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8 CONDENSATION 8.1 Introduction Condensation occurs when a saturated or superheated vapor – pure or multicomponent – comes into contact with an object, such as a wall or other contaminant, that has a temperature below the saturation temperature. In most applications involving the condensation of a vapor, heat is transferred to a solid wall adjacent to the vapor. If the solid wall temperature is below the equilibrium saturation temperature at the system pressure, a liquid droplet embryo may form at this solid-vapor interface. This condensation is referred to as heterogeneous nucleation of a liquid droplet embryo. Heterogeneous liquid droplet nucleation is nucleation of a vapor droplet embryo at the interface of a metastable vapor phase and another phase; this second phase is usually solid and is naturally held at a lower temperature than the vapor. A metastable vapor is one that is supercooled below its equilibrium saturation temperature at the system pressure (see Chapter 2). Figure 8.1 is a flowchart schematic showing the different modes by which a liquid droplet embryo can form. It can be seen from this figure that the embryos can also be formed homogeneously. Homogeneous nucleation of a liquid droplet occurs entirely within a supercooled vapor. The liquid droplet is completely surrounded by supercooled vapor and is not attached to a lower temperature wall, as is the case in the heterogeneous process. The heterogeneous condensation occurs when vapor condenses on the cooled surface as a thin film or as droplets, depending on whether the surface is wettable or nonwettable with the condensate. If the surface is wettable with the condensate, a continuous condensate film forms on the surface and filmwise condensation occurs [see Fig. 8.2(a)]. Conversely, if the surface is not wettable with the condensate, a series of condensate droplets form on the surface (i.e., dropwise condensation occurs) [see Fig. 8.2(b)]. Dropwise condensation is preferred over filmwise condensation, because its resulting overall heat transfer coefficient is higher by as much as an order of magnitude. This difference occurs because the cooler wall surface always has an area that is not covered by condensate. The resistances encountered at the liquid-vapor interface, along with the conduction through the liquid itself, are removed and the heat transfer is increased significantly. Therefore, in industrial applications it is wise to Chapter 8 Condensation 581 Condensation Liquid droplet nucleation Homogeneous Heterogeneous Liquid droplet nucleation occurring entirely within a supercooled vapor Liquid droplet nucleation occurring at the interface of a metastable vapor and another phase (usually solid) at a lower temperature Figure 8.1 Flowchart of the different modes of condensation. (a) Filmwise condensation Figure 8.2 Heterogeneous condensation. (b) Dropwise condensation introduce conditions that promote dropwise condensation. The dropwise condensation can be promoted by taking one or more of the following steps: 1. Introduce a nonwetting agent into the vapor that will eventually deposit on the cooling surface to break up wetting conditions. 2. Apply grease or waxy products that are poor wetting agents to the cool wall surface in order to promote nonwetting conditions. 3. Permanently coat the wall surface with a low surface energy or noble metal. 582 Transport Phenomena in Multiphase System Figure 8.3 Homogeneous condensation: (a) condensation on small contaminant particles in the vapor mixture, (b) condensation on liquid droplets, and (c) condensation of vapor bubbles. If a tiny, sufficiently supercooled contaminant, which is introduced to the vapor, condensate will form on the contamination in the middle of the vapor. This is an example of homogeneous condensation that is different from the heterogeneous condensation mentioned above, in that it relies on a solid, liquid, or even vapor contaminant to initiate condensation. This type of condensation produces a mist-like quality and is depicted in Fig. 8.3 (a). Two more examples of homogeneous condensation are shown in Figs. 8.3 (b) and (c). When liquid is introduced into vapor through a nozzle, liquid droplets are formed; vapor condenses on the surface of these droplets suspended in a vapor phase [see Fig. 8.3 (b)]. When vapor bubbles are introduced into the bulk liquid, as shown in Fig. 8.3 (c), the vapor bubbles surrounded by the cold liquid shrink and eventually collapse due to condensation. In many industrial applications the saturated vapor to be condensed is in fact a miscible binary vapor mixture. In a multicomponent vapor, the saturation temperature is referred to as the dew point. As will be described below, this binary mixture does condense differently than a pure vapor and has lower heat transfer and condensation rates (Fujii, 1991). Fig. 8.4 (a) shows the transverse distributions of temperature T and mass fractions 1v and 2v in the condensate film, vapor boundary layer, and bulk vapor of a steady condensation process consisting of a binary miscible vapor mixture in contact with a vertical cooled wall; the phase equilibrium diagram of this condensation process is shown in Fig. 8.4 (b). Dew point line refects the point at which the binary vapor/gas begins to condense. When there is less 1 and more 2, this occurs at a higher temperature. The boiling point line reflects the point at which the binary liquid begins to boil or when the binary vapor is completely condensed. When a binary vapor mixture is cooled below its saturation temperature by contact with a cold wall, the less volatile component 2 (with higher saturation temperature) condenses more than the volatile component 1. In other words, as the vapor mixture cools, the component with the higher saturation temperature at the system saturation pressure will condense first. If it is assumed that the bulk vapor mixture is kept at a constant density, the volatile component (with lower saturation temperature) must become very dense (while remaining in its vapor form) at the liquid-vapor interface. Meanwhile, the less volatile component condenses into the liquid phase Chapter 8 Condensation 583 T T∞ T∞ Tsat T Tw T Tw T ωv ω1v’ + ω2v’ =1 ω2v’ ω1v’ y Boiling point line 0 1Aδ ω1v’ ω1vδ 1 ω1v (b) (a) Figure 8.4 Temperature and mass fraction in the condensation of a binary miscible vapor mixture: (a) T and distribution in the condensate film and vapor boundary layer; (b) variation of T and on a diagram of phase equilibrium (Fujii, 1991; Reproduced with kind permission of Springer Science and Business Media). first. This can be seen in Fig. 8.4 (a), where the volatile component increases in mass at the interface, while the less volatile component is at its lowest mass at the interface. Gravity or drag forces constantly flush the surface of the wall, so the dense mass of the volatile component is removed by convection forces. However, due to this buildup of the volatile component at the interface, a larger resistance to heat flow is produced in a multicomponent system than in a pure vapor condensation process. Due to the difference in saturation properties of the binary mixture components, the temperature drop across the interface is larger than for a pure vapor condensation process. Therefore, the interfacial resistance for the condensation of a binary vapor mixture usually cannot be neglected, as is sometimes the case for the condensation of pure vapors. This interfacial resistance will, however, be developed below for a pure vapor. Figure 8.4(b) illustrates condensation of a binary mixture of miscible vapors on a phase equilibrium chart. Initially the vapor is at T’ and it is placed in contact with a cooler wall. The vapor temperature decreases until it reaches Tsat, at which point the condensation process begins. The dew point line on the diagram is the saturation temperature for various concentrations of the two components in the mixture. The less volatile component in the mixture (component 2 in this case) will condense faster than the other. At the temperature of the interface between 584 Transport Phenomena in Multiphase System Variation of and T in the concentration boundary layer Dew point line Bulk flow of binary vapor mixture Thermal boundary layer in vapor Liquid film Concentration boundary layer Figure 8.5 Condensation of immiscible fluids. the liquid and the vapor phases, the mass fraction of each phase can be read off the tie line between the dew point and boiling point lines. As can be seen, the condensate film will be much heavier with the less volatile component, and the vapor will have a higher concentration of the more volatile component. Other applications involve the condensation of vapors of partially or completely immiscible liquids such as water or organic compounds. During condensation of vapors of some immiscible liquids, the condensate will form in a combination of dropwise and filmwise liquid. One component will condense as a liquid film with droplets of the other component floating within or on top of it. This mixture can be seen in Fig. 8.5 and is caused by the different surface tension forces of the two components in relation to the vapor and solid wall. Figures 8.6 and 8.7 show the phase diagrams for binary mixtures with a miscibility gap and with completely immiscible liquids, respectively. The point where the dew point lines meet the immiscible liquid regions is the eutectic point, which occurs at the Pure comp. 2 Vapor region Dew point line Dew point line Miscible liquid region heavy w/ comp. 2 Liquid-vapor region heavy w/ comp. 2 Boiling point line Immiscible liquid region 0 mass fraction of component 1 (ω1) Liquid-vapor region heavy w/ comp. 1 Miscible liquid region heavy w/ comp. 1 Pure comp. 1 585 Figure 8.6 Phase diagram of liquids with a miscibility gap. Temperature Chapter 8 Condensation Pure comp. 2 Vapor region Pure comp. 1 Dew point line Dew point line Temperature Liquid-vapor region heavy with component 2 Boiling point line Immiscible liquid region 0 Liquid-vapor region heavy with component 1 mass fraction of component 1 (ω1) Figure 8.7 Phase diagram of completely immiscible pure liquid. eutectic temperature for a given pressure. Condensation may occur in several forms for these cases. If the wall temperature is greater than the eutectic temperature or if the condensate film is sufficiently thick, the interface temperature must also be above the eutectic temperature. This case is similar to the condensation of miscible liquids: a homogenous condensate film will form, and the concentrations of each component in the vapor and liquid phases can be read off the tie line corresponding to the interface temperature. When the wall temperature is less than the eutectic temperature and the condensate film is thin, the interface temperature will be equal to the eutectic temperature. As condensation occurs, two immiscible liquid phases form. These liquid phases are in equilibrium with the remaining vapor, which will have the eutectic composition. In Section 8.2, the mechanisms of dropwise condensation will be discussed, starting with the critical equilibrium radius for the initial formation of a liquid droplet embryo. This discussion will show that the critical equilibrium radius is dependent on the contact angle of the liquid droplet with the wall surface, and on the interfacial surface tensions found in the liquid droplet. A detailed discussion on heat transfer resistance in the condensation process will be presented, with emphasis on the dropwise condensation process. Section 8.3 begins with an introduction to the filmwise condensation problem, which is followed by the generalized governing equations for laminar film condensation of a binary vapor. The classical Nusselt analysis of laminar film condensation on a vertical wall is discussed, and the generalized condensation process for a flowing vapor reservoir for both laminar and turbulent film condensation will be addressed. Turbulent film condensation in a tube with both cocurrent and countercurrent vapor flow is also presented, as are empirical correlations for filmwise condensation 586 Transport Phenomena in Multiphase System configurations. Section 8.3 closes with a discussion of the effect of noncondensable gases and flooding limit in the condensation process. Section 8.4 presents analyses of four nongravitational condensate removals. In a microgravity environment, the major challenge is how to thin the condensate film in order to improve the heat transfer coefficient. Forced convection condensation in an annular tube with suction at the inner porous wall is discussed first, followed by discussions of condensation heat transfer with forced shear, centrifugal force, and capillary force in a microgravity environment. Film condensation in porous media is discussed in Section 8.5, beginning with gravitydominated film condensation, followed by discussion of the effect of surface tension on film condensation in porous media. 8.2 Dropwise Condensation 8.2.1 Dropwise Condensation Formation Theories Several theories have been proposed to explain the mechanism of dropwise condensation. The first model, proposed by Eucken (1937), has been supported by many experimental studies, such as that by McCormick and Baer (1963). It states that liquid droplets form only heterogeneously at nucleate sites; if they are formed with a radius exceeding that of equilibrium, they will continue to grow and then join with surrounding droplets. Once the mass of the condensate reaches a critical point, it will be removed from the surface by gravitational forces or by drag forces produced by the surrounding gas. As droplets are removed, the surface is wiped clean of condensate and the process restarts at the nucleate sites. This periodic cleaning constitutes the advantage of dropwise condensation over filmwise condensation, as there is no resistance to heat transfer through the condensate when the condensate layer is removed; and thus the heat transfer rate increases greatly. The second approach postulates that between drops there exists a thin and unstable liquid film on a solid surface. As the condensation process continues and the thin film grows thicker, the film reaches a critical thickness – estimated to be in the order of 1μm – at which point it breaks up into droplets. Condensation then continues in the dry areas between the recently-ruptured droplets, and on top of the already-formed droplets. The majority of new condensate does occur on the wall surface because there is less resistance to heat conduction than if the new condensate formed on the already-existing droplets. These new condensate droplets are then drawn to the neighboring droplets by surface tension effects, producing a new thin film. This film will then grow and rupture at the critical thickness, and the process will repeat continuously. Dropwise condensation takes place if the condensate cannot wet the surface. The wettability can be measured by a contact angle defined as (see Section 5.3.1) Chapter 8 Condensation 587 cos θ = σ sv − σ sA σ Av (8.1) where σ sv , σ sA , and σ Av are surface tensions between solid-vapor, solid-liquid, and liquid-vapor interfaces. When the contact angle, , is greater than 90°, the condensate cannot wet the surface and dropwise condensation occurs. The criterion for dropwise condensation is the critical surface tension σ cr which is characteristics of the surface alone. If the surface tension between liquid-vapor interface σ Av is greater than σ cr , dropwise condensation occurs (Shafrin and Zisman, 1960). Critical surface tensions for selected solid surfaces are given in Table 8.1. It can be seen that the critical surface tensions for all solids listed in Table 8.1 are below the surface tension of water at 1 atm ( σ Av = 58.91 × 10−3 N/m ). Therefore, a metal surface, on which film condensation usually occurs, can be coated with another substance with lower critical surface tension to promote dropwise condensation. Table 8.1 Critical surface tension for selected solid surfaces (Shafrin and Zisman, 1960)a Solid Surface Kel-F ® Nylon Platinum with perfluorobutyric acid monolayer Platinum with perfluorolauric acid monolayer Polyethylene Polystyrene Polyvinyl Chloride Teflon ® a Reprinted with permission from American Chemical Society. σ cr (10-3N/m) 31 46 10 6 31 33 39 18 8.2.2 Critical Droplet Radius for Spontaneous Growth and Destruction As mentioned before, upon formation in their nucleation sites, droplets grow only if they form with a radius that exceeds the equilibrium radius. The analysis that leads to the definition of the critical equilibrium radius is presented below. A good place to start this derivation lies in the Gibbs free energy minimum principle. The Gibbs free energy, G, arises from one of the Legendre transformations for closed systems states, which states that G = G (T , p ) = E − TS + pV (8.2) where T, p, V, E, S are the system temperature, pressure, volume, internal energy, and entropy, respectively. Equation (8.2) can be differentiated to obtain dG = dE − TdS − SdT + pdV + Vdp (8.3) Assuming the only work term is of the pdV type, and neglecting potential and kinetic terms, the fundamental thermodynamic relationship is (see Section 2.2) dE ≤ TdS − pdV (8.4) 588 Transport Phenomena in Multiphase System Figure 8.8 Contact angle and equilibrium of surface tensions of a liquid droplet embryo. Substituting eq. (8.4) into eq. (8.3) yields dG ≤ − SdT + Vdp (8.5) Assuming that the system pressure and temperature are held at fixed values, the above expression reduces to dG ≤ 0 (8.6) Along with the assumption that the system pressure and temperature are fixed, G becomes the availability of the system, Ψ: Ψ = G = E + p0V − T0 S (8.7) where p0 and T0 are the reservoir pressure and temperature, respectively. Availability is defined as the maximum amount of work one can get from a system as it comes into equilibrium with a large reference environment with pressure p0 and temperature T0. It follows from this analysis that for equilibrium to occur in the system, dΨ = 0. Furthermore, stable equilibrium corresponds to a minimum value of Ψ. Carey (1992) considered a system with supersaturated vapor adjacent to a vertical wall that has an initial temperature and pressure, Tv and pv. The initial availability Ψo of this system is given by Ψ 0 = mtotal g v + ( Asv ) I σ sv (8.8) where mtotal is the total mass of the system, and gv is the specific Gibbs free energy of the vapor phase and is a function of Tv and Pv. The second term on the right-hand side represents the contribution of the work done by surface tension at solid-vapor interface on the availability [see eq. (2.228)], and (Asv)I is the initial surface area shared by the solid and vapor of the system. Now consider that the wall temperature is lowered and droplets begin to form on the surface of the wall. The total availability of the system is the sum of the availabilities of the liquid Ψ A , the vapor Ψv, and the interface ΨI. The expressions for each are as follows: Ψ A = mA [ g A + ( pv − pA )vA ] (8.9) where the second term in the brackets on the right-hand side corrects for the difference between the vapor and liquid, and Ψ v = (mtotal − mA ) g v (8.10) Chapter 8 Condensation 589 Ψ I = AAvσ Av + ( Asv ) f σ sv + AsAσ sA (8.11) where the three terms on the right-hand side represent the work done by surface tension at the liquid-vapor, solid-vapor, and solid-liquid interfaces. (Asv)f is the difference between the initial surface area shared by the solid-vapor interface and the surface area shared by the solid-liquid interface. (8.12) ( Asv ) f = ( Asv ) I − AsA The areas are calculated directly from Fig. 8.8: AAv = 2π r 2 (1 − cos θ ) 2 2 (8.13) AsA = π r (1 − cos θ ) (8.14) The volume of the liquid droplet is π r3 VA = (2 − 3cos θ + cos3 θ ) (8.15) 3 The change of the total availability of the system due to formation of the condensate embryo is then ΔΨ = ( Ψ A + Ψ v + Ψ I ) − Ψ 0 = [ mA g A + VA ( pv − pA ) + (mtotal − mA ) g v = mA ( g A − g v ) + VA ( pv − pA ) + AAvσ Av + ( Asv ) f σ sv + AsAσ sA º − [ mtotal g v − ( Asv ) I σ sv ] ¼ + AAvσ Av + [( Asv ) f − ( Asv ) I ]σ sv + AsAσ sA (8.16) Substituting eq. (8.12) into eq. (8.16), the change of availability becomes ΔΨ = mA ( g A − g v ) + VA ( pv − pA ) + AAvσ Av + AsA (σ sA − σ sv ) (8.17) Substituting Young’s equation, eq. (8.1), into eq. (8.17), one obtains ΔΨ = mA ( g A − g v ) + VA ( pv − pA ) + ( AAv − AsA cos θ )σ Av (8.18) Substituting eqs. (8.13) and (8.14) into eq. (8.18), the following expression is obtained for a change in availability of the system: ΔΨ = Ψ − Ψ 0 (8.19) = mA ( g A − g v ) + VA ( pv − pA ) + 4π r 2σ Av F where 2 − 3cos θ + cos3 θ F= (8.20) 4 Considering that the pressure in the liquid droplet is related to the pressure in the vapor phase by pA − pv = 2σ Av / r and substituting eq. (8.15) into eq. (8.19), one obtains 4 ΔΨ = Ψ − Ψ 0 = mA ( g A − g v ) + π r 2σ Av F (8.21) 3 The mass of the embryo is 590 Transport Phenomena in Multiphase System VA 4 = π r3F (8.22) vA 3vA Substituting eq. (8.22) into eq. (8.21), one obtains 4 4 ΔΨ = Ψ − Ψ 0 = π r 3 F ( g A − g v ) + π r 2σ Av F (8.23) 3vA 3 If the embryo droplet has the exact equilibrium radius, re, in which the liquid droplet is in thermodynamic and mechanical equilibrium with the surrounding vapor, g A ,e = g v ,e (8.24) Therefore, the change of availability at equilibrium becomes 4 ΔΨ e = π re2σ Av F (8.25) 3 The change of availability near the equilibrium radius can be obtained by expanding eq. (8.21) in the form of a Taylor series, i.e., ∂ΔΨ 1 ∂ 2 ΔΨ ΔΨ = ΔΨ e + (8.26) (r − re ) + (r − re ) 2 + " ∂r e 2 ∂r 2 e mA = The derivative of ΔΨ with respect to r can be found from eq. (8.23), i.e., ∂ΔΨ 4 8 § ∂g ∂g · 4 = π r 3 F ¨ A − v ¸ + π r 2 F ( g A − g v ) + π rσ Av F (8.27) ∂r ∂r ¹ vA 3vA 3 © ∂r The second order derivative can be found by differentiating eq. (8.27), i.e., § ∂2 g ∂2 g · 8 4 ∂ 2 ΔΨ § ∂g ∂g · π r 3 F ¨ 2A − 2v ¸ + π r 2 F ¨ A − v ¸ = 2 3vA ∂r ∂r ¹ vA ∂r ¹ © ∂r © ∂r (8.28) 8 8 + π rF ( g A − g v ) + πσ Av F vA 3 Since the pressure and temperature of the vapor phase, pv and Tv , are fixed, the Gibbs free energy for the vapor phase remains the same near the equilibrium: g v = g v ,e (8.29) i.e., ∂g v =0 (8.30) ∂r ∂ 2 gv =0 (8.31) ∂r 2 While the temperature of the liquid droplet is the same as the vapor temperature ( TA = Tv ), the pressure in the liquid droplet is related to the pressure in the vapor phase by pA = pv + 2σ Av / r . Therefore, the Gibbs free energy for the liquid phase is g A = g A ,e + dg A (8.32) where Chapter 8 Condensation 591 dg A = vA dpA = − i.e., 2vAσ Av dr r2 (8.33) 2v σ ∂g A = − A 2 Av r ∂r and (8.34) ∂ 2 g A 4vAσ Av = (8.35) r3 ∂r 2 Substituting eqs. (8.30), (8.31), (8.34), and (8.35) into eqs. (8.27) and (8.28), one obtains ∂ΔΨ 4 2 = π r F ( gA − gv ) (8.36) ∂r vA ∂ 2 ΔΨ 8 = −8π Fσ Av + π rF ( g A − g v ) 2 ∂r vA At equilibrium, eqs. (8.36) and (8.37) become ∂ΔΨ =0 ∂r e ∂ 2 ΔΨ = −8π Fσ Av ∂r 2 e (8.37) (8.38) (8.39) Substituting eqs. (8.38) and (8.39) into eq. (8.26), the change of availability near the equilibrium radius becomes 4 ΔΨ = π re2σ Av F − 4πσ Av F (r − re ) 2 + " (8.40) 3 Figure 8.9 Variation of the system availability with droplet radius. 