapter 8 Boiling 667 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press active nucleation sites, the surface area between these sites retains the liquidsolid contact that characterizes the natural convection regime. Convection remains the primary mechanism of heat transfer in this so-called “isolated bubble” regime. As we shall see, however, the character of this convection is markedly different from that of the natural convection encountered at lower excess temperatures (ΔT < 5 °C for water). As the excess temperature increases beyond point B in Fig. 8.3, additional nucleation sites become active and more bubbles are generated. The higher density of bubbles leads to their interaction with each other. Bubbles from separate sites now merge to form columns and slugs of vapor, thus decreasing the overall contact area between the heating surface and the saturated liquid. Consequently, the slope of the boiling curve begins to decrease and the heat flux ′′ eventually reaches a maximum value, qmax , referred to as the critical heat flux. The critical heat flux, which marks the upper limit of the nucleate boiling regime, reaches a value of approximately 106 W/m2 for water at an excess temperature of about ΔTc= 30 oC. The nucleate boiling regime is most desirable for many industrial applications because of its high heat flux at relatively low levels of excess temperature (5 oC ≤ΔT≤30 oC for water). However, certain circumstances are required to avoid nucleate boiling, such as wicked heat pipes (Faghri, 1995). As the temperature increases beyond the critical heat flux point, the rate of bubble generation exceeds the rate of bubble detachment from the heater surface. Bubbles from an increasing number of sites merge to form continuous vapor films over portions of the surface, further decreasing the contact area between the heating surface and the saturated liquid. These vapor films are not stable, however, they can detach from the surface, leading to restoration of contact with the liquid and resumption of nucleate boiling. Under these unstable conditions, the surface temperature may fluctuate rapidly, so the excess temperature shown on the ΔT-axis of Fig 8.3 between points C and D should be regarded as an average value. Since the boiling in this regime combines unstable film with partial-nucleate boiling types, it is referred to as the region of transition boiling. When the excess temperature becomes high enough to sustain a stable vapor film, ′′ the heat flux reaches its minimum value, qmin . This point, known as the Leidenfrost temperature, marks the upper temperature limit of the transition boiling regime. At temperatures above the Leidenfrost temperature, the bulk liquid and the heating surface are completely separated by a stable vapor film, so boiling in this regime is known as film boiling. The phase change in film boiling occurs at a liquid-vapor interface, instead of directly on the surface, as in the case of nucleate boiling. Thermal energy from the heating surface reaches the liquidvapor interface by convection in the vapor film as well as by direct radiation to the interface. In the film boiling regime, the surface heat flux becomes a monotonically increasing function of the excess temperature, because radiation heat transfer from the solid surface to the liquid plays a significant role at high surface temperature. Pool boiling continues in this regime until the surface 668 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press temperature reaches the maximum allowable temperature of the heating surface (2042 K for platinum, for example). Beyond that point, the heating surface can melt in a potentially catastrophic failure. If protective insulation is provided, however, as in the case of refractory metals, for example, it is possible for film ′′ boiling heat flux to exceed the critical heat flux, qmax . The boiling curve presented in Fig. 8.3 and described above assumes that the surface temperature is independently controlled and that the heat flux is the dependent variable. However, direct control over the surface temperature is not always possible. For example, when electric heating provides thermal energy to the solid, the controllable parameter is heat flux. Surface temperature then becomes the dependent variable, and heat flux becomes the independent variable. If the experiment of gradually increasing the added thermal energy is repeated using constant heat flux instead of constant temperature control, the resulting boiling curve matches that of controlled-temperature up to the critical heat flux, ′′ qmax . When the surface heat flux is increased slightly above the critical heat flux, however, portion C-D-E of the boiling curve is bypassed and the surface temperature increases abruptly to point E of the stable film boiling regime (Nukiyama, 1934). This abrupt increase in surface temperature is undesirable because TE usually exceeds the melting point of the solid material. The critical heat flux can thus serve as the warning point above which burnout can occur; consequently, the critical point is sometimes referred to as the burnout point. If the pool boiling curve is defined by decreasing the controlled heat flux from an initial point in the film boiling regime (point E, for example), a characteristic identical to Fig. 8.3 appears down to the point of minimum heat flux. After that point, a continued decrease of q′′ yields a second hysteresis path that leads immediately to the nucleate boiling regime between points A and B. In this case, the transition boiling regime and a portion of the nucleate boiling regime are bypassed. Therefore, the transition regime that is observed when temperature control is used to define the pool boiling curve is unavailable in the controlledheat-flux case. 8.3 Nucleate Boiling As was indicated in the previous section, nucleate boiling is the most desirable regime in most industrial applications (with few exceptions) because relatively high rates of heat transfer can be achieved with relatively low temperature differentials. Because vapor bubbles exert a major influence on the hydrodynamic and heat transfer characteristics of the nucleate boiling regime, a thorough understanding of how bubbles form and how they influence heat transfer is essential. Three stages of bubble production, each of which will be discussed in the following subsections, are defined as follows: (1) initiation or nucleation, (2) growth, and (3) detachment. The section then closes with a Chapter 8 Boiling 669 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press summary of modeling approaches that have been developed to describe heat transfer in nucleate boiling. 8.3.1 Nucleation and Inception Nucleation, or bubble initiation, in typical industrial applications is characterized by the cyclic formation of vapor bubbles at preferred sites on the solid heating surface of the system. Any surface, regardless of how highly polished, contains surface irregularities – micro-cracks, cavities, or boundaries between solid crystals – that can trap small gas pockets. These serve as nucleation sites for vapor bubbles. The trapped gas and/or vapor, known as the bubble embryo, grows by acquiring mass via evaporation from the nearby liquid. It grows until it reaches a critical size, at which point the forces tending to cause separation of bubbles from the heating surface overcome the adhesive forces. When the force imbalance favors the separating forces, the bubble is released from the surface. This process, by which solid surface imperfections with entrapped gases promote the formation of bubble embryos, is known as heterogeneous nucleation. Pure liquids that have been thoroughly degassed may still experience bubble formation at molecular vapor clusters within the bulk liquid, (i.e., away from any solid surface) by a process known as homogeneous nucleation. However, the degree of superheat required for homogeneous nucleation is substantially larger than that for heterogeneous nucleation. The common observation of bubble formation at the low superheat indicated by point A of Fig. 8.3, coupled with the repetitive formation of bubbles at selected points on heater surfaces, confirms that heterogeneous nucleation is by far the more common process. Since the trapped gases and/or vapors are central to the nucleation event, a consideration of the thermodynamics of vapor bubbles immersed in a liquid is appropriate. We will find that it justifies the expectation that the presence of gas or vapor pockets enhances bubble formation. We begin by considering a vapor bubble, including the possibility of noncondensable gas, immersed in its own liquid. The Laplace-Young equation must hold in order for the vapor/gas and liquid to be in equilibrium: 2σ pg + pv − p = (8.1) Rb where Rb is the radius of the vapor bubble. Equation (8.1) indicates that the pressure of the vapor and gas must be greater than the pressure of the liquid to maintain a balance of forces in the vapor-interface-liquid system. Although these events occur on a molecular scale, we may attempt to visualize the physical implications of this imbalance of pressures across the boundary separating vapor and liquid phases. The higher vapor pressure in the bubble causes an increased number of molecules to strike the interface where they are absorbed by the liquid phase. To maintain the mass balance that equilibrium requires, there must be a corresponding increase in the number of molecules emitted through the interface 670 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press from the liquid. Under the given constraints, this can be accomplished only by increasing the temperature of the liquid side of the interface. Consequently, the liquid near the bubble’s surface must be superheated to a temperature above the saturation temperature that corresponds to the prevailing bulk liquid pressure. To put these qualitative observations on a quantitative basis, consider that the vapor pressure for a curved bubble interface can be expressed as (Faghri and Zhang, 2006)   −2σρ v pv = pv , sat (T ) exp  (8.2)   pv , sat (T ) Rb ρ     where pv, sat (T ) is the normal saturation pressure corresponding to temperature, T. Since 2σρ v / [ pv, sat (T ) Rb ρ ]  1 , eq. (8.2) can be approximated as   2σρ v pv  pv, sat (T ) 1 −   pv, sat (T ) Rb ρ    (8.3) Combining eqs. (8.1) and (8.3) yields 2σ  pg + pv, sat (T ) − p  ρv  1 +  Rb  ρ   (8.4) Recalling the fundamental relationship between pressure and temperature differences in the two-phase region, the Clapeyron equation, hv dp = dT T (1/ ρ v − 1/ ρ ) (8.5) (8.6) (8.7) and assuming that the fluid obeys the ideal gas law, i.e., p = ρ RgT and that ρ  ρ v , we arrive at h dp = v 2 dT p RgT For a pressure change from p to pv, sat (T ), the corresponding saturation temperature changes from Tsat to T . So by integrating eq. (8.7) between these limits, we obtain ln pv , sat (T ) p = hv T − Tsat RgTsat T (8.8) Substituting eq. (8.4) into eq. (8.8) gives RgTsat T  2σ  ρ v  pg  ΔT = T − Tsat = ln 1 + 1 + −  hv  p Rb  ρ  p    (8.9) Considering that 2σ / ( p Rb ) − pg / p  1 and again that ρ  ρ v , eq. (8.9) can be further simplified as RgTsat T  2σ  ΔT = T − Tsat = − pg  (8.10)  p hv  Rb  Chapter 8 Boiling 671 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press For the case with no noncondensable gas ( pg = 0 ), eq. (8.10) can be rearranged to yield the critical vapor bubble radius: 2σ Tsat (8.11) Rb = hv ρ v ΔT This critical radius is necessary for a bubble to exist at ΔT above the saturation temperature that corresponds to the prevailing vapor pressure. Any bubble with a radius less than the size given by eq. (8.11) will collapse, but a bubble with a radius equal to Rb will grow in a spontaneous fashion. It should be noted from eq. (8.10) that the size of the equilibrium vapor nucleus becomes smaller as the superheat (ΔT) increases. This identifies the basic mechanism for the increase of q ′′ with rising excess temperature between points A and C of Fig. 8.3, namely, the increased bubble density that results from the activation of more and more nucleation sites at ever-smaller surface imperfections. Equation (8.10) shows that the presence of noncondensable gas reduces the superheat required to generate bubbles. The bubble growth is spontaneous if the excess temperature is greater than the value given in eq. (8.10). Any real surface always has some small cavities with very small inclined angles where liquid can only partially fill the cavities, so opportunities for the formation of gas or vapor pockets are common. Equation (8.10) gives the superheat required to initiate nucleate boiling from a pre-existing nucleus with a radius of Rb. Griffith and Wallis (1960) suggested that it is also applicable for nucleate boiling from a surface with a micro cavity whose mouth radius is Rb. Alternative studies by Mizukami (1975), Nishio (1985), and Wang and Dhir (1993a) suggested that the Figure 8.4 Dimensionless modified curvature versus dimensionless volume of vapor bubble nucleus in a spherical cavity (Wang and Dhir, 1993a). 672 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press superheat required to initiate nucleate boiling is determined by the instability of vapor nuclei in the cavity. If the curvature of the liquid-vapor interface, K, increases with increasing vapor volume ( dK / dV * > 0 , where V* is dimensionless vapor volume), the vapor nucleus is stable. On the other hand, the vapor bubble embryo or nucleus is unstable if K decreases with increasing vapor volume. For the nucleate boiling to initiate from the wall, the wall superheat must be sufficient to ensure the vapor nuclei are unstable. Figure 8.4 shows the dependence of the nondimensional modified curvature of the interface on the dimensionless volume of the vapor nucleus in a spherical cavity with a mouth angle of 30°. The noncondensable gas is not present and the results at different contact angles are shown. The dimensionless volume of the vapor nucleus is defined as V * = Vb / Vc where Vb is the volume of the nucleus and Vc is the volume of the nucleus at point A, where the curvature is zero. It can be seen from Fig. 8.4 that the nucleus is stable between points A and D, since dK / dV * > 0 . When the contact angle is less than 90°, the maximum curvature, Kmax, is 1. For the cases in which θ > 90 , the maximum curvature becomes sin θ . Nucleation occurs beyond the point where the curvature reaches its maximum; the corresponding superheat is (Wang and Dhir, 1993a) 4σ Tsat ΔT = T − Tsat = (8.12) K max pv hv Dc where Dc is the diameter of the cavity mouth. Example 8.1 A steam bubble with a radius of 5 μm is surrounded by liquid water at 120 ˚C. Will this bubble grow or collapse? Solution: The liquid pressure is p = 1 atm = 1.013 × 105 Pa. The saturation temperature at this pressure is Tsat = 100 o C = 373.15 K. The properties of water at this temperature are σ = 58.9 × 10−3 N/m, hv = 2251.2 kJ/kg, and ρ v = 0.5974 kg/m3 . Therefore, the critical bubble radius can be determined from eq. (8.11), i.e., 2σ Tsat Rb = hv ρ v ΔT = 2 × 58.9 × 10 −3 × 373.15 = 1.63 × 10−6 m = 1.63 μm 2251.2 × 103 × 0.5974 × (120 − 100) Since the radius of the bubble, 5 μm, is greater than the critical bubble radius, the bubble will grow. Figure 8.5 shows an idealized model of a surface crack containing a vapor pocket as it expands under the influence of evaporation. The vapor-liquid interface is idealized as spherical in shape. As the vapor pocket expands upward in the surface crack, its interface takes a concave shape and the radius of Chapter 8 Boiling 673 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press φ Figure 8.5 Variation of bubble radius as the bubble grows within and emerges from an idealized cavity. curvature, Rb, increases as the interface moves toward the lip of the cavity. Once the incipient bubble reaches the lip of the interface, the radius of curvature begins to decrease. It reaches its minimum value (approximately) as the liquid- vapor interface becomes a hemisphere with a radius equal to that of the cavity mouth, Rc. Beyond this point, any further evaporation pushes the liquid-vapor interface out of the surface cavity, and the interface radius of curvature begins to increase again. Consideration of this sequence of radius variation, along with eq. (8.10), indicates that the superheat required to form a vapor bubble from this cavity can be determined by substituting Rb = Rc into eq. (8.10). In normal circumstances, of course, this variation deviates from the idealized model in Fig. 8.5 and is a function of cavity angle φ and contact angle θ. Figure 8.6 Liquid microlayer under a vapor bubble at a nucleation site. 674 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press In reality, the shape of a vapor bubble is not hemispherical, and there is a microlayer (a very thin liquid layer, see Fig. 8.6) that is responsible for more than half of heat transfer in some cases. Plesset and Sadhal (1979) provided the average liquid film thickness δ in terms of bubble lift time t0 and liquid kinematic viscosity by the following equation δ = (3ν t0 )1/ 2 8 7 (8.13) The contribution of the microlayer to the overall heat transfer obviously depends on the fraction of area occupied by the microlayer. Despite its significant importance, there is no detailed treatment of the microlayer related to nucleate boiling. 8.3.2 Bubble Dynamics The second stage of bubble production is the growth stage. In real applications, the conditions under which growth occurs can be complex. The temperature field of the liquid near the heating surface must be nonuniform for heat to be transferred within the liquid. Several types of forces influence bubble growth, including the inertia of the surrounding liquid, shear forces at the interface, surface tension, and the pressure difference between the vapor and the liquid. The energy exchanged includes heat transfer by conduction and convection as well as latent heat of vaporization. The process is by definition transient, the terms in the mass, momentum, and energy equations constantly shifting over time. So we cannot expect that modeling the process of bubble growth will yield simple solutions. Nevertheless, insight can be gained from a process of analysis followed by synthesis. The growth of a bubble can be classified into two groups (both of them will be discussed below): one in which the bubble grows in a quiescent superheated liquid of infinite extent (homogeneous); another in which the bubble is attached and grows on a heated wall (heterogeneous). Finally the dynamics of bubble growth within the superheated liquid droplet will also be discussed in this subsection. Homogeneous Bubble Growth Bubble growth within a superheated liquid drop will be considered first. Once a growing vapor bubble’s radius reaches that unstable equilibrium, it grows spontaneously. During the early stage, when the bubble radius is small, the Laplace-Young equation indicates that the pressure differential across the interface is at its maximum value. The resulting high interfacial velocity leads to significant inertia terms in the momentum equation. Meanwhile, the temperature of the interface is close to the superheat temperature of the surrounding liquid, so heat transfer into the vapor bubble experiences the highest driving temperature Chapter 8 Boiling 675 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press differential that occurs at any time during the process. As a result, early bubble growth tends to be limited by inertia, or the exchange of momentum between the vapor and liquid phases. In the physical model shown in Fig. 8.7, the instantaneous radius and the liquid-vapor interface velocity at time t are R(t) and dR/dt, respectively. For an incompressible, radially symmetric inviscid flow, the continuity equation in the spherical coordinate system is 1 ∂ (r 2 u) =0 r 2 ∂r (8.14) where u is the radial velocity in the liquid phase. Assuming ρ v  ρ , this velocity at the liquid-vapor interface is equal to the growth rate, i.e., u( R, t ) = dR dt (8.15) Integrating eq. (8.14) over the interval of (R, r), where r is an arbitrary location in the liquid phase, one obtains u(r , t ) = dR  R   dt  r  2 (8.16) The momentum equation for the surrounding liquid is ρ  ∂u  ∂p  ∂u +u  = − ∂r  ∂r  ∂t ∂p   =− dt  ∂r  (8.17) Substituting eq. (8.16) into eq. (8.17) yields 2 2  2 ρ R 4  dR  2 ρ   dR  2d R 2R  −  +R r 2   dt  dt 2    r5 (8.18) Integrating eq. (8.18) from the bubble surface (r=R) to infinity ( r → ∞ ) yields R d 2 R 3  dR  1 +  = [ p ( R) − p (∞) ] ρ dt 2 2  dt  2 (8.19) Figure 8.7 Homogeneous nucleation of a vapor bubble in superheated liquid. 676 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Substituting eq. (8.1) with pg = 0 into eq. (8.19) gives R d 2 R 3  dR  1 + = 2 dt 2  dt  ρ 2 2σ    pv − p (∞) − R    (8.20) which is referred to as Rayleigh’s equation. During the inertia-controlled stage of bubble growth, the surface tension term 2σ / R is negligible compared to pv − p (∞). The pressure difference pv − p (∞) can be evaluated by using the Clausius-Clapeyron equation, i.e., ρ h [T − Tsat [ p (∞)]] (8.21) pv − p (∞) = v v ∞ Tsat [ p (∞)] Substituting eq. (8.21) into eq. (8.20) and neglecting the surface tension term 2σ / R in eq. (8.20), one can obtain an ordinary differential equation for the bubble radius: 2 ρ v hv T∞ − Tsat [ p (∞)] d 2 R 3  dR  (8.22) R 2+  = 2  dt  ρ dt Tsat [ p (∞)] which is subject to the following initial condition: R = 0, t = 0 (8.23) The instantaneous bubble radius can be obtained by solving eq. (8.22):  2 ρ v hv T∞ − Tsat [ p (∞)]  2 R(t ) =  t Tsat [ p (∞)]   3 ρ 1 (8.24) which indicates that the radius of the bubble is a linear function of time during the early stage of the bubble’s growth. During the later stage of bubble growth, the inertial force becomes insignificant as the pressure difference pv − p (remember Laplace-Young’s equation) diminishes and the interfacial motion slows. At the same time, the temperature difference between the vapor inside the bubble and the interface diminishes. With a lower level of forcing fraction, heat transfer becomes the process that limits bubble growth. ∂T ∂T α  ∂  2 ∂T  +u  = 2 (8.25) r  ∂t ∂r r ∂r  ∂r  where the radial velocity, u, is obtained by eq. (8.16). The initial and the boundary conditions for eq. (8.25) are T (r , 0) = T (∞), T ( R, t ) = Tsat ( pv ), T (∞, t ) = T (∞) (8.26) The energy balance at the liquid-vapor interface is k ∂T ∂r = ρ v hv r=R dR dt (8.27) (8.28) The instantaneous bubble radius is R(t ) = 2C R α  t where CR is a constant. Chapter 8 Boiling 677 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press For a large Jakob number (corresponding to low pressure), CR can be obtained by (Plesset and Zwick, 1954) π and the Jakob number is defined as ρ  c p [T (∞) − Tsat ( pv ) ] Ja = ρ v hv For a small Jakob number (corresponding to high pressure) CR = Ja 2 CR = 3 Ja (8.29) (8.30) (8.31) Equation (8.28) indicates that the bubble radius is proportional to the square root of time. Bubble growth in the two limiting cases described above has been analyzed. The inertia-controlled bubble growth model ignored heat transfer, which is only valid in the early stage of bubble growth. In the heat transfer-controlled bubble growth model, on the other hand, the effect of inertia on bubble growth was neglected; this assumption is appropriate only during the late stage of bubble growth. A comprehensive bubble growth model must include both inertia and heat transfer and provide a smooth transition between the two regimes. For bubble growth in an initially uniformly superheated liquid, Mikic et al. (1970) obtained the following expression of the relationship between bubble radius and time: R+ = 3 3  2 + +2 2 (t + 1) − (t ) − 1 3  (8.32) where R + and t + are nondimensional radius and time, respectively, defined as R+ = RA B2 t+ = tA 2 B2 1 (8.33) and  b [T (∞) − Tsat ( p (∞)) ] hv ρ v  2   A=  ρ Tsat ( p (∞))      12α   2 B=  Ja π ρ c p [T (∞) − Tsat ( p (∞))] 1 (8.34) (8.35) (8.36) ρ v hv where b in eq. (8.34) is a geometric parameter with a value of 2/3 for a perfect spherical bubble. Equation (8.32) provides a generalized solution for the bubble radius because it can be reduced to eq. (8.24) for small t+ and to eq. (8.28) for larger t+. Prosperetti and Plesset (1978) developed a more advanced model for vapor bubble growth in a superheated liquid. Ja = 678 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Example 8.2 A vapor bubble is initiated and grows in liquid water at 120 ˚C. Find the time at which the bubble sizes predicted by the inertiacontrolled model and the heat transfer-controlled model are the same. What is the bubble size obtained by both models? Solution: The saturation temperature at 1 atm is Tsat = 100 o C = 373.15 K . The properties of water at this temperature are σ = 58.9 × 10−3 N/m, hv = 2251.2 kJ/kg, ρ = 958.77 kg/m3 , ρv = 3 0.5974 kg/m , c p = 4.216 kJ/kg-K, and k = 0.68 W/m-K. The thermal diffusivity is α  = k / ( ρ c p ) = 1.68 × 10−7 m 2 /s . The Jakob number is obtained from eq. (8.30), i.e., ρ  c p [T (∞) − Tsat ( pv ) ] 958.77 × 4.216 × (120 − 100) Ja = = = 60.11 ρ v hv 0.5974 × 2251.2 and CR for the heat transfer-controlled model can be obtained from eq. (8.29): π π The bubble sizes in the inertia-controlled model and heat transfercontrolled model can be obtained from eqs. (8.24) and (8.28), respectively. The time at which the bubble sizes obtained by both models are the same can be found by equalizing eqs. (8.24) and (8.28):  2 ρ v hv T∞ − Tsat [ p (∞)]    t = 2C R α  t Tsat [ p (∞)]   3 ρ 1/ 2 CR = 3 Ja = 3 × 60.11 = 58.7 i.e.,  2 ρ v hv T∞ − Tsat [ p (∞)]  2 t = 4C Rα    Tsat [ p (∞)]   3 ρ  2 0.5974 × 2251.2 × 103 120 − 100  = 4 × 58.7 2 × 1.68 × 10−7  ×  958.77 373.15  3 = 4.62 × 10 −5 s The bubble radius obtained is R = 2C R α  t = 2 × 58.7 × 1.68 × 10−7 × 4.62 ×10−5 = 3.27 × 10−4 m = 0.327 mm Heterogeneous Bubble Growth The bubble growth discussed so far is limited to bubbles in an extensive, uniformly superheated liquid. For a vapor bubble attached to a heating surface, growth occurs in a nonuniform temperature field. The criteria for initiation and growth of a bubble near a wall were analyzed by Han and Griffith (1965). They assumed that a surface cavity, which serves as a nucleation site, has a Chapter 8 Boiling 679 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press hemispherical vapor cap on it. As a result, the bubble can grow only if the thermal layer adjacent to the nucleation site is sufficiently thick. A relation between cavity size and surface temperature elevation was derived on the basis of an assumed relation between the thermal layer thickness and the nucleus size at incipience. For saturation conditions, they found the maximum and minimum cavity sizes that can be active under a constant temperature: ( Rc ) max,min   12σ Tsat = 1 ± 1 −  3   δρ v hv (Tw − Tsat )   δ  1/ 2      (8.37) where δ is the thermal layer thickness. Howell and Siegel (1966) examined bubble characteristics on single nucleation sites of known diameter (0.1~1mm) on highly polished surfaces in pool boiling (see Fig. 8.8). Since the bubbles were large and slow-growing, only surface tension and buoyancy forces were important at departure. If the vapor bubble extends through the liquid thermal layer ( Rc > δ ) as shown in Fig. 8.8 (a), evaporation occurs from the portion of the nucleus surface within the thermal layer while condensation occurs over the remaining portion of the nucleus. For the vapor bubble to grow, the amount of evaporative heat transfer must be greater than that of the condensation, which leads to the following criterion: Tw − Tsat > 4σ Tsat , Rc > δ ρ v hvδ (8.38) If the nucleus is wholly contained in the thermal layer, as shown in Fig. 8.8(b), the temperature required for the bubble to grow is 2σ Tsat 1 Tw − Tsat > , Rc < δ (8.39) ρ v hv Rc 1 − Rc / (2δ ) For the growth rate of a vapor bubble attached to the wall, Mikic et al. (1970) suggested that eqs. (8.32) – (8.36) are applicable, provided that the geometric parameters b = π / 7 and T (∞) are replaced by the heating surface temperature Tw. As noted before, the microlayer plays a very important role during bubble growth near a heated surface (Fig. 8.6). van Stralen et al. (1975a) proposed the (a) Rc > δ (b) Rc < δ Figure 8.8 Criteria for growth of hemispherical bubble nucleus (Howell and Siegel, 1966). 680 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press following relation for bubble growth using microlayer theory, which is applicable to both inertia and heat transfer controlled region for pure liquids: R(t ) = R1 (t ) R2 (t ) R1 (t ) + R2 (t ) (8.40) where R1 (t ) = 0.8165 ρ v hv (Tw − Tsat ) exp[−(t / td )1/ 2 ] t ρTsat (8.41)    t 1/ 2  T − T   1/ 2 sat  R2 (t ) = 1.954  R* exp  −    + ∞  Ja(α  t )   t d   Tw − Tsat          t 1/ 2    −1/ 6  +0.373Pr exp  −     Ja(α  t )1/ 2   td        R2 (t d ) R* = 1.3908 − 0.1908 Pr−1/6 Ja(α  t d )1/ 2 1/ 2 (8.42) (8.43) R* can be determined if the departure time td is known. van Stralen et al. (1975b) compared the above equation with experimental bubble growth rate data for water over a range of temperature, and obtained good agreement. Bubble Growth within Superheated Liquid Droplets Two immiscible liquids of different volatility are mixed together such that one liquid is dispersed in the form of droplets within the other, so that the droplets may be heated by direct contact across the liquid/liquid interface. Heat can be supplied by conduction, convection, nucleate boiling, or film boiling. When nucleation sites are not present, single-phase conduction or convection can exist beyond the normal saturation state of liquid 1; this is termed superheating. This state can exist as long as the liquid does not come into contact with a vapor phase with which it is in equilibrium. The upper limit to the temperature of the liquid at a given pressure before a phase change occurs is called the superheat limit. At the superheat limit, homogeneous nucleation will begin to occur inside liquid 1 and form a vapor droplet. These vapor bubbles are in metastable equilibrium with the surrounding liquid, and subsequent growth of the initial bubbles will complete the phase transition. The two steps in phase change related to superheated liquid droplets are (Avedisian, 1986): 1. The initial stage, during which microscopic bubbles form in the droplet 2. A second or bubble growth stage, in which the initial bubble grows as the liquid droplet vaporizes Bubble growth within superheated liquid droplets is important in applications such as preparation of emulsified liquids, fuel coolant interactions in postulated nuclear reactor accidents, and heat exchangers. In these cases, droplets of the volatile liquid are dispersed in another stagnant liquid. Heating the field liquid Chapter 8 Boiling 681 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press leads to superheating of the droplets, homogeneous nucleation, and bubble growth. Bubble growth in a volatile droplet suspended in an immiscible nonvolatile liquid was developed by Avedisian and Suresh (1985) by solving the coupled energy and momentum equations for the temperature fields in both liquids. A numerical solution for a two-phase droplet modeled as a vapor bubble growing from the center of the liquid is presented. When the properties of the two liquids are very different, the bubble growth rate can experience a significant change when the thermal boundary layer extends into nonvolatile liquid, as shown in Fig. 8.9. Consider the following conditions for superheat bubble growth. At t = 0 a vapor bubble appears in liquid 1. The initial temperatures of liquids 1 and 2 are the same, and there is no motion between the droplet and the liquid. The bubble begins to grow due to a perturbation caused by a slight reduction in ambient pressure or an increase in ambient temperature. Continuous growth of the bubble completely consumes liquid 1 until the mass of the droplet is entirely vaporized and only a vapor bubble in liquid 2 is left. The radius of the final bubble is −1/3 (8.44) R f = (1 − ε ) So where ε is the density ratio, and So is the initial overall drop radius. To formulate an equation of motion for a spherical bubble growing from the center of a spherical droplet, assume a vapor bubble growing from the center of a volatile liquid suspended in a nonvolatile liquid, as shown in Fig. 8.9. The velocity, Liquid 2 Bubble Liquid 2 Bubble S Rδ Boundary Layer Edge Liquid 1 S R δ Boundary Layer Edge Liquid 1 To Tsat R R+δ To Tδ Tsat R S R+δ (a) (b) Figure 8.9 Schematic illustration of bubble growth in a droplet suspended in an immiscible fluid. (a) Early stage where thermal boundary layer (δ) is within the droplet; (b) later stage when boundary layer extends into liquid 2 (Avedisian and Suresh, 1985; Reproduced with permission from Elsevier). 682 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press pressure, and temperature in the two liquids are governed by the continuity, momentum, and energy equations for one-dimensional and unsteady conditions, given as 1∂ 2 ( r vi ) = 0 r 2 ∂r ∂vi ∂v  1 ∂  2 ∂vi 1 ∂pi +ν i i = − +ν i  2 r ∂t ∂r ρ ∂r  r ∂r  ∂r ∂Ti ∂T 1 ∂  2 ∂Ti  +ν i i = α i 2 r ∂t ∂r r ∂r  ∂r    (8.45)  2vi   − r2    (8.46) (8.47) where i = 1 and 2, vi is the radial velocity, pi is the pressure within liquid i, Ti is the temperature within liquid i, and αi is the thermal diffusivity of liquid i. The boundary and initial conditions are (8.48) T1 ( r , 0 ) = T2 ( r , 0 ) = To T1 ( S , t ) = T2 ( S , t ) T1 ( R, t ) = Tv ( t ) ∂T1 ∂r ∂T2 ∂r (8.49) (8.50) (8.51) S ,t k1 = k2 S ,t T2 ( ∞, t ) = To R ( 0 ) = Ro (8.52) (8.53) (8.54) where k1 and k2 are the thermal conductivity of liquid 1 and 2, respectively, R is the bubble radius and S is the droplet radius. The interfacial energy balance around the bubble yields k1 ∂T1 ∂r  = ρ v hv R R ,t  R (0) = 0 (8.55)  where R is the velocity of the bubble wall. To and Ro are the initial conditions of the limit of superheat and initial unstable bubble radius. These terms, defined by the critical nucleus state for homogeneous nucleation, are intrinsic properties of liquid 1 and are functions of ambient pressure for a given nucleation rate. Radial velocity is continuous in this problem, because there is no mass transfer across the liquid’s interface. The velocity can be found by integrating eq. (8.45) and applying a mass balance around the bubble. R2  R (8.56) r2 where ε is the density ratio (1 − ρ v / ρ1 ) . Equation (8.56) is the same as bubble vi = ε growth in an infinite medium. The Rayleigh equation for this problem is obtained by integrating eq. (8.46) over r twice: from R to S and again from S to ∞. Chapter 8 Boiling 683 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press  β R  ∂vi ∂v  1 ∂  2 ∂vi 1 ∂pi + vi i = − + vi  2  r ∂r ρ ∂r  r ∂r  ∂r  ∂t  2vi  − r2    dr  (8.57) where β = S or ∞. pv − pR1 = When r = R, 2σ 1 R − 2μ1 ∂v1 ∂r (8.58) r=R and when r = S pS1 − pS 2 = 2σ 12 ∂v − 2 ( μ2 − μ1 ) 1 S ∂r (8.59) r=S The equation of motion for a bubble growing in the center of a droplet is found by combining eqs. (8.56) and (8.55), integrating twice, and substituting eqs. (8.58) and (8.59):   R R2  R4  R R3     RR + 2 R 2  1 − ε  − ε 1 − ε 4  + 4v1 1 − μ 3     2 S R S S  2σ 1  σ 12 R  = − 1 +  ρ1ε ερ1 R  σ 1 S  where ε is the density ratio (1 − ρ 2 / ρ1 ) , μ is the viscosity ratio pv − po (8.60) (1 − μ2 / μ1 ) , σ12 is the liquid 1/ liquid 2 interfacial tension, and po is pressure at infinity in liquid 2. Figure 8.10 shows the evolution of pressure, temperature and average radius for a bubble growing in an n-octane droplet at Ja = 10 [ ρ1c p1 (To − Tsat ) / ( ρ v hv )] , 2 where R = R / So and τ = tα1 / So . The early stages of bubble growth in a 540 15.0 520 10-1 T0 10 -2 pv(T0) 10-3 R R Tv (K) Tsat (p0) 500 12.5 pv(atm) 10.0 10-4 Tv 480 10-5 pv p0 = pv(Tsat) 7.5 460 10-6 10-6 10-11 10-10 10-9 10-8 10-7 τ Figure 8.10 Early time variation of vapor pressure, vapor temperature and radius of a bubble growing in a superheated n-octane droplet. Initial conditions correspond to the kinetic limit of superheat of octane at the indicated pressure (Avedisian and Suresh 1985; Reproduced with permission from Elsevier). 