592 Transport Phenomena in Multiphase System It can be seen from the above equation that Δψ is at its maximum at r = re, and, therefore, it is shown once again that at the equilibrium radius the droplet is in unstable equilibrium (see Fig. 8.9). However, to maintain equilibrium dΨ = dG < 0, i.e., the system tends naturally to achieve the lowest Gibbs free energy value. Therefore, as the droplet increases in radius, the availability of the system decreases and the droplet is in equilibrium. Also, it should be pointed out that if a droplet forms with a radius smaller than equilibrium it spontaneously destroys itself. This can also be seen by the minimum Gibbs free energy principle. If the liquid droplet forms with a radius below the equilibrium radius and tries to grow, ΔΨ > 0; therefore, the system is increasing in availability, which is highly unstable. For the availability to decrease and approach equilibrium, the liquid droplet would need to continuously decrease in size until it disappears. This phenomenon can also be seen in Fig. 8.9. The unstable equilibrium radius is at the top of the curve. If a droplet with such a radius loses one molecule, it will need to continuously decrease in size to meet the ΔΨ< 0 criterion. If the droplet gains one molecule, it will need to continuously increase in size to meet the ΔΨ< 0 criterion. An expression to determine the minimum equilibrium radius of a newly formed droplet that will not spontaneously disappear was developed in Section 2.6.3; the result is expressed by eq. (2.259), i.e., 2σ vA rmin = (8.41) Rg T ln[ pv / psat (T )] Considering the Clapeyron equation (2.168) with Tv = T0 , T = Tw , and TvTw ≈ Tw2 , the expression for minimum equilibrium radius size becomes 2vAσ Tv D rmin = min = (8.42) 2 hAv (Tv − Tw ) where hAv is the latent heat of energy for the vapor-to-liquid conversion, and σ is the surface tension of the condensing fluid. Example 8.1 Saturated steam at 1 atm is in contact with a cold wall at 95 ÛC. Estimate the minimum equilibrium size for dropwise condensation. Solution: The saturated vapor temperature at 1 atm is Tv = 100 o C, and the latent heat of vaporization is hAv = 2251.2 kJ/kg . The surface tension is σ = 58.91 × 10−3 N/m. The specific volume of liquid water, determined at Tave = (Tv + Tw ) / 2 = 97.5 o C , is vA = 1.041 × 10−3 m3 /kg. The minimum radius of the liquid droplet is obtained from eq. (8.42): 2vAσ Tw 2 × 1.041× 10−3 × (95 + 273.15) = = 6.8 × 10−8 m = 0.068 m rmin = 2251.2 × 103 × (100 − 95) hAv (Tv − Tw ) Chapter 8 Condensation 593 8.2.3 Thermal Resistances in the Condensation Processes The condensation process must overcome a series of thermal resistances for the heat and mass transfer to occur. These resistances include the thermal resistance found in the vapor, thermal resistance encountered during the phase change from vapor to liquid, resistance caused by capillary depression of the equilibrium saturation temperature at the interface, thermal resistance found in the liquid phase, and thermal resistance found at the wall where heat is conducted from the surface into the wall. The mode of conduction into the wall depends on whether dropwise condensation or filmwise condensation occurs, as will be discussed below. In short, when filmwise condensation occurs uniformly along the surface, the heat flux can simply be found from a straight application of Fourier’s Law of conduction into a solid. However, if dropwise condensation occurs, conduction into the wall is constricted around the individual droplets and cannot occur uniformly over the solid wall. However, some resistances can be neglected in relation to others, except in special cases. These individual resistances and their importance to the overall resistance will be discussed in this section. Figure 8.10 shows the resistance to heat flow associated with both filmwise and dropwise condensations, in which Rw is resistance resulting from conduction of heat through the cold wall, Rliquid is resistance resulting from heat conduction through liquid film or a droplet, Rcap is the resistance resulting from capillary depression of the equilibrium saturation temperature, Rδ is interfacial thermal resistance, Rv is resistance resulting from heat transfer in the vapor phase, and Rconv is the convective thermal resistance for the area not covered by the droplet in the dropwise conduction. Overall, the thermal resistance associated with vapor will be the same for both dropwise and filmwise condensation, as is also the case for the interfacial resistance and capillary depression resistance. The conduction resistances found in dropwise and filmwise condensation are different, i.e., heat Figure 8.10 Schematic of the resistance to heat flow in the condensation process: (a) filmwise condensation; (b) dropwise condensation. 594 Transport Phenomena in Multiphase System is conducted differently through liquid droplets as compared to a liquid film. Conduction through an individual liquid droplet is a function of the size of the droplet (an expression that takes into account all droplets found on the wall surface will also be presented), while the conduction through a thin liquid film is a function of the film thickness and thus a function of position on the wall. The thin liquid film will be discussed in great detail in the next section. It is assumed in this discussion that the wall temperature is held at a constant temperature, Tw, and therefore the overall temperature drop for the area covered by the droplet is as follows: ΔTtotal = Tvapor − Tw = ΔTvapor + ΔTδ + ΔTcap + ΔTdroplet (8.43) where the temperature differences are for the vapor, interface δ , capillary depression of the equilibrium saturation temperature cap, and conduction through the droplet, respectively. Resistance in the Vapor The thermal resistance found in vapor can usually be ignored except in special cases, because it is usually an order of magnitude less than the other resistances found in the condensation process. This low contribution to the overall resistance is the result of its ability to mix extremely well if either free or forced convection is present; this in turn allows the heat and mass transfer towards the cooler surface to easily occur. However, if the vapor is superheated, this mixing process is severely limited, and the resistance of the vapor phase would have to be considered in the form of conduction through vapor. In that case, the controlling temperature difference would be (Tv –Tsat), i.e., the temperature difference between the bulk superheated vapor and the saturated temperature at the liquid-vapor interface. Other cases in which the thermal resistance of the vapor would have to be taken into account include the condensation of vapor mixtures (described above in binary mixtures) and vapor mixtures that include an inert gas (noncondensable gas). The inert gas effectively insulates the conduction of heat through a vapor. However, in this discussion the thermal resistance of the vapor will be ignored because it is usually negligible except for the above cases. Interfacial Resistance The next resistance encountered in the thermal path from vapor to wall is that found at the vapor-liquid interface. The high heat transfer coefficients associated with the condensation process make it possible to achieve a large heat transfer rate with a small temperature drop (Faghri, 1995). This is necessary because the temperature drop at the vapor-liquid interface in a condensation process is very small. This resistance is found in both filmwise and dropwise condensation and the expressions are identical. Chapter 8 Condensation 595 The heat flux at the interface can be obtained by eq. (5.152), i.e., Mv § pv vAv · 2α · § hA2v · ′′ § qδ = ¨ (8.44) ¸ ¨1 − ¸ (Tv − TA ) ¸¨ 2hAv ¹ © 2 − α ¹ © Tv vAv ¹ 2π RuTv © where α is accommodation coefficient. The corresponding heat transfer coefficient across the interface is obtained by eq. (5.154), i.e., 2 ′′ qδ Mv § pv vAv · § 2α · § hAv · hδ = =¨ (8.45) ¸ ¨1 − ¸ ¸¨ (Tv − TA ) © 2 − α ¹ © Tv vAv ¹ 2π RuTv © 2hAv ¹ For most systems the second term in the last parentheses is very small compared to unity and, therefore, can be neglected to obtain the following: 2 ′′ qδ Mv § 2α · § hAv · (8.46) =¨ hδ = ¸ ¸¨ (Tv − TA ) © 2 − α ¹ © Tv vAv ¹ 2π RuTv The temperature drop across the interface of a single liquid droplet – assuming it is hemispherical – is qd ΔTδ = (8.47) hδ (π D 2 / 2) where π D 2 / 2 = AAv is the surface area of the hemispherical liquid droplet. Resistance Due to the Capillary Depression of the Equilibrium Saturation Temperature According to Graham and Griffith (1973), a resistance exists due to the slight depression of the equilibrium interface temperature below that of the normal saturation temperature for a droplet of diameter D. Assuming that a droplet forms with the minimum equilibrium diameter and grows spontaneously to any nonstable diameter D, we can replace the temperature difference Tsat – Tw with ΔTcap and the minimum droplet diameter Dmin with the actual size of the droplet D in eq. (8.42) to get the following expression for the temperature drop across the capillary depression: 4v σ T ΔTcap = A sat (8.48) hAv D Combining eq. (8.48) with the minimum equilibrium droplet size expression, eq. (8.42), one obtains (T − Tw ) Dmin (8.49) ΔTcap = sat D 596 Transport Phenomena in Multiphase System Resistance Due to Conduction through the Droplet The conduction of heat through either a liquid droplet or liquid film is usually the controlling factor in resistance to heat flow. This is due directly to the fact that the largest temperature drop in the condensation process occurs in the liquid film or droplet, even though the conduction path is relatively short in comparison to the other heat flow lengths in the condensation process. This leads to high resistances and low heat transfer coefficients. Graham and Griffith (1973) also developed an expression for the conduction of heat through a single droplet of diameter D from the liquid-vapor interface to the wall. As the heat flux travels through the droplet from interface to wall, the planar area normal to the heat flux varies due to the droplet’s spherical shape. Also, the distance that the heat flux has to travel through the droplet depends on where it entered the droplet at the liquid-vapor interface. Therefore, Graham and Griffith (1973) took into account these variations in the development of the heat flux through a single droplet. This heat flux can be written in terms of the temperature drop through the droplet due to conduction as follows: qd ( D / 2) ΔTdroplet = (8.50) 4π kA ( D / 2) 2 8.2.4 Heat Transfer Coefficient for Dropwise Condensation By substituting the expressions for temperature drops through the interface, eq. (8.47); capillary depression, eq. (8.49); and liquid droplets, eq. (8.50), into eq. (8.43) and neglecting the temperature drop in the vapor phase, the temperature drop is obtained: 2qd qd D ΔTtotal = Tsat − Tw = + (Tv − Tw ) + (8.51) 2 hδ π D Dmin 2kAπ D The heat flux through a single droplet can then be given by rearranging eq. (8.51), i.e., § π D2 · (1 − Dmin / D ) qd = ¨ (8.52) ¸ ΔTtotal (1/ hδ + D / 4kA ) ©2¹ To obtain an expression for the total heat transfer through all of the droplets, one must integrate over the total number of droplets and the whole size distribution of the droplets. To do this, we must know the droplet size distribution equal to the number of droplets with diameters between D and D + dD per unit area of surface. Therefore, to obtain the expression for total heat flux ′′ ′′ we multiply qδ times the number density nD dD of droplets of size D and integrate over the whole range of droplet size (Carey, 1992): πΔTtotal Dmax (1 − D / D ) ′′ 2 q′′ = (8.53) ³Dmin nD D (1/ hδ +min / 4kA ) dD D 2 Chapter 8 Condensation 597 The total heat transfer coefficient for the area covered by the liquid droplets can be found for the dropwise condensation process: (1 − Dmin / D ) π Dmax ′′ hdrops = ³ nD D 2 dD (8.54) Dmin 2 (1/ hδ + D / 4kA ) In dropwise condensation, the type of conduction that occurs at the wall is a direct result of the constricted heat flow around and between the large droplets on the surface of the wall. In filmwise condensation, conduction from the liquidsolid interface would be obtained by applying Fourier’s Law through the liquid film. However, in dropwise condensation, the resistance to thermal transport found in the liquid droplets is much higher than the resistance found in the liquid-free area between the droplets. Conduction through the walls would prefer to initiate at the bare areas. Therefore, the droplets effectively “squeeze” or “constrict” the heat flow toward the small bare areas. Mikic (1969) developed an expression for the resistance encountered in this constriction: § 1 · Dmax nD D 2 dD ′′ Rcons = ¨ (8.55) ¸ ³Dmin [1 − f ( D)] © 3π kA ¹ where f(D) is the fraction of the surface area covered by droplets with diameters greater than D. As can be seen above, this expression is a total expression for resistance, because it takes into consideration the full range and number of droplets. Finally, if this resistance is combined and assumed to be in series with all the other resistances described in previous sections, the total heat transfer coefficient, including conduction through the wall (in this development, the constriction conduction mode was used due to the dropwise condensation discussion), is found to be §1 · + Rcons ¸ htotal = ¨ (8.56) ¨h ¸ © drops ¹ This theoretical model cannot be used to predict the heat transfer coefficient ′′ for dropwise condensation unless we know the droplet size distribution nD . Determination of droplet size is very difficult, especially for droplets smaller than 10 m, which are major contributors to the heat transfer in dropwise condensation. Among the possible combinations of fluids and surfaces, steam and wellpromoted copper surfaces have been investigated extensively. Griffith (1983) recommended the following correlation for prediction of the heat transfer coefficient for dropwise condensation of steam: ­51104 + 2044Tsat 22 o C < Tsat < 100 o C ° (8.57) h=® 255510 Tsat > 100 o C ° ¯ Example 8.2 On a clear winter night, dropwise condensation occurred on the inner surface of window glass at a temperature of 5 °C. The air −1 598 Transport Phenomena in Multiphase System temperature in the room is 25 °C and the relative humidity in the room is 80%. Estimate the condensation rate using Griffith’s correlation. Solution: The saturation pressure of the water vapor at 25 °C is psat = 0.03596 bar. The partial pressure of the water vapor in an air with 80% humidity is pwater = 0.8 × 0.03596 = 0.02877 bar . Therefore, the dew point is the saturation temperature corresponding to pwater, i.e., Tsat = 22.1 °C. The latent heat of vaporization at this temperature is hAv =2448.8 kJ/kg. Assuming eq. (8.57) is valid for dropwise condensation, the heat transfer coefficient is h = 51104 + 2044Tsat = 51104 + 2044 × 22.1 = 96276.4 W/ o C-m 2 The condensation rate is then q′′ h(Tsat − Tw ) 96276.4 × (22.1 − 5) m′′ = = = = 0.672 kg/m 2 -s hAv hAv 2448.8 × 103 8.3 Filmwise Condensation 8.3.1 Regimes of Filmwise Condensation As in the case of external heterogeneous dropwise condensation, filmwise condensation occurs when a cold wall surface is in contact with a vapor near saturation conditions. Filmwise condensation on a vertical surface occurs when the liquid phase fully wets the surface, whereas in dropwise condensation the liquid incompletely wets the solid surface. The condensation process begins with vapor condensing directly on the wall surface. However, in contrast with dropwise condensation, after the wall is initially wetted it remains covered by a thin film of condensate. After that point, condensation occurs only at the liquid-vapor interface. Therefore, the condensation rate is directly a function of the rate at which heat is transported across the liquid film from the liquid-vapor interface to the wall. Figure 8.11 shows three distinct regimes of filmwise condensation on a vertical wall. These regimes are proceeding in order from the top of the wall (x = 0): laminar, wavy, and turbulent. The Reynolds number is defined as Reδ = 4Γ / μA , where Γ is mass flow rate of condensate per unit width. At the top of the wall, where the film is thinnest, the laminar regime exists. As the condensation process proceeds, more and more condensation appears on the surface and the liquid condensate is pulled downward by gravity. As the condensate moves downward, the film becomes thicker. The first sign of transition to a non-laminar regime appears as a series of regular ripples or waves of condensate. This regime is called the wavy regime and is considered neither laminar nor turbulent. It is characterized by consistent, regular Chapter 8 Condensation 599 Re § 30 Re § 1800 Figure 8.11 Flow regimes of film condensate on a vertical wall. series of waves in time. Finally, if the wall is long enough, the film thickness becomes so great that irregular ripples in both time and space will appear which is identified as turbulent flow regime. The laminar regime was first rigorously analyzed by Nusselt (1916). Because many simplifying assumptions were made, this analysis provided a closed-form solution. This classical analysis was a very good building block for later studies that gradually chipped away at the assumptions made by Nusselt by employing numerical methods. The classical laminar flow condensation analysis will be presented in this section, followed by some of the later studies that improved Nusselt’s model. The improvements presented here will include the consideration of noncondensable vapors in the condensation process and the effect of vapor flow (Nusselt assumed a stagnant vapor reservoir). The wavy and turbulent regimes are obviously much more difficult to solve than the laminar, and numerical methods are required to obtain an acceptable solution. However, the reasoning behind these regimes will be presented because the overall heat transfer rate from the vapor reservoir to the cooled wall is dominated by contributions from the wavy and turbulent sections. In fact, most industrial applications require that the walls in surface condensers are of a sufficient length and have the surface modified in order to guarantee wavy and turbulent regimes. 8.3.2 Modeling for Laminar Film Condensation of a Binary Vapor Mixture Knowledge of binary vapors is of the utmost practical importance, because pure vapor is rarely present in everyday life or in industrial applications. Consider the steady laminar film condensation of a binary vapor on a smooth, vertical, cooled 600 Transport Phenomena in Multiphase System Figure 8.12 Physical model and coordinate system for condensation of a binary vapor mixture (Fujii, 1991; Reproduced with kind permission of Springer Science and Business Media). wall. The bulk vapor mixture flows downwards and parallels the vapor boundary layer and condensate film flows, which are also flowing downwards due to the effects of gravity. The physical model of this condensation of a binary vapor is shown schematically in Fig. 8.12, where x is the distance measured along the flat plate starting from the leading edge and y is the normal distance from the plate. This figure depicts the condensate film boundary layer directly adjacent to the wall with thickness δ=δ(x). Directly adjacent to this condensate film is the binary vapor boundary layer, with thickness δ v = δ v ( x), which develops between the condensate film and bulk vapor flow. The velocity components of the x- and y-directions are u and v respectively. The temperature is denoted by T and the pressure of the system is a constant p. The mass fraction of a component is denoted by ω . The subscripts A , v, δ , w, and ∞ denote the liquid, vapor, interface, wall, and bulk conditions, respectively. Finally, the subscript 1 denotes the component of the binary vapor with the lower boiling point. The following assumptions were made to describe the laminar film condensation (Fujii, 1991): 1. Tw, Tδ , ω1vδ , Tv∞ , ω1v∞ , uvδ , uAδ , and uv∞ are independent of x. 2. The condensate film and binary vapor boundary layers both develop from the leading edge of the vertical surface, x=0. 