684 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press droplet are similar to growth in a semi-infinite medium. The initial state corresponds to the homogeneous nucleation limit and the asymptotic temperatures and pressures correspond to saturation conditions. Sideman and Isenberg (1967) developed an analytical, closed-form solution for boiling a droplet in another immiscible liquid with the following assumptions: 1. No radial convection 2. Constant and uniform temperature of liquid 1 3. Inviscid and uniform translatory motion of liquid 2 around liquid 1 droplet This closed form solution is valid only after boundary layer penetration of liquid 2. During early bubble growth the assumptions are not valid, because the temperature field is transient and the boundary layer is entirely in liquid 1. The analytical solution for boiling of a droplet in another immiscible liquid is given as S = (1 − ε ) −1 2   9 1/ 2   1/ 2 3 1 − ε   (1 − ε ) JaPe τ − 1   2π        1 3 (8.61) where S = S / So , Pe is the Peclet number (U ∞ So / α 2 ) , and Ja is the Jakob number. 8.3.3 Bubble Detachment A vapor bubble attached to a heating surface is subjected to different forces. While surface tension and liquid inertia forces tend to prevent such a bubble from detaching, buoyancy and dynamic forces act directly on the bubble to lift it from the heating surface. When the bubble grows large enough, the detachmentoriented forces will dominate and the bubble will detach from the heating surface. The inertial force is a strong function of liquid superheat, which, as indicated by eq. (8.12), is inversely proportional to the size of the cavity. Therefore, a bubble formed from a small cavity grows faster than one formed from a large cavity. The bubble size at departure can be found by performing a force balance on the bubble. For a small cavity size ( rc < 10 μ m ), the bubble size at departure is dominated by a balance between buoyancy and liquid inertial forces. When the cavity size is large, the bubble grows more slowly, thus the inertial force becomes unimportant. Instead, a force balance between buoyancy and surface tension dominates the bubble size at departure. Fritz (1935) was the first to correlate the bubble departure diameter by balancing buoyancy and surface tension forces, and he proposed the following equation: Chapter 8 Boiling 685 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Db = 0.0208θ 2σ g( ρ  − ρ v ) (8.62) where θ is the contact angle measured in degrees. Further experimental investigations indicated that eq. (8.62) can provide a correct length scale for the bubble diameter at departure, but significant deviation of the bubble diameter at departure has been reported. Cole and Rohsenow (1969) proposed much better correlations for the bubble diameter at departure: For water: Db = 1.5 × 10−4 5 2σ ( Ja* ) 4 g( ρ  − ρ v ) (8.63) For other fluid: Db = 4.65 × 10−4 5 2σ ( Ja* ) 4 g( ρ  − ρ v ) (8.64) where ρ  c p Tsat (8.65) ρ v hv At high-heat flux, the bubble departure diameter can be obtained from (Gorenflo et al., 1986) Ja * =  Ja 4 h2v  Db = C1   g 1/3 1/ 2  2π   1 + 1 +     3Ja     4 /3 (8.66) where the constant C1 obtained by fitting experimental data for boiling of R-12, R-22, and propane are 14.7, 16.0, and 2.78, respectively. The bubble release frequency fb is related to the waiting time between detachment of the one bubble and initiation of the next bubble, tw, and the growth time, tg, of a bubble before its detachment. Attempts to obtain bubble release frequency by prediction of tw and tg were rarely successful because (a) these models generally did not consider evaporation from both the base and surface of bubbles; (b) cavity size variations significantly alters the growth time; (c) bubble activity, fluid flow, and heat transfer near a nucleation site can influence both bubble growth and waiting time; and (d) bubble shapes always change during growth (Kandlikar et al., 1999). While the bubble release frequency depends distinctly upon the bubble diameter at departure, which differs from site to site, it is a constant for each individual nucleation site. It is recommended that the correlation have the form n of fb Db = const, where n is the exponent. Ivey (1967) suggested that the value of exponent n for inertia-controlled bubble growth is 2. When heat transfer controls bubble growth, the value of n is equal to 1/2. Malenkov (1971) proposed the most comprehensive correlation for the product of bubble release frequency and bubble diameter at departure: 686 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press fb Db = Vb   1 π 1 −   1 + Vb ρ v hv / q′′  (8.67) where Vb is the bubble departure velocity and is calculated by Db g( ρ  − ρ v ) 2σ (8.68) Vb = + 2( ρ + ρ v ) Db ( ρ + ρ v ) Eastman (1984) developed an analysis related to dynamics of bubble departure that is presented here. During nucleate boiling, the forces that hold the bubble to the wall (negative) are larger than the forces that pull the bubble from the wall (positive) when the bubble is very small. As the bubble grows, the positive forces grow faster than the negative forces, until the total force becomes positive and pulls the bubble away from the surface. The negative forces are surface tension and drag force. The positive forces are buoyancy (for an upward facing surface), internal pressure, and inertia. These forces and directions are shown in Fig. 8.11. The five forces that act on a bubble during growth from a heated wall, are: internal pressure, surface tension, buoyancy, drag, and inertia. Analyses show that buoyancy is related to the gravitational acceleration and the inertial force is dependent on the deceleration of the bubble. The bubble departure can be determined by a force balance: (8.69) Fd + Fs = Fi + Fp + FB where Fi and Fp are liquid inertial force and pressure force, respectively. The surface tension force, Fs, is caused by the attraction of the liquid to the surface FB Rb Figure 8.11 Forces acting on a vapor bubble growing on a heating surface. Chapter 8 Boiling 687 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press that acts around the perimeter of the bubble base. The surface tension force is proportional to the fluid surface tension, σ, and the contact angle, θ, i.e. Fs = 2π Rbσ sin θ (8.70) where Rb is the base radius. The base contact angle goes to 90° as the bubble nears departure. When the bubble is growing in a viscous fluid, it will be subject to drag force. In realistic applications, this force is negligible for most fluids. A very rough estimate of this force was made by Keshock and Siegel (1962), and this approximation will be used here. In estimating drag force, it is also assumed that the bubble is spherical and growing away from the wall at a velocity equal to the change of its radius with time, i.e., dR/dt. The drag force can be calculated as: 2 ρ  dR  2 (8.71) Fd = C d   πR 2  dt  where drag coefficient Cd can be calculated by using the following experimental correlation Cd = 45 Re dR (8.72) where (8.73) R μ dt For a spherical bubble, which is submerged in a stagnant fluid, the buoyancy force is equal to the weight of the fluid displaced. So the buoyancy force is 4π R3 FB = (8.74) ( ρ − ρ v ) g Re = 3 2 ρ Vapor inertia is negligible for most of the bubble growth; however, the inertia for the liquid surrounding the bubble needs to be accounted for. The liquid around the bubble is moved by the growth of the bubble. The affected mass of the fluid is that occupied by 11/16 of the bubble volume (Han and Griffith, 1965). The growth rate of the bubble decreases after the initial unbinding. The inertia of the liquid works against the decrease in velocity and tries to pull the vapor away from the surface. According to Newton’s second law, the inertial force can be approximated as follows, based on the above argument: Fi = 11 d2 R π R3 ρ  2 6 dt (8.75) The pressure force results from the contribution of the dynamic excess vapor pressure and capillary pressure. It can be written as follows:  2σ  2 (8.76) Fp =  + pv  π Rb R  where the vapor pressure is given below 2σ pv − p = R (8.77) 688 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 10 8 6 Buoyancy Force (105 N) 4 2 .5 .6 .7 .8 Radius in .9 .10 cm Total -4 -6 -8 -10 Surface Tension Internal Pressure Inertia -2 .11 .12 .13 Figure 8.12 Forces acting on a bubble in saturated water with 1-g acceleration (Eastman, 1984). 10 8 6 4 Internal Pressure Buoyancy Inertia .10 .11 .12 .13 .14 .15 .16 .17 .18 Force (105 N) 2 -2 -4 -6 -8 -10 Radius in cm Total Surface Tension Figure 8.13 Forces acting on a bubble in saturated water under 0.229-g acceleration (Eastman, 1984). The departure radius is assumed to be the radius at which the total force changed signs from negative to positive. Equations (8.70) – (8.77) were used to calculate the departure radius for saturated water for 1g and 0.229g, respectively. The forces acting on a bubble versus its radius are shown in Figs. 8.12 and 8.13. Analysis shows that buoyancy is directly related to the local gravitational acceleration. The inertial force is independent of gravity but dependent on the deceleration of the bubble. Chapter 8 Boiling 689 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 8.3.4 Nucleate Site Density The knowledge of distribution of nucleation sites is an important factor in determining the boiling characteristics of a surface under specific operating conditions. The number density of sites, or total number of active sites per unit area, is a function of contact angle, cavity half angle, and heat flux (or superheat) (Fig. 8.6), i.e., ′′ N a = f (θ , φ , ΔT , fluid properties) (8.78) Equation (8.11) indicated that for a given local heat flux or superheat, a cavity will be active if Rmin is greater than Rb, 2σ Tsat Rmin ≥ (8.79) hv ρ v ΔT Obviously each cavity on a real surface has a specific Rmin that is a function of geometry and the contact angle. Considering eqs. (8.78) and (8.79), one expects that as the wall superheat increases, Rmin decreases and the number of active sites having cavity radii greater than Rmin increases. Lorenz et al. (1974) counted total active sites/cm2 on a #240 (sand paper) finished copper surface for different working fluids as a function of Rmin, which is shown in Fig. 8.14. Figure 8.14 Number density of active sites for boiling on a copper surface (Lorenz et al., 1974). 690 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Kocamustafaogullari and Ishii (1983) have correlated various existing ′′ experimental data of N a for water on a variety of surfaces and pressure ranges from 1 to 198 atm by  D  −0.44  ′′ N a = D  c  F  Dd     2 d 1/ 4.4 (8.80) where  ρ − ρv  F = 2.157 × 10 −7     ρv  −3.2 Dc = 4σ [1 + ( ρ  / ρ v ) ] / p ⋅ exp hv (Tv − Tsat ) / ( RgTvTsat )  − 1   Dd = 0.0208θ  ρ − ρv  ⋅ 0.0012    g( ρ  − ρ v )  ρv  {   ρ − ρv   1 + 0.0049     ρv    0.9 4.13 (8.81) } (8.82) (8.83) σ where Rg is the gas constant for the vapor. Wang and Dhir (1993a, 1993b) have studied number density for boiling of water at 1 atm on a mirror-finished copper surface, and they provided a mechanistic approach for relating the cavities that are present on the surface to the cavities that actually nucleate. 8.3.5 Bubble Growth and Merger Although nucleate boiling has been a subject of extensive studies, a fundamental understanding of the dynamics of bubble growth, departure, and merger is still lacking. Most existing analytical models did not consider how the flow and temperature field are influenced by the bubbles’ motion. While early efforts to develop numerical simulations of bubble growth during partial nucleate boiling did not consider the merger of the vapor bubbles (e.g., Lee and Nydahl, 1989; Welch, 1998; Son et al., 1998), single and multiple bubble mergers during nucleate boiling have been numerically simulated recently, and the results compared favorably with experimental observations (Son et al., 2002; Mukherjee and Dhir, 2004; Dhir, 2005). The numerical simulation of the bubble merger process on a single nucleation site during pool boiling, performed by Son et al. (2002), will be introduced in this subsection. The transport phenomena in the nucleate boiling occurred at different scales, and for this reason, Son et al. (2002) divided the computational domain into macro and micro regions, as shown in Fig. 8.15. The macro region includes the vapor bubble and the surrounding liquid, while the micro region is the thin film region beneath the bubble. Evaporation in the micro region was solved using lubrication theory. The phase change and macro region were solved using a level set formulation. Chapter 8 Boiling 691 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press y=Y Macro region Vapor g T = Tsat Liquid θ y r r=R wall Micro region δ0 h/2 δ r r = R0 wall r = R1 Figure 8.15 Macro and micro region in numerical simulation (Son et al., 2002) Assuming laminar and constant properties, the conservation of mass in the microlayer is ∂δ q′′ = v − (8.84) ρ hv ∂t where the liquid velocity normal to the film surface, v , is obtained by integrating the continuity equation in the cylindrical coordinate system, i.e., v = − y 1∂ δ ru dy r ∂r 0 (8.85) Neglecting inertial and gravity effects, the momentum equation in the thin film is ∂p ∂ 2 u = μ ∂r ∂y 2 (8.86) Assuming conduction is the mechanism for heat transfer across the thin film, the heat flux is (8.87) δ where Tδ is the temperature at the liquid-vapor interface. The evaporating heat flux can also be written using a modified Clausis-Clapeyron equation: 2 q′′ =  πR T gv      1/ 2 q′′ = k Tw − Tδ ρ v h2v  Tv Tδ − Tv +  ( p − pv )Tv   ρ hv  (8.88) 692 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The pressures in the liquid and vapor are related by momentum balance at interface (Chapter 5) p = pv − σ K − pd + q′′2 ρ v h2v (8.89) where surface tension, σ , is a function of temperature, and K is curvature: K= 1 ∂  ∂δ r r ∂r  ∂r  2  ∂δ   1+    ∂r    (8.90) and pd = Aδ-3 is the disjoining pressure. The combination of eqs. (8.84) - (8.88) yields a fourth order ordinary differential equation in the following form: δ ′′′′ = f (r , t , δ , δ ′, δ ′′, δ ′′′) (8.91) which is subject to the following boundary conditions: δ = δ 0 ; δ ′ = δ ′′′ = 0 at r = R0 (8.92) δ = h / 2; δ ′ = tan θ ; δ ′′ = 0 at r = R1 (8.93) where δ0 is the nonevaporating film thickness for which Son et al. (2002) used δ 0 = 6 × 10−10 m , h / 2 is the distance to the first computational node for the level set function from the wall, and θ is the apparent contact angle. R1 is obtained from the solution of the macro region. As for the macro region, the level set formulation is used to accommodate the effect of liquid-vapor phase change.The interface separating the two phases is captured by a level set function, φ , which is defined as a signed distance from the interface: a negative sign for the vapor phase and a positive sign for the liquid phase. The governing equations for the macro region are  ∂V  + V ⋅∇V  = −∇p + ρ g − ρ gβ (T − Tsat ) ∂t   −σ K ∇H + ∇ ⋅ μ∇V + ∇ ⋅ μ∇V T ρ (8.94)  ∂T  + V ⋅∇T  = ∇ ⋅ k∇T for H >0  ∂t  T = Tsat ( pv ) for H =0 ρ c p  (8.95) (8.96)  ∇ρ + Vmicro ρ2 where the properties are evaluated by ∇⋅V = m′′ ρ = ρv + ( ρ − ρv ) H , − μ −1 = μv 1 − +( μ−1 − μv 1 ) H , and k −1 = k−1 H . Due to the discontinuity at the liquid-vapor interface, the step function H is defined by φ ≤ −1.5h 0  H = 0.5 + φ / (3h ) + sin[2πφ / (3h)] / (2π ) φ ≤ 1.5h 1 φ > 1.5h  (8.97) where h is grid spacing. The source term included in eq. (8.96) is obtained by conservation of mass and energy at the interface, Chapter 8 Boiling 693 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press m′′ = ρ (Vδ − V ) = k ∇T hv (8.98)  and the rate of volume production from microlayer Vmicro is obtained from R1 k (T − T )  δ w  Vmicro =  rdr R0 ρ h δΔV v v micro (8.99) where ΔVmicro is the vapor side control volume near the micro region. The level set function, φ , is advanced by ∂φ (8.100) = −Vδ ⋅∇φ ∂t and reinitialized by solving ∂φ ∂t = φ0 φ + h2 2 0 (1 − ∇φ ) (8.101) The boundary conditions for the macro region are ∂φ = − cos θ at y = 0 u = v = 0, T = Tw , ∂y (8.102) (8.103) (8.104) ∂v ∂T ∂φ at r = 0, R = = ∂r ∂r ∂r ∂u ∂v ∂φ = = =0, T = Tsat , at y = Y ∂y ∂y ∂y u= The grid size used by Son et al. (2002) was 26 μm; the time step was 8 × 10−3 ms, which satisfies Δt ≤ h / ( u + v ) because of the explicit treatment of the convection terms. Figure 8.16 shows the bubble growth pattern during one cycle for ΔT = 10 K and waiting time of 1.28 ms. The computational domain was 2.5×10 mm2 (R×Y). It can be seen that the bubble merger occurred twice during one cycle at t =1.4 ms and t = 4.5 ms. The merged bubble departed from the surface at t = 5.5 ms, with no more bubble merger during this cycle. Son et al. (2002) also compared the predicted bubble merger pattern with the experimental observations and the agreement was very good. The bubble dynamics and heat transfer associated with lateral bubble merger during transition from partial to fully developed nucleate boiling were studied numerically by Mukherjee and Dhir (2004). Multiple bubble mergers in a line and in a plane were simulated, and the bubble dynamics and wall heat transfer are compared to those for a single bubble. The results show that merger of multiple bubbles significantly increases the overall wall heat transfer, which is caused by trapping the liquid layer between the bubble bases during merger and the drawing of cooler liquid toward the wall during the post-merger contraction. While these numerical investigations provided insights about bubble formation, growth, and mergers in nucleate boiling, the results are as good as the knowledge of the distribution of the nucleation sites. 