3. Condensation takes place only at the vapor-liquid interface. In other words, no condensation takes place within the binary vapor boundary layer in the form of a mist or fog. 4. Both temperature and velocity are continuous at the vapor-liquid interface. 5. The condensate is miscible. Chapter 8 Condensation 601 6. The physical properties of the system are assumed to be constant with respect to concentration and temperature except in the case of buoyancy terms. 7. The density of the condensate liquid is assumed to be much greater than that of the binary vapor ( ρA  ρv ). 8. The vapor mixture can be treated as an ideal gas, and the thermal diffusion is negligible. Before the governing equations are presented, an explanation is needed for the physical conditions that occur during this process. When a binary vapor makes contact with a cooled vertical wall, the less volatile component of the mixture (the component with the higher boiling point) begins to condense first and in greater quantity. Since the system must maintain the total mass concentration to maintain equilibrium, the volatile component becomes much denser at the liquid-vapor interface. During this process, the bulk vapor with a constant mass concentration is steadily supplying the liquid-vapor interface. Therefore, the vapor boundary layer develops with a very high concentration of the volatile component at the liquid-vapor interface, which is quickly diluted to the concentration of the bulk vapor. The governing equations for the laminar film condensation of a binary vapor mixture can be given by taking the above assumptions into account and using boundary layer analysis. For the condensate film, the continuity, momentum and energy equations are: ∂uA ∂vA (8.58) + =0 ∂x ∂y uA ∂uA ∂u ∂ 2u 1 dp + vA A = ν A 2A + g − ρA dx ∂x ∂y ∂y uA (8.59) ∂TA ∂T ∂ 2T (8.60) + vA A = α A 2A ∂x ∂y ∂y For the vapor boundary layer, the continuity, momentum, energy, and species equations are: ∂uv ∂vv (8.61) + =0 ∂x ∂y § ρ· ∂uv ∂u ∂ 2u + vv v = ν v 2v + g ¨ 1 − v∞ ¸ ∂x ∂y ∂y ρv ¹ © 2 ∂T ∂T ∂T ∂ω ∂T uv v + vv v = α v 2v + Dc p12 1v v ∂x ∂y ∂y ∂y ∂y uv uv (8.62) (8.63) (8.64) ∂ω1v ∂ω ∂ 2ω1v + vv 1v = D ∂x ∂y ∂y 2 602 Transport Phenomena in Multiphase System where ν is the kinematic viscosity, ρ is the density, k is the thermal conductivity, and D is the diffusivity between components 1 and 2. The second term on the right-hand side of eq. (8.63) represents the contribution of mass concentration gradient to the energy balance, which is usually negligible. The isobaric specific heat difference of the binary vapor, c p12 , is a weighted average of the isobaric specific heats of the two individual components; it is nondimensional as follows: c p1v − c p 2 v c p1v − c p 2 v c p12 = = (8.65) c p1vω1v + c p 2 vω2 v c pv This isobaric specific heat term is assumed to be constant even though it has a larger fluctuation then the other physical properties, because overall the second term on the right-hand side of eq. (8.63) is much smaller than the first. Other definitions of the terms in eqs. (8.58) – (8.64) are as follows, beginning with the mass fractions ω1v and ω2 v in the vapor phase: ρ ω1v = 1v (8.66) ρv ω2 v = where ρ2v ρv (8.67) (8.68) Therefore, it follows from eqs. (8.66) – (8.68) that a relationship between ω1v and ω2 v can be written as follows: ω1v + ω2v = 1 (8.69) Note that only the continuity equation for component 1 of the vapor is given. However, it should be immediately recognized from eq. (8.69) that if the mass fraction of one component is known at any point in space, then the mass fraction of the other components can easily be determined. The partial pressures of the system are determined by the following simple expressions: p1 § M 1ω2 v · = ¨1 + ¸ p © M 2ω1v ¹ −1 ρv = ρ1v + ρ 2v (8.70) −1 p2 § M 2ω1v · = ¨1 + (8.71) ¸ p © M 1ω2 v ¹ where M1 and M2 are the molecular masses of components 1 and 2, respectively. The boundary equations for the above-generalized governing equations at the surface of the cold wall are as follows: uA = 0, y = 0 (8.72) vA = 0 , y = 0 (8.73) TA = Tw , y = 0 (8.74) Chapter 8 Condensation 603 These boundary conditions represent no-slip at the wall and the continuity of temperature at the wall. The boundary conditions at locations far from the cold wall are uv = uv∞ , y → ∞ (8.75) Tv = Tv∞ , y → ∞ (8.76) ω1v = ω1v∞ , y → ∞ (8.77) These boundary conditions represent conditions in the constant bulk binary vapor. Finally, the boundary conditions that exist at the liquid-vapor interface ( y = δ ) are given as follows: uAδ = uvδ = uδ (8.78) § ∂uA · § ∂uv · ¨ μA ¸ = ¨ μv ¸ ∂y ¹δ © ∂y ¹δ © ª § dδ ª § dδ ·º ·º ′′ « ρ A ¨ uA dx − vA ¸ » = « ρv ¨ uv dx − vv ¸ » = m′′ = m1′′ + m2 ¹ ¼δ ¬ © ¹¼δ ¬© TAδ = Tvδ = Tδ (8.79) (8.80) (8.81) § ∂TA · § ∂Tv · (8.82) ¨ kA ¸ = hAv m′′ + ¨ kv ¸ © ∂y ¹δ © ∂y ¹δ ω1v = ω1vδ (8.83) where μ is the dynamic viscosity, k is the thermal conductivity and hAv is the ′′ ′′ latent heat of condensation. Finally, m′′ = m1 + m2 is the condensation mass flux perpendicular to the vertical plate; it is a dependent of the location along the xaxis. Equations (8.78), (8.81), and (8.83) are the boundary conditions that refer to the continuity of velocity, temperature, and mass fraction at the liquid-vapor interface, respectively. Equation (8.79) recognizes that the shear stresses in the liquid and vapor layers are equal at the liquid-vapor interface. Equation (8.82) is an energy balance at the interface, which relates the heat transferred to and from the interface with that released by the latent heat. Equation (8.80) is the overall condensation mass continuity at the interface. The mass flux of the ′′ noncondensable component 2, m2 x , at interface in eq. (8.80), is not zero because the noncondensable component 2 may dissolve into the condensate. The mass fluxes of the vapor in the binary vapor system are [see eq. (1.100)] ∂ω ′′ ′′ ′′ m1 = − ρ D12 1 + ω1 (m1 + m2 ) (8.84) ∂y ∂ω ′′ ′′ ′′ m2 = − ρ D21 2 + ω2 (m1 + m2 ) (8.85) ∂y and the molar fluxes of the vapor in the binary vapor system are [see eq. (1.101)] ∂c ′′ ′′ n1 = − D12 1 + x1nT (8.86) ∂y 604 Transport Phenomena in Multiphase System ∂c2 ′′ (8.87) + x2 nT ∂y where ci is molar concentration (Kmol/m3) of component i (i =1, 2). If both components can be treated as ideal gases, the molar fraction is identical to the ′′ ′′ ′′ partial pressure obtained from eqs. (8.70) and (8.71). nT = n1 + n2 is the total molar flux of component 1 and 2. Equations (8.86) and (8.87) are often expressed in terms of partial pressure, i.e., p D ∂p1 ′′ ′′ + nT 1 (8.88) n1 = − RuT ∂y p p D ∂p2 ′′ ′′ + nT 2 (8.89) n2 = − RuT ∂y p where Ru is the universal gas constant. Equations (8.86) – (8.89) are valid in the binary vapor boundary layer ( δ ≤ y ≤ δ + Δ ). Also, the mass fraction of component 1 in the condensate film can be found from the following expression: ′′ m1x ω1A = (8.90) ′′ ′′ m1x + m2 x ′′ n2 = − D21 If the noncondensable component cannot be dissolved into the liquid film, ω1A will be unity. Effects of noncondensable gas on film condensation will be discussed in Section 8.3.7. 8.3.3 Filmwise Condensation in a Stagnant Pure Vapor Reservoir Laminar Flow Regime The classical analysis of laminar film condensation on a vertical or inclined wall was performed by Nusselt (1916). The physical conditions of laminar film condensation have been shown in Fig. 8.13. As is the case in any heat transfer analysis, the final goal is to obtain the heat transfer coefficient and the corresponding Nusselt number for the heat transfer device under consideration. Therefore, the objective of this section is to find the heat transfer coefficient and the Nusselt number for the laminar flow regime in film condensation on a vertical surface. The classical Nusselt analysis requires many assumptions in order to achieve a closed-form solution of the boundary layer type momentum equations. These assumptions include the following: 1. The flow is laminar. 2. Fluid properties are constant. 3. Subcooling of the liquid is negligible in the energy balance, i.e., all condensation occurs at the saturation temperature corresponding to the pressure in the liquid film near the wall. Chapter 8 Condensation 605 Figure 8.13 Overview of the control volume under consideration in the Nusselt analysis. 4. Inertia and convection effects are negligible in the boundary layer momentum and energy equations, respectively. 5. The vapor is assumed stagnant, and therefore, shear stress is considered to be negligible at the liquid-vapor interface. 6. The liquid-vapor interface is smooth, i.e., condensate film is laminar and not in the wavy or turbulent stages. Since the vapor phase is stationary, the governing equations for the vapor phase are no longer needed. The continuity equation for the liquid phase is the same as eq. (8.58). The momentum equation in the liquid phase is eq. (8.59), in which the pressure gradient may be approximated in terms of conditions outside the liquid film, as required by boundary layer approximation. It follows that pressure in the liquid film satisfies dpA dpv = = ρv g (8.91) dx dx Substituting eq. (8.91) into eq. (8.59), the momentum equation becomes § ∂u ∂u · ∂ 2u ρA ¨ u + v ¸ = μA 2 + g ( ρA − ρv ) (8.92) ∂y ¹ ∂y © ∂x 606 Transport Phenomena in Multiphase System The left-hand side term represents the inertia effects in the slender film region, while the right-hand terms are the effects due to friction and the sinking effect. As stated by Assumption #4 of the above list, the inertia term is negligible in this analysis. Therefore, the above equation simplifies to ∂ 2u g (8.93) = ( ρv − ρA ) ∂y 2 μA Integrating this equation twice with respect to y and using the boundary conditions of nonslip at the wall (u=0 at y=0) and zero shear at the liquid-vapor interface ( ∂u / ∂y = 0 at y = δ, Assumption #5) yields the following: ( ρA − ρv ) g § y2 · yδ − ¸ (8.94) ¨ μA 2¹ © It can be seen from this equation that the downward velocity component is directly dependent on both the x and y coordinates due to the varying (increasingly thick in the x-direction) film thickness. The mass flow rate per unit width of surface, , of this liquid film at any point can be found by integrating the above velocity profile across the liquid film thickness and multiplying by the liquid density, i.e., δ ρ ( ρ − ρ v ) gδ 3 Γ = ρA ³ udy = A A (8.95) 0 3μ A where it can now be seen that the mass flow rate is a function of the x coordinate. Recalling that the heat transfer across the liquid film is by conduction only, and assuming that no subcooling exists at the liquid-vapor interface (Assumption #3), Fourier’s Law can be used to obtain the heat flux across the film thickness: k (T − Tw ) q′′ = A sat (8.96) u ( x, y ) = δ where Tsat and Tw are the saturation temperature and wall temperature, respectively. The heat transfer rate per unit width for the control volume shown in Fig. 8.13 is given as k ΔT dq′ = A dx (8.97) δ where ΔT = Tsat − Tw . Since no subcooling in the liquid film exists, the latent heat effects of condensation dominate the process. Therefore, it can be said that dq′ = hAv d Γ (8.98) where d Γ is found by differentiating the expression for mass flow rate per unit surface, eq. (8.95). It is found to be ρ ( ρ − ρ v ) gδ 2 dΓ = A A dδ (8.99) μA Chapter 8 Condensation 607 This expression and the expression for heat flow dq′ across the control volume are then substituted into the energy equation, eq. (8.98), and the following is found: kA μA ΔT dδ (8.100) = δ3 dx ρ A ( ρ A − ρv ) ghAv Finally, δ can be found by integrating the above and using the boundary condition that δ = 0 at x = 0: ª 4kA μA xΔT º δ =« » ¬ ρ A ( ρA − ρ v ) ghAv ¼ The local heat transfer coefficient hx is found to be 1/ 4 (8.101) ª ρ ( ρ − ρv ) ghAv º k q′′ = A = kA3/ 4 « A A hx = » Tsat − Tw δ 4 μA xΔT ¬ ¼ 1/ 4 (8.102) The local Nusselt number, Nux, of the laminar film condensation process on a vertical plate is found to be h x ª ρ ( ρ − ρv ) ghAv x 3 º (8.103) Nu x = x = « A A » kA ¬ 4kA μA ΔT ¼ It is also desirable to obtain the average heat transfer coefficient and Nusselt number for a plate of length L. The mean heat transfer coefficient can be defined as 1L h = ³ hx ( x) dx (8.104) L0 Substituting the expression for hx , eq. (8.102), into eq. (8.104) and integrating, the following expressions are obtained: ª ρ ( ρ − ρv ) ghAv L3 º hL = 0.943 « A A Nu = » μA kA ΔT kA ¬ ¼ 1/ 4 1/ 4 (8.105) The local and average heat transfer coefficient can also be nondimensionalized in term of Reynolds number Reδ , defined as 4Γ (8.106) Reδ = μA Substituting eq. (8.95) into eq. (8.106), the Reynolds number for laminar film condensation becomes 4 ρ ( ρ − ρ ) gδ 3 Reδ = A A 2 v (8.107) 3μA Substituting eqs. (8.106) and (8.101) into eqs. (8.103) and (8.105), the following nondimensional correlations are obtained: hx kA ª º μA2 « » ¬ ρA ( ρA − ρv ) g ¼ 1/ 3 − = 1.1Reδ 1/ 3 (8.108) 608 Transport Phenomena in Multiphase System ª º μA2 −1/ 3 (8.109) « » = 1.47 Reδ ρA ( ρA − ρv ) g ¼ ¬ where the bracket term to the left of the equal sign, together with its exponent (1/3), is the characteristic length. The above Nusselt analysis assumes no subcooling in the liquid condensate. This assumption can be removed by making an energy balance at the interface that takes subcooling of the liquid into account, as follows: dq′ kA ΔT dΓ d δ ρA c pA u (Tsat − T ) dy (8.110) = = hAv + δ dx dx dx ³0 The final term of the right-hand side takes into account a temperature gradient across the liquid condensate film. Substituting the velocity profile in eq. (8.94) and using a linear temperature profile, Tsat − T y =1− (8.111) δ Tsat − Tw to evaluate eq. (8.110), the energy balance can be written as kA ΔTA dΓ = hA′v (8.112) δ dx where ­ ½ ° 3 ª c pA (Tsat − Tw ) º ° hA′v = hAv ®1 + « (8.113) »¾ hAv ° 8¬ ° ¼¿ ¯ Equation (8.112) is identical to the energy balance in eq. (8.97) and (8.98) except that hAv has been replaced by hA′v . Rohsenow (1956) improved this analysis even more by including the effects of convection in the liquid along with liquid subcooling, and thereby developing the following: ­ ½ ª c pA (Tsat − Tw ) º ° ° hA′v = hAv ®1 + 0.68 « (8.114) »¾ hAv ° ° ¬ ¼¿ ¯ Finally, if it is desirable to include the effects of vapor superheat in the above analysis, the latent heat obtained by eqs. (8.113) or (8.114) can be further modified by adding c pv (Tv − Tsat ) . h kA 1/ 3 The above analysis can also be applied for condensation outside a vertical tube if δ / D  1 , where D is the diameter of the tube. For an inclined wall with an inclination angle of θ (the angle between the wall and the vertical direction), the component of the gravitational acceleration along the inclined wall is It follows from eqs. (8.102) and (8.105) that g cosθ . hx ∝ (cos θ )1/ 3 and h ∝ (cosθ )1/ 4 . Chapter 8 Condensation 609 Wavy Condensate Regime Wavy flows of thin liquid films have higher heat transfer coefficients than smooth thin films, due to mixing action and an increase in the interfacial surface area. It has been shown by experiments that even during well defined laminar film flow, the film surface (liquid-vapor interface) can be wavy. These waves can occur on film flow over a rough wall or a polished wall. This wave formation can even occur when the vapor reservoir is stagnant; however, the waves can be more pronounced when the vapor reservoir has an average velocity, which leads to a higher shear stress at the liquid-vapor interface. These small disturbances can amplify under specific circumstances, i.e., when the film reaches a critical thickness and produces full waves that are regular with respect to time and therefore are not considered turbulent. These waves can lead to an improvement in the heat transfer coefficient by as much as 50%, compared to that of the above laminar Nusselt analysis. The mechanisms of heat transfer enhancement by wave include enlargement of liquid-vapor interfacial area and decrease in mean film thickness. The flow regime of the liquid condensate is determined by the Reynolds number defined by eq. (8.107). While experimental observations of the condensate indicate that the laminar flow regime usually becomes wavy in the general vicinity of Reδ = 30 , Brauer (1956) suggested that the Reynolds number for onset of waves is related to the Archimedes number by Reδ > 9.3 ArA1/ 5 (8.115) where Archimedes number is defined as ArA = ρAσ 3 / 2 μA2 g 1/ 2 ( ρA − ρv )3/ 2 (8.116) The wavy laminar film remains so until approximately Reδ = 1800 , where it becomes turbulent. In heat transfer analysis of the wavy film regime, there is no one theory reliable for calculating the heat transfer across the wavy film. Many numerical analyses have been performed and different coefficients have been found for the heat transfer coefficient and corresponding Nusselt number. Kutateladze (1982) gave the following correlation for the mean Nusselt number of film condensation on a vertical plate where wave effects are present: Re k h = 2 A 1/ 3 1.22 δ , 30 ≤ Reδ ≤ 1800 (8.117) (ν A / g ) Reδ − 5.2 To use eq. (8.117) to determine the heat transfer coefficient, it is first necessary to know the Reynolds number, which depends on the mass flow rate of condensate per unit width, Γ , as indicated by eq. (8.107). The mass flow rate of the condensate can be determined by q′ hL(Tsat − Tw ) Γ= = (8.118) hA′v hA′v 610 Transport Phenomena in Multiphase System Substituting eq. (8.118) into eq. (8.106), one obtains 4hL(Tsat − Tw ) Reδ = (8.119) hA′vν A Rearranging eq. (8.119) yields Reδ hA′vν A h= (8.120) 4 L(Tsat − Tw ) Combining eq. (8.117) and (8.120) yields the Reynolds number for film condensation with waves as follows: 1/ 3 ª 3.7 LkA (Tsat − Tw ) § g · º Reδ = « 4.81 + (8.121) ¨ 2¸ » μA hA′v « ©ν A ¹ » ¬ ¼ After the Reynolds number is determined by eq. (8.121), the heat transfer coefficient can be determined by eq. (8.117). 0.82 Turbulent Film Regime If one follows the film condensate farther down the vertical wall, the film reaches a critical thickness where the waves become irregular with respect to both time and space. This is where the turbulent film regime exists and can exist even with a stagnant vapor reservoir. It follows that this regime occurs when the Reynolds number, Reδ , becomes a critical value that is much higher than that of the wavy regime. As stated in the previous section, the condensate film becomes turbulent at approximately Reδ = 1800. The heat transfer rate in this turbulent regime is much larger than in laminar and wavy flow. The turbulent region in film condensation, as in any turbulent flow, is extremely difficult to model, and all accurate results come from empirical correlations or from detailed numerical modeling. For turbulent flow of condensation on a vertical plate, Labuntsov (1957) recommended the following empirical correlation: ª º μA2 −0.5 0.25 (8.122) « » = 0.023Reδ PrA , PrA ≥ 10 ρA ( ρA − ρv ) g ¼ ¬ Chun and Seban (1971) obtained the following experimental correlation for local turbulent heat transfer coefficient for evaporation of water from a vertical wall, which is applicable for local heat transfer coefficient for condensation: hx kA hx § ν A2 · ¨¸ kA © g ¹ 1/ 3 1/ 3 = 3.8 × 10−3 Re0.4 PrA−0.65 , Re > 5800 PrA−1.06 (8.123) Butterworth (1983) obtained the average heat transfer coefficient for film condensation that covers laminar, wavy laminar, and turbulent flow by combining eqs. (8.109), (8.117), and (8.122) as follows Chapter 8 Condensation 611 Reδ kA , 1 < Reδ ≥ 7200 (8.124) 1/ 3 0.75 (ν / g ) 8750 + 58PrA−0.5 (Reδ − 253) The Reynolds number, Reδ , is needed in order to use eq. (8.124) to determine the heat transfer coefficient for turbulent film condensation. Equation (8.120) was obtained by energy balance and it is valid for all film condensation regimes. Combining eqs. (8.124) and (8.120), the Reynolds number for turbulent flow is obtained: h= 2 A ª 0.069 Lk Pr 0.5 (T − T ) § g ·1/ 3 º A A sat w − 151PrA0.5 + 253» Reδ = « (8.125) ¨ 2¸ μA hA′v « » ©ν A ¹ ¬ ¼ which can be used together with eq. (8.124) to determine the heat transfer coefficient for turbulent film condensation. Example 8.3 Saturated steam at 1 atm condenses on a vertical wall with a height of L= 1 m and width of b= 1.5 m. The surface temperature of the vertical wall is 80 ÛC. What are the average heat transfer and condensation rates? Solution: The saturation temperature of steam at 1 atm is Tsat =100 ÛC. The vapor density at this temperature is ρv = 0.5974 kg/m3 , and the latent heat of vaporization is hAv = 2251.2kJ/kg . The liquid properties evaluated at T f = (Tsat + Tw ) / 2 = 90 ÛC are 4/3 ρA = 965.3 kg/m3 , c pA = 4.206 kJ/kg-K , μA = 0.315 × 10−3 kg/m-s, kA = 0.675 W/m-K , and ν A = μA / ρA = 0.326 × 10−6 m 2 /s. . The revised latent heat of vaporization is ­ ª c pA (Tsat − Tw ) º ½ ° ° hA' v = hAv ®1 + 0.68 « »¾ hAv ° ¬ ¼° ¯ ¿ ­ ª 4.206 × (100 − 80) º ½ = 2251.2 × ®1 + 0.68 « » ¾ = 2308.4 kJ/kg 2251.2 ¬ ¼¿ ¯ Assuming the film condensation is laminar (as will be verified later), the heat transfer coefficient can be obtained from eq. (8.105), i.e., ª ρ ( ρ − ρv ) g kA3 hAv º h = 0.943 « A A » μA ΔT L ¬ ¼ 1/ 4 ª 965.3 × (965.3 − 0.5974) × 9.8 × 0.6753 × 2308.4 × 103 º = 0.943 × « » 0.315 × 10−3 (100 − 80) × 1 ¬ ¼ 2 = 5340.2W/m -K The heat transfer rate is then q = hLb(Tsat − Tw ) = 5340.2 × 1 × 1.5 × (100 − 80) = 1.602 × 105 W 1/ 4 612 Transport Phenomena in Multiphase System The condensation rate is q 1.602 × 105 m= = = 0.0694 kg/s hA′v 2308.4 × 103 The assumption of laminar film condensation is now checked by obtaining the Reynolds number defined in eq. (8.106), i.e., 4Γ 4 m 4 × 0.0694 Reδ = = = = 588 μA μA b 0.315 × 10−3 × 1.5 which is greater than 30 and below 1800. This means that the assumption of laminar film condensation is invalid and it is necessary to consider the effect of waves on the film condensation. It should be kept in mind the above Reynolds number of 588 is obtained by assuming laminar film condensation. For film condensation with wavy effects, the Reynolds number should be obtained from eq. (8.121), i.e., 1/ 3 ª 3.7 LkA (Tsat − Tw ) § g · º Reδ = « 4.81 + ¨ 2¸ » μA hA′v « ©ν A ¹ » ¬ ¼ 0.82 1/ 3 ª ·º 3.7 × 1 × 0.675 × (100 − 80) § 9.8 = « 4.81 + ¨ ¸» 0.315 × 10−3 × 2308.4 × 103 © (0.326 × 10−6 ) 2 ¹ ¼ « » ¬ = 730.9 which confirms that the film condensation is in the wavy regime. The heat transfer coefficient is obtained from eq. (8.117), i.e., Re k h = 2 A 1/ 3 1.22 δ (ν A / g ) Reδ − 5.2 0.82 0.675 730.9 = 7160 W/m 2 -K 1/ 3 −6 2 [(0.326 × 10 ) / 9.8] 730.91.22 − 5.2 The heat transfer rate is then q = hLb(Tsat − Tw ) = 7160 × 1 × 1.5 × (100 − 80) = 2.15 × 105 W The condensation rate is q 2.15 × 105 = = 0.0931 kg/s m= hA′v 2308.4 × 103 which is much higher than the condensation rate obtained by assuming laminar film condensation. = Chapter 8 Condensation 613 8.3.4 Effects of Vapor Motion Laminar Condensate Flow It was assumed in Section 8.3.3 that the vapor reservoir was stagnant. This assumption was made in order to simplify the analysis of the heat transfer across a thin condensing film. In most real systems, however, the effect of vapor motion must be taken into account. This vapor motion can be due to free (natural effects) or forced (mechanical effects) convection processes. The analysis is conducted in the same way in both cases except that with vapor motion the shear stress at the liquid-vapor interface cannot be assumed to be zero. In the analysis below, the vapor will be considered as having a downward motion. The analysis follows the same outline as the Nusselt analysis presented above for a case where the vapor reservoir is stagnant, producing zero interfacial shear. In other words, referring to Fig. 8.12, the shear stress at the y = δ location of the interfacial control volume will now have a finite real value of shear stress. The boundary layer momentum equation for liquid is once again given as § ∂u dp ∂u · ∂ 2u ρA ¨ u + v ¸ = − A + μA 2 + ρA g (8.126) dx ∂y ¹ ∂y © ∂x where dp / dx has a value different from the Nusselt analysis, because another pressure gradient imposed by the motion of the adjacent vapor exists along with the hydrostatic pressure gradient in the liquid, i.e., dpA § dp · = ρv g + ¨ ¸ (8.127) dx © dx ¹v where ρv is the vapor density and the subscript v denotes vapor motion. For the sake of convenience, this superimposed pressure gradient can be combined into a * fictitious density, ρv , denoted as 1 § dp · (8.128) ¨¸ g © dx ¹v Substituting eq. (8.128) into eqs. (8.127) and (8.126) produces § ∂u ∂u · ∂ 2u ρA ¨ u + v ¸ = μA 2 + g ( ρA − ρv* ) (8.129) ∂y ¹ ∂y © ∂x The left-hand side of eq. (8.129) represents the inertia effects of the slender film region, while the right-hand side gives the effects of friction and the pressure gradient in the liquid. It is assumed once again that the inertia terms are negligible compared to the other terms in this analysis. Therefore, the above equation simplifies to ∂ 2u g * (8.130) = ( ρv − ρA ) ∂y 2 μA ρv* = ρ v + 614 Transport Phenomena in Multiphase System Integrating eq. (8.130) twice with respect to y, using the nonslip boundary conditions at the wall (u = 0 at y = 0), and using the new boundary condition of a finite shear stress at the liquid-vapor interface ( ∂u / ∂y = const at y = δ), the following is obtained: * ( ρ − ρv ) g § y2 · τ y (8.131) u ( x, y ) = A yδ − ¸ + δ ¨ μA 2 ¹ μA © where τ δ is the shear stress at the interface. In this equation it can be seen once again that the downward velocity component depends directly on both the x and y coordinates due to the varying (increasingly thick in the x-direction) film thickness. In addition, u is now also a function of the interfacial shear stress. The mass flow rate per unit width of surface, Γ , of this liquid film at any point can now be found by integrating the velocity profile across the liquid film thickness and multiplying by the liquid density – δ ρ ( ρ − ρv* ) gδ 3 τ δ ρAδ 2 (8.132) Γ = ρA ³ udy = A A + 0 3μ A 2μA where it can now be seen that the mass flow rate is also a function of the x coordinate. The previously described procedure for Nusselt analysis is followed verbatim, using eq. (8.132) for the mass flux across the control volume. Since it is still assumed that heat transfer across the liquid film is by conduction only, and that no subcooling exists at the liquid-vapor interface, heat flux across the film thickness can be obtained using Fourier’s law: k (T − Tw ) q′′ = A sat (8.133) δ where Tsat and Tw are the saturation temperature and wall temperature respectively (ΔT = Tsat – Tw). Rewriting eq. (8.133) in terms of heat transfer rate for the control volume shown in Fig. 8.10, the heat flow is given as k ΔT dq′ = A dx (8.134) δ Since no subcooling of the liquid film exists, the latent heat effects of condensation dominate the process. Thus dq′ = hAv d Γ (8.135) where d Γ can be found by differentiating the above expression for mass flow rate per unit surface area, with the result * § ρ ( ρ − ρ v ) gδ 2 τ δ ρ Aδ · dΓ = ¨ A A + (8.136) ¸ dδ μA μA ¹ © Substituting this expression and the expression for heat flow across the control volume, dq, into the conservation of mass and energy equation, the following is found: Chapter 8 Condensation 615 kA μA ΔT dδ = * dx ρA ( ρ A − ρv ) ghAvδ 3 + τ δ ρA hAvδ 2 (8.137) Finally, δ can be found by integrating the above and using the boundary condition that δ = 0 at x = 0. ª 4 k μ x ΔT º 4τ δ δ 3 AA » (8.138) =« δ4 + * * 3 ( ρ A − ρ v ) g « ρA ( ρ A − ρ v ) ghAv » ¬ ¼ This equation can be nondimensionalized by using the following dimensionless numbers (Rohsenow et al., 1956): δ* = δ LF (8.139) (8.140) (8.141) § x · 4c pA ΔT x* = ¨ ¸ © LF ¹ PrA hAv * τδ = * LF ( ρ A − ρv ) g τδ where ª º μA2 » LF = « (8.142) * « ρA ( ρA − ρv ) g » ¬ ¼ is the characteristic length of the non-dimensional problem. Equation (8.138), which relates film thickness to the vertical location on the plate, can be rewritten as follows: 3* 4 x* = (δ * ) 4 + (δ * ) τ δ (8.143) 3 It directly follows that the mean Nusselt number and Reynolds number for laminar flow with finite vapor shear can be expressed as follows: * * * º hª μA2 4 (δ ) 2 (δ ) τ δ = + (8.144) Nu = L « » * 3 x* kA ¬ ρ A ( ρ A − ρ v ) g ¼ x* 2* 4Γ 4 * 3 Reδ = = (δ ) + 2 (δ * ) τ δ (8.145) μA 3 From the above it can be seen that in order to solve for the average Nusselt number or Reynolds number, the dimensionless numbers x* and δ* are needed. These numbers are directly dependent on each other; therefore, an iterative procedure is required to solve for the heat transfer parameters for each and every point. Additional complications follow from the fact that δ depends on position x and either the variation of heat removal rate across the film with respect to x or the varying temperature drop across the film thickness. In other words, unless the problem statement is oversimplified, the only easy way to solve this problem is numerically. 1/ 3 3 2 1/ 3 616 Transport Phenomena in Multiphase System For the case in which the gravitational force is negligible compared with the interfacial shear force imposed by the co-current vapor flow, Butterworth (1981) recommended the following correlation for local heat transfer coefficient: − Nu * = 1.41Reδ 1/ 2 (τ δ+ )1/ 2 (8.146) x where the modified local Nusselt number is defined as ª º μA2 « » « ρA ( ρA − ρv ) g » ¬ ¼ and the dimensionless interfacial shear stress is h Nu = x k * x 1/ 3 (8.147) (8.148) 2/3 ª ρ A ( ρ A − ρ v ) μA g º ¬ ¼ Butterworth (1981) also recommended the following expression for the cases where both gravity and interfacial vapor shear are significant: 2 2 h = (hshear + hgrav )1/ 2 (8.149) where hgrav is heat transfer coefficient for gravity-dominated film condensation – determined with eqs. (8.103) or (8.108) – and hshear is heat transfer coefficient for shear-dominated film condensation, eq. (8.146). Lin and Faghri (1998) developed a model for predicting the condensation heat transfer coefficient for annular flow in rotating stepped-wall heat pipes. The theoretical result is compared with experimental data. The effect of vapor shear drag on the condensation heat transfer is discussed (see Problem 8.22). Turbulent Condensate Flow τ δ+ = ρAτ δ The velocity and pressure in turbulent flow experience large fluctuations and they can be expressed u = u + u′ (8.150) v = v + v′ (8.151) p = p + p′ (8.152) where the bar notation denotes the mean value averaged over time and the prime notation denotes velocity fluctuations. The boundary-layer equation for forced turbulent flow along a planar surface is · ∂u ∂u 1 ∂p 1 ∂ § ∂u u (8.153) +v =− + − ρ u ′v′ ¸ ¨μ ρ ∂x ρ ∂y © ∂y ∂x ∂y ¹ where the bar notation denotes the mean value averaged over time and the prime notation denotes velocity fluctuations. Introducing the following notation: ∂u − ρ u ′v′ = ρε (8.154) ∂y Chapter 8 Condensation 617 which is known as the eddy shear stress. ε is an empirical function known as the momentum eddy diffusivity; it is a flow property, not a fluid property. A close look at eq. (8.153) shows that the shear stress expression has changed from the normal laminar flow form. The apparent shear stress expression for turbulent flow can be written as follows: ∂u − ρ u ′v′ τ app = μ (8.155) ∂y Substituting eq. (8.154) into eq. (8.155) gives ∂u ∂u ∂u τ app = μ (8.156) + ρε = ρ (ν + ε ) ∂y ∂y ∂y which is the shear stress expression for turbulent flow consisting of both laminar and turbulent portions. If the following is defined ε + =1+ eq. (8.156) can be simplified to ε ν (8.157) ∂u (8.158) ∂y which shows that when ε+ = 1, the problem simplifies to the laminar case. Further, it can be said that when ε /ν  1, then ε+ = ε /ν , and therefore eq. (8.158) becomes simplified to represent the fully turbulent case. The turbulent regime in film condensation with vapor motion is very difficult to model, so empirical correlations are often used to predict the heat transfer coefficient. Figure 8.14 was adapted from Rohsenow et al. (1956) and shows the variation of τ app = ρε +ν Re ,tr * τ δ = 0, 2.5, 5, 10, 20, 50 Transition points Re Figure 8.14 Variation of the mean film condensation heat transfer coefficient with Reynolds * number and τ δ as predicted by Rohsenow et al. (1956). 618 Transport Phenomena in Multiphase System the average film condensation heat transfer coefficient with Reynolds number * and the nondimensional number τ δ . The figure shows both the laminar and turbulent flow regimes, demonstrating that the above expressions for Reynolds number and average Nusselt number give a good solution up to approximately Reδ = 1100 at low nondimensional shear stress values. At that point the heat transfer rate rises sharply in response to the transition to turbulent film flow. It appears that wavy flow at these low shear stress values does not contribute to any change in the flow’s average heat transfer coefficient and Reynolds number. However, higher shear stress numbers allow for a more gradual change (shallow gradient) from laminar to fully turbulent flow, in turn allowing a wavy flow to exist. This shallow gradient shows neither the abrupt steep rise of a turbulent flow nor the consistent downward gradient characteristic of Nusselt flow. Instead, it is a combination of the two types of flow. The Reynolds numbers at the transition points shown in Fig. 8.13 were presented by Rohsenow et al. (1956) as the following (marked on the plot as solid circles): § ρ· * § ρ· *3 Reδ ,tr = 1800 − 246 ¨1 − v ¸ τ δ + 0.667 ¨1 + v ¸ (τ δ ) (8.159) © ρA ¹ © ρA ¹ Rohsenow et al. (1956) extended Seban’s (1954) falling film condensation analysis for turbulent flow from zero vapor shear to finite vapor shear cases to arrive at the following expression §g· h = 0.065Pr (τ δ ) kA ¨ 2 ¸ (8.160) © vA ¹ which may be used to predict the average heat transfer coefficient beyond the transition point into turbulent flow. It is found to agree well with a relationship obtained by Carpenter and Colburn (1951). 1/ 2 A * 1/ 2 1/ 3 1/ 3 8.3.5 Turbulent Film Condensation We will now consider turbulent flow for a specific application – turbulent condensate flow in a circular tube, as shown in Fig. 8.15. Turbulent film condensation occurs at the inner surface of the circular tube. While the liquid condensate flows downward due to gravity, the vapor flows either downward (cocurrent vapor flow) or upward (countercurrent vapor flow). Faghri (1986) proposed a method of predicting the average film thickness, the local heat transfer coefficient, and the overall heat transfer coefficient for turbulent film condensation in a tube with interfacial shear stress caused by cocurrent and countercurrent vapor flow. In a fashion similar to that of Nusselt condensation, the inertia terms are neglected and the only forces included are body, pressure, and viscous forces. This particular model takes into account the decrease in the stream flow rate due to condensation. Chapter 8 Condensation 619 Figure 8.15 Physical model of the condensation phenomena in contact with flowing vapor. To obtain an expression of the shear stress in the liquid film, a control volume with radius r and height Δx as shown in Fig. 8.15 is considered. For the case of countercurrent flow, a force balance results in dp · § (8.161) p (π r 2 ) + ρ v gVv + ρA gVA = ¨ p + Δx ¸ π r 2 + τ ( 2π r Δx ) dx ¹ © where the volume of the vapor and the liquid portion of the control volume are as follows: 2 Vv = ( R − δ ) πΔx (8.162) VA = ( R − y ) π Δx − Vv (8.163) Substituting eqs. (8.162) and (8.163) into eq. (8.161) and dividing through by Δx to solve for the shear stress, the following is obtained: 2 The shear stress at the liquid-vapor interface can be found by letting y = δ in eq. (8.164), i.e., R −δ § dp · (8.165) τδ = ¨ ρv g − ¸ dx ¹ 2© Substituting eq. (8.165) into eq. (8.164), the shear stress at any radius, τ , can be related to the shear stress at the liquid film surface, τ δ , by (R −δ ) R− y§ dp · τ= ¨ ρA g − ¸ − ( ρA − ρv ) g 2© 2( R − y) dx ¹ 2 (8.164) τ= ª 2 R(δ − y ) − δ 2 + y 2 º R− y τ δ + ( ρA − ρv ) g « » 2( R − y ) R −δ ¬ ¼ (8.166) 620 Transport Phenomena in Multiphase System Assuming the tube radius is much greater than the condensate film thickness forming on the inner surface, the curvature of the liquid film can be neglected and the resulting analysis would be applicable to condensation between two flat plates. Taking this into account, eq. (8.165) reduces to R§ dp · (8.167) τ δ = ¨ ρv g − ¸ dx ¹ 2© If it is further assumed that ρA  ρv , eq. (8.166) would reduce to (8.168) which can be written into a generalized form that includes the case of cocurrent flow, i.e., τ = ±τ δ + ρA g (δ − y ) (8.169) where the + sign denotes downward vapor flow (cocurrent flow) and the − sign denotes upward vapor flow (countercurrent flow). The shear stress at the wall, y = 0, is τ w = ±τ δ + ρA gδ (8.170) The velocity profile in the liquid film can be found from the following differential equation when all axial terms and the curvature are neglected: dª du º (8.171) «( vA + ε m ) dy » + g = 0 dy ¬ ¼ where εm is the momentum eddy diffusivity and is a time-measured flow property that adjusts the viscosity term for turbulent flow. Assuming that the wall is impermeable and that the interfacial shear stress is known, the boundary conditions for this problem are given as u = 0, y = 0 (8.172) ∂u (8.173) −μ = ±τ δ , y =δ ∂y Integrating eq. (8.171) twice with respect to y and applying boundary conditions specified by eqs. (8.172) and (8.173), the velocity profile in the liquid film is obtained: y [ g (δ − y ) ± τ δ / ρ A ] u=³ dy (8.174) 0 vA + ε m The liquid Reynolds number obtained by the following expression: δ udy 4Γ (8.175) Reδ = = 4 ρA ³ τ = τ δ + ρA g (δ − y ) μA 0 μA To generalize the problem statement, the following nondimensional variables are defined: −1 / 3 δuf yu f § ν A2 · τ (ν g ) −2 / 3 * * + , δ = δ ¨ ¸ , y+ = , τδ = δ A δ= ρA νA νA ©g¹ Chapter 8 Condensation 621 xu f Du f ε u + , ε m = 1 + m , D+ = u + = u , x+ = f νA vA vA where u f is the fractional velocity, defined as 1/ 2 (8.176) §τ · uf = ¨ w ¸ (8.177) © ρA ¹ Applying these nondimensional variables to eqs. (8.170), (8.174), and (8.175), their nondimensional forms are obtained as follows: u 3 B τ δ+ u f ( vA g ) f u =³ + y+ 0 2/3 (1 − gν − ( vA g ) δ + = 0 + m (8.178) (8.179) (8.180) A ε δ y+ / u3 ) f dy + Reδ = 4³ u + dy + 0 It should be noted that if τ δ+ = 0 and u 3 = ν A gδ + then eq. (8.178) would reduce f to the nonsheared film case (classical Nusselt analysis). We will now consider the thermal side of the problem. An energy balance ′′ can be written for the case of constant heat flux at the wall ( qw ). This energy balance also assumes that heat transfer across the liquid film is dominated by conduction, so the convective terms can therefore be neglected. §v ε · dT d (Γ / μA ) ′′ = qw = μA hAv ρ A c pA ¨ A + m ¸ (8.181) dx © Pr Prt ¹ dy where Prt is the turbulent Prandtl number, which will be discussed thoroughly towards the end of this subsection. Equation (8.181) can be nondimensionalized to obtain d (Γ / μA ) = NT dx + where −1 ­ δ + ª Pr § Pr ½ ° ° + ·º t − 1 + ε m ¸ » dy + ¾ ® ³0 « ¨ ¹¼ ° ¬ Prt © Pr ° ¯ ¿ −1 (8.182) (8.183) hAv Pr Finally, the local heat transfer coefficient can be found directly from eq. (8.181): −1 ­« ½ ′′ qw ° δª ° § ν A ε m ·º » hx = (8.184) = ® ³ « ρAc pA ¨ + ¸» dy ¾ Tsat − Tw ° 0 « ° » © Pr Prt ¹» ¬ ¼ ¯« ¿ Nondimensionalizing eq. (8.184) as a Nusselt number, the following is obtained: NT = (Tsat − Tw ) c pA −1 622 Transport Phenomena in Multiphase System −1 ­ δ + ª Pr § Pr ½ hδ ° ° + ·º Nu x = x = δ + ® ³ « ¨ t − 1 + ε m ¸ » dy + ¾ (8.185) 0 kA ¹¼ ¬ Prt © Pr ° ° ¯ ¿ An average heat transfer coefficient is desirable in many practical + applications. A modified Nusselt number Nu x is related to Nux by −1 h §v Nu = x ¨ A kA © g The average modified relation: + x uf · −1/ 3 (8.186) ¸ = Nu x + (ν A g ) δ ¹ Nusselt number is then found from the following 1/ 3 1 § x+ · Nu + = ³ Nu + d ¨ + ¸ (8.187) 0 ©L ¹ The dimensionless shear stress at the interface can be written as ªf §ρ · d (Γ / μ A ) º * (8.188) τ δ = (ν A g ) −2 / 3 u 2 (uv+ + uA+,δ ) « E ¨ v ¸ (uv+ + uA+,δ ) + » f dx + ¼ ¬ 2 © ρA ¹ where f E is the friction factor for vapor flow, which is different for upflows and downflows. The friction factor for vapor flow can be obtained by modifying the friction factor for single-phase flow, f, to accommodate the two-phase nature of the flow. It is also different for ripple ( Reδ ≤ 75 ) and roll wave ( Reδ > 75 ) regimes, i.e., + ­ Reδ ≤ 75 ° f [1 + 0.045( Mg − 5.9)] fE = ® (8.189) −0.2 + ° f [1 + 0.045Rev ( Mg − 5.9)] Reδ > 75 ¯ where 1/2 ­ 0.6 § ν ·§ ρ τ · °0.78Reδ ¨ A ¸¨ A δ ¸ Reδ ≤ 75 ° © ν v ¹© ρv τ c ¹ Mg + = ® (8.190) 1/ 2 § ν A ·§ ρA τ δ · ° 0.7 Reδ > 75 ¸ ° 0.50 Reδ ¨ ¸¨ © ν v ¹© ρv τ c ¹ ¯ The characteristic stress is given by τδ § 1 2 τ w · =¨ + ¸ τ c © 3 3 τδ ¹ −1 (8.191) To calculate the velocity distribution and the heat transfer coefficient, a definition is required for εm and Prt from an appropriate turbulent model. In modeling of εm, it is customary to divide the flow into two regions – the inner region, where the turbulent transport is dominated by the wall, and another wavelike region that is directly adjacent. Faghri (1986) used a combination of the Szablewski (1968) and Van Driest models to obtain the following expression: Chapter 8 Condensation 623 ­ 2τ 1 1° ε = + ®1 + 0.64 y + 2 2° τw ¯ + m ª § § y + τ ·º ª § τ · ·º ¨ ¸ «1 − exp ¨ − + ¸ » «exp ¨ −1.66 ¨1 − ¸ ¸ » « © τ w ¹ ¹» © A τ w ¹» « © ¬ ¼¬ ¼ 2 ½ ° ¾ ° ¿ (8.192) 2 where A+ = 25.1 and τ / τ w = 1 − y + ( gν A ) / u 3 . f This profile represents the eddy diffusivity in the inner layer closest to the wall ( 0 ≤ y + ≤ 0.6δ + ), where the influence of the wall is important. In the outer layer ( 0.6δ + ≤ y + ≤ δ + ) the eddy viscosity is assumed to be constant, with a continuous transition to the inner layer. Finally, because the turbulent transport near the liquid-vapor interface is quite different from that near the wall, the turbulent Prandtl number, Prt, cannot be assumed to be constant. Faghri (1986) used the following expression (Habib and Na, 1974) for the analysis of turbulent transfer in pipes: 1 − exp(− y + / A+ ) Prt = (8.193) ª1 − exp − y + Pr / B+ º ¬ ¼ where ( ) B + = ¦ ci ( log10 Pr ) i =1 5 i −1 (8.194) and c1 = 34.96; c2 = 28.79; c3 = 33.95; c4 = 6.3; c5 = -1.186. * The solution procedure begins with guessing an initial value of τ δ for the initial values of δ 0+ and the initial vapor flow, which specifies the initial value of * Rev = 4mv /(π D μv ) . Based on the values of τ δ and δ 0+ , the fractional velocity, u f , is then obtained by solving eq. (8.178). The dimensionless eddy diffusivity, + ε m , is obtained from eq. (8.188). Equations (8.179) and (8.180) are integrated numerically to obtain the velocity profile and the liquid Reynolds number. An updated dimensionless shear stress at the interface can be obtained from eq. (8.188). The process is repeated until the ReA values between two consecutive iterations differ by less than 0.5%. The local convective heat transfer coefficient can be obtained from eqs. (8.186) and (8.187). The above procedure can be repeated for different x until heat transfer coefficients are obtained at all locations. Heat transfer in the condenser sections of conventional and annular twophase closed thermosyphon tubes has been studied analytically by Faghri et al. (1989). The method involved extending Nusselt theory to include the variation of the shear at the vapor-liquid film interface. Harley and Faghri (1994) presented a transient two-dimensional condensation in a thermosyphon that accounts for conjugate heat transfer through the wall and the falling condensate film. The complete transient two-dimensional conservation equations are solved for the vapor flow and pipe wall, and the liquid film was modeled using a quasi-steadystate Nusselt-type solution. 