694 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press t = 1.4 msec 7.5 t = 2.2 msec t = 4.2 msec t = 4.5 msec y(mm) 5.0 2.5 0.0 7.5 t = 4.9 msec t = 5.5 msec t = 6.3 msec t = 6.8 msec y(mm) 5.0 2.5 0.0 7.5 t = 7.6 msec t = 9.4 msec t = 13.8 msec t = 32.2 msec y(mm) 5.0 2.5 0.0 0.0 r(mm) 2.5 0.0 r(mm) 2.5 0.0 r(mm) 2.5 0.0 r(mm) 2.5 Figure 8.16 Bubble growth and demerger pattern for ΔT = 10K and waiting time of 1.28 ms (Son et al. 2002). 8.3.6 Heat Transfer in Nucleate Boiling In general, heat flux and heat transfer coefficient during evaporation and nucleate boiling can be correlated with the driving temperature difference (Tw – Tsat) according to the following equations: m q′′ = c1 ( Tw − Tsat ) (8.105) and since q′′ = h ( Tw − Tsat ) one can get h = c2 ( Tw − Tsat ) m −1 m −1 m = c2 ( Tw − Tsat ) = c3 q′′P n (8.106) (8.107) The values of m, n and P are shown in Table 8.1 and are almost constant with weak dependencies on liquid properties and heated surfaces. However, c1, c2 h = c3 q′′ Chapter 8 Boiling 695 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 8.17 Temperature profile above a heating surface during nucleate boiling (Stephan, 1992; Reprinted with kind permission of Springer Science and Business Media). and c3 depend greatly upon the properties of liquid and heated surface material and geometry. For evaporation, one can use free convection correlation and therefore 1/ 4 h = c2 ( Tw − Tsat ) Laminar flow (8.108) h = c2 ( Tw − Tsat ) 1/3 Turbulent flow (8.109) Table 8.1 Heat transfer comparison of evaporation and nucleate pool boiling* q′′ = c1 ( Tw − Tsat ) Evaporation Nucleate boiling Laminar Turbulent m = 5/4 m = 4/3 m=4 m h = c2 ( Tw − Tsat ) n = 1/4 n = 1/3 n=3 n h = c3 q′′P P= 1/5 P= 1/4 P= 3/4 *c1, c2, and c3 are constants, not functions of ΔT = Tw − Tsat . A typical temperature distribution during nucleate boiling under 1 atm over a horizontal plate at 109.1 °C is shown in Fig. 8.17. It can be seen that the difference between wall temperature and the liquid temperature, Tw − T , is significantly greater than the difference between the liquid temperature and saturation temperature, T − Tsat . The driving force of heat transfer is Tw − Tsat , which is used to define the heat transfer coefficient. It is found that the heat transfer coefficient is approximately proportional to the third power of this temperature difference, i.e., 3 (8.110) h = c2 ( Tw − Tsat ) Figure 8.18 shows the comparison of equations shown in Table 8.1 with experimental results for evaporation and nucleate pool boiling of water at 100 °C from a horizontal surface. Clearly, there are two distinct and separate ranges, one for evaporation (up to q′′ = 104 W / m 2 ) and one for nucleate boiling ( q′′ > 104 W / m 2 ). The power law equation works for both regions a and b with different exponents and coefficients as shown in Table 8.1 and Figure 8.18 (P = 696 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 8.18 Evaporation and nucleate boiling of water at 100 °C on a heated surface (Stephan, 1992; Reprinted with kind permission of Springer Science and Business Media). 3/4 for region a and P = 1/5 for region b). Cao et al. (1989) used m = 1 for evaporation inside a spot heated heat pipe with high working fluid temperature and m = 1.215 for evaporation inside a block heated heat pipe with moderate working fluid temperature. As can be seen from Table 8.1, n = 3 corresponding to P = 0.75 in eq. (8.107). The range of P for nucleate boiling is generally between 0.6 and 0.8, which corresponds to 2.5 ≤ m ≤ 5 or 1.5 ≤ n ≤ 4 (e.g., the well known Rohsenow equation (8.115) for nucleate boiling has m =3 and n =2). For low-boiling point liquids such as helium, the value of P can be as small as 0.5 (m = 2 and n = 1). Heat transfer in nucleate boiling processes has been described through a variety of heat transport models, all of which have limitations of some sort because they were developed under ideal conditions. For nucleate boiling on an upward-facing horizontal surface in the isolated bubble region, heat transfer mechanisms include (1) transient heat conduction from the heating surface to the adjacent liquid, (2) natural convection on inactive areas of the heating surface, and (3) evaporation from the micro layer underneath the vapor bubble (Forster and Greif, 1959). The liquid adjacent to the heating surface becomes superheated, is pushed outward by the vapor bubble, and then mixes with the bulk liquid. An expression of the heat flux during partial nucleate boiling is (Kandlikar et al., 1999)  C2 C2 π2 2 2 ′′ ′′ q′′ = 1 π (k ρ c p ) fb Db N a ΔT +  1 − 1 N aπ Db  hnc ΔT + N a Db hevp ΔT (8.111) 2  2  4 where the three terms on the right-hand side represent the contributions of transient conduction, natural convection, and micro layer evaporation. In order to use eq. (8.111) to calculate heat transfer, it is necessary to know the proportionality constant C1 for bubble area influence, the bubble diameter at departure Db, the bubble release frequency fb, the number density of the nucleate Chapter 8 Boiling 697 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ′′ site N a , and average heat transfer coefficient of natural convection and evaporation, hnc and hevp . Mikic and Rohsenow (1969) justified eq. (8.111) without the last term by using empirical correlations for several of these parameters. Mikic and Rohsenow’s (1969) correlation was recently revisited by Bergles (2005), who concluded that the Mikic-Rohsenow correlation is the ultimate pool-boiling model because all of the parameters can be obtained by various empirical correlations and only one experimental observation – the cavity size distribution. The problem is that it is impossible to get closer than one empirical constant since the nucleation site distribution is unknown a priori. The third term on the right-hand side of eq. (8.111) was introduced by Judd and Hwang (1976) to account for microlayer evaporation at the base of the bubble. The above nucleate boiling model considers only isolated bubbles. The analysis breaks down when bubbles coalesce and become slugs and columns. In addition, the above model includes expressions of bubble departure diameter Db and frequency fb, which are only correlations for most cases. There may be a high degree of uncertainty associated with the resulting correlations for heat transfer. Complete and reliable mathematical models for heat transfer in nucleate boiling have yet to be developed. Consequently, empirical correlations based on experimental data are widely used to predict heat transfer in nucleate boiling. Jakob and Linke (1935) proposed one of the earliest and most popular models. Their model assumes that bubble growth and departure cause the convective heat transfer of liquid from the solid surface. This reasoning allowed Rohsenow (1952) to construct the correlation for nucleate boiling in the following single-phase forced convection type: Nu = 1 Re1− m Pr1− n C s , (8.112) where C s , is an empirical constant that accounts for the effect of different combinations of heating surface and liquid. The Nusselt number is defined as  h σ Nu =   k  ( ρ  − ρ v ) g  1/ 2   q′′ σ =   k (Tw − Tsat )  ( ρ − ρ v ) g  1/ 2 (8.113) where the bracket term at the right-hand side, together with its exponent (1/2) is the characteristic length. The Reynolds number using the same characteristic length is  q′′ Re =   ρ  hv  1 σ     ( ρ − ρ v ) g  ν  1/ 2 (8.114) where q′′ / ( ρ hv ) is superficial velocity of the liquid. The heat flux in nucleate boiling, q′′, is proportional to (Tw − Tsat )3 ; therefore, the exponent m in eq. (8.112) is chosen to be 1/3. The exponent n is, however, dependent on the specific combination of the surface and the liquid. 698 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Substituting eqs. (8.113) and (8.114) into eq. (8.112), the following empirical correlation is obtained (Rohsenow, 1952):  g( ρ  − ρ v )  q′′ = μ hv   σ   1/ 2  c p (Tw − Tsat )   n  C s , hv Pr    3 (8.115) which applies to clean surfaces and is insensitive to the shape and orientation of the surface, which is obviously an approximation. It is still by far the most widely used empirical correlation for heat transfer in nucleate boiling. The coefficient C s , and the exponent n depend on the liquid-surface combination, and their representative values are shown in Table 8.2. It is necessary to point out that the error of heat flux estimated by eq. (8.115) can be as high as ±100%. If the heat flux is specified and it is necessary to estimate the temperature of the heated surface, one can rearrange eq. (8.115) to obtain   (8.116)   under these circumstances, the error is within ±25%. Table 8.2 Values of C s and n for various combinations of surfaces and fluids for eqs. (8.115) and (8.116) Tw − Tsat   σ    g( ρ  − ρ v )  (Rohsenow, 1952). Surface material Brass Chromium Chromium Copper Copper Copper Copper Copper Copper Nickel Platinum Stainless steel Stainless steel Stainless steel Stainless steel Surface finish Fluid Water Benzene Ethyl alcohol Carbon tetrachloride Isopropanol n-Pentane n-Pentane Water Water Water Water Water Water Water Water  q′′ h = v Prn C s ,  c p  μ hv  1/ 2 1/3 C s 0.0060 0.0101 0.0027 0.0130 0.0130 0.0049 0.0154 0.0130 0.0068 0.0060 0.0130 0.0130 0.0060 0.0130 0.0058 Lapped Polished Polished Scored Chemically etched Ground and polished Mechanically polished Teflon pitted n 1.0 1.7 1.7 1.7 1.7 1.7 1.7 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 In addition to Rohsenow’s empirical correlation presented here, there are numerous other empirical correlations for heat transfer in nucleate boiling. Cooper (1984) proposed an empirical correlation for the heat transfer coefficient for pool nucleate boiling as a function of reduced pressure and surface roughness: 0.12 − 0.4343ln Rp h = 55 pr (−0.4343ln pr )−0.55 M −0.5q′′0.67 (8.117) where pr is reduced pressure defined as pr = p / pc , Rp is the surface roughness in μm (it can be set to 1.0 μm for unknown roughness), and M is molecular mass of the fluid. Equation (8.117) is valid for reduced pressure between 0.001 and Chapter 8 Boiling 699 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Table 8.3 Critical pressure and reference heat flux (Gorenflo, 1993). Fluid Acetone Argon (on Cu) Argon (on Pt) Ammonia Benzene n-Butane n-Butanol i-Butanol Carbon Dioxide (at triple point) Chloromethane Diphenyl Ethane Ethanol Freon®-11 Freon®-12 Freon®-13 Freon®-22 Freon®-23 Freon®-113 Freon®-114 Freon®-115 Freon®-123 Freon®-134a Freon®-152a Freon®-502 n-Heptane n-Hexane Hydrogen Methane Neon Nitrogen (on Cu) Nitrogen (on Pt) Oxygen (on Cu) Oxygen (on Pt) n-Pentane i-Pentane Propane n-Propanol i-Propanol RC-318 Sulfur Hexafluoride Tetrafluoromethane Toluene Water Chemical Formula (CH3)2CO Ar Ar NH3 C6H6 CH3CH2CH2CH3 C4H10O C4H10O CO2 CH3Cl C12H10O2S C2H6 C2F3Cl3 CFCl3 C2F3Cl3 CClF3 CHF2Cl CHF3 C2F3Cl3 CCl2FCF3 C2ClF5 CHCI2CF3 CF3CH2F CHF2CH3 CHF2Cl/C2ClF5 C7H16 C6H14 H2 CH4 Ne N2 N2 O2 O2 C5H12 C5H12 CH3CH2CH3 C3H8O C3H8O C4F8 SF6 CF4 C7H8 H2O Molecular Mass (kg/kmol) 58.08 39.95 39.95 17.03 78.11 58.12 74.12 74.12 44.01 50.49 154.2 30.07 46.07 137.4 120.9 104.5 86.47 70.02 187.4 170.9 154.5 152.9 102.0 66.05 111.6 100.2 86.18 2.02 16.04 20.18 28.02 28.02 32.00 32.00 72.15 72.15 44.10 60.1 60.1 200.0 146.1 88.00 92.14 18.02 Pcrit (bar) 47.0 49.0 49.0 113.0 48.9 38.0 49.6 43.0 73.8 66.8 38.5 48.8 63.8 44.0 41.6 38.6 49.9 48.7 34.1 32.6 31.3 36.7 40.6 45.2 40.8 27.3 29.7 12.97 46.0 26.5 34 34 50.5 50.5 33.7 33.3 42.4 51.7 47.6 28.0 37.6 66.8 41.1 220.6 h0 (W/m2-K) 3,950 8,200 6,700 7,000 2,750 3,600 2,600 4,500 5,100 4,400 2,100 4,500 4,400 2,800 4,000 3,900 3,900 4,400 2,650 2,800 4,200 2,600 4,500 4,000 3,300 3,200 3,300 24,000 7,000 20,000 10,000 7,000 9,500 7,200 3,400 3,400 4,000 3,800 3,000 4,200 3,700 4,400 2,650 5,600 700 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 0.9, and molecular masses ranging from 2 to 200. For horizontal copper cylinders, Cooper (1984) recommended that the boiling heat transfer coefficient can be obtained by multiplying the results obtained from eq. (8.117) by 1.7. However, Thome (2004) pointed out that this correction is unnecessary for the boiling of a refrigerant on copper tubes. While Cooper (1984) attempted to a develop correlation that is valid for all fluids, Gorenflo (1993) proposed a fluid-specific correlation that includes the effect of the surface roughness, i.e., ′′ h = h0 FPF (q′′ / q0 )nf ( Rp / Rp 0 )0.133 , 0.0005<pr < 0.95 (8.118) where pr = p / pcrit is the reduced pressure and h0 is a reference heat transfer coefficient at fixed reference conditions: pr0 = 0.1, Rp0 = 0.4 μm, and ′′ q0 = 2.0 × 104 W/m 2 ; it is different for different fluids and is listed in Table 8.3. For the fluid that is not listed, h0 can be obtained by the experimental results or other correlations. The pressure correction factor is 0 2  1.73 pr .27 + [6.1 + 0.68 / (1 − pr )] pr , for water  FPF =  0.27 1.2 pr + 2.5 pr + pr / (1 − pr ), for all fluids except water and helium  (8.119) and the exponent in eq. (8.118), nf, decreases with increasing pr: 0 0.9 − 0.3 pr .3 , for water  nf =  0.15  0.9 − 0.3 pr , for all fluids except water and helium  (8.120) The surface roughness Rp is in μm. For the cases with unknown surface roughness, Rp is set as 0.4 μm. 8.4 Critical Heat Flux In nucleate boiling, heat flux increases and reaches a maximum value with increasing surface temperature. Further increase in the surface temperature results in decreasing heat flux because the transition from nucleate boiling to film boiling takes place. The maximum heat flux that can be obtained by nucleate boiling is referred to as critical heat flux (CHF). In the case of controlled heat flux, a slight increase of heat flux beyond the CHF can cause the surface temperature to rise to a value exceeding the surface material’s maximum allowable temperature. The CHF condition can cause a drop of one or two orders of magnitude in the value of heat transfer coefficient as compared with the value obtained in nucleate boiling (Collier and Thome, 1996). This in turn can cause severe damage or meltdown of the surface. The CHF is also referred to as burnout heat flux for this reason. The value of the critical heat flux is affected by hydrodynamic instabilities, wetting criteria, heat capacity, heater geometry, heating method, and pressure. The CHF phenomenon has received considerable attention in the past, and different mechanisms have been proposed to interpret its cause. Heat transfer in nucleate boiling involves (1) heat conduction through liquid from the heating Chapter 8 Boiling 701 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press surface to the liquid-vapor interface, (2) evaporation at the liquid-vapor interface, and (3) the escape of vapor from the heating surface. The critical heat flux phenomena occur when either generation or escape of vapor is restricted. Based on the assumption that CHF proceeds from the vapor escape limit, Zuber (1959) proposed one of the earliest successful models. This model was later refined by Lienhard and Dhir (1973). In the late stage of nucleate boiling, vapor bubbles join together to form vapor columns and slugs. The vapor columns and slugs are subject to the Helmholtz instability (Section 5.6.1), which occurs when small disturbances in the interface between counter-flowing liquid and vapor are amplified. As a result, the columns and slugs become distorted and block the liquid from wetting the heating surface. A vapor blanket forms over part of or the entire heating surface, and separates the heating surface and the liquid. Zuber (1959) postulated that the CHF is reached when the Helmholtz instability appears in the interface of the large vapor columns leaving the heating surface. These vapor columns are assumed to be in a square array. To determine the centerline spacing of the vapor columns, Lienhard and Dhir (1973) used the most dangerous wavelength in the two-dimensional wave pattern for the Taylor instability of the interface between a horizontal semi-infinite liquid region above a layer of vapor, λD . The diameter of each column is assumed to be λD / 2. For vertical liquid and vapor flow, the critical Helmholtz velocity is (Carey, 1992) σα ( ρ  + ρ v ) uc = u − uv = (8.121) ρ ρ v where the wave number α and the wavelength λD have the following relationship: 2π (8.122) α= λD Substituting eq. (8.122) into eq. (8.121) and considering ρ  ρ v , the critical Helmholtz velocity becomes 2πσ uc = (8.123) ρ v λD Since the density of the vapor is significantly lower than that of the liquid, the upward vapor column velocity is much higher than the downward liquid velocity. Therefore, the critical Helmholtz velocity can be approximated as uc  uv = ′′ qmax As ρ v hv Ac (8.124) where As/Ac is the ratio of the total surface area to the area of the surface occupied by the vapor column. Since the spacing of the centerline of the vapor column is λD and the diameter of the vapor column is λD / 2, the area ratio is As λD 2 16 = = (8.125) Ac π (λD / 2) 2 / 4 π 702 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Substituting eq. (8.125) into eq. (8.124), the critical Helmholtz velocity becomes uc = ′′ 16qmax πρ v hv (8.126) Combining eqs. (8.123) and (8.126), the critical heat flux is obtained: πρ h 2πσ ′′ qmax = v v (8.127) ρ v λD 16 The most dangerous wavelength, λD , as obtained by the Raleigh-Taylor instability analysis, was given by eq. (5.219), i.e., 3σ (8.128) λD = 2π ( ρ − ρv ) g Substituting eq. (8.128) into eq. (8.127), the critical heat flux becomes  σ g( ρ  − ρ v )  4 ′′ qmax = 0.149 ρ v hv   2 ρv   1 (8.129) Equation (8.129), which was obtained by Lienhard and Dhir (1973), is identical to the result of Zuber’s (1959) original model except that the constant on the right-hand side of the latter correlation is π / 24 = 0.131. Kutateladze (1948) obtained the critical heat flux by using a dimensional analysis based on the similarity between critical heat flux and column flooding as described by the Helmholtz instability. He also suggested that the constant should be 0.131. However, Lienhard and Dhir (1973) showed that the constant of 0.149 could provide better agreement with the available experimental data. The critical heat fluxes predicted by eq. (8.129) reasonably agreed with the experimental data for water, hydrocarbons, cryogenic liquids, and halogenated refrigerants. The experimentally measured CHF for the boiling of liquid metal, on the other hand, is two to four times higher than that predicted by eq. (8.129). This discrepancy is due to a strong conduction-convection mechanism of liquid metal that the above model did not account for. Equation (8.129) indicates that the critical heat flux is proportional to g1/4, which means the critical heat flux will be reduced under microgravity conditions. Equation (8.129) is applicable to saturated boiling. The critical heat flux increases with subcooling of the liquid according to Zuber et al. (1961) by the following correlation ′′ 1/ 2   qmax 5.3(Tsat − T ) = 1+ ( k ρ c p )  σ ( ρρ−2 ρv )g  ′′ qmax, sat ρ v hv v   −1/8  ( ρ − ρ v ) g    σ   1/ 4 (8.130) Equation (8.129) is applicable to an infinite horizontal heater surface, since there is no characteristic length in the expression. In order to use eq. (8.129), which was obtained by assuming equality in spacing between vapor columns and the most dangerous wavelength, λD , the length scale of the heater surface, L, Chapter 8 Boiling 703 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press must be greater than 2λD . Considering eq. (8.128), this criterion can be written in terms of confinement number: σ / [( ρ − ρ v ) g] 1 Co = (8.131) ≤ L 4π 3 An extensive summary of the critical heat fluxes for various geometric configurations can be found in Lienhard and Dhir (1973) and Lienhard and Hasan (1979). There has been some debate regarding how surface conditions affect the value for critical heat flux. Some experimental results (in accordance with Zuber’s theory) show that there is no relationship between the two. To be more specific, surface-roughening techniques such as scoring and sandblasting have been found not to greatly influence the critical heat flux. In contrast, some experiments have shown that certain surface conditions (such as oxidation and deposition) do seem to increase the value of critical heat flux by increasing wettability of the fluid. The critical heat flux is significantly reduced on essentially nonwetting surfaces. In general, if the surface condition can directly increase the wettability of the fluid, then the value of the critical heat flux may be increased. In other circumstances, no general conclusion will apply in every instance. Example 8.3 An electrically heated 5 mm platinum wire boils water at atmospheric conditions. Find the maximum heat flux and surface temperature that can be attained in the nucleate boiling regime. Solution: The saturation temperature of water at 1 atm is Tsat = 100 o C and the properties are as follows: hv = 2251.2 kJ/kg, σ= 3 3 −3 ρ = 958.77 kg/m , ρ v = 0.5974 kg/m , 58.9 × 10 N/m, c p = 4.22 kJ/kg-K, Pr = 1.76 , and μ = 279 × 10−6 N-s/m 2 . The critical heat flux can be found from eq. (8.129), i.e.,  σ g( ρ  − ρ v )  4 ′′ qmax = 0.149 ρ v hv   2 ρv    58.9 × 10−3 × 9.8 × (958.77 − 0.5974)  4 = 0.149 × 0.5974 × 2251.2 × 10 ×   The 0.5974 2   2 = 1257.3kW/m 3 1 1 surface temperature can be estimated from eq. (8.116). For water boiling on the platinum surface, we have C s , f = 0.013 and n = 1 , i.e., Tw = Tsat  h q′′ + v Prn C s ,   μ hv c p   3   σ    g( ρ  − ρ v )     1 2 1 704 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 13    2  2251.2 1257.3 58.9 × 10−3 1  = 100 + × 1.76 × 0.013 × ×   279 × 10−6 × 2251.2  9.8 × (958.77 − 0.5974)   4.22     o = 120.9 C. 1 8.5 Transition Boiling and Minimum Heat Flux 8.5.1 Transition Boiling The transition boiling regime is probably the least studied region of the boiling curve because: (1) transition boiling is less important, practically speaking, than nucleate boiling, and (2) the mechanism of transition is very complicated and experimental investigation is very difficult. In this region, the liquid periodically contacts the heating surface creating large amounts of vapor and forcing the liquid away from the surface. The combined result is an unstable vapor film or blanket. Then the film collapses, allowing the liquid to contact the heating surface again. It is a region with alternate nucleate and film boiling, and is bypassed if the boiling curve experiment is performed with a controlled-heat flux setting. It is only obtainable by experiments with controlled surface temperature, which are very rare in practical applications. The boiling curve in Fig. 8.3 shows that the excess temperature ΔT for transition boiling at one atmosphere is about 30 to 120 °C. The limited conditions conducive to this regime in turn limit its practical appeal to many investigators. However, investigation of transition boiling can help the experimenter to avoid burnout caused by excessive surface temperature. It is also possible to reduce the heat flux fluctuations that result from unstable dry regions in contact with the heating surface. Witte and Lienhard (1982) reported the existence of a hysteresis in transition boiling and suggested that there are two distinct boiling curves: the nucleatetransition boiling curve, obtained by increasing the excess temperature from nucleate boiling, and the film-transition boiling curve, created by decreasing excess temperature from film boiling (see Fig. 8.19). If the liquid wets the heating surface very well, which is generally presumed to be the case, the boiling curve does not progress immediately into the transition regime once the excess temperature exceeds its value corresponding to the critical heat flux. Witte and Lienhard (1982) suggested that the low contact angle made it easier for the liquid to spread over the surface, which promotes vigorous nucleation instead of the expected transition boiling. The curve also deviated from the conventional pool boiling curve when film boiling was initiated first, and the curve was then traversed by decreasing the excess temperature. As the point of minimum heat flux is approached, vapor is not produced as rapidly. The vapor layer becomes less stable, and the film breaks down. The lower the contact angle (i.e., the better Chapter 8 Boiling 705 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 8.19 Transition boiling curves. the wetting characteristics of the liquid), the higher the temperature or heat flux at which the curve departs from film boiling. In this case, the boiling curve departs from film boiling without first reaching the minimum heat flux. Recent experimental work in transition boiling did not support the existence of a hysteresis, because there was no evidence of two distinctive boiling curves. In an experimental work by Hohl et al. (1996), the entire boiling curve for FC-72 under steady and transient conditions was measured. The heater surface was a horizontal circular flat plate of 34 mm in diameter, coated with a 20 μm nickel layer. The surface temperature was controlled via feedback control. The typical boiling curve is shown in Fig. 8.20. As is evident, the boiling curves are identical regardless of the direction of the temperature change. The contradicting evidence with regard to the existence of hysteresis in the boiling curve could be due to different geometric configurations and the temperature control system. Another possible cause is contamination, since some combinations of liquid and solid surface can affect wettability. At this time, there is no universal criterion for the existence of hysteresis in transition boiling. Heat transfer in the transition boiling regime is very challenging to predict because it is the most complex regime in pool boiling. Contradiction about the existence of a hysteresis makes accurately predicting heat transfer even more difficult. Transition boiling can be viewed as a combination of unstable nucleate boiling and unstable film boiling at different locations, or at different times at the same location. Therefore, the heat flux for transition boiling can be expressed as (Berenson, 1962) ′′ ′′ q′′ = Fq + (1 − F )qv (8.132) where F is the average proportion of the heating surface in contact with liquid at any given moment. It can be estimated by the following correlation: 706 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 8.20 Boiling curves of saturated FC 72 (Hohl et al., 1996).   ΔT F = exp  −2.2 + 2 ΔTCHF   (8.133) ′′ where ΔTCHF is the excess temperature at critical heat flux (CHF), qmax . The heat flux during the vapor contact can reasonably be assumed to be equal to the minimum heat flux, i.e., ′′ ′′ qv = qmin (8.134) The heat flux during the liquid contact, q′′ , is time-dependent. Its timeaveraged value can be correlated as ′′ ′′ ′′ 1 − 0.18qmin / qmax q = (8.135) ′′ 0.82ΔT / ΔTCHF qmax ′′ The minimum heat flux, qmin , in the above correlations needs to be determined in the next subsection. 8.5.2 Minimum Heat Flux The minimum heat flux point in the boiling curve is the boundary between the transition boiling regime and the film boiling regime. The minimum heat flux, ′′ qmin , is reached when the heat flux is equal to the minimum vapor formation rate that can sustain a stable vapor film over the heating surface. In controlled-heat flux pool boiling, nucleate boiling is reestablished when the heat flux falls below the minimum heat flux. The minimum heat flux is referred to as the Leidenfrost point. The vapor in the vapor film can be released by regular generation of vapor bubbles at some point and time. The minimum heat flux is related to the vapor bubble release rate, and can be estimated by (Zuber, 1959) Chapter 8 Boiling 707 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ′′ ′′ qmin = eb nb f min (8.136) where eb represents the energy per vapor bubble, nb′′ is the number of vapor bubbles released per unit area and per release cycle, and fmin is the minimum number of cycles per second needed to compensate for normal collapse rate. The energy per bubble is assumed to be equal to the latent heat carried away by the bubble with a radius equal to λD / 4 and released at the node of the Taylor wave: 3 4π  λD  (8.137) eb =   ρ v hv 34 Zuber (1959) postulated that the vapor generated at the liquid-vapor interface was released at the node and antinode of a two-dimensional Taylor wave pattern, or once per half cycle, i.e., (8.138) 2 λD The bubble release frequency, fmin, is f min = C1 β max (8.139) where β max is the frequency of the most rapidly growing disturbance obtained by an interfacial stability analysis:  4( ρ − ρ v )3 g3  (8.140) β max =   3  27( ρ + ρ v ) σ  The minimum heat flux can be obtained by substituting eqs. (8.137) – (8.139) into eq. (8.136) and using eq. (8.128) to obtain λD , i.e.,  gσ ( ρ  − ρ v )  4 ′′ qmin = C ρ v hv  (8.141) 2  ( ρ + ρ v )  where C = C1 (π 2 / 12)(4 / 3)3 is a new constant. Berenson (1961) recommended C 1 ′′ nb = 2 = 0.09 by fitting the experimental data for pool boiling so that eq. (8.141) becomes  gσ ( ρ − ρ v )  4 ′′ qmin = 0.09 ρ v hv  2  ( ρ + ρv )  1 (8.142) Equation (8.142) provides minimum heat flux for a horizontal surface; it is accurate within approximately 50% for most fluids at moderate pressure and is less accurate for higher pressure. Lienhard and Witte (1985) provided a much more detailed analysis on this subject. Example 8.4 Estimate the minimum heat flux for pool boiling of water at one atmosphere. When pressure increases, how will the minimum heat flux change? 708 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Solution: The properties of water at 1 atm can be found in Example 8.3. The minimum heat flux for pool boiling of water can be obtained from eq. (8.142), i.e.,  gσ ( ρ − ρ v )  4 ′′ qmin = 0.09 ρ v hv  2  ( ρ + ρv )   9.8 × 58.9 ×10−3 × (958.77 − 0.5974)  4 = 0.09 × 0.5974 × 2251.2 ×   (957.85 + 0.5974) 2   = 18.95 kW/m 2 1 1 To estimate the effect of pressure on the minimum heat flux, it is necessary to simplify eq. (8.142). Considering the fact that ρ  ρ v , eq. (8.142) can be simplified as  gσ  4 ′′ qmin = 0.09 ρ v hv    ρ  1 Since the vapor density, ρ v , is a strong function of pressure, i.e., it increases significantly with increasing pressure, it is expected that the ′′ minimum heat flux qmin also increases with increasing pressure. 8.6 Film Boiling 8.6.1 Film Boiling Analysis When the excess temperature increases beyond the minimum heat flux point, boiling enters the film boiling regime, characterized by a continuous vapor film covering the entire surface. The major thermal resistance is in this vapor layer. Unlike the vapor film in the transition boiling regime, that in the film boiling regime is stable. Since the liquid is separated from the heating surface by a stable vapor film, heat transfer in this regime can be obtained by analyzing evaporation at the liquid-vapor interface. Figure 8.21 shows film boiling from a heated vertical flat plate in a motionless pool of subcooled liquid. Vapor generated at the liquid-vapor interface flows upward due to buoyancy force. Drag force from the vapor causes the adjacent liquid to flow upward as well. Therefore, a velocity boundary layer forms in the liquid phase adjacent to the vapor film. Successful solution of the film boiling problem requires solutions of vapor and liquid flow as well as heat transfer in the vapor phase and at the interface. Since the liquid phase is subcooled, there is also heat transfer in the liquid. The vapor layer thickness is usually small compared to the length scale of the vertical flat plate, so boundary layer approximations are applicable to the vapor film. It is further assumed that Chapter 8 Boiling 709 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 8.21 Film boiling on a vertical surface. the vapor flow is laminar and two-dimensional. The continuity, momentum, and energy equations in the vapor layer are ∂uv ∂vv + =0 ∂x ∂y (8.143) (8.144) (8.145) uv ∂uv ∂u ∂ 2 uv g( ρ − ρ v ) + vv v = ν v + ρv ∂x ∂y ∂y 2 uv ∂Tv ∂T ∂ 2Tv + vv v = α v ∂x ∂y ∂y 2 Vapor flow drags the liquid adjacent to the vapor film, but the rest of the liquid phase is stationary. Therefore, boundary layer theory can be applied to describe the motion of the liquid, i.e., ∂u ∂v + =0 ∂x ∂y (8.146) (8.147) u ∂u ∂u ∂ 2 u + v  = ρ gβ  (T − T∞ ) + ν  ∂x ∂y ∂y 2 There will also be a thermal boundary layer in the liquid phase, and its temperature satisfies u ∂T ∂T ∂ 2T + v  = α  ∂x ∂y ∂y 2 (8.148) The boundary conditions at the heated wall (y = 0) are (8.149) The boundary conditions at the liquid-vapor interface ( y = δ ) are necessary to couple the fluid flows in the liquid and vapor phases. The velocities for both phases are the same due to the nonslip condition, i.e. y = 0 : uv = vv = 0, T = Tw 710 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 8.22 Mass balance at the liquid-vapor interface. (8.150) The shear stresses calculated from the liquid and vapor phases are the same as required by force balance, i.e.,  ∂u   ∂u  μ  = μ  , y =δ  ∂y  v  ∂y  y = δ : uv = u (8.151) The mass balance at the liquid-vapor interface can be obtained by applying