624 Transport Phenomena in Multiphase System 8.3.6 Other Filmwise Condensation Configurations Other configurations for external condensation have been investigated over the years. One of the most popular is external condensation on a vertical cylinder. If the thickness of the liquid film is smaller than the diameter of the cylinder by at least an order of magnitude, the heat transfer and Nusselt number expressions for a vertical cylinder are the same as those for a vertical plate. Nusselt analysis for laminar film condensation for a vertical plate can also be applied to condensation on a plate at an angle (with respect to the vertical) by replacing g with g cosθ : ª ρ ( ρ − ρv ) g cos θ hA′v L3 º hL L Nu = = 0.943 « A A » kA μA kA (Tsat − Tw ) ¬ ¼ 1/ 4 (8.195) When the cold surface is curved, the tangential direction of the gravity varies along the condensate film. The empirical correlations for other configurations also have almost the same forms as that of a vertical plate, except the leading coefficient and the characteristic length differ. For example, for laminar film condensation on a horizontal cylinder, the leading coefficient changes from 0.943 to 0.729 and the characteristic length changes from L to D, the diameter of the cylinder, i.e., ª D 3 hAv g ( ρA − ρv ) º hD D Nu = = 0.729 « » kA ¬ kAν A (Tsat − Tw ) ¼ 1/ 4 (8.196) which can be obtained by applying Nusselt analysis (see Problem 8.11). For laminar film condensation on a sphere, the average heat transfer coefficient can also be obtained by Nusselt analysis (see Problem 8.14). ª D 3 hA′v g ( ρA − ρv ) º hD D Nu = = 0.815 « » kA ¬ kAν A (Tsat − Tw ) ¼ 1/ 4 (8.197) Another expression of great importance in heat exchanger design is that of a vertical column of n horizontal tubes. The average heat transfer coefficient for all n cylinders can be obtained by: ª D 3 hA′v g ( ρ A − ρv ) º h D ,n D Nu = = 0.729 « » kA ¬ nkAν A (Tsat − Tw ) ¼ 1/ 4 (8.198) which indicates that the average heat transfer coefficient for all n cylinder, hD ,n , is related to the average heat transfer coefficient for the first tube of this array, hD , by hD (8.199) n1/ 4 This calculation would lead to a lower average heat transfer coefficient for an array than that found for a single tube. In practice, however, it has been found that these calculations significantly underestimate the heat transfer capabilities of h D ,n = Chapter 8 Condensation 625 this system, due to splashing effects that occur as the condensate (in the form of sheet or droplets) falls from one tube to a tube underneath. For condensation on an upward-facing horizontal surface of a finite size, the condensate in the central region flows toward the edge where it is spilled (Bejan, 1991). For condensation over a long horizontal strip with a width of L, the average heat transfer can be obtained by ª ρ ( ρ − ρ v ) ghA′v L3 º hL Nu = = 1.079 « A A » kA « μA kA (Tsat − Tw ) » ¬ ¼ 1/ 5 (8.200) where the exponent on the right-hand side becomes 1/5. The heat transfer coefficient for film condensation over an upward-facing horizontal circular disk with a diameter of D is ª ρ ( ρ − ρv ) ghA′v D 3 º hD Nu D = = 1.368 « A A » kA « μA kA (Tsat − Tw ) » ¬ ¼ 1/ 5 (8.201) Example 8.4 Saturated acetone at Tsat = 60 ÛC condenses on the outside of a copper tube with a diameter of D = 3.0 cm . The outer surface temperature of the copper tube is Tw = 40 ÛC. Find the heat transfer coefficient and the rate of condensation per unit length of the tube. Solution: The vapor properties are evaluated at saturation temperature Tsat = 60 ÛC. The vapor density at this temperature is ρ v = 2.37 kg/m3 , and the latent heat of vaporization is hAv = 517 kJ/kg . The liquid properties evaluated at T f = (Tsat + Tw ) / 2 = 50 ÛC are ρ A = 756.0 kg/m 3 , c pA = 2255 J/kg-K , μA = 0.248 × 10−3 kg/m-s, kA = 0.172 W/m-K , and ν A = μA / ρA = 0.328 × 10−6 m 2 /s. The revised latent heat of vaporization is ­ ½ ª c pA (Tsat − Tw ) º ° ° hA′v = hAv ®1 + 0.68 « »¾ hAv ° ¬ ¼° ¯ ¿ ­ ª 2.255 × (60 − 40) º ½ = 517 × ®1 + 0.68 « » ¾ = 547.7 kJ/kg 517 ¬ ¼¿ ¯ The heat transfer coefficient can be obtained from eq. (8.196), i.e., ª ( ρ − ρv ) g kA3 hA′v º h = 0.729 « A » ν A ΔTD ¬ ¼ 1/ 4 ª (756.0 − 2.37) × 9.8 × 0.1723 × 547.7 × 103 º = 0.729 × « » 0.328 × 10−6 (60 − 40) × 0.03 ¬ ¼ 2 = 2331.3 W/m -K 1/ 4 626 Transport Phenomena in Multiphase System The heat transfer rate per unit width is then q′ = h π D(Tsat − Tw ) = 2331.3 × π × 0.03 × (60 − 40) = 4394.4 W The condensation rate per unit width is q′ 4394.4 m′ = = = 0.00802 kg/s-m hA′v 547.7 × 103 The correlations that we discussed thus far have been limited to cases in which the flow of the condensate is driven by gravity. When the vapor is forced to flow over the cooler surface, the vapor interacts with the condensate and drags liquid in the vapor flow direction. For condensation on the outside of a horizontal tube in crossflow, the heat transfer coefficient is affected by both free-steam velocity of vapor, u∞ , and gravitational force (Shekriladze and Gomelauri, 1966), so that 1/ 2 ª§ ghA′v μA D · º hD 1/ 2 = 0.64 Re D «1 + ¨1 + 1.69 2 (8.202) Nu = ¸» ¨ ¸ kA u∞ kA (Tsat − Tw ) ¹ » «© ¬ ¼ where Re D = u∞ D /ν A is based on the viscosity of liquid. Equation (8.202) is valid for a Reynolds number up to 106. For laminar film condensation on a horizontal flat plate in a parallel stream of saturated vapor, the average heat transfer coefficient is 1/ 2 ª PrA § ρ v μv · º hL 1.508 1/ 2 Nu = (8.203) = 0.872 Re L « + ¨ ¸» 3/ 2 kA Ja © ρ A μA ¹ » « (1 + Ja / PrA ) ¬ ¼ where L is the length of the flat plate along the vapor flow direction, Re L = U ∞ L /ν A is based on the viscosity of liquid, and Ja = c pA (Tsat − Tw ) / hAv is 1/ 3 1/ 2 the Jakob number. Equation (8.203) is valid for ρA μA / ρv μv = 10  500 and Ja / PrA = 0.01  1 . 8.3.7 Effects of Noncondensable Gas The physical model and governing equations for binary vapor (condensable vapor and noncondensable gas) have been discussed in Section 8.3.2. It can be seen from Fig. 8.12 that heat and mass transfer in the vapor phase must be studied, in addition to heat transfer in the liquid film. There exists a boundary layer in the vapor phase ( δ < y < δ + Δ ), in which the partial pressure of the condensable vapor, pv, decreases from a constant value pv ,∞ at y = δ + Δ to the value pv,δ at y = , where the vapor is condensing to a liquid. The condensing vapor must diffuse through this vapor boundary layer to the liquid-vapor interface. The partial pressure of the noncondensable gas, pg, on the other hand, increases from its reservoir value, pg ,∞ , to the value pg,δ at the liquid-vapor Chapter 8 Condensation 627 y Vapor ωv ,∞ , pv ,∞ T∞ liquid Tδ ′′ mv v Tsat ( pv ,∞ ) ωv ,δ pv ,δ x Figure 8.16 Mass transfer in the equivalent laminar film. interface. At any point in space and time the summation of the partial pressures of this binary system must equal the constant total pressure. pv + pg = p (8.204) The partial pressure of the condensing gas decreases as it approaches the phase interface, and its corresponding saturation temperature Tsat(pv) also falls. Depending on the noncondensable gas content, the temperature at the interface can be much lower than if no such gas were present. The temperature difference across the interface would also be lower as a result, which would lead to a lower overall heat transfer coefficient. This clearly demonstrates the benefit of removing as much noncondensable gas from the system as possible. However, systematic purity cannot always be achieved, and the noncondensable gas content must be taken into account. The mass transfer in the vapor boundary layer can be rigorously analyzed by solving the boundary-layer type of governing equations discussed in Section 8.3.2. Hewitt et al. (1994) presented an alternative model based on the concept of equivalent laminar film – a layer in the vapor phase in which the mass fraction of the condensable vapor, v, varies linearly from v, at the liquid-vapor interface to ωv ,∞ at y = δ + δ v (see Fig. 8.16). Meanwhile, the partial pressure of the condensable vapor changes from pv ,δ to pv ,∞ in the equivalent laminar film. Assuming that the noncondensable gas cannot be dissolved into the liquid condensate, the molar flux of the condensable vapor at any point within the equivalent laminar layer can be obtained by [see eq. (8.86)] ∂c ′′ ′′ (8.205) nv = − Dvg v + nv xv ∂y 628 Transport Phenomena in Multiphase System which can be rearranged to obtain Dvg ∂cv ′′ (8.206) nv = − 1 − cv / cT ∂y where cT is total molar concentration of vapor and noncondensable gas. Integrating eq. (8.206) over the equivalent laminar layer and considering ′′ and cT are constants, one obtains nv δ +δ v cv ,∞ 1 ′′ nv ³ dy = − Dvg cT ³ dcv (8.207) cv ,δ c − c δ T v Equation (8.207) can be rearranged to yield § c − cv ,δ · § cT − cv ,δ · cD ′′ T vg ln ¨ T nv = (8.208) ¸ = cT hm ,G ln ¨ ¨c −c ¸ ¨c −c ¸ ¸ δv T v ,∞ ¹ T v ,∞ ¹ © © where Dvg hm ,G = (8.209) δv is mass transfer coefficient (m/s). Since the thickness of equivalent laminar film, δ G , is still unknown at this point, the mass transfer coefficient can be approximately related to the heat transfer coefficient, hG, through the Lewis equation, i.e., 1 hG (8.210) hm ,G = ρvg c p ,vg where cp,vg is the constant-pressure specific heat of the vapor-gas mixture. If the condensable vapor and noncondensable gas can be treated as ideal gas, ′′ the molar flux in eq. (8.208) can also be expressed in term of mass flux, mv , and partial pressure of the condensable vapor, pv, in the mixture (see Problem 8.27) p − pv ,δ ′′ mv = ρvg hm ,G ln (8.211) p − pv ,∞ The energy balance across a differential control volume at the liquid-vapor interface, as shown in Fig. 8.17, is (Stephan, 1992) ′′ hA (Tδ − Tw ) = mv hAv + hG (Tvg − Tδ ) (8.212) where hG is the heat transfer coefficient from the vapor-gas mixture to the liquidvapor interface, and where Tδ is the temperature at the liquid-vapor interface. Substituting eqs. (8.210) and (8.211) into the energy balance equation (8.212) and using hvg = ξ hG , which corrects for the fact that vapor does not flow along the wall but stops at the liquid-vapor interface (where ξ is a correction factor), the following is obtained (Stephan, 1992): º p − pv ,δ h ªh (8.213) + ξ (Tvg − Tδ ) » Tδ − Tw = G « Av ln hA « c p ,vg p − pv ,res » ¬ ¼ Chapter 8 Condensation 629 Condensate liquid Vapor ′′ mv hAv hA (Tδ − Tw ) hG (Tvg − Tδ ) Interface Figure 8.17 Energy balance at the liquid-vapor interface for film condensation on a vertical plate including the effects of non-condensable gases. In the case of small inert gas content, the above equation will reduce to p − pv ,δ º h ªh (8.214) Tδ − Tw = G « Av ln » hA « c p ,vg p − pv ,∞ » ¬ ¼ In the best-case scenario where there is no noncondensable gas in the vapor, the heat flux across the liquid film would be as follows: k q′′ = A (Tsat − Tw ) (8.215) δ However, if a noncondensable gas is present, the temperature drop across the film would be lower and the heat flux would be as follows: k ′′ qvg = A (Tδ − Tw ) (8.216) ′′ where qvg is the heat flux across the liquid film in the presence of a noncondensable gas. The ratio of the heat fluxes obtained by eqs. (8.216) and (8.215) is ′′ qvg Tδ − Tw = ≤1 (8.217) q′′ Tsat − Tw Substituting the expression for temperature difference across a liquid film in the presence of a noncondensable gas – eq. (8.214) – into the above ratio, the following is obtained: ′′ qvg p − pv ,δ hG hAv = ln (8.218) q′′ (Tsat − Tw ) hA cvg ,∞ p − pv ,∞ δ 630 Transport Phenomena in Multiphase System ′′ qvg q ′′ Tvg Mass fraction of noncondensable gas Figure 8.18 Falling film condensation of steam with non-condensable gas (Collier and Thome, 1994; Reprinted with permission from Oxford University Press). or, substituting mcv from eq. (8.211): ′′ qvg ′′ mv hAv (8.219) = q′′ (Tsat − Tw )hA It can be seen from the above expression that for large (Tsat – Tw), the velocity or mass flow rate mcv , must be made sufficiently large to acquire a large heat transfer coefficient for the heat transfer from the vapor-gas mixture to the ′′ liquid-vapor interface. This must be done in order to make qvg not too small and therefore remove the undesirable effects of the noncondensable gas as much as possible. Figure 8.18 shows the effects of non-condensable gas (air) in saturated steam on falling film condensation under both stagnant and forced convection conditions. It can be seen that the effect of non-condensable gas on condensation in a stagnant vapor is very significant. For forced convective condensation, the effect of non-condensable gas is still significant, but is much weaker than its effect on the stagnant vapor condensation. Therefore, the effect of noncondensable gas on condensation can be minimized by allowing vapor to flow. Example 8.5 A mixture of 20% steam and 80% air at 100 ÛC and 1 atm flows across a horizontal cylinder. The diameter of the cylinder is 0.1 m and the velocity of the mixture is 30 m/s. The condensation rate on the ′′ cylinder is mv = 0.02 kg/m2-s. The properties of the mixture are Chapter 8 Condensation 631 ρvg = 0.944 kg/m3 , μvg = 8.2 × 10−6 N-s/m 2 , Dvg = 3.64 × 10−5 m 2 /s , respectively. What is the temperature at liquid-vapor interface? If the temperature of the tube is 80 ÛC, what is the percentage of heat transfer reduction due to the existence of noncondensable gas? Solution: The mass fraction of the steam is ωv ,∞ = 0.9 . The molecular M v = 18.02 kg/kmol and mass of water and air are M g = 28.96 kg/kmol . The partial pressure of the steam can be obtained from eq. (8.70), i.e., pv ,∞ ª M (1 − ωv ,∞ ) º = p «1 + v » M g ωv , ∞ » « ¬ ¼ 5 −1 ª 18.02 × (1 − 0.9) º = 1.013 × 10 × «1 + = 0.9475 × 105 kPa 28.96 × 0.9 » ¬ ¼ The Reynolds number of the mixture is ρvg u∞ D 0.944 × 30 × 0.1 = = 34536 Re D = μvg 8.2 × 10−6 −1 The Schmidt number of the mixture is μvg 8.2 × 10−6 = = 0.239 Sc = ρvg Dvg 0.944 × 3.64 × 10−5 The empirical correlation for forced convective heat transfer across a cylinder is (see Table 1.9) ª § Re D ·5 / 8 º 0.62 Re1/ 2 Pr1/ 3 D × «1 + ¨ Nu D = 0.3 + ¸» [1 + (0.4 / Pr) 2 / 3 ]1/ 4 « © 282000 ¹ » ¬ ¼ Analogy between mass and heat transfer gives us 4/5 0.62 Re1/ 2 Sc1/ 3 D Sh D = 0.3 + [1 + (0.4 / Sc) 2 / 3 ]1/ 4 ª § Re D ·5 / 8 º «1 + ¨ ¸» « © 282000 ¹ » ¬ ¼ 4/5 ª § 34536 ·5 / 8 º «1 + ¨ ¸ » = 108.1 « © 282000 ¹ » ¬ ¼ The mass transfer coefficient is therefore Sh D Dvg 108.1 × 3.64 × 10−5 hm,G = = = 0.03935 m/s D 0.1 The partial pressure of the vapor at the liquid-vapor interface, pv ,δ , can be obtained from eq. (8.211) 0.62 × 345361/ 2 × 0.2391/ 3 = 0.3 + [1 + (0.4 / 0.239) 2 / 3 ]1/ 4 4/5 632 Transport Phenomena in Multiphase System § mv ′′ pv ,δ = p − ( p − pv ,∞ ) exp ¨ ¨ρ h © vg m ,G · ¸ ¸ ¹ 0.02 § · = 1.013 × 105 − (1.013 × 105 − 0.9475 × 105 )exp ¨ ¸ 0.944 × 0.03935 ¹ © = 0.900 × 105 Pa The interfacial temperature, T , is the saturation temperature corresponding to the above partial pressure. It can be found from the Clapeyron-Clausis equation (2.168). p h §1 1· (8.220) ln v ,δ = − Av ¨ − ¸ p Rg © Tδ Tsat ¹ where hAv = 2251.2 kJ/kg and Rg = 0.4615 kJ/kg-K for water at 1 atm. Equation (8.220) can be rearranged to obtain −1 Rg pv ,δ · §1 − Tδ = ¨ ln ¸ p¹ © Tsat hAv §1 0.4615 0.900 × 105 · D =¨ − ln ¸ = 369.80K = 96.65 C 373.15 2251.2 1.013 × 105 ¹ © The ratio of heat fluxes with and without noncondensable gas can be obtained from eq. (8.217) ′′ qvg Tδ − Tw 96.65 − 80.0 = = = 0.8325 q′′ Tsat − Tw 100.0 − 80.0 In other words, the heat transfer is decreased by 16.75% due to the presence of noncondensable gas. −1 8.3.8 Flooding or Entrainment Limit The flooding limit occurs due to the instability of the liquid film generated by a high value of interfacial shear, which is a result of the large vapor velocities. For example, in a vertical closed two-phase thermosyphon, the condensate liquid film flows down the walls and evaporates at the bottom. When a large velocity opposite to the liquid flow occurs due to evaporation, the flow of the condensate liquid film can stop. The vapor shear hold-up prevents the condensate from returning to the evaporator and leads to a flooding condition in the condenser section. This causes a partial dryout of the evaporator, which results in wall temperature excursions or in limiting the operation of the system. Flooding can also happen in cocurrent two-phase open systems. There are two major fundamental semi-empirical correlations for the prediction of flooding limit of open two-phase systems. The first is the Wallis Chapter 8 Condensation 633 correlation (1969), which is characterized by a balance between the inertia and hydrostatic forces. The second is the Kutateladze two-phase flow stability criterion, in which the inertia, buoyancy, and surface tension forces are balanced (Kutateladze, 1972). The Wallis empirical correlation is based on results from open channel water-gas experiments. Two coefficients in the Wallis correlation must be determined by experiment because they are dependent upon the design of the pipe. The shortcoming of the Wallis correlation is that the effect of surface tension is not taken into account. Surface tension is of great importance to the hydrodynamic and heat transfer characteristics of gas-liquid systems. Physically, increasing the surface tension means that a higher pressure difference can be sustained across a film surface without forming waves. In the Kutateladze correlation, the effect of the diameter of the pipe is not included. For small tubes, the diameter of the vapor passage plays an important role in flooding characteristics. It was shown by Wallis and Makkenchery (1974) that the Kutateladze criterion produces a good correlation of the results for pipes with large diameters, but for small pipes the effect of the diameter should be considered. Various efforts have been made by investigators to extend the existing semiempirical correlation from two-phase open systems to thermosyphons. Bezrodnyi (1978) proposed a correlation similar to Kutateladze’s with the Kutateladze number being determined by the vapor pressure and other properties for thermosyphons. Tien and Chung (1978) combined the Kutateladze and the Wallis correlations to account for the diameter of the pipe and surface tension effects. This correlation resulted in agreement with experimental data for certain types of working fluids, but large deviations were found when water was used. Faghri et al. (1989) improved the existing semi-empirical correlations to predict the flooding limit for thermosyphons by including the effect of diameter, surface tension, and working fluid properties. This is the most general flooding correlation in existence for thermosyphons, and therefore a detailed discussion is presented here. The Wallis correlation (1969), in which experimental data were correlated for packed beds and countercurrent flow in tubes, is represented by the following empirical equation (j ) * 12 v * + m ( jA ) 12 = Cw −1 2 (8.221) (8.222) where ji* = ji ρi1 2 ª gD ( ρA − ρ v ) º ¬ ¼ , ( i = A, v ) in which m and Cw are empirically determined dimensionless constants and are functions of fluid properties. The quantity Cw also depends on entrance and exit geometries. jl and jv are the liquid and vapor volumetric flow rates divided by the total cross-sectional area of the thermosyphon. These volumetric fluxes represent ratios of the momentum fluxes of the components to the buoyant forces. Values * of m and c were traditionally determined from graphs of jv 1 2 as a function of 634 Transport Phenomena in Multiphase System jA*1 2 for open systems. For most cases, values of m = 1.0 and Cw = 0.7 and 1.0 are reported in the literature. * Wallis suggested that the liquid film would always flow upward if jv > 1 and * * would flow downward, wetting a dry wall below it, if jv < 0.5 . Thus, jv = 1 is considered to be the case of total flooding (j * A = 0 ) . For thermosyphons, as * * opposed to open systems, jv and jA are related due to the fact that it is a closed system under steady state conditions. m (8.223) jv = v ρv A m jA = A (8.224) ρA A where mv = mA = q hAv , A is the total cross-sectional area of the thermosyphon and q is the heat rate. Combining eqs. (8.221) – (8.224) and rearranging will result in an equation for the flooding limit based on an extension of the Wallis correlation for open systems to thermosyphons. 