conservation of mass for a control volume shown in Fig. 8.22, i.e., dδ dδ    − ρv  =  ρu − ρv  (8.152)  ρu  dx v  dx  The temperature at the liquid-vapor interface is equal to the saturation temperature: Tv = T = Tsat , y = δ (8.153) The energy balance at the liquid-vapor interface is ∂T ∂T  dδ  ′′ −kv v + qwR = ρ v  uv − vv  − k  (8.154) ∂y  dx  ∂y ′′ where qwR is heat flux due to radiation from the heat wall to the interface. The boundary condition in the liquid far from the heated surface is u → 0 , y → ∞ (8.155) For a case in which the liquid is saturated ( T = Tsat ), the first term on the right-hand side of eq. (8.147), representing buoyancy force in the liquid phase, will disappear. The energy equation (8.148) for the liquid phase will not be needed. The similarity solution for the film boiling in a saturated liquid was obtained by Koh (1962) and will be presented here. Defining the stream functions in the vapor and liquid phases as ∂ψ v ∂ψ v ∂ψ  ∂ψ uv = , vv = − ; u = , v = −  (8.156) ∂y ∂x ∂y ∂x Chapter 8 Boiling 711 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press the continuity equations (8.143) and (8.146) are automatically satisfied by the stream functions defined by eq. (8.156). To obtain the similarity solution, the following similarity variables are defined:  g ρ − ρv  η = 2    4ν v ρ v  1/ 4 y x1/ 4 x 3/ 4 f (η ) (8.157) (8.158) (8.159)  g ρ − ρv  ψ v = 4ν v  2    4ν v ρ v  T − Tsat θ= Tw − Tsat 1/ 4 where η is the independent similarity variable and f (η ) is the nondimensional stream function in the vapor phase. Substituting eqs. (8.157) and (8.158) into eq. (8.156), the velocity components in x- and y- directions are respectively uv =  g ρ − ρv  ∂ψ v ∂ψ v ∂η = = 4ν v  2   ∂y ∂η ∂y  4ν v ρ v  1/ 4 1/ 2 x1/ 2 f '(η ) x −1/ 4 [η f '(η ) − 3 f (η ) ] (8.160)  g ρ − ρv  ∂ψ v ∂ψ v ∂η vv = − =− =νv  2   ∂x ∂η ∂x  4ν v ρ v  (8.161) Substituting eqs. (8.160) – (8.161) into eq. (8.144), the momentum equation in the vapor phase becomes f ′′′ + 3 ff ′′ − 2( f ′) 2 + 1 = 0 (8.162) Substituting eqs. (8.157) and (8.159) into eq. (8.145), the nondimensional energy equation in the vapor layer is obtained: θ ′′ + 3 Prv f θ ′ = 0 (8.163) The momentum equation in the liquid phase can also be transformed into an ordinary differential equation: F "'+ 3FF "− 2( F ') 2 = 0 (8.164) where the similarity variables in the liquid phase are  g ρ − ρv  ξ = 2   ρv   4ν  1/ 4 y x1/ 4 x 3/ 4 F (ξ ) (8.165) (8.166)  g ρ − ρv  ψ  = 4ν   2   ρv   4ν  1/ 4 Therefore, the film boiling problem can be described with a set of ordinary differential equations (8.162) – (8.164). To solve these ordinary differential equations, the boundary conditions specified by eqs. (8.149) – (8.155) must be rewritten in terms of similarity variables. Substituting the nondimensional variables defined by eqs. (8.157) – (8.159) and eqs. (8.165) – (8.166) into eq. (8.149) – (8.155), the boundary conditions in terms of nondimensional variables become 712 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press f (0) = f '(0) = 0 θ (0) = 1  ρ μ 2 F (ξδ ) =  v v  f (ηδ )  ρ μ  F '(ξδ ) = f '(ηδ ) ρ μ  F "(ξδ ) =  v v  f "(ηδ )  ρ  μ  θ (ηδ ) = 0 1 2 1 (8.167) (8.168) (8.169) (8.170) (8.171) (8.172) F '(∞) = 0 (8.173) The film boiling problem is now described by a set of three ordinary differential equations with their corresponding boundary conditions. The film boiling problem is not closed because there is no equation that governs the thickness of the vapor film. It is therefore necessary to develop another equation to find the vapor film thickness. This additional equation can be found by performing the energy balance at the interface. In the control volume shown in Fig. 8.23, the mass flow rate per unit width of the surface at x is Γ v ( x ) =  ρ v uv dy 0 δ (8.174) (8.175) The mass flow rate at x+dx is obtained by Γ v ( x + dx ) = Γ v ( x ) + dΓ v dx dx Figure 8.23 Energy balance at the liquid-vapor interface. Chapter 8 Boiling 713 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press The increase of vapor mass flow rate is due to evaporation at the interface and is related to heat transfer as dq′ = hv dΓ v = hv d dx (  ρ u dy ) dx δ 0 v v (8.176) On the other hand, heat transfer at the interface can be obtained by applying Fourier’s law at the interface, i.e., dq′ = kv ∂T ∂y dx y =δ (8.177) If all the heat transferred to the system goes into latent heat of vaporization, one can combine eqs. (8.176) and (8.177) to obtain the energy balance at the liquid-vapor interface. ∂T ∂y = y =δ hv d δ ρ v uv dy kv dx 0 (8.178) Equation (8.178) can be rewritten in terms of the similarity variables defined in eqs. (8.157) – (8.159), i.e., 3 f (ηδ ) c p , v (Tw − Tsat ) = (8.179) θ ′(ηδ ) hv Prv At this point, the film boiling problem is mathematically closed and can be solved numerically. This is a boundary value problem that can be solved by Figure 8.24 Dimensionless velocity and temperature profiles ( Prv = 1.0, [( ρ v μv ) /( ρ  μ )] = 0.01 ; Koh, 1962). 714 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press using the shooting method. The velocity profile in the vapor and liquid, as well as the temperature profile in the vapor layer – obtained by Koh (1962) for the case of Prv = 1.0, ( ρ v μv ) / ( ρ μ ) = 0.01 – is shown in Fig. 8.24. The local heat transfer coefficient is 1 ∂T ∂θ hx = − kv = −kv (8.180) Tw − Tsat ∂y y=0 ∂y y=0 Substituting eq. (8.157) into eq. (8.180), the local heat transfer coefficient becomes  g ρ − ρv  hx = −kv  2    4ν v ρ v  1/ 4 θ '(0) x1/ 4 1/ 4 (8.181) It follows that the local Nusselt number is of the form Nu x =  ( ρ − ρ v ) gx 3  hx x = [−θ ′(0)]    2 kv  4 ρ vν v  (8.182) where θ ′(0) is the first-order derivative of nondimensional temperature at η = 0 . It is a function of Prv and ( ρ v μv ) / ( ρ μ ) , and it can be obtained by numerical solution of the problem. The effect of subcooling on boiling from a vertical plate was studied by Sparrow and Cess (1962). Figure 8.25 shows the effect of subcooling on Nusselt number for film boiling of water with Tsat = 467 °F from a vertical plate at Tw = 767 °F. It can be seen that liquid subcooling can have a significant effect on the Nusselt number. When the liquid subcooling is very large, the Nusselt number approaches that for the cases of free convection. Figure 8.25 Effect of subcooling on Nusselt number for film boiling from a vertical plate (Sparrow and Cess, 1962). Chapter 8 Boiling 715 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Forced convection laminar film boiling from a horizontal constant temperature plate for the cases of saturated and subcooled liquids is solved by liquid and vapor boundary layer equations by Cess and Sparrow (1961a, b). The result for saturated film boiling is presented below: μ 1 1/ 2 K = Nu Re −1/ 2  v 2  μ  −1/ 2  μ v 1/ 2  1 + π Nu Re    μ      1/ 2 (8.183) where ρ μ  Prv hv K = v v   ρ μ  c p , v (Tw − Tsat ) hx Nu = k ux Re = ∞ (8.184) (8.185) (8.186) ν where x is distance from leading edge, and u∞ is liquid forced velocity far from the plate. Equation (8.183) is valid for 10−3 ≤ K ≤ 1 . Gravity is negligible in the above analysis since forced convection was the driving force and therefore the result for u∞ = 0 is not meaningful. Liquid subcooling will increase both heat transfer and plate shear. The thermophysical properties of both liquid and vapor in the above analysis were assumed to be independent from the temperature. The influence of constantproperties assumption was examined by Nishikawa et al. (1976) by numerically solving the two-phase boundary layer equations. They confirmed that, except at high system pressure, the results based on the constant-properties assumption reasonably agreed with the results based on variable properties. Nakayama (1986) also provided additional analysis related to forced convective film subcooled boiling for cases of constant surface temperature. The analysis developed above do not account for heat transfer due to radiation from the heating surface across the vapor layer to the liquid. When the surface temperature is very high, the radiation across the vapor film contributes more and more to the overall heat transfer rate from the heating surface to the liquid. The rate of vapor production increases and the vapor layer thickens when radiation is significant. A detailed analysis of the radiation heat transfer must be performed in order to accurately predict the film boiling heat transfer. Since the vapor is virtually transparent to infrared radiation emitted by the heating surface and liquid, the vapor can be treated as a nonparticipating medium; this assumption significantly simplifies the modeling of radiation heat transfer. The vapor films in film boiling are usually thin, allowing the radiation heat transport to be modeled as the radiation exchange between two parallel plates. Heat transfer due to radiation acts independently from the heat transfer from conduction and convection. The total heat transfer can be found by adding the contribution from radiation directly to the contributions from conduction and convection. Since the heating surface and the interface are treated as two parallel 716 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press plates with different temperatures, the contribution from radiation can be obtained by 4 σ (T 4 − Tsat ) ′′ qrad = SB w (8.187) 1 εs εI where σ SB is the Stefan-Boltzmann constant, and ε s and ε I are the emissivity of the heating surface and the interface, respectively. The effective heat transfer coefficient due to radiation is then obtained by 2 ′′ qrad σ (T 2 + Tsat )(Tw + Tsat ) = SB w hrad = (8.188) + 1 −1 εs εI The overall heat transfer coefficient for film boiling can be obtained by (Bromley, 1950) h  h = hcon  con  h 1/3 Tw − Tsat 1 + 1 −1 + hrad (8.189) where hcon is the convective heat transfer coefficient determined by eq. (8.181). Equation (8.189) is not convenient for determining the overall heat transfer coefficient h, since h appears on both sides of the equation. Bromley (1950) further recommended that eq. (8.189) could be simplified to h = hcon + 0.75hrad (8.190) The difference between the overall heat transfer coefficient obtained by eqs. (8.189) and that obtained by eq. (8.190) is within 5%. Film boiling near an immersed body, such as a cylinder or sphere, can be solved by the integral approximation method. The average Nusselt number for film boiling on a horizontal cylinder (Bromley, 1950) or sphere (Lienhard and Dhir, 1973) is given by  ρ g( ρ  − ρ v )h′v D 3  4 hD Nu = =C v  kv μv kv   1 (8.191) where  0.4c pv (Tw − Tsat )  h′v = hv 1 +  hv   (8.192) The correlation constants C for horizontal cylinders and spheres are 0.62 and 0.67, respectively. The vapor properties are evaluated at the film temperature, T f = (Tw + Tsat ) / 2. Equation (8.191) is valid for 0.8 < λc / D ≤ 8 where λc = 2π {σ / [( ρ − ρ v ) g]}1/ 2 is critical wavelength, or as long as the diameter of the horizontal cylinder or sphere is much greater than the vapor film thickness. Another useful geometric configuration is film boiling over a horizontal, large flat plate, with the dense liquid overlaying a less-dense vapor. Vapor leaves the film in the form of detaching bubbles, which determines the heat transfer rate. Berenson (1961) performed an analysis of film boiling on a horizontal heat surface in which he assumed that the bubbles are released at the node of the Chapter 8 Boiling 717 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Taylor wave. The correlation resulting from the Taylor instability analysis is given by 14 3    kv g ρ v ( ρ  − ρ v )h′v   g( ρ  − ρ v )  2  h = 0.425     σ    μv (Tw − Tsat )     1 (8.193) which is in good agreement with experimental results obtained by using npentane and carbon tetrachloride. Klimenko (1981) correlated the film boiling from a horizontal plate 0.19 Ga ( ρ / ρ − 1) 1/3 Pr1/3 f Ga ( ρ / ρ v − 1) < 108 (Laminar) 1  v v    Nu =  1/ 2 1/3 0.0086 Ga ( ρ / ρ v − 1)  Prv f 2 Ga ( ρ / ρ v − 1) > 108 (Turbulent)    (8.194) where 1  f1 =  −1/3 0.89Ja v  1  f2 =  −1/ 2 0.71Ja v  Ja v ≥ 0.71 Ja v < 0.71 Ja v ≥ 0.5 Ja v < 0.5 (8.195) (8.196) where Nu = hλc k c p , v (Tw − Tsat ) (8.197) (8.198) (8.199) Ja v = hv gλc3 2 2ν v Ga = are Nusselt, Jakob, and Galileo numbers, respectively. All vapor properties are evaluated at (Tw + Tsat ) / 2. Equation (8.194) is valid for 7 ×104 < Ga( ρ  / ρ v − 1) < 3 × 108 , 0.69 < Prv < 3.45, 0.14 < Ja v < 32.26, 0.0045 < p / pcrit < 0.98 and up to 21.7g. The effect of plate size can be accounted for by L / λc > 5 h 1, (8.200) = 0.67 h∞ 2.9(λc / L ) , L / λc < 5 where L is the minimum dimension of the horizontal plate. Example 8.5 Film boiling at 1 atm occurs on the surface of a 2 cm-diameter sphere made out of polished copper. The surface temperature of the sphere is Tw = 360 °C. The emissivity of the sphere surface is ε w = 0.05, but the liquid-vapor interface can be treated as a black body. Find the heat transfer coefficients due to convection and radiation, and the overall heat transfer coefficient. 718 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Solution: The liquid density and latent heat are evaluated at Tsat = 100 o C , i.e., ρ = 958.77 kg/m3 , hv = 2251.2 kJ/kg . The properties of the vapor must be evaluated at the mean temperature of the vapor, (Tw + Tsat ) / 2 = 230 o C, i.e., ρ v = 0.4381 kg/m3 , μv = −5 2 1.715 × 10 N-s/m , kv = 0.0341 W/m-K , and c pv = 1.986 kJ/kg-K . The corrected latent heat is obtained from eq. (8.180):  0.4c pv (Tw − Tsat )  h′v = hv 1 +  hv    0.4 × 1.986 × (360 − 100)  = 2251.2 × 1 +  = 2457.7 kJ/kg 2251.2   The Nusselt number for film boiling can be obtained from eq. (8.191), i.e.,  ρ g( ρ  − ρ v )h′v D 3  4 Nu = 0.67  v  μv kv    0.4381× 9.8 × (958.77 − 0.4381) × 2457.7 ×10 × 0.02  = 0.67 ×   = 408.6 1.715 × 10−5 × 0.0341   3 3 1 4 1 The heat transfer coefficient due to film boiling is hcon = kv Nu 0.0341× 408.6 = = 696.7 W/m 2 -K D 0.02 The radiative heat transfer coefficient is 2 σ (T 2 + Tsat )(Tw + Tsat ) hrad = SB w 1 / εw +1 / ε I −1 = 5.67 ×10−8 × (6332 + 3732 ) × (633 + 373) = 1.54 W/m-K 1 / 0.05 + 1 / 1.0 − 1 The overall heat transfer coefficient can be obtained from eq. (8.190), i.e., h = hcon + 0.75hrad = 696.7 + 0.75 × 1.54 = 697.8W/m-K 8.6.2 Direct Numerical Simulation of Film Boiling In additional to the above analytical results about film boiling, numerical solutions on this topic were also reported in the recent literature (Esmaeeli and Tryggvason, 2004). Son and Dhir (1997) simulated film boiling for both twodimensional and axisymmetric flows using a moving body-fitted coordinate system, which is limited to modest deformation of the phase boundary. This limitation was later overcome by Juric and Tryggvason (1998) using a front tracking method and Son and Dhir’s (1997) using a level set method. The subcooled film boiling was solved by Banerjee and Dhir (2001) using a level set Chapter 8 Boiling 719 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press method. The Volume of Fluid (VOF) method was employed to simulate film boiling by Welch and Wilson (2000). In addition to the above work that use one method, the front tracking method was also combined with the finite difference method (Esmaeeli and Tryggvason, 2001; 2003) and level contour technique (Shin and Juric, 2002). The direct numerical simulation of film boiling conducted by Esmaeeli and Tryggvason (2004) will be presented here. The continuity, momentum, and energy equations for each phase in the boiling flow are ∂ρ (8.201) + ∇ ⋅ ( ρ V) = 0 ∂t ∂ρ V + ∇ ⋅ ( ρ VV ) = −∇p + ρ g + ∇ ⋅ μ (∇V + ∇V T ) ∂t ∂ρ c p T + ∇ ⋅ ( ρ c p VT ) = ∇ ⋅ (k∇T ) ∂t (8.202) (8.203) where the viscosity dissipation has been neglected in eq. (8.203). At the liquidvapor interface, the following jump conditions must be satisfied:  ρ (V − VI ) ⋅ n = ρ v (Vv − VI ) ⋅ n = m ′′ (8.204)  m ′′(Vv − V ) = ( τ v − τ  ) ⋅ n − ( pv − p )I ⋅ n + σ K n (8.205)  ′′ m ′′hv = qI = kv ∂T ∂T − k ∂n v ∂n (8.206)  where V and Vv are the fluid velocities at the liquid and vapor sides of the interface, and VI is the interfacial velocity. The above momentum and energy equations and their jump conditions can be rewritten as the following “single-field” form: ∂ρ V + ∇ ⋅ ( ρ VV ) = −∇p + ρ g + ∇ ⋅ μ (∇V + ∇V T ) + σ  δ (r − rI ) K n I dAI I ∂t (8.207) ∂ρ c p T ∂t  T + ∇ ⋅ ( ρ c p VT ) = ∇ ⋅ (k∇T ) − 1 − (cv − c ) sat   δ (r − rI )q′′dAI I hv  I  (8.208) where δ is a two- or three- dimensional delta-function, r is the position vector and rI is the position of the interface. The subscript I in eqs. (8.207) and (8.208) represents interface. Although both liquid and vapor can be treated as incompressible flows, the incompressible assumption near the interface is invalid. In order to rewrite the continuity equation and the jump condition into a single-field equation, the velocity field can be rewritten as V = Vv H + V (1 − H ) (8.209) where H is a Heaviside function that has a value of one in vapor but zero in the liquid phase. The gradient of the Heaviside function can be expressed in terms of the interfacial properties: 720 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press ∇H =  δ (r − rI )n I dAI I (8.210) which is zero everywhere except at the interface. The divergence of the velocity defined in eq. (8.209) is (8.211) Substituting eq. (8.210) into eq. (8.211) and considering liquid and vapor are incompressible ( ∇Vv = ∇V = 0 ), one obtains ∇ ⋅ V = H ∇ ⋅ Vv + (1 − H )∇ ⋅ V + (Vv − V ) ⋅∇H ∇ ⋅ V =  δ (r − rI )(Vv − V ) ⋅ n I dAI I (8.212) Eliminating VI in eq. (8.204) yields 1 1  q′′  1 1  − = − (Vv − V ) ⋅ n = m ′′    ρ v ρ  hv  ρ v ρ  (8.213) Substituting eq. (8.213) into eq. (8.212), the single-field continuity equation becomes ∇⋅V = q′′  1 1 ′′ −   δ (r − rI )qI dAI  hv  ρ v ρ  I (8.214) Esmaeeli and Tryggvason (2004) solved the single-field governing equations (8.207), (8.208), and (8.214) using a second- order space-time accurate fronttracking/finite difference method on a staggered grid. They obtained a numerical solution of film boiling on a horizontal surface where a thin vapor film separates the liquid and the heated surface. Phase change takes place at the liquid-vapor interface. Evolution of a liquid vapor interface and velocity field during film boiling is shown in Fig. 8.26. The properties of the liquid and vapor correspond (a) t = 0 (b) t = 8.38 (c) t = 16.76 Figure 8.26 Film boiling on a heated horizontal surface (Pr = 4.2, Gr = 17.85, Ja = 0.064, ρ v / ρ  = 0.21, μv / μ = 0.386, cv / c = 1.83; ; Esmaeeli and Tryggvason, 2004; Reprinted with permission from Elsevier). Chapter 8 Boiling 721 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press to the properties of water at psat = 169 bar, with the exception of the Grashof number. Initially, a quiescent liquid pool rests on a hot horizontal plate blanketed by a thin vapor film, as shown in Fig. 8.26(a). The initial temperatures of liquid and vapor are at the saturation temperature. As the horizontal plate is heated, liquid vapor phase change takes place at the interface and a large vapor bubble is formed as is shown in Fig. 8.26(b). As the heating continues, the neck of the bubble shrinks and the vapor bubble rises due to buoyancy effect [see Fig. 8.26(c)]. 8.6.3 Leidenfrost Phenomena A drop of liquid is introduced to a solid surface of constant temperature. If the temperature of the solid is around the boiling point of the liquid, the drop will rapidly boil and evaporate. However, if the solid is held at a temperature much higher than the boiling point of the liquid, a thin film of vapor forms between the solid and the liquid (see Fig. 8.27). A drop that floats on its own vapor in this way is called a Leidenfrost drop, named after the German physician, Johann Gottlob Leidenfrost, who first reported the phenomenon in 1756. The thin vapor film keeps the liquid and solid from coming into direct contact with each other thus inhibiting bubble nucleation. The film also acts as an insulator for the liquid above it, thereby slowing the evaporation process such that, for Liquid Drop Vapor Film 2R λ δ Hot Plate a. Semi-spherical drop (R<Lc) 2R Liquid Drop Vapor Hot Plate b. Puddle liquid drop (R>Lc) λ h δ Figure 8.27 Schematic of a Leidenfrost drop hovering over a solid hot surface with radius R and vapor film thickness δ. 722 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press example, a droplet of water with radius of 1 mm can float for an entire minute over a metallic surface at 200 oC to evaporate completely. A variety of technological applications involve imposing liquid drops on a hot surface, therefore making the Leidenfrost phenomenon an important topic of study. Examples of such applications include spray cooling of hot metal during metallurgical production, film cooling of a rocket nozzle, mist flow heat transfer in evaporators, fuel droplet vaporization in fuel-injected engines, and reflooding of a nuclear reactor core after an accidental loss of coolant. The Leidenfrost Effect can be easily observed by placing a few drops of water on a hot frying pan (not nonstick). When the pan reaches the Leidenfrost temperature, the water droplets will bead up and “dance” across the surface due to the immediate vaporization of the bottom of the liquid drop. Physicists have hypothesized that firewalkers (people who walk barefooted across hot beds of coal) used the Leidenfrost phenomenon to their advantage to impress crowds. However, after several injurious experiments, they lost faith in their argument and decided instead that the entertainers completed this feat due to a combination of other hidden conditions (such as a layer of insulating ash). To explain the Leidenfrost phenomenon, the lifetime τ of a water droplet and an n-heptane droplet are shown as a function of the solid plate temperature in Figs. 8.28 and 8.29, respectively. The lifetime of the water droplet decreases to around 200 ms at temperatures below 100 oC. As the plate temperature increases from 100 oC to 150 oC, the vapor film begins to form and a considerable increase in the droplet’s lifetime is observed. The temperature at which the longest droplet lifetime is recorded is known as the Leidenfrost temperature. This temperature, and correspondingly the maximum droplet lifetime, is dependent on a number of factors including solid roughness, liquid purity, and the manner in which the drop is placed on the solid. At temperatures greater 100 Evaporation time(s) 80 60 40 20 0 0 Transition Boiling Leidenfrost Film Boiling 100 TLeid 200 Tw,°C 300 400 Figure 8.28 Lifetime of a millimetric water droplet of radius Ro = 1mm deposited on a Duralumin plate of temperature T [Reused with permission from Anne-Laure Biance et al., Physics of Fluids, 15, 1632 (2003), Copyright 2003, American Institute of Physics]. Chapter 8 Boiling 723 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 10 8 Evaporation time(s) 6 4 2 TLeid 0 60 100 140 Tw,°C 180 220 260 Figure 8.29 Lifetime of an n-heptane droplet of radius Ro = 0.75 mm deposited on a stainless steel surface of temperature T (Chandra and Avedisian, 1991). than the Leidenfrost temperature, the lifetime of the droplet gradually decreases. The same trend is true for n-heptane for which the Leidenfrost temperature is 200 o C. Leidenfrost drops are considered nonwetting and exhibit levitation characteristics. The capillary length Lc for a liquid drop is defined as σ Lc = (8.215) ρg where σ is the liquid surface tension and ρ is the liquid density. If the radius R is smaller than the capillary length, the drop is spherical except for a flattened bottom [Fig. 8.27(a)]. Contact is taken as the region of the drop interface parallel to the solid surface. The relationship between the contact size λ, radius R, and capillary length Lc is found by combining the correlation established by Mahadevan et al. (1999) with the geometric Hertz relation λ~ R2 Lc (8.216) If the radius R is larger than the capillary length, the drop forms a puddle flattened by gravity [Fig. 8.27(b)]. In this case, the contact size λ is related only to the radius R: 724 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press λ~R (8.217) The puddle’s thickness h is found by balancing the surface tension and the hydrostatic force, yielding h = 2 Lc (8.218) It has been observed that the radius R of a Leidenfrost drop of water must be on the order of 1 cm. In cases where the radius R exceeded 1 cm, a bubble of vapor rises at the center and bursts at the upper interface due to a RayleighTaylor instability at the lower interface (Biance et al., 2003). The largest radius a droplet of water can have without bubbles forming is named the critical radius Rc and it is linearly related to the capillary length Lc and to the puddle height according to the following two equations Rc = 3.84 Lc (8.219) Rc = 1.92h (8.220) Biance et al. (2003) presented a simple dimensional scaling and analysis by combining force and energy balance to show the lifetime of a droplet for R < Lc and R > Lc, which is presented below. The thickness of the vapor film δ varies with time due to the evaporation process of the liquid drop on top of it. For a puddle, R > Lc [Fig. 8.27(b)], an energy balance yields the following equation for the evaporation rate: dm kv ΔT 2 πλ = dt hv δ (8.221) where πλ2 is the contact zone surface area, kv is the vapor thermal conductivity, ΔT/δ is the temperature gradient, m is the mass of the liquid drop and ΔT = T∞ − Tsat . Assuming fully developed vapor flow between the liquid drop and the solid surface [Fig. 8.27(b)], a force balance yields the following solution dm 2πδ 3 = ρv Δp (8.222) 3μ v dt where Δp is the pressure drop imposed by the liquid drop and μv is the gas viscosity. By equating eqs. (8.221) and (8.222) and using λ from eq. (8.217) with Δp = 2 ρ gLc , one obtains the following equation for vapor film thickness for R > L c: δ =  3kv ΔT μ v    4hv ρ v ρ gLc  1/ 4 R1/ 2 (8.223) For small drops, R < Lc [Fig. 8.25(a)], contact size is found using eq. (8.216), and the pressure drop acting on the vapor film is the Laplace pressure 2σ/R. Equations (8.224) and (8.225) predict the forms of evaporation rate and vapor film thickness for small drops, respectively: dm kv ΔT 2 ~ R dt hv R (8.224) Chapter 8 Boiling 725 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press δ ~  kv Δ T μ v ρ  g   2  hv ρ vσ  1/3 R 4/3 (8.225) For both small drops and puddles, the film thickness δ increases monotonically with the drop radius. To determine the lifetime of a Leidenfrost drop of a particular radius, we must assume that the radius and thickness are related by the quasi-steady conditions developed above [Eqs. (8.221) – (8.225)]. For a puddle R > Lc assuming m = π R 2 h ρ ,  1 R(t ) = Ro 1 −   τ 2 (8.226) 1/ 4 1/ Ro 2 where Rο is the radius at t = 0 and the lifetime τ is τ = 2  4 ρ Lc hv    kv Δ T  3/ 4  3μ v     ρv g  (8.227) Equation (8.226) is the line in Fig. 8.30 and shows good agreement with the experimental data for water droplets for two different wall temperatures. The time dependence of the film thickness is derived from the above equations and gives the following linear relationship, which decreases as the time approaches the lifetime of the drop (as seen in Figure 8.31). 2  3k ΔT μ v Ro  δ (t ) =  v   4hv ρ v ρ gLc  1/ 4  t 1 −   τ (8.228) Figure 8.30 Radius of a water droplet placed on a duralumin plate and filmed from above as a function of time. The fitted lines are predicted by eq. (8.226) [Reused with permission from Anne-Laure Biance et al., Physics of Fluids, 15, 1632 (2003), Copyright 2003, American Institute of Physics]. 726 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press 160 140 120 100 δ( μ m ) 80 60 40 20 0 0 50 100 t (s) 150 200 250 Figure 8.31 Thickness of the vapor film of a water droplet placed on a duralumin plate and filmed from above as a function of time. The fitted line is predicted by eq. (8.228) [Reused with permission from Anne-Laure Biance et al., Physics of Fluids, 15, 1632 (2003), Copyright 2003, American Institute of Physics]. Similarly, for smaller drops, R < Lc, evaporation occurs over the whole drop surface as described earlier. Therefore, assuming m = 4π R3 ρ / 3 , the time dependence of the radius is  t R(t ) = Ro 1 −   τ 1/ 2 (8.229) where τ~ ρ  hv kv ΔT 2 Ro (8.230) Equation (8.229) fits well with the trend seen in Fig. 8.30, where the radius variations occur more quickly toward the end of the drop’s life. A comparison of eqs. (8.227) and (8.230) reveals that the lifetime of a small drop is slightly more dependent on changes in temperature and much more on radius than the lifetime of a puddle. The above analysis was performed using a simple, one-dimensional analysis of evaporation of a Leidenfrost droplet. A numerical solution of the twodimensional analysis for two boundary conditions was performed by Nguyen and Avedisian (1987). The first case is a horizontal surface maintained at a constant temperature, and in the second case, the surface is insulated while the surrounding gas is heated. In the two-dimensional analysis, vapor flows out of the droplet in the radial direction and the streamlines bend due to the presence of the wall. This results in the creation of a pressure field that lifts the droplet off the surface. The droplet continues to levitate until it has completely evaporated. Chapter 8 Boiling 727 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press For both cases, assume the droplet shape to be spherical (which is valid for droplets in most industrial sprays and droplets with diameters of 100 μm or less). Also, take the evaporation process to be quasi-steady and properties to be constant. Radiative effects, spatial nonuniformities of temperature in the liquid, internal liquid motion, and buoyancy induced flow are all neglected. The continuity, momentum, energy, and species equations for the steady case in the vapor phase are ∇⋅V = 0 (8.231) μv 2 1 V ⋅∇V = − ∇p + ∇V (8.232) ρv ρv V ⋅∇T = α v ∇ 2T (8.233) 2 V ⋅ ∇ωi = D∇ ωi (8.234) where i = 1 is the droplet and i = 2 is the inert ambient, such that ω1 + ω2 = 1. Performing a force balance and neglecting the effects of droplet acceleration caused by variations in levitation height, the weight of the droplet is found to be equal to the net force acting on the droplet due to viscous stress and pressure, as follows (8.235)  j ⋅ τ ' dA = g( ρ − ρv ) ⋅ V where τ ' is the total stress tensor and j is the unit vector in the vertical direction. The following boundary conditions are used: Solid surface V = 0  ∂V  n = ∂ω1 = 0  ∂n ∂n  T = Tw (case 1)   ∂T = 0 (case 2)   ∂n (8.236) Droplet surface T = Tsat   ω = ω1s  (8.237) Ambient V → 0  ω1 → 0 T → T  ∞ (8.238) Equations (8.236) – (8.238) were classified in the bispherical coordinate system, which is made up of a family of spheres (β = constant) that are each orthogonal to a family of spindle-shaped surfaces (α = constant). The droplet evaporation time is obtained by the following mass balance: 4d π ( ρ R3 ) = ρ v  uβ dA 3 dt (8.239) where a droplet is the sphere β = βo and the solid surface corresponds to β = 0. 728 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Figure 8.32 Temporal variation of droplet temperature for n-heptane evaporating over an isothermal surface and an adiabatic surface. Initial droplet temperature is 300 K, temperature of the isothermal surface is 600K, temperature of the inert ambient is 600 K, and the initial droplet radius is 25 μm (Nguyen and Avedisian, 1987; Reprinted with permission from Elsevier). Droplet lifetime (s) Surface temperature (K) Figure 8.33 Variation of droplet lifetime of n-heptane due to changes in surface temperature of the isothermal solid plate, initial droplet and ambient temperature is 300 K and the initial droplet radius is 25 μm (Nguyen and Avedisian, 1987; Reprinted with permission from Elsevier). Variation in the droplet temperature and evaporation time based on the above two-dimensional analysis are presented in Figs. 8.32 and 8.33 for n-heptane. While predictions for the lifetime and time dependence of the radius and vapor film have been made, as seen above, other characteristics associated with the Leidenfrost phenomenon remain an enigma. The rapid formation of the vapor Chapter 8 Boiling 729 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press film, the natural vibration of the drops, and the ability of a drop to bounce when thrown at a solid hot surface are all examples of areas where further study is required. References Avedisian, C.T., 1986, “Bubble Growth in Superheated Liquid Droplets,” Encyclopedia of Fluid Mechanics, Chapter 8, Gulf Publishing Company, Houston, TX. Avedisian, C.T., and Suresh, K., 1985, “Analysis of Non-Explosive Bubble Growth within a Superheated Liquid Droplet Suspended in an Immiscible Liquid,” Chemical Engineering Science, Vol. 40, pp. 2249-2259. Banerjee, D., and Dhir, V.K., 2001, “Study of Subcooled Film Boiling on a Horizontal Disc: Part I - Analysis,” ASME Journal of Heat Transfer, Vol. 123, pp. 271-284. Berenson, P.J., 1961, “Film Boiling Heat Transfer from a Horizontal Surface,” ASME Journal of Heat Transfer, Vol. 83, pp. 351-356. Berenson, P.J., 1962, “Experiments on Pool-Boiling Heat Transfer,” International Journal of Heat and Mass Transfer, Vol. 5, pp. 985-999. Bergles, A.E., 2005, “Bora Mikic and Pool Boiling,” Proceedings of the 2005 ASME Summer Heat Transfer Conference, San Francisco, CA. Biance, A.L., Clanet, C., and Quere, D., 2003, “Leidenfrost Drops,” Physics of Fluids, Vol. 15, pp. 1632-1637. Bromley, L.A., 1950, “Heat Transfer in Stable Film Boiling,” Chemical Engineering Progress, Vol. 46, pp. 221-227. Cao, Y., Faghri, A., and Mahefkey, E.T., 1989, “The Thermal Performance of Heat Pipes with Localized Heat Input,” International Journal of Heat and Mass Transfer, Vol. 32, No. 7, pp. 1279-1287. Carey, V.P., 1992, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corp., Washington, D.C. Cess, R.D., and Sparrow, E.M., 1961a, “Film Boiling in a Forced-Convection Boundary-Layer Flow,” ASME Journal of Heat Transfer, Vol. 83, pp. 370-376. Cess, R.D., and Sparrow, E.M., 1961b, “Subcooled Forced-Convection Film Boiling on a Flat Plate,” ASME Journal of Heat Transfer, Vol. 83, pp. 377-379. Chandra, S., and Avedisian, C.T., 1991, “On the Collision of a Droplet with a Solid Surface,” Proceedings of Royal Society of London A, Vol. 432, pp. 13-41. 