2 q Cw hAv gD ( ρA − ρv ) ρv = (8.225) 2 A ª1 + ( ρ v ρ A )1 4 º ¬ ¼ Tien and Chung (1978) extended the Kutateladze correlation for cases with jA = 0 by an analogy of the Wallis correlation. According to the Kutateladze correlation 12 12 (8.226) ( K v ) + ( K A ) = Ck where Ck = 3.2 and K i = ji ρi1 2 ª gσ ( ρA − ρ v ) º ¬ ¼ −1 4 , ( i = A, v ) (8.227) The dimensionless group K (Kutateladze number) is a balance between the dynamic head, surface tension, and gravitational force. Comparing the Kutateladze correlation, eq. (8.227), with the Wallis correlation, eq. (8.221), the following relation is found assuming the two are identical. §C · D=¨ k ¸ © Cw ¹ 4 g ( ρA − ρv ) σ (8.228) The critical wavelength of the Taylor instability is: λcrit = 2π  2π 3 ( ) g ( ρA − ρv ) σ (8.229) Setting eq. (8.229) (characteristic length) equal to eq. (8.228) and choosing the upper limit in eq. (8.228) results in Chapter 8 Condensation 635 Ck = 3.2 Cw (8.230) If we let Cw = 1.0, eq. (8.230) gives Ck = 3.2 . K = Ck2 = 3.2 is really the Kutateladze number for total flooding ( K A = 0 ), which does not consider the effect of diameter as mentioned in the preceding section. According to the experimental results of Wallis and Makkenchery (1974), the Kutateldaze number decreases as the dimensionless diameter decreases, which is called the Bond number. This trend requires that Ck is in terms of the Bond number º §C · ª σ Bo = D = ¨ k ¸ « » © Cw ¹ « g ( ρ A − ρ v ) » ¬ ¼ or Ck (8.232) = Bo1 4 Cw With reference to the variation of the Kutateladze number versus the Bond * number in the paper by Wallis and Makkenchery (1974) with jv = 1.0 , the function y = tanh x is introduced to account for the effect of the diameter on the 4 12 (8.231) flooding limit. If we let x = Bo1 4 , eq. (8.232) results in Ck = 3.2 tanh Bo1 4 , with Cw = 1.0 Cw or 12 12 (8.233) ( K v ) + ( K A ) = 3.2 tanh Bo1 4 = Ck This correlation was found to be highly accurate in predicting the flooding limit with water as the working fluid. The experimental deviations from this correlation are with 15%. For other fluids the deviations are more significant. It is noticed that for different working fluids, the variation of the ratio of the density of the liquid to the density of the vapor is quite different within the same temperature range. Using experimental results for different working fluids, the following correlation is proposed §ρ · K = Ck2 = ¨ A ¸ tanh 2 Bo1 4 = R′ tanh 2 Bo1 4 © ρv ¹ or, for the maximum heat transfer rate, ª º qmax = KhAv A ¬ gσ ( ρA − ρv ) ¼ 14 −2 0.14 (8.234) −1 4 ª −1 4 º (8.235) ¬ ρv + ρA ¼ Equation (8.235) (Faghri et al., 1989) is a combination of the correlations developed by Tien and Chung (1978) and Imura et al. (1983). In the Tien and Chung correlation, the argument of the hyperbolic tangent in eq. (8.233) is 0.5Bo1/4, rather than Bo1/4, and R′ = 1 . 636 Transport Phenomena in Multiphase System 5 Cw = 1.0, Wallis (1969) 4 §ρ · R′ = ¨ v ¸ © ρA ¹ 1/ 4 R = 3.2, Faghri et al. (1989) 3 K Cw = 0.7, Wallis (1969) 2 R = 2.0, Faghri et al. (1989) 1 Tien and Chung (1978) 0 0 5 15 10 Bond Number, Bo 20 25 Figure 8.19 Variation of the modified Kutataldze number versus the Bond number for a closed two-phase thermosyphon (Faghri et al., 1989). Figure 8.19 shows the variation of the modified Kutateladze number with respect to the Bond number for R′ = 3.2 and 2.0. The symbols in Fig. 8.19 represent experimental data taken for various fluids. The lines in Fig. 8.19 are the predictions of the Kutataladze number made by various investigators. For the case of R′ = 3.2 , the variation of K is similar to Wallis’ experimental data (1974) * of water with jv = 1.0, which is the criterion for Faghri’s derivation. Fig. 8.19 shows that most of the experimental data of water with R′ = 2.81  4.34 are around the curve R′ = 3.2 , and the data of R113, ethanol and methanol with R′ = 1.62  2.43, are around the R′ = 2.0 curve. For the experimental data of R113, ethanol and methanol, the results are also close to Tien and Chung’s prediction and the Wallis correlation with Cw = 0.7, but a large deviation occurs with water as the working fluid. Choosing the upper or lower limit for Cw will make a significant difference in the prediction of the critical heat flux. Figure 8.19 illustrates good agreement between the results of the Faghri et al. (1989) correlation and the experimental data of the conventional thermosyphon for different working fluids. For annular thermosyphons, it is recommended to modify eqs. (8.234) and (8.235) to predict the maximum heat transfer rate with the diameter D being replaced by the hydraulic diameter. Chapter 8 Condensation 637 8.4 Nongravitational Condensate Removal One of the more intriguing applications in the future of heat transfer is condensation in microgravity environments. Space missions will require condensers that can perform at least an order of magnitude higher than today’s technology. Constraints on the new designs include stringent weight requirements, compactness, and, of course, heat transfer performance. The condensate removal poses the biggest challenge in a microgravity environment, since gravity cannot help in the removal process. It is already known that thinner condensate film correlates to better heat transfer rates due to conduction. Four methods have been proposed for the removal of condensate in a microgravity environment; these include (1) suction through a porous wall; (2) forced vapor shear at the liquid-vapor interface; (3) centrifugal force; and (4) capillary force. 8.4.1 Condensation in a Tube with Suction at the Porous Wall In condensation, both vapor shear and suction through a porous wall directly reduce the film thickness on the wall and, therefore, significantly increase the heat transfer. Faghri and Chow (1988) incorporated both of these effects into their investigation, which is described in detail below. Faghri and Chow (1988) investigated a system using an annular pipe with its inner wall made of a porous material (See Fig. 8.20). In this system, steam flows through a porous pipe and cooling liquid flows through the annular region between a solid outer pipe and the porous inner pipe. The steam condenses onto the inner wall of the porous material, because the cooling liquid keeps the temperature of the porous wall below that of the steam’s saturation temperature. A small pressure drop is maintained between the steam and the cooling flow in order to drive the condensate through the porous material to the annular region, where it mixes with the cooling liquid and is swept away to a heat exchanger. Figure 8.20 Physical model of condensation in an annular tube with suction at the inner porous wall. 638 Transport Phenomena in Multiphase System The Nusselt analysis for laminar film condensation was extended to flow inside a tube with a constant-temperature porous wall. The shear stress at the liquid-vapor interface decreases due to condensation. Assuming that the properties of the fluid remain constant and that the curvature of the condensate film can be neglected, the following nondimensional momentum and energy boundary layer equations can be written: ∂u * ∂ 2u* A = − PrA *2A (8.236) ∂y * ∂y ∂T * ∂ 2T * = − *2 ∂y * ∂y (8.237) where the suction velocity at wall v ≅ vw = const. The nondimensional variables in eqs. (8.236) and (8.237) are defined as follows: yv T − Tw u y* = − w u* = − A T * = (8.238) vw Tsat − Tw αA The inertial and convective effects are approximated in eqs. (8.236) and (8.237), respectively; therefore the results are the asymptotic behavior of the conservation of momentum and energy equations as PrA → ∞. The boundary conditions for a co-current liquid-vapor flow are as follows: At y* = 0 (8.239) u * = 0, T * = 0 * * At y = δ T * = 1, N2 = τ δ αA 2 μA vw (8.240) where N 2 in eq. (8.240) is the dimensionless shear stress at the liquid-vapor interface and δ* is the dimensionless film thickness given as −v δ δ* = w (8.241) αA Integrating eq. (8.236) and applying boundary conditions, eqs. (8.239) and (8.240), the dimensionless velocity profile is obtained: § δ* · ª § − y* · º * (8.242) uA = N 2 exp ¨ ¸ PrA «exp ¨ ¸ − 1» © PrA ¹ ¬ © PrA ¹ ¼ The dimensionless temperature profile across the liquid film can be obtained by integrating eq. (8.237) twice and applying appropriate boundary conditions, eqs. (8.239) and (8.240), i.e., T= * e− y − 1 e −δ − 1 * * (8.243) Chapter 8 Condensation 639 At the liquid-vapor interface, an energy balance is needed to equate the latent heat given off by the condensation process and the heat conducted from the interface into the liquid film, i.e., dδ ∂T ­ ½ ρ hAv ®vw − ³ uA dy ¾ = − k (8.244) 0 dx ∂y y =δ ¯ ¿ The assumptions inherent in eq. (8.244) are that the vapor is not superheated and that the condensate flow is laminar. It is also assumed that the condensate film is not subcooled. Equation (8.244) can be nondimensionalized as shown here: * ª −e −δ º d δ* * * (8.245) uA dy = Ja « −δ * » −1 dx* ³0 « e − 1» ¬ ¼ where c pA (Tsat − Tw ) Ja = (8.246) hAv xv (8.247) x* = − w αA Using u from eq. (8.242) to evaluate eq. (8.245), we obtain dφ1 = φ2 dx* where ­ ½ § δ* · δ* N 2 PrA2 ° ° φ1 = − 1¾ exp ¨ − ® ¸+ * /Pr exp ( −δ A)° ° © PrA ¹ PrA ¿ ¯ and * * A (8.248) (8.249) ª e −δ º (8.250) φ2 = Ja « » −1 −δ * «1 − e » ¬ ¼ Equations (8.248) – (8.250) show that δ* is a function of x*, and the integration of eq. (8.248) would also lead to this expression. As a primary step, a few definitions that will be incorporated into the final numerical solution procedure to generalize this problem must be shown. To start, the mass flow rate of the vapor is related to the mass flow rate of the condensate, including the suction force at the porous wall. Performing a mass balance and writing it in terms of the Reynolds numbers of the vapor and liquid, the resulting equation is as follows: μ μ x* Rev ,e − Rev = A ( ReA − ReA ,e ) + 4 A (8.251) μv μv PrA where uD Re v = v h (8.252) νv 640 Transport Phenomena in Multiphase System 4φ1 (8.253) μA PrA and e indicates the entering condition at x = 0. Also, the shear stress at the interface is due to both the friction between the liquid and the vapor and the momentum gained by condensing liquid from the faster-moving vapor. A relationship for dimensionless shear stress at the interface, N2, is given as follows: C f ρv 1 * 2 Ja e −δ * (8.254) N2 = ( uv − uA*,δ ) + Pr 1 − e−δ * ( uv* − uA*,δ ) 2 ρ A PrA A where the dimensionless vapor velocity is Re ν * uv = − v v (8.255) Re w ν A and the radial Reynolds number at the wall is given as Dv Re w = h w (8.256) ReA = 0 4 ³ ρ A uA dy δ = νA The dimensionless axial velocity of the liquid at the liquid-vapor interface is given by § δ* · ­ § δ* · ½ ° ° * (8.257) uA ,δ = − N 2 exp ¨ PrA ®exp ¨ − ¸ ¸ − 1¾ PrA ¹ ¯ PrA ¹ ¿ ° ° © © The frictional coefficient in eq. (8.254) depends on whether the vapor flow is laminar or turbulent, i.e., Rev ≤ 2300 ­ 16 / Rev Cf = ® (8.258) −0.2 ¯0.046 Rev (1 + 850 F ) Rev > 2300 where F is an empirical coefficient recommended by Henstock and Henratty (1976) that is based on the average shear stress around the circumference of the annular flow. 0.9 [(0.707 Re0.5 )2.5 + (0.0379 ReA ) 2.5 ]0.4 ( μA / μv ) A F= (8.259) ( ρA / ρv )0.5 Re0.9 v Now assuming that δ * / PrA << 1, which is a very good assumption for most liquids, and also assuming that the dimensionless shear stress, N2, is constant, eqs. (8.249) – (8.250) can be approximated by 2 N 2δ * (8.260) φ1 = 2 Ja φ2 = * − 1 (8.261) δ Substituting eqs. (8.260) and (8.261) into eq. (8.248) yields * 2 dδ δ * Ja δ* + = dx* N 2 N 2 (8.262) Chapter 8 Condensation 641 Figure 8.21 Variation of δ+ along the tube for Rev,e = 5000 (Faghri and Chow, 1988). Figure 8.22 Variation of δ+ along the tube for Rev,e = 50000 (Faghri and Chow, 1988). 642 Transport Phenomena in Multiphase System Integrating eq. (8.262) with a boundary condition of δ* = 0 at x* = 0, one obtains an expression relating δ* to x* that must be solved numerically. 2 1 Ja (8.263) x* = − N 2δ * − N 2 Jaδ * + N 2 Ja 2 ln 2 Ja − δ * Faghri and Chow (1988) presented results from the iterative calculations of the above set of equations for steam condensing at one atmosphere. Porous wall temperatures of both 85 °C and 70 °C were considered. The liquid properties were assumed to be constant along the porous wall. Figures 8.21 and 8.22 show these results in the form of a nondimensional film thickness, δ + = δ / D = −δ * /(Re w PrA ) , as a function of the normalized axial distance, x + = x / D = − x* /(Re w PrA ) , for Rev,e = 5000 and 50000. The results were presented with these particular normalized numbers in order to show them in terms of the Reynolds number of the suction at the wall instead of the velocity at the wall caused by suction, as was done in δ* and x*. The two specifications of Ja = 0.02797 and Ja = 0.0559 correspond to the wall temperatures of 85 °C and 70 °C, respectively. Three Reynolds numbers of –0.79, –1.6, and –5.0 for the suction at the wall were also specified in Figs. 8.21 and 8.22. These plots show that increased heat transfer rates are made possible by employing suction at the porous wall. This is clearly seen by the reduction in film thickness. Heat transfer improvement is low for small wall suction rate but increases significantly for higher suction rates. In other words, for the same values of the vapor Reynolds number at the pipe inlet, vapor condenses much faster in pipes with suction at the wall than pipes with no suction. 8.4.2 Annular Condensation Heat Transfer in a Microgravity Environment In condensation application, the main resistance to heat transfer comes from conduction across the condensate film. Therefore, thinning the condensate film is crucial for better heat transfer in a microgravity environment. This thinning is difficult for two main reasons: (1) no gravity is present to help flush the condensate away, and (2) lightweight designs are critical, so bulky pumps and blowers are not feasible. As mentioned above, one mechanism of condensate removal in a microgravity environment is vapor shear at the liquid-vapor interface. Faghri and Chow (1991) proposed the use of an annular pipe as a condenser, with vapor shear as the driving force for condensate removal. The physical model they investigated is shown in Fig. 8.23. Here, saturated vapor enters the annular region at x = 0 while the inner and outer walls are held at constant temperatures Tw,i and Tw,o respectively. Condensate films form on both laminar and turbulent flows. A pressure drop occurs along the x-axis due to the friction at the walls. Chapter 8 Condensation 643 Also, the vapor temperature, density, and viscosity were all assumed to be constant. An annulus would be more efficient than a conventional pipe due to the larger heat transfer surface area. A shorter pipe therefore would also be possible and would help with meeting the lightweight space requirement. The vapor velocity would be greater in an annular pipe than in a conventional pipe with the same outer diameter and vapor mass flow rate, thus increasing the vapor shear at the liquid-vapor interface. Analysis of this annular system begins by assuming that the film thicknesses, δi at inner wall and δo at outer wall are small compared to the inner diameter of the annulus, Di. A momentum equation, neglecting inertial term, can be written for the velocity of the liquid film on either the inner or outer wall. d 2u dp μA 2A − =0 (8.264) dy dx The boundary conditions for the liquid film on the inner wall are uA ,i = 0, y=0 (8.265) ∂uA ,i ∂y = τ δ ,i , μA y = δi (8.266) Figure 8.23 Physical model for convective condensation in an annulus. 644 Transport Phenomena in Multiphase System After integrating eq. (8.264) twice and applying the boundary conditions, the liquid-vapor interface velocity for the inner liquid film is τ 1 dp δ i2 (8.267) uA ,i ,δ = δ ,i δ i − μA μA dx 2 This is similarly true for the outer wall with boundary conditions: uA ,o = 0, y = Do (8.268) (8.269) ∂y The outer liquid film interface velocity is τ 1 dp δ o2 (8.270) u A , o ,δ = δ , o δ o − μA μA dx 2 The inner and outer walls liquid Reynolds numbers are determined as follows: δ i u dy Γ ρ ª δ 2 dp δ i3 º ReA ,i = i = ρ A ³ A ,i = A «τ δ ,i i − (8.271) » 0 2 dx 3 ¼ μi μA μA2 ¬ ª δ 2 dp δ o3 º (8.272) τ δ ,o o − « » Do −δ o μi μA 2 dx 3 ¼ ¬ where Γ is the mass flow rate per unit width. A momentum balance is needed for the vapor region, to help in determination of the overall pressure drop in the annular region. dp 4 (τ δ ,o Do + τ δ ,i Di ) = (8.273) dx Do2 − Di2 Neglecting convection in the liquid films allows for a simple energy balance expression for the inner and outer walls, respectively: d ReA ,i kA (Tsat − Tw,i ) N Ja i = = = 1i (8.274) PrA δ i δ i dx hAv μAδ i ReA ,o = ∂uA ,o = τ δ ,o , y = Do − δ o μA Γo = ρA ³ Do uA ,o dy = ρA μA2 N Ja o = 1o (8.275) PrA δ o δ o dx hAv μAδ o where Ja is the Jacob number c pA (Tsat − Tw ) / hAv and N1 is the ratio of the Jacob = = number to the liquid Prandtl number. After the momentum and energy balances, the only balance that needs to be defined is the mass balance between the liquid and vapor. mv ,e = mv + mA ,i + mA ,o (8.276) where e indicates the entering condition to the annular region, x = 0. Rewriting eq. (8.276) in terms of the Reynolds number, the following is a definition of the mass balance: d ReA ,o kA (Ts − Tw,o ) Chapter 8 Condensation 645 Rev , h = Rev ,e ,h − 4 where μA K * μ 1 ReA ,i − 4 A Re * μv 1 + K μv 1 + K * A , o K* = (8.277) Di (8.278) Do The total shear stress is the sum of the shear stress due to friction with no mass transfer, and the shear stress due to the liquid’s momentum gain due to the condensing of the faster-moving vapor into the slower-moving liquid. This combined shear stress drives the removal of condensate from the condenser in microgravity. The expression for the inner and outer liquid-vapor shear stress is therefore as follows: C d ReA ,i 2 τ δ ,i = f ρv ( uv − uA ,δ ,i ) + μA (8.279) ( uv − uA,δ ,i ) 2 dx C d ReA ,o 2 τ δ ,o = f ρv ( uv − uA ,δ ,o ) + μA (8.280) ( uv − uA,δ ,o ) 2 dx As in the above section on condensation with suction at the wall, C f / 2 for both laminar and turbulent flows is given as follows: A ­ Rev , h ≤ 2300 Cf ° Rev ,h (8.281) =® 2° −0.25 ¯0.085Rev ,h (1 + 850 F ) Rev ,h > 2300 where A is a constant that is a function of K*, F is an empirical coefficient based on the average shear stress around the circumference of the annular flow recommended by Henstock and Hanratty (1976), and Rev ,h is the Reynolds number based on hydraulic diameter. Defining the following normalized parameters u δ δ u x uv+ = v ; uA+ = A ; x + = ; δ i+ = i ; δ o+ = o Do Do Do uv , e uv , e p+ = τD τD Do p ; N 2 i = δ ,i o ; N 2 o = δ , o o μ A uv , e μ A uv , e μ A uv , e (8.282) eqs. (8.267), (8.270) – (8.275), (8.279), and (8.280) can be nondimensionalized as follows Normalized liquid velocity profiles: 2 δ i+ dp + + 2 uA ,i ,δ = N 2iδ i − (8.283) 2 dx + 2 δ o+ dp + + 2 (8.284) uA ,o ,δ = N 2 oδ o − 2 dx + 646 Transport Phenomena in Multiphase System Normalized wall liquid Reynolds numbers: 2 1 Rev ,e ,h vv 1 Rev ,e , h vv dP + +3 ReA ,i = N 2iδ i+ − δi 2 (1 − K * ) vA 3 (1 − K * ) vA dx + ReA ,o = 2 1 Rev ,e ,h ν v 1 Rev ,e, h ν v dp + +3 δo N 2 oδ o+ − 2 (1 − K * ) ν A 3 (1 − K * ) ν A dx + (8.285) (8.286) Normalized pressure gradient: dp + 4 ( N 2 o + N 2i ) = 2 dx + 1− K* ( ) (8.287) Normalized energy balance: d ReA ,i dx d ReA ,i + + = = N1i δ i+ N1i (8.288) (8.289) dx δ i+ Normalized liquid-vapor shear stress: C f § 1 · μv d Rel ,i + (8.290) N 2i = Rev ,e ,h ( uv+ − uA+,δ ,i ) + ( uv − uA+,δ ,i ) ¨ *¸ dx + 2 © 1 − K ¹ μA C f § 1 · μv d ReA ,o + N 2o = (8.291) Rev ,e,h ( uv+ − uA+,δ ,o ) + ( uv − uA+,δ ,o ) ¨ *¸ dx + 2 © 1 − K ¹ μA Equations (8.277) and (8.283) – (8.291) are used to find the condensate film thicknesses δ o+ and δ i+ , N2i, N2o, ReA ,i , ReA ,o and Rev,h as functions of the normalized axial length x+. The local Nusselt numbers at the inner and outer walls are respectively given as h ( x) Do 1 Nui ( x) = i =+ (8.292) kA δi h ( x) Do 1 Nuo ( x) = o =+ (8.293) δo kA Results of this analysis were presented by Faghri and Chow (1991) for steam condensing at one atmosphere. Here, the results were reproduced for equal inner and outer wall temperatures at 70 °C. The liquid properties were approximated as the average between the vapor temperature and the inner and outer wall temperatures. The ratio of inner to outer tube diameters was set to be K* = 0.5. Figure 8.24 (a) shows two plots: one for the annular case described above, and another for a circular tube with the same diameter as the annular tube’s outer wall. The Reynolds number of the vapor at the entrance was set to Rev,e,h = 2000. The plots are labeled A2000 and C2000, respectively, where the A and C denote annular and conventional tubes, respectively. As can be seen from the plots, the film thickness, δ+, as a function of axial length x+, is smaller for the annulus than Chapter 8 Condensation 647 for the circular pipe. This leads to a higher condensation rate for the annulus. The same pattern can be seen in Fig. 8.24(b), where the Reynolds numbers of the vapor at the entrance are Rev,e,h = 3333 and 5000. These results are not surprising, because the vapor flow is faster in an annulus than in a circular pipe of similar outer diameter, which leads to a thinning of the condensate film. (a) ΔT = 30ºC. Rev,e,h = 2000. (b) ΔT = 30ºC. Rev,e,h = 3333 and 5000. Figure 8.24 Forced condensation in an annular and conventional tube (Faghri and Chow, 1991; Reprinted with permission from Elsevier). 648 Transport Phenomena in Multiphase System 8.4.3 Condensation Removal by a Centrifugal Field via a Rotating Disk Another method to create artificial gravity involves use of a centrifugal field that is, a rotating disk. The problem studied here addresses a cooled rotating disk in a large quiescent body of pure saturated vapor, as shown in Fig. 8.25. The liquid forms a continuous film on the disk, and the fluid in this film will move radially outward due to the centrifugal force. Sparrow and Gregg (1959) investigated this problem, which is presented below. Conservation equations for mass, momentum in the r-, -, and z-directions and energy for an incompressible, constant-property liquid are derived below. 