730 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Cole, R., and Rohsenow, W.M, 1969, “Correlations for Bubble Departure Diameters for Boiling of Saturated Liquid,” Chemical Engineering Progress, Vol. 65, pp. 211-213. Collier, J.G., and Thome, J.R., 1996, Convective Boiling and Condensation, 3rd ed., Oxford University Press, Oxford, UK. Cooper, M.G., 1984, “Heat Flow Rates in Saturated Nucleate Boiling – A Wide Ranging Examination using Reduced Properties,” Advances in Heat Transfer, Vol. 16, pp. 157-239, Academic Press, Princeton, NJ. Dhir, V., 2005, “Mechanistic Prediction of Nucleate Boiling Heat Transfer – Achievable or a Hopeless Task,” Max Jakob Lecture in 2005 ASME Summer Heat Transfer Conference, San Francisco, CA. Eastman, R.E., 1984, “Dynamics of Bubble Departure,” Thermophysics Conference, Snowmass, CO, AIAA-1984-1707. AIAA 19th Esmaeeli, A., and Tryggvason, G., 2001, “Direct Numerical Simulations of Boiling Flows,” Proceedings of the Fourth International Conference on Multiphase Flow, ICMF-2001, New Orleans, LA. Esmaeeli, A., and Tryggvason, G., 2003, “Computations of Explosive Boiling in Microgravity,” Journal of Scientific Computing, Vol. 19, pp. 163-182. Esmaeeli, A., and Tryggvason, G., 2004, “Computations of Film Boiling. Part I: Numerical Method,” International Journal of Heat and Mass Transfer, Vol. 47, pp. 5451-5461. Faghri, A., 1995, Heat Pipe Science and Technology, Taylor & Francis, Washington, DC. Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA. Friedel, L., 1979, “Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two-Phase Pipe Flow,” European Two-Phase Flow Group Meeting, Paper E2, Ispra, Italy. Fritz, W., 1935, “Maximum Volume of Vapor Bubbles,” Phys. Z. Vol. 36, pp. 379-384. Forster, D.E., and Greif, R., 1959, “Heat Transfer to a Boiling Liquid – Mechanism and Correlation,” ASME Journal of Heat Transfer, Vol. 81, pp. 4353. Gorenflo, D., 1993, “Pool Boiling,” VDI Heat Atlas, VDI-Verlag, Düsseldorf. Gorenflo, D., Knabe, V., and Beiling, V., 1986, “Bubble Density on Surface with Nucleate Boiling – Its Influence on Heat Transfer and Burnout Heat Flux at Elevated Saturation Pressure,” Proceedings of the 8th International Heat Transfer Conference, Vol. 4, pp. 1995-2000, San Francisco, CA. Chapter 8 Boiling 731 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Griffith, P., and Wallis, J.D., 1960, “The Role of Surface Conditions in Nucleate Boiling,” Chemical Engineering Progress Symposium, Ser. 56, No. 30, pp. 4963. Han, Y.Y., and Griffith, P., 1965, “The Mechanism of Heat Transfer in Nucleate Pool Boiling. I – Bubble Initiation, Growth and Departure,” International Journal of Heat Mass Transfer, Vol. 8, pp. 887-904. Hohl, R., Auracher, H., Blum, J., and Marquardt, W., 1996, “Pool Boiling Heat Transfer Experiments with Controlled Wall Temperature Transients,” 2nd European Thermal Science and 14th UIT National Heat Transfer Conference, Rome, pp. 1647-1652. Howell, J.R., and Siegel, R., 1966, “Incipience, Growth, and Detachment of Boiling Bubbles in Saturated Water from Artificial Nucleation Site of Known Geometry and Size,” Proceedings of the 3rd International Heat Transfer Conference, Chicago, IL, Vol. 4, pp. 12-23. Ivey, H.J., 1967, “Relationship Between Bubble Frequency, Departure Diameter and Rise Velocity in Nucleate Boiling,” International Journal of Heat and Mass Transfer, Vol. 10, pp. 1023-1040. Jakob, M., and Fritz, W., 1931, “Versuche uber den Verdampfungsvorgang,” Forsch. Ingenieurwes, Vol. 2, pp. 435-447. Jakob, M., and Linke, W., 1933, “Der Warmeubergang von einer waagerechten Platte an siedendes Wasser,” Forsch. Ingenieurwes, Vol. 4, pp. 75-81. Jakob, M., and Linke, W., 1935, “Heat Transmission in the Evaporation of Liquids at Vertical and Horizontal Surfaces,” Physik Z., Vol. 36, pp. 267-280. Judd, R.L., and Hwang, K.S., 1976, “A Comprehensive Model for Nucleate Boiling Heat Transfer Including Microlayer Evaporation,” ASME Journal of Heat Transfer, Vol. 98, pp. 623-629. Juric, D., and Tryggvason, G., 1998, “Computations of Boiling Flow,” International Journal of Multiphase Flow, Vol. 24, pp. 387-410. Kandilikar, S.G., Dhir, V.K., and Shoji, M., 1999, Handbook of Phase Change: Boiling and Condensation, Taylor and Francis, Philadelphia, PA. Keshock, E.G., and Siegel, R., 1962, Forces Acting on Bubbles in Nucleate Boiling Under Normal and Reduced Gravity Conditions, NASA TN D-2299. Klimenko, V.V., 1981, “Film Boiling on a Horizontal Plate–New Correlation,” International Journal of Heat and Mass Transfer, Vol. 24, pp. 69-79. Kocamustafaogullari, G., and Ishii, M., 1983, “Interfacial Area and Nucleation Site Density in Boiling Systems,” International Journal of Heat and Mass Transfer, Vol. 26, pp. 1377–1387. 732 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Koh, J.C.Y., 1962, “Analysis of Film Boiling on Vertical Surfaces,” ASME Journal of Heat Transfer, Vol. 84, p. 55-62. Kutateladze, S.S., 1948, “On the Transition to Film Boiling under Natural Convection,” Kotloturbostroenie, No. 3, pp. 10-12. Lee, R.C., and Nydahl, J.E., 1989, “Numerical Calculation of Bubble Growth in Nucleate Boiling from Inception Through Departure,” ASME Journal of Heat Transfer, Vol. 111, pp. 474-479. Lienhard, J.H., and Dhir, V.K., 1973, Extended Hydrodynamic Theory of the Peak and Minimum Heat Fluxes, NASA CR-2270. Lienhard, J.H., and Hasan, M.Z., 1979, “On Predicting Boiling Burnout with the Mechanical Energy Stability Criterion,” ASME Journal of Heat Transfer, Vol. 101, pp. 276-279. Lienhard, J.H., and Witte, L.C., 1985, Rev. Chemical Engineering, Vol. 3, pp. 187-280. Lorenz, J.J., Mikic, B.B., and Rohsenow, W.M., 1974, “The Effect of Surface Conditions on Boiling Characteristics,” Proceedings of 5th International Heat Transfer Conference, Vol. 1, pp. 35-39. Mahadevan, L., and Pomeau, Y., 1999, “Rolling Droplets,” Physics of Fluids, Vol. 11, pp. 2449-2453. Malenkov, I.G., 1971, “The Frequency of Vapor Bubble Separation as Function of Bubble Size,” Fluid Mech. Sov. Res., Vol. 1, pp. 36-42. Mikic, B.B., and Rohsenow, W.M., 1969, “A New Correlation of Pool-Boiling Data Including the Effect of Heating Surface Characteristics,” Journal of Heat Transfer, Vol. 91, pp. 245-250. Mikic, B.B., Rohsenow, W.M., and Griffith, P., 1970, “On Bubble Growth Rate,” International Journal of Heat and Mass Transfer, Vol. 13, pp. 657-666. Mizukami, K., 1975, “Entrapment of Vapor in Re-Entrant Cavities,” Letters in Heat and Mass Transfer, Vol. 2, pp. 279-284. Mukherjee, A., and Dhir, V.K., 2004, “Study of Lateral Merger of Vapor Bubble during Nucleate Pool Boiling,” ASME Journal of Heat Transfer, Vol. 126, pp. 1023-1039. Nakayama, A., 1986, “Subcooled Forced Convection Film Boiling on Plane and Axisymmetric Bodies in the Presence of Pressure Gradient,” AIAA Journal, Vol. 24, pp. 230-236. Nguyen, T.K., and Avedisian, C.T., 1987, “Numerical Solution for Film Evaporation of a Spherical Liquid Droplet on an Isothermal and Adiabatic Surface,” International Journal of Heat and Mass Transfer, Vol. 30, pp. 14971509. Chapter 8 Boiling 733 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Nishikawa, K., Ito, T., and Matsumoto, K., 1976, “Investigation of Variable Thermophysical Property Problem Concerning Pool Film Boiling from Vertical Plate with Prescribed Uniform Temperature,” International Journal of Heat and Mass Transfer, Vol. 19, pp. 1173-1181. Nishio, A., 1985, “Stability of Pre-Existing Vapor Nucleus in Uniform Temperature Field,” Transactions of JSME, Series B, Vol. 54-303, pp. 18021807. Nukiyama, S., 1934, “The Maximum and Minimum Values of Heat Q Transmitted From Metal to Boiling Water Under Atmospheric Pressure,” Journal of Japanese Society of Mechanical Engineering, Vol. 37, pp. 367-374 (1934) (translated in International Journal of Heat and Mass Transfer, Vol. 9, pp. 14191433 (1966)). Prosperetti, A., and Plesset, M.S., 1978, “Vapor Bubble Growth in a Superheated Liquid,” Journal of Fluid Mechanics, Vol. 85, pp. 349-368. Plesset, M.S., and Sadhal, S.S., 1979, “An Analytical Estimate of the Microlayer Thickness in Nucleate Boiling,” ASME Journal of Heat Transfer, Vol. 101, pp. 180-182. Plesset, M.S., and Zwick, S.A., 1954, “The Growth of Vapor Bubble in Superheated Liquids,” Journal of Applied Physics, Vol. 25, pp. 493-500. Rohsenow, W.M., 1952, “A Method for Correlating Heat-Transfer Data for Surface Boiling of Liquids,” Transactions of ASME, Vol. 74, pp. 969-976. Shin, S., and Juric, D., 2002, “Modeling Three-Dimensional Multiphase Flow Using a Level Contour Reconstruction Method for Front Tracking without Connectivity,” Journal of Computational Physics, Vol. 180, pp. 427-470. Sideman, S. and Isenberg, J., 1967, “Direct Contact Heat Transfer with Change of Phase: Bubble Growth in Three-Phase Systems,” Desalination, Vol. 2, pp. 207-214. Son, G., and Dhir, V.K., 1997, “Numerical Simulation of Saturated Film Boiling on a Horizontal Surface,” ASME Journal of Heat Transfer, Vol. 119, pp. 525533. Son, G., and Dhir, V.K., 1998, “Numerical Simulation of Film Boiling near Critical Pressures with a Level Set Method,” ASME Journal of Heat Transfer, Vol. 120, pp. 183-192. Son, G., Ramanujapu, N., and Dhir, V.K., 2002, “Numerical Simulation of Bubble Merger Process on a Single Nucleation Site during Pool Nucleate Boiling,” ASME Journal of Heat Transfer, Vol. 124, pp. 51-62. Sparrow, E.M., and Cess, R.D., 1962, “The Effect of Subcooled Liquid on Laminar Film Boiling,” ASME Journal of Heat Transfer, Vol. 84, pp. 149-156. 734 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press Stephan, K., 1992, Heat Transfer in Condensation and Boiling, Springer-Verlag, Berlin. Thome, J.R., 2004, Engineering Data Book III, Wolverine Tube, Inc., Huntsville, AL. van Stralen, S.J.D., Sohal, M.S, Cole, R. and Sluyter, W.M., 1975a, “Bubble Growth Rates in Pure and Binary Systems: Combined Effects of Relaxation and Evaporation Microlayers,” International Journal of Heat and Mass Transfer, Vol. 18, pp. 453-467. van Stralen, S.J.D., Cole, R., Sluyter, W.M., and Sohal, M.S, 1975b, “Bubble Growth Rates in Nucleate Boiling of Water at Subatmospheric Pressures,” International Journal of Heat and Mass Transfer, Vol. 18, pp. 655-669. Wang, C.H., and Dhir, V.K., 1993a, “On the Gas Entrapment and Nucleation Density during Pool Boiling of Saturated Water,” ASME Journal of Heat Transfer, Vol. 115, pp. 670-679. Wang, C.H., and Dhir, V.K., 1993b, “Effect of Surface Wettability on Active Nucleation Site Density During Pool Boiling of Water on a Vertical Surface,” ASME Journal of Heat Transfer, Vol. 115, pp. 659-669. Welch, S.W.J., 1998, “Direct Simulation of Vapor Bubble Growth,” International Journal of Heat and Mass Transfer, Vol. 41, pp. 1655-1666. Welch, S.W.J., and Wilson, J.J., 2000, “A Volume of Fluid Based Method for Fluid Flows with Phase Change,” Journal of Computational Physics, Vol. 160, pp. 662-682. Witte, L.C., and Lienhard, I.H., 1982, “On the Existence of Two ‘Transition’ Boiling Curves,” International Journal of Heat and Mass Transfer, Vol. 25, pp. 771-779. Zuber, N., 1959, “Hydrodynamic Aspects of Boiling Heat Transfer,” USAEC Report AECU-4439. Zuber, N., Tribus, M., and Westwater, J.W., 1961, “The Hydrodynamic Crisis in Pool Boiling of Saturated and Subcooled Liquids,” International Development in Heat Transfer: Proceedings of 1961-62 International Heat Transfer Conference, Boulder, CO, pp. 230–236. Problems 8.1 8.2 In saturated pool boiling at 1.013 × 105 Pa, what are the liquid temperatures at which vapor bubbles with diameters of 0.1 and 1 mm can exist and grow? For vapor bubble growth in an initially uniformly superheated liquid, the relationship between the bubble radius and time is expressed by eq. (8.32). Chapter 8 Boiling 735 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press At the early stage of bubble growth, the relationship between the bubble radius and time is expressed by eq. (8.24). Show that eq. (8.32) can be reduced to eq. (8.24) for the early stage of bubble growth. 8.3 At the late stage of radius and time is relationship between can be reduced to eq. bubble growth, the relationship between the bubble expressed by eq. (8.28). Show that the general the bubble radius and time expressed by eq. (8.32) (8.28) for the late stage of bubble growth. 8.4 A vapor bubble is initiated at t = 0 and grows in liquid water at 120 ˚C. Find the bubble size at time t = 0.1 ms using different models and compare the results. Estimate the bubble departure diameter for water at 1 atm. Redo the problem for water at 2 atm and discuss the effect of pressure on the bubble departure diameter and frequency. Use an example from industry to explain why it is very important to keep heat flux below critical heat flux. A mechanically polished thin-wall stainless tube with a diameter of 3.5 mm and a length of 100 mm is immersed into nearly saturated water at 1 atm. The electric current passes through the stainless tube to supply the heat. Estimate the heat transfer coefficient when the electric power is 100 W. A mechanically polished stainless steel pan with a diameter of 30 cm is filled with water and placed on top of the heating unit. The top surface of the water is exposed to ambient air at 1 atm pressure. Find the inner surface temperature of the pan that is necessary to generate vapor at a rate of 2.3 kg/h. If the heat flux in Problem 8.8 is 50% of the critical heat flux, find (a) the rate of vapor generation, and (b) the inner surface temperature at the bottom of the pan. 8.5 8.6 8.7 8.8 8.9 8.10 Estimate the maximum heat flux for nucleate boiling of water at p = 1.013 × 105 Pa on the moon, where the gravitational acceleration is g/6. 8.11 Estimate the minimum heat flux for pool boiling of water at two atmospheric pressures. Compare your results with those in Example 8.4 and discuss the effect of pressure on the minimum heat flux. ′′ 8.12 The critical heat flux of water at 1 atm is qmax = 1258.7 kW/m 2 , which o occurs at ΔT = 30 C . Boiling enters the transition boiling regime when the excess temperature is increased further. Estimate the heat flux when the excess temperature is equal to 50 ˚C. 8.13 If the convection terms in eqs. (8.144) and (8.145) can be neglected, find the vapor film thickness and heat transfer coefficient for film boiling over a vertical flat plate. The liquid adjacent to the vapor layer can be treated as 736 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press stationary and the effect of radiation from the heating surface to the interface can be neglected. 8.14 Film boiling of a saturated liquid on a vertical plate is described by a set of ordinary differential equations (8.162) – (8.164), and corresponding boundary conditions, eqs. (8.167) – (8.173). Write a computer program to solve the film boiling problem and compare your results with those in Fig. 8.24. 8.15 The similarity solution for film boiling presented in Section 8.6.1 describes a case in which the liquid is saturated. Perform a scale analysis of the governing equations and corresponding boundary conditions for subcooled boiling, eqs. (8.143) – (8.155), and identify the dominant nondimensional parameters for boiling of subcooled liquid on a vertical plate. 8.16 Define appropriate similarity variables for subcooled boiling and reduce the problem to a set of ordinary differential equations. Obtain the necessary boundary conditions for the ordinary differential equations to mathematically close the problem. 8.17 Film boiling occurs on an electrically heated 1.27 mm platinum wire placed horizontally in the water at atmospheric condition. The surface temperature of the wire is 754 °C. The emissivity of the platinum wire and the liquid vapor interface can be assumed as ε s = ε i = 1. Find the power dissipated per unit length of the wire. 8.18 Film boiling at 1 atm occurs on the surface of a 2 cm diameter sphere made out of polished copper. The surface temperature of the sphere is Tw = 800 °C. The emissivity of the sphere surface is ε w = 0.05 , but the liquid-vapor interface can be treated as a black body. Find the heat transfer coefficients due to convection, radiation, and the overall heat transfer coefficient. Compare your results with those of Example 8.5 and discuss the effect of radiation at different temperatures. 8.19 Specify the governing equations and corresponding boundary conditions for film boiling of saturated liquid on an isothermal cylinder with diameter D and temperature Tw. The effect of radiation from the cylinder surface to the liquid-vapor interface can be neglected. 8.20 If the inertia terms in the governing equations for both vapor and liquid phases are negligible and μv / μ  0 , obtain the vapor film thickness for film boiling in Problem 8.19. 8.21 Block-heated heat pipes are normally used to transport energy from one place to another for the purpose of energy conservation or electronic component cooling. The evaporator and condenser segments are normally separated, which is similar to conventional heat pipes except for the localized heating. The working temperature is comparatively low or Chapter 8 Boiling 737 Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press moderate, which means that radiation heat transfer is not important in the analysis of these heat pipes. Consider the heat pipe shown in Fig. P8.1. It has an evaporator section of length LE, a wall thickness δ , an outside radius R, and a block-heated area of width WH and length LH. An examination of the reported data shows that a power-law boiling relation is appropriate for relating the heat flux to the evaporating temperature drop in ′′ a heat pipe, i.e., qe = a(T − Ts )b . Assume the wall thickness is much smaller than the radius of the heat pipe and that variation of temperature along the thickness direction of the heat pipe is negligible. Specify the governing equation and corresponding boundary conditions to describe temperature distribution in the heat pipe wall. Figure P8.1 738 Advanced Heat and Mass Transfer Amir Faghri, Yuwen Zhang, and John Howell Copyright © 2010 Global Digital Press