1∂ 1 ∂Vφ ∂Vz (8.294) + ( rVr ) + + =0 r ∂r r ∂φ ∂z § DV V 2 · § 2 ∂V Vr · ∂p (8.295) ρ ¨ r − φ ¸ = − + μ ¨ ∇ 2Vr − 2 φ − 2 ¸ ¨ Dt ∂r r¸ r ∂φ r ¹ © © ¹ § DV V V · § 1 ∂p 2 ∂V Vr · ρ¨ φ + r φ ¸= − (8.296) + μ ¨ ∇ 2Vφ + 2 r − 2 ¸ r¹ r ∂φ r ∂φ r ¹ © © Dt DVz ∂p ρ = − + μ∇ 2Vz (8.297) Dt ∂z DT ρcp = k ∇ 2T (8.298) Dt z r Tw Figure 8.25 Constant temperature cooled rotating disk in a large quiescent body of saturated vapor. Chapter 8 Condensation 649 where D ∂ Vφ ∂ ∂ = Vr + + Vz ∂r r ∂φ ∂z Dt and ∂2 1 ∂ 1 ∂2 ∂2 + + 2 2+ 2 ∂r 2 r ∂r r ∂φ ∂z The boundary conditions at the wall, z = 0, are Vr = 0 , Vφ = rω , Vz = 0 , T = Tw ∇2 = (8.300) (8.301) (8.299) where is the angular velocity. The boundary conditions at the liquid-vapor interface, z = , are τ zr = 0 , τ zφ = 0 , T = Tsat (8.302) The governing equations are transformed from partial differential equations into ordinary differential equations using similarity transformation variables. The new independent variable is §ω · η =¨ ¸ z ©v¹ and the new dependent variables are V F (η ) = r rω Vφ G (η ) = rω Vz H (η ) = 1/ 2 (ω v ) 1/ 2 (8.303) (8.304) (8.305) (8.306) (8.307) P (η ) = p μω Tsat − T Tsat − T = (8.308) Tsat − Tw ΔT Transforming eqs. (8.294) – (8.298) using the above variables, they become H ′ = −2 F (8.309) 2 2 F ′′ = HF ′ + F − G (8.310) G ′′ = HG ′ + 2 FG (8.311) P′ = H ′′ − HH ′ (8.312) θ ′′ = ( Pr ) Hθ ′ (8.313) Combining eqs. (8.309), (8.310), and (8.311) results in 2 H ′′′ = HH ′′ − ( H ′ ) / 2 + 2G 2 (8.314) G ′′ = HG ′ − H ′G (8.315) The boundary conditions in terms of the new variables are given as H = H ′ = G = θ = 0, η = 0 (8.316) θ (η ) = 650 Transport Phenomena in Multiphase System H ′′ = G ′ = θ = 0, η = ηδ (8.317) (dimensionless condensate layer thickness) to physical To relate quantities, an energy balance is created: δ δ § ∂T · 2 (8.318) hlv ³ ρ 2π rVr dz + ³ ρ 2π rVr c p (Tsat − T ) dz = k ¨ ¸ πr 0 0 ∂z ¹ z = 0 © The first term on the left-hand side represents energy released as latent heat; the second term is the energy dissipated by subcooling of the condensate. The right-hand side is the heat transferred from the condensate to the disk over a span of r = 0 to r = r. In terms of the defined variables, the energy balance becomes c p ΔT H (ηδ ) (8.319) = Pr hlv θ ′ (ηδ ) Local heat flux to the disk may be computed from Fourier’s law § ∂T · (8.320) q′′ = ¨ k ¸ © ∂z ¹ z = 0 In terms of the transformed variables, the equation for q′′ becomes §ω · q′′ = −k (Tsat − Tw ) ¨ ¸ ( ∂θ ∂η )η = 0 ©v¹ The definition of the local heat transfer coefficient is q′′ h≡ Tsat − Tw Substituting eq. (8.321) into eq. (8.322) and rearranging gives 1/ 2 1/ 2 (8.321) (8.322) §v· h¨ ¸ © ω ¹ = − ∂θ ∂η (8.323) ( )η =0 k Evaluating eqs. (8.319) and (8.323) from numerical solutions, the heat-transfer results for high Prandtl numbers have been plotted in Fig. 8.26. 1.1 Pr 1.0 § v ·2 1 h¨ ¸ w ¹ ª c p T/hlv º 4 © k « Pr » .9 ¬ ¼ 1 100 10 1 .8 .001 .01 .1 1.0 cp T hlv Figure 8.26 Heat-transfer results for high Prandtl number fluids. Chapter 8 Condensation 651 Inspection of the figure reveals that for small values of c p ΔT / hAv < 0.1, the results are represented by § · Pr = 0.904 ¨ (8.324) ¸ ¨ c ΔT h ¸ Av ¹ ©p For the limiting case of negligible inertia and heat convective effects, the following limiting relationship is derived (see Problem 8.24) 1/ 2 1/ 2 § c p ΔT / hAv · §ω · (8.325) δ ¨ ¸ = 1.107 ¨ ¸ Pr ©v¹ © ¹ Laminar flow of the condensate is expected when Re = r 2ω / v ≤ 3 × 105 . The heat transfer coefficient for a rotating disk, hrot, is related to the heat transfer coefficient for a vertical plate, hvert, only under the influence of gravity by hrot § 8 xω 2 · =¨ (8.326) ¸ hvert © 3g ¹ Condensation on a rotating cone was studied by Sparrow and Hartnett (1961). The uppermost apex was found to have close to the same coefficient as a rotating disk since hcone 12 = ( sin φ ) (8.327) hdisk where φ is the half-angle of the cone. A vertical tube spinning about its own axis was experimentally studied by Nicol and Gacesa (1970). At low angular velocity, the overall heat-transfer coefficient measurements were correlated by: 0.0943, We ≤ 250 ­ Nu =® (8.328) 0.39 14 3 ª gL hAv ρA / vA kA (Tsat − Tw ) ¼ º ¯0.00923We , We > 250 ¬ for L/D = 10, where We = ρAω 2 D 3 / 4σ is the Weber number and Nu = hD / k . This correlation holds until the Nusselt number is tripled. At high angular velocities, the effect of gravity is not as important and the correlation becomes Nu = 12.26We0.496 (8.329) 14 §v· h¨ ¸ ©ω ¹ k 1/ 2 1/ 4 8.4.4 Condensation by Capillary Action in a Heat Pipe In a zero gravity environment, capillary action is one mechanism of condensate removal. Heat pipes fall under the category of capillary driven devices. Gasloaded heat pipes have been applied in many diverse fields, and are useful when the temperature of a device must be held constant while a variable heat load is dissipated. In this section, a noncondensable gas-loaded heat pipe modeled by 652 Transport Phenomena in Multiphase System Harley and Faghri (1994) is presented below where the effect of capillary and noncondensable action is applied simultaneously. The physical configuration and coordinate system of the gas-loaded heat pipe studied is shown in Fig. 8.27. Gas-loaded heat pipes offer isothermal operation for varying heat loads by changing the overall thermal resistance of the heat pipe. As the heat load increases, the vapor temperature and total pressure increase in the heat pipe. This increase in total pressure compresses the noncondensable gas in the condenser, increasing the surface area available for heat transfer, which maintains a nearly constant heat flux and temperature. Vapor Space The conservation equations for transient, compressible, two-species flow for mass, momentum, energy, and species in vapor space are as follows: ∂ρ 1 ∂ ∂ (8.330) + ( ρ rv ) + ( ρ w) = 0 ∂t r ∂r ∂z DV 1 ρ = −∇p + μ∇ ( ∇ ⋅ V ) + μ∇ 2 V (8.331) 3 Dt §2 · Dp DT ρcp − ∇ ⋅ k ∇T − ∇ ⋅ ¨ ¦ Dd c pjT ∇ρ j ¸ − − μΦ = 0 (8.332) Dt © j =1 ¹ Dt where the subscript j denotes either vapor (v) or gas (g). Dρ g − ∇ ⋅ Dgv ∇ρ g = 0 Dt T’ Noncondensable gas Tg,i pg,i (8.333) r z Wick v w Rv Wall Rw Ro pv,a Tv,a pv,i Evaporator Le Adiabatic La Condenser Lc,a Lc L Figure 8.27 Noncondensable heat pipe configuration. Chapter 8 Condensation 653 and 2 ª§ ∂v · 2 § v · 2 § ∂w · 2 º § ∂v ∂w · 2 2 ª 1 ∂ ∂w º Φ = 2 «¨ ¸ + ¨ ¸ + ¨ ( rv ) + » ¸ »+¨ + ¸− « ∂z ¼ «© ∂r ¹ © r ¹ © ∂z ¹ » © ∂z ∂r ¹ 3 ¬ r ∂r ¬ ¼ (8.334) Furthermore, v and w are the radial and axial vapor velocities, p is the total mixture pressure, μ is the mass-fraction-weighted mixture viscosity, c p is the specific heat of the mixture, k is the thermal conductivity of the mixture, Dd is the self-diffusion coefficient for both vapor and gas species, Dgv is the mass diffusion coefficient of the vapor-gas pair, g is the density of the noncondensable gas, and the mixture density is ρ = ρ g + ρv . The partial gas density is determined from the species equation, and the vapor density is found from the ideal gas relation using the partial vapor pressure. The two choices in species conservation formulation are mass and molar fraction. Molar fraction offers the possibility of a direct simplification in the formulation by the assumption of constant molar density. The assumption is valid over a wider range of temperature and pressure than the corresponding assumption of constant mass density. However, when molar fractions are used, the momentum equation must be written in terms of molar-weighted velocities. The resulting equation cannot be written in terms of the total material derivative, and is significantly more difficult to solve. A benefit of the general equation formulation is its allowance for variable properties. Typical of compressible gas applications, the density is related to the temperature and pressure through the equation of state ρ R uT (8.335) p= M where M is the molecular weight of the vapor-gas mixture and Ru is the universal gas constant. In the species equation, Dgv is a function of pressure and temperature. For a vapor-gas mixture of sodium-argon, Harley and Faghri (1994) used the following relationship for Dgv −1 Dgv = 1.3265 × 10−3 T 3 / 2 ( p ) (8.336) where T is in degrees Kelvin, p is in N/m2, and Dgv is in m2/s. Following a similar procedure, the variable diffusion coefficient formulation for the sodiumhelium pair is −1 Dgv = 1.2795 × 10−3 T 3 / 2 ( p ) (8.337) The interspecies heat transfer that occurs through the vapor-gas mass diffusion was modeled with a self-diffusion model. In the present model, however, the self-diffusion coefficient, Dd, was assumed constant at the initial temperature of the heat pipe. 654 Transport Phenomena in Multiphase System Wick The solid structure in the wick is saturated with the working fluid. The condensate is pumped to the evaporator through capillary action. The liquid A velocity is taken to be the intrinsic phase-averaged velocity vA through the porous wick, which is assumed to be isotropic and homogeneous. For simplicity, A is dropped for velocity. Furthermore, the working fluid and wick structure are assumed to be in local thermal equilibrium. The continuity, momentum, and energy equations for the liquid saturated wick are ∂w 1∂ (8.338) ( rvA ) + A = 0 ∂z r ∂r ∂v · 1 ∂vA 1 § ∂vA vA + + wA A ¸ ε ∂t ε 2 ¨ ∂r ∂z ¹ © (8.339) 1 ∂pA vA vA ν A ª 1 ∂ § ∂vA · vA ∂ 2 vA º r =− − +« −+ K ρ A ∂r ε ¬ r ∂r ¨ ∂r ¸ r 2 ∂z 2 » © ¹ ¼ ∂w · 1 ∂wA 1 § ∂wA vA + + wA A ¸ ε ∂t ε 2 ¨ ∂r ∂z ¹ © (8.340) 1 ∂pA vA wA ν A ª 1 ∂ § ∂wA · wA ∂ 2 vA º r =− − +« − + K ρA ∂r ε ¬ r ∂r ¨ ∂r ¸ r 2 ∂z 2 » © ¹ ¼ ∂Tl ∂TA ∂TA 1 ∂ § ∂TA · ∂ § ( ρ c p )eff ∂t + vA ∂r + wA ∂z = r ∂r ¨ rkeff ∂r ¸ + ∂z ¨ keff ∂∂TA · (8.341) ¸ z¹ © ¹ © where is the porosity of the wick and K is the permeability. Wall In the heat pipe wall, heat transfer is described by the transient twodimensional conduction equation ª 1 ∂ § ∂T · ∂ 2T º ∂T = kw « (8.342) ρ w c pw ¨r ¸+ 2 » ∂t ¬ r ∂r © ∂r ¹ ∂z ¼ where the subscript w denotes the heat pipe wall material. Boundary Conditions At the end caps of the heat pipe, the no-slip condition for velocity, the adiabatic conduction for temperature, and the overall gas conservation conditions are imposed Chapter 8 Condensation 655 ∂T ∂ρ g = = 0, z = 0 (8.343) ∂z ∂z ∂T v=w= = 0, ρ g = ρ g , BC , z = L (8.344) ∂z where g,BC is iteratively adjusted to satisfy overall conservation of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas in the heat pipe. If the total mass is found to be less than the mass initially present in the pipe, the boundary value is increased by 10% of the previous value. Conversely, if the calculated mass is larger than that initially present, the boundary value is decreased. This ensures the conservation of the overall mass to within a preset tolerance, which is 1% in the present formulation. The symmetry of the cylindrical heat pipe requires that the radial vapor velocity and the gradients of the axial vapor velocity, temperature, and gas density be zero at the centerline: ∂w ∂T ∂ρ g = = = 0, r = 0 (8.345) v= ∂r ∂r ∂r The liquid-vapor interface ( r = Rv ) is impermeable to the noncondensable gas mg = Sδ ADgv ∇ρ g + ρ g Sδ V = 0, r = Rv (8.346) v=w= where m g is the mass flow rate of gas, A is the cross-sectional area of the heat pipe and S is the surface area of the liquid-vapor interface. This formulation of mg accounts for both the convective and diffusive noncondensable gas mass fluxes at the liquid-vapor interface. To ensure saturation conditions in the evaporator section (and part of the adiabatic section since the exact transition point is determined iteratively), the Clausius-Clapeyron equation is used to determine the interface temperature as a function of pressure. The interface radial velocity is then found through the evaporation rate required to satisfy heat transfer requirements. The no-slip condition is still in effect for the axial velocity component. At r = Rv for z ≤ Le + La : §1 Ru p· Tsat = ¨ − ln v ¸ © T0 M v hAv p0 ¹ ∂Tv · ∂TA § ¨ keff ∂r − kδ ∂r ¸ ¹ vδ = © hAv + c pδ Tsat ) ρδ ( w=0 −1 (8.347) (8.348) (8.349) 656 Transport Phenomena in Multiphase System where kδ , c pδ and ρδ are the vapor-gas mixture properties at the liquid-vapor interface. In eq. (8.347), the saturation temperature of the vapor is found from the partial vapor pressure. A solution of the momentum equation gives the total mixture pressure, but the partial vapor pressure can be found using the local gas density: M § ρg · (8.350) pv = p ¨1 − ¸ Mv © ρ¹ which was derived assuming a mixture of ideal gases following Dalton’s model for mixtures. At the liquid-vapor interface in the active portions of the condenser section, vapor condenses and releases its latent heat energy. This process is simulated by applying a heat source at the interface grids in the condenser section. The interface velocity can be obtained through a mass balance between the evaporator and condenser section, allowing for inactive sections of the condenser. At r = Rv for z > Le + La : qso = − ( hAv + c pδ Tδ ) ( ρ − ρ g ) vg (8.351) Due to the conjugate nature of the solution procedure, the boundary condition between the wick and the wall is automatically satisfied. In addition to the equality of temperature, this condition requires the equality of the heat fluxes into and out of the wick-wall interface: ∂T ∂T = keff (8.352) kw , r = Rw ∂r ∂r At the outer pipe wall surface, the boundary conditions depend on both the axial position and the mechanism of heat transfer being studied. In the evaporator, a constant heat flux is specified. In the condenser, a radiative boundary condition is imposed. ­q′′ evaporator °e ∂T ° kw adiabatic (8.353) =®0 ∂r r = RO ° 4 4 condenser ° σε r (Tw − T∞ ) ¯ where is the Stefan-Boltzman constant and r is the emissivity. Initial Conditions There is no motion of either the gas or vapor, and the noncondensable gas is evenly distributed throughout the vapor space by diffusion. The initial temperature of the heat pipe is above the free-molecular/continuum-flow transition temperature for the specific heat pipe vapor diameter. Chapter 8 Condensation 657 Figure 8.28 Temperature profiles for the gas-loaded heat pipe with Qin = 451W: (a) transient wall temperature profile; (b) transient centerline temperature profile (Harley and Faghri, 1994). The gas-loaded heat pipe experimentally studied by Ponnappan (1989) was simulated using the above analysis, with results shown in Fig. 8.28. It can be seen that the wall and vapor temperatures decreased significantly in the condenser section due to the presence of the noncondensable gas. 8.5 Film Condensation in Porous Media 8.5.1 Overview Film condensation occurs when the temperature of a vertical, impermeable, and wettable wall next to a porous medium saturated with vapor falls below saturation temperature (Fig. 8.29). In addition to gravity-driven downward liquid flow, the liquid also infiltrates the vapor region due to capillary force. The latter will create a two-phase region between the liquid film and the vapor region, where both condensate and vapor are present. It is assumed that the vapor 658 Transport Phenomena in Multiphase System y x Tw A A Tsat T∞ + Av y2 Liquid film region Twophase region g Vapor region Figure 8.29 Film condensation in a porous medium. temperature is equal to the saturation temperature, i.e., there is no superheat in the vapor phase. The temperature in the liquid region is below saturation temperature, while the temperature in the two-phase region is at saturation temperature. The discussion in this section is limited to the case where the thickness of the liquid film is much greater than the diameter of the pore size. This is referred to as a thick-film region, and the local volume average is applicable (Kaviany, 1995). If the liquid film thickness is less than or comparable to the pore size, the local volume average will no longer be applicable and a direct simulation at the pore level must be performed. The dominant forces in the condensation process are gravitational and capillary forces, and the latter dictates the thickness of the two-phase region, Av . The ratio of gravity and capillary forces is measured by Bond number: g ( ρA − ρv ) K / ε Bo = (8.354) where K and are, respectively, permeability and porosity. When Bo  1 , condensation in a porous medium is dominated by both gravitational and capillary force. The condensation is dominated by capillary force when Bo < 1 . When gravity dominates, Bond number will be greater than 1, and there will be no two-phase region, in which case the analysis will be significantly simplified. In the following two subsections, an analysis of gravity-dominated condensation will be discussed first, followed by a discussion of the effect of surface tension on the condensation process. σ Chapter 8 Condensation 659 8.5.2 Gravity-Dominated Film Condensation on an Inclined Wall When condensation is dominated by gravity, the effect of surface tension is negligible, and consequently, no two-phase region exists. Condensation along an inclined wall in a porous medium (See Fig. 8.30) will be discussed in this subsection. A porous medium saturated with dry vapor at its saturation temperature, Tsat, is bounded by an inclined impermeable wall with a temperature Tw ( Tw < Tsat ). Since the wall temperature is below saturation temperature, film condensation occurs on the inclined wall and the condensate flows downward due to gravity. It is assumed that the condensation is gravity-dominated and, therefore, that the liquid and vapor are separated by a sharp interface, not a twophase region. In addition, the following assumptions are made by Cheng (1981) in order to obtain an analytical solution: 1. The condensate film is very thin compared to the length of the inclined wall ( δ A  L ) so that boundary layer assumption is valid. 2. The properties for the porous medium, liquid, and vapor are independent from temperature. 3. The inclination angle, φ , is small enough for the gravity component in the normal direction of the surface to be negligible. 4. Darcy’s law is valid for both liquid and vapor phases. 5. The saturation temperature, Tsat, is constant. y x φ A Tsat T∞ Vapor region Liquid g Tw Figure 8.30 Gravity dominated film-condensation on an inclined wall in a porous medium. 660 Transport Phenomena in Multiphase System Under these assumptions, the continuity, momentum, and energy equations for the liquid layer are ∂uA ∂vA + =0 (8.355) ∂x ∂y K uA = ( ρA − ρ v ) g cos φ (8.356) μA ∂TA ∂T ∂ 2T + vA A = α A 2A ∂x ∂y ∂y The boundary conditions at the wall are vA = 0 , y = 0 TA = Tw , y = 0 At the interface, the boundary conditions are TA = Tsat , y = δ A uA (8.357) (8.358) (8.359) (8.360) § dδ · (8.361) m′′ = ρA ¨ uA A − vA ¸ , y = δ A © dx ¹ ∂T , y = δA (8.362) m′′hAv = − kmA ∂y where m′′ is mass flux of condensate across the interface, and kmA is thermal conductivity of the porous medium saturated with liquid. Combining of eqs. (8.361) and (8.362) yields ∂T § dδ · ρA hAv ¨ uA A − vA ¸ = −kmA , y = δA (8.363) ∂y © dx ¹ Introducing stream function ∂ψ ∂ψ uA = , vA = − (8.364) ∂y ∂x and the following similarity variables: y η = RaAx (8.365) x f (η ) = ψ α A RaAx TA − Tsat Tw − Tsat (8.366) (8.367) θ (η ) = where RaAx = ( ρ A − ρ v ) g cos φ Kx μAα A (8.368) the governing equations and the corresponding boundary conditions become f ′ =1 (8.369) Chapter 8 Condensation 661 2θ ′′ + f θ ′ = 0 f (0) = 0 θ (0) = 1 θ (ηδ ) = 0 1 Ja Aθ ′(ηδ ) = − f (ηδ ) 2 where (8.370) (8.371) (8.372) (8.373) (8.374) (8.375) x is the dimensionless liquid film thickness and c pA (Tsat − Tw ) Ja A = (8.376) hAv is Jakob number that measures the degree of subcooling at the wall. Integrating eq. (8.369) and considering eq. (8.371), one obtains f =η (8.377) which can be substituted into eqs. (8.370) and (8.374) to get 2θ ′′ + ηθ ′ = 0 (8.378) 1 Ja Aθ ′(ηδ ) = − ηδ (8.379) 2 The solution of eq. (8.378) with eqs. (8.372) and (8.373) as boundary conditions is erf (η / 2) θ (η ) = 1 − (8.380) erf(ηδ / 2) where the dimensionless film thickness can be obtained by substituting eq. (8.380) into eq. (8.379): §η2 · §η · πηδ (8.381) Ja A = exp ¨ δ ¸ erf ¨ δ ¸ 2 ©4¹ ©2¹ The heat flux at the wall is k (T − Tw ) Ra Ax § ∂T · ′′ qw = −kmA ¨ A ¸ = mA sat θ A′ (0) (8.382) x © ∂y ¹ y = 0 and the local Nusselt number is 1 ′′ qw x RaAx/ 2 = Nu x = (8.383) kmA (Tsat − Tw ) π erf(ηδ / 2) where ηδ is function of Jakob number, Ja A , as indicated by eq. (8.381). Cheng (1981) recommended that eq. (8.383) can be approximated using §1 1· Nu x = ¨ +¸ © 2 JaA π ¹ 1/ 2 ηδ = RaAx δA Ra1/ 2 Ax (8.384) 662 Transport Phenomena in Multiphase System In practical application, the average Nusselt number is often of the interest. It can be obtained by integrating eq. (8.384): hL § 1 2· Nu = =¨ +¸ kmA © JaA π ¹ 1/ 2 1/ RaAL2 (8.385) 8.5.3 Effect of Surface Tension on Condensation in Porous Media The analysis in the preceding subsection is valid for gravity-dominated condensation in porous media ( Bo  1 ). When the condensation is gravitycapillary forces dominated ( Bo  1 ) or capillary force dominated ( Bo  1 ), there will be a two-phase region that is saturated by a mixture of liquid and vapor, as shown in Fig. 8.29. The fraction of liquid in the pore space is defined as saturation: γA = where ε A and ε are volume fraction of the liquid and porosity in the porous media. The effect of surface tension on condensation in porous media was studied by Majumdar and Tien (1990) and their work will be briefly described below. The continuity equation for the two-phase region is § ∂u ∂v · § ∂u ∂v · ρA ¨ A + A ¸ + ρv ¨ v + v ¸ = 0 (8.387) © ∂x ∂y2 ¹ © ∂x ∂y2 ¹ where y2 is measured from the interface between liquid and the two-phase region (see Fig. 8.29). The mass fluxes for liquid and vapor are governed by Darcy’s law, as follows: KK rA ′′ ∇pA (8.388) mA = − εA ε (8.386) νA ′′ mv = − KK rv νv ∇pv (8.389) where K rA and K rv are relative permeabilities (dimensionless) for liquid and vapor phases, respectively. Their products with the permeability K represent the permeability for liquid and vapor flow in the porous media. Since the density of the vapor is much lower than that of the liquid, the pressure variation in the liquid phase is very insignificant. Also, the change of capillary pressure is due mainly to change of liquid pressure. Consequently, the vapor flow is negligible compared to the liquid flow, so that eq. (8.387) is reduced to ∂uA ∂vA + =0 (8.390) ∂x ∂y2 Chapter 8 Condensation 663 The velocity components in the x- and y- directions are KK rA ( ρA − ρ v ) g uA = K r A u D = μA (8.391) KK rA ∂pA KK rA ∂pcap = (8.392) μA ∂y2 μA ∂y2 where uD = K ( ρ A − ρv ) g / μA is Darcian velocity. The capillary pressure can be written as vA = − f (s) (8.393) K /ε where f(s) is a Leverett’s function (see Section 4.6.6): f ( s ) = 1.417(1 − s ) − 2.120(1 − s ) 2 + 1.263(1 − s )3 (8.394) and s is dimensionless saturation defined as γ − γ Ai s= A (8.395) 1 − γ Ai where γ Ai is irreducible saturation, below which liquid flow will not occur. The relative permeability in eqs. (8.391) and (8.392) is obtained from K rA = s 3 (8.396) Substituting eqs. (8.391) – (8.396) into eq. (8.390), the following form of the dimensionless continuity equation is obtained: 2 ∂s ª σ / K / ε º ª ∂2s º § ∂s · (8.397) 3 +« (3 f ′ + sf ′′) ¨ ¸ + sf ′ 2 » = 0 »« ∂x ¬ ( ρ A − ρv ) g ¼ « ∂y2 » © ∂x ¹ ¬ ¼ which is subjected to the following boundary conditions: s = 0, x = 0 (8.398) s = 1, y2 = 0 (8.399) s = 0, y2 → ∞ (8.400) Introducing the following similarity variable: ª ( ρ − ρv ) g º η = y2 « A » ¬ (σ / K / ε ) x ¼ eqs. (8.397) – (8.400) are transformed to 3η s′ − 2(3 f ′ + sf ′′) s′2 s′′ = 2 sf ′ 2 pcap = σ (8.401) (8.402) s = 1, η = 0 (8.403) s = 0, η → ∞ (8.404) The numerical solution of eq. (8.402) was obtained using the Runge-Kutta method and it is concluded that η = 1.296 can be chosen to determine the thickness of the two-phase region. 664 Transport Phenomena in Multiphase System For the liquid region, the model in the preceding subsection overpredicts the heat transfer coefficient because of slip in the velocity at the wall. They presented a new model based on the following assumptions: (a) the boundary layer assumption applies in the liquid film, (2) convection terms in the energy equation are negligible, (3) subcooling in the liquid is negligible, and (4) the fluid properties are constants. The governing equations for the liquid film in dimensionless form are ∂uA+ ∂vA+ + =0 (8.405) ∂x + ∂y + ∂ 2uA+ + 1 − uA+ = 0 ∂y +2 (8.406) ∂ 2θ A =0 (8.407) ∂y +2 where the dimensionless variables are defined as T − Tw δ u v x y , y+ = , δ A+ = A , θ = (8.408) uA+ = A , vA+ = A , x + = uD uD Tsat − Tw K K K Compared with eq. (8.356) in Cheng’s (1981) model, eq. (8.406) allows for nonslip conditions at the wall. The boundary conditions at the wall are uA+ = vA+ = θ = 0 , y + = 0 (8.409) The boundary conditions at the interface between the liquid film and the twophase region require that the velocity and shear stress in these two regions match, which makes the solution of the condensation problem very challenging. Majumdar and Tien (1990) proposed three models to handle the boundary condition at the interface between the liquid and the two-phase regions; two of them are discussed below. Model 1. At the interface between the liquid and the two-phase region, the shear stress is zero, i.e., ∂uA+ / ∂y + = 0 at y = δ A+ , which is the same as in classical Nusselt analysis. The velocity profile in the liquid layer is (8.410) uA+ = 1 − cosh y + + tanh δ A+ sinh y + The liquid layer thickness can be obtained from an energy balance at the interface, and the result is dδ + Ja 0.373 (1 − sech 2δ A+ ) A + (1 − sechδ A+ ) =+A (8.411) + + δ A Ra K dx Box where K 3 / 2 ( ρA − ρv ) g Ra K = (8.412) μAα e is the Rayleigh number based on permeability, and α e = keff /( ρA c pA ) is effective thermal diffusivity. Analytical solution of eq. (8.411) is not possible and it must be solved numerically. Chapter 8 Condensation 665 Figure 8.31 Comparison of results of models 1 and 2 with experiments: (a) aluminum foam metal, (b) polyurethane foam (Majumdar and Tien, 1990). Model 2. This model also employs eq. (8.406) to obtain the velocity in the liquid layer, except that the boundary condition at y + = δ A+ is changed to uA+ = 1 . Although it is not as rigorous as Model 1, it is an improvement over Cheng (1981) because it uses a nonslip condition at the wall. The velocity profile in the liquid layer is uA+ = 1 − cosh y + + coth δ A+ sinh y + (8.413) 666 Transport Phenomena in Multiphase System and the overall energy balance at the interface is § · d δ A+ Ja 1 0.373 + =+A 1− ¨ +¸ + Box + δ A Ra K © 1 + cosh δ A ¹ dx (8.414) which also needs to be solved numerically. In the liquid region, eq. (8.407) will yield a linear temperature profile, and the local Nusselt number is x+ Nu x = + (8.415) δA Figure 8.31 shows comparison between the results predicted from Models 1 and 2, together with the experimental results by White and Tien (1987) for constant porosity media. The parameter R in the figure is defined as Ra K σ K ε R= = (8.416) Bo μAα e which reflects the ratio of surface tension force and viscous force. It can be seen from Fig. 8.31(a) that for aluminum foam, the agreement between Model 1 and experimental results is very good, but Model 2 significantly overpredicts the heat transfer rates. This is expected because Model 2 is not as rigorous as Model 1. The slight overprediction of Model 1 may be attributed to the fact that shear stress at the interface is neglected. For polyurethane foam, on the other hand, Model 1 also significantly overpredict heat transfer rate. This overprediction is due to the fact that surface tension drag at the liquid region interface is neglected in Model 1. Since R in Fig. 8.31 (b) is one order of magnitude higher than that in Fig. 8.31 (a), it is expected that surface tension plays a more significant role in Fig. 8.31 (b). 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Graham, C., and Griffith, P., 1973, “Drop Size Distribution and Heat Transfer in Dropwise Condensation,” International Journal of Heat and Mass Transfer, Vol. 16, pp. 337-346 Griffith, P., 1983, “Dropwise Condensation,” in Heat Exchange Design Handbook, edited by E.U. Schlunder, Vol. 2, Chapter 2.6.5, Hemisphere Publishing, New York, NY. 668 Transport Phenomena in Multiphase System Habib, I.S., and Na, T.P., 1974, “Prediction of Heat Transfer Pipe Flow with Constant Wall Temperature,” ASME Journal of Heat Transfer, Vol. 96, pp. 253254. Harley, C., and Faghri, A., 1994, “Transient Two-Dimensional Gas-Loaded Heat Pipe Analysis,” ASME Journal of Heat Transfer, Vol. 116, pp. 716-723. Henstock, W.H., and Hanratty, T.S., 1976, “The Interfacial Drag and the Height of the Wall Layer in Annular Flows,” AIChE Journal, Vol. 22, pp. 990-1000. Hewitt, G.F., Bott, T.R., Shires, G.L., 1994, Process Heat Transfer, CRC Press, Boca Raton, FL Imura, H., Sasaguchi, K., and Kozai, H., 1983, “Critical Heat Flux in a Closed Two-Phase Thermosyphon,” International Journal of Heat and Mass Transfer, Vol. 26, pp. 1181-1188. Kaviany, M., 1995, Principles of Heat Transfer in Porous Media, SpringerVerlag, New York, NY. Kutateladze, S.S., 1972, “Elements of Hydrodynamics of Gas-Liquid Systems,” Fluid Mechanics – Soviet Research, Vol. 1, pp. 29-50. Kutateladze, S.S., 1982, “Semi-empirical Theory of Film Condensation of Pure Vapors,” International Journal of Heat and Mass Transfer, Vol. 25, pp. 653-660. Labuntsov, D.A., 1957, “Heat Transfer in Film Condensation of Pure Steam on Vertical Surfaces and Horizontal Tubes,” Teploenergetika, Vol. 4, pp. 72-80. Lin, L., and Faghri, A., 1998, “Condensation in a Rotating Stepped Wall Heat Pipe with Hysteretic Annular Flow,” AIAA Journal of Thermophysics and Heat Transfer, Vol. 12, No. 1, pp. 94-99. Majumdar, A., Tien, C.L., 1990, “Effects of Surface Tension on Film Condensation in a Porous Medium,” ASME J. Heat Transfer, Vol. 112, pp. 751757. McCormick, J.L., and Baer, E., 1963, “On the Mechanism of Heat Transfer in Dropwise Condensation,” Journal of Colloid Science, Vol. 18, pp. 208-216. Mikic, B.B., 1969, “On Mechanism of Dropwise Condensation,” International Journal of Heat and Mass Transfer, Vol. 12, pp. 1311-1323. Nicol, A.A. and Gaceda, M., 1970, “Condensation of Steam on a Rotating Vertical Cylinder,” ASME Journal of Heat Transfer, Vol. 92, pp. 144-152. Nusselt, W., 1916, “Die Oberflächenkondensation des Wasserdampfes,” Z. Vereins deutscher Ininuere, Vol. 60, pp. 541-575. Ponnappan, R., 1989, “Studies on the Startup Transients and Performance of a Gas Loaded Sodium Heat Pipe,” WRDC-TR-89-2046, Wright-Patterson AFB, OH. Chapter 8 Condensation 669 Rohsenow, W.M., 1956, “Heat Transfer and Temperature Distribution in Laminar Film Condensation,” Transactions of ASME, Vol. 78, pp. 1645-1648. Rohsenow, W.M., Webber, J.H., and Ling, T., 1956, “Effect of Vapor Velocity on Laminar and Turbulent Film Condensation,” Transactions of ASME, Vol. 78, pp. 1637-1643. Seban, R., 1954, “Remarks on Film Condensation with Turbulent Flow,” Transaction of the ASME, Vol. 76, pp. 299-303. Shafrin, E.G., and Zisman, W. A., 1960, “Constitutive relations in the wetting of low energy surfaces and the theory of the retraction method of preparing monolayers,” Journal of Physical Chemistry, Vol. 64, pp. 519-524. Shekriladze, I.G., and Gomelauri, V.I., 1966, “Theoretical Study of Laminar Film Condensation of Flow Vapour,” International Journal of Heat and Mass Transfer, Vol. 9, pp. 581-591. Sparrow, E.M. and Gregg, J.L., 1959, “A Theory of Rotating Condensation,” Transactions of ASME, Vol. 81, 113-120. Sparrow, E. M. and Hartnett, J. P., 1961, “Condensation on a Rotating Cone,” ASME Journal of Heat Mass Transfer, Vol. 83, pp. 101-102. Stephan, K., 1992, Heat Transfer in Condensation and Boiling, Springer-Verlag, Berlin. Szablewski, W., 1968, “Turbulent Parallelstromunjen,” Zeitshr. Ang. Math Mech., Vol. 48, p. 35. Tien, C.L., and Chung, K.S., 1978, “Entrainment Limits in Heat Pipes,” Proceedings of the 3rd International Heat Pipe Conference, Palo Alto, California, pp. 36-40. Wallis, G., 1969, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, NY. Wallis, G.B., and Makkenchery, S., 1974, “The Hanging Film Phenomenon in Vertical Annular Two-Phase Flow,” ASME Journal of Fluids Engineering, Vol. 96, pp. 297-298. White, S.M., and Tien, C.L., 1987, “An Experimental Investigation of Film Condensation in Porous Structures,” presented at the 6th International Heat Pipe Conference, Grenoble, France. Problems 8.1. For a liquid-vapor phase change problem, explain how the interface affects heat transfer. Describe the conditions under which the interfacial heat transfer resistance is not negligible. 670 Transport Phenomena in Multiphase System 8.2. Perform an energy balance for a control volume that includes a liquidvapor interface, and show that the conservation of mass at the liquid-vapor interface in film condensation is [see eq. (8.80)] ª § dδ ·º ª § dδ ·º « ρA ¨ uA dx − vA ¸ » = « ρv ¨ uv dx − vv ¸ » , ¹¼ I ¬ © ¹¼ I ¬© y =δ 8.3. A beverage can with diameter of 65 mm and a height of 120 mm is removed from a refrigerator with a temperature of 4 °C and placed in a room at temperature of 30 °C and relative humidity of 75%. What is the condensation rate if the condensation can be assumed to be dropwise? If the relative humidity is changed to 50%, how will condensation rate change? 8.4. A square vertical plate 0.5-m long separates saturated steam at 2 atm from 50 °C air blowing parallel to the plate at 2 m/s. 100 W is transferred from the plate to the air by convective cooling. What is the condensation rate assuming dropwise condensation? q =100 W Air @ 1 atm Saturated Steam @ 2 atm Condensate u’=2 m/s T’=50 °C Figure P8.1 8.5. A bathroom mirror is initially at 20 °C. As a bath is drawn, the room temperature increases to 30 °C and the relative humidity increases to 90%. Calculate the heat transfer rate for a single droplet of condensate 1 mm in diameter. Discuss which thermal resistances are most significant. 8.6. Many assumptions are made in Nusselt analysis. Discuss the reasonability of these assumptions and their expected effects on heat transfer. 8.7. For film condensation of steam at atmospheric pressure, find the distance from the top of a vertical plate at which a laminar flow would become wavy. What is the average heat transfer coefficient for this plate? The heat removal rate can be assumed to be 500 kW/m2. Chapter 8 Condensation 671 8.8. Assuming that the plate in the previous problem has a length of 5 cm, and recognizing that this exceeds the maximum laminar length found in the previous problem, find the average heat transfer coefficient for the plate by using both the Kutateladze correlation (which assumes that wavy flow exists along the length at some point) and the classical Nusselt analysis. Compare the two answers. 8.9. For a vertical pipe at 25 °C with a length of 2 m in a saturated steam environment of 100 °C, Find the average heat transfer coefficient of an external turbulent film condensation. What is the corresponding mass flow rate? 8.10. Assume that an interfacial shear stress of 0.5 N/m2 exists in the film condensation on a vertical plate with the physical properties of Problem 8.7. Find the corresponding average heat transfer coefficient for this configuration. The imposed pressure gradient that drives the flow is 2000 N/m3. 8.11. The effect of vapor flow on film condensation was considered in Section 8.3.4 by introducing a pressure gradient imposed by the motion of the adjacent vapor and a shear stress at the interface. A more rigorous analysis requires consideration of the vapor flow using boundary layer theory. Write the complete governing equations for both liquid and vapor flow and give the corresponding boundary conditions. 8.12. A pure vapor at its saturation temperature, Tsat, flows downward across a horizontal cylindrical tube with a wall temperature of Tw (Tw<Tsat). The downward vapor velocity is u∞ . Write the governing equations and corresponding boundary conditions for the convective condensation problem. The pressure gradient in the vapor flow can be determined by assuming potential vapor flow. 8.13. When a stagnant pure vapor condenses on the outer surface of a horizontal cylinder, the condensate generated from the top of the cylinder flows along its surface and eventually drips from the bottom of the cylinder. The driving force of the liquid condensate flow is the tangential component of the gravitational force along the surface of the cylinder. Perform Nusselt analysis to get the average heat transfer coefficient and compare your result with eq. (8.196). 8.14. Condensation on the outer surface of a vertical cylinder is often estimated using a correlation for condensation on a vertical plate. This treatment is valid only if the thickness of the liquid film is much smaller than the radius of the vertical cylinder. For condensation over a thin vertical cylinder, the effect of curvature of the cylindrical surface on the condensation must be considered. This requires that the governing equations be written in the cylindrical coordinate system. Obtain the film thickness for film condensation on the outer surface of a cylinder using Nusselt’s theory. 672 Transport Phenomena in Multiphase System 8.15. A saturated vapor condenses on the outside surface of a constanttemperature cone (see Fig. P8.2). Neglecting vapor shear stress at the liquid-vapor interface and the subcooling effect, obtain a single differential equation relating local film thickness to the temperature difference, the fluid properties, and the distance from the apex of the cone (hint: make assumptions similar to those made for the Nusselt Theory). Figure P8.2 8.16. During laminar film condensation on the outer surface of a sphere, the condensation starts from the top of the sphere and the condensate eventually drips from the bottom of the sphere. When the condensate flows down along the spherical surface, both the driving force and the width of the liquid film vary. Derive the average heat transfer coefficient for this case and compare your result with eq. (8.197). 8.17. For laminar film condensation occurring on a vertical surface, write down an appropriate set of governing equations and boundary conditions that could be solved to predict the variation of the film thickness and heat transfer coefficient with x for the following different boundary conditions. a. The cold surface temperature Tw is constant. b. There is a constant heat flux at the cold surface. c. The back of the cold surface is subjected to convective cooling with a coolant temperature of T∞ and heat transfer coefficient of hc. The thickness of the cold wall can be neglected. 8.18. The process of condensation inside a vertical tube is complex. Build a reasonable model; write governing equations and boundary conditions for the model. 8.19. The effect of liquid subcooling was neglected in the Nusselt analysis. Starting from the energy equation for liquid film with convection terms accounted for, show that the energy balance in the liquid film is eq. (8.110). If the temperature profile in the liquid can be assumed to be linear Chapter 8 Condensation 673 like eq. (8.111), show that the Nusselt analysis is still valid provided that the latent heat, hAv , is replaced by hA′v as defined in eq. (8.113). 8.20. A pure vapor flows into a horizontal channel formed by two parallel plates (see Fig. P8.3) with uniform velocity ( u∞ ) and temperature ( T∞ ). The bottom plate is maintained at a temperature, Tw, below the saturation temperature, while the upper plate is adiabatic. Condensation occurs on the surface of the bottom plate, and the condensate is dragged by the vapor and flows to the positive x-direction. The thickness of the liquid film is much smaller than the distance between the two plates ( δ  H ). Both velocity and temperature distribution in the liquid film can be assumed to be linear. The shear stress at the liquid-vapor interface can be estimated by ′′ τ δ = 0.5 f ρv (uv − uδ ) 2 + mδ (uv − uδ ) , where the fraction coefficient f = C Re n (C and n are constants), uδ is the axial velocity at interface, and ′′ mδ is rate of condensation. Obtain the ordinary differential equation that governs the liquid film thickness. u v0 , T v0 Adiabatic plate y Vapor boundary layer Liquid film Cold plate (Tw) Figure P8.3 δ x 8.21. A condenser is made of a vertical array of 200 horizontal tubes, each 0.5 m long and with diameters of 10 mm. If the condenser is meant to condense saturated steam at 0.5 bars and the cooling tubes are at 25 °C, what is the condensation rate for a single tube of the condenser? 8.22. The condenser section of a rotating heat pipe is shown in Fig. P8.4. The liquid condensate is removed by centrifugal force, and the effect of gravity can be neglected. The temperature of the condenser wall uniformly equals Tw, and the vapor is saturated. The effect of vapor flow on the film condensation can be neglected. Assuming the circumferential velocity and temperature gradient are negligible, find the velocity profile in the liquid condensate. 674 Transport Phenomena in Multiphase System Figure P8.4 8.23. Heat conduction in the condenser wall was neglected in the above problem by assuming the entire condenser wall is at a uniform temperature. If the outer surface of the condenser wall is maintained at Tw, repeat the above problem while considering heat conduction in the condenser wall. Heat conduction in the axial direction can be neglected for simplicity. Figure P8.5 8.24. In a rotating heat pipe, thermal energy transport relies on the evaporation and condensation of a small amount of working fluid, whereas the condensate return relies on the centrifugal force. In the cylindrical section, including condenser and adiabatic zones with a step at the position connecting the evaporator, the condensate return is maintained by centrifugal forces acting on the liquid film with varying film thickness, which results in a hydrostatic pressure change in the film. The coordinate system for annular flow is shown in Fig. P8.5. It can be assumed that (1) the condensate film flow is laminar, (2) the liquid film thickness is much smaller than the heat pipe’s inner radius, (3) the radial heat flux is uniform, (4) inertial and convective effects in the liquid are neglected, and (5) the gravitational force is negligible compared to the centrifugal force. Obtain the liquid film thickness along the axial direction. Chapter 8 Condensation 675 8.25. Under an ideal situation, no liquid accumulates at the bottom of the thermosyphon (see Fig. P8.6). It is assumed that (1) the vapor condensation is filmwise with no interfacial waves, (2) liquid subcooling is negligible, (3) inertial effects in the liquid are negligible, (4) convection effects in the liquid are negligible, (5) the liquid film thickness is much smaller than the vapor space radius, and (6) the vapor density is much smaller than the liquid density. Specify the governing equations for flow and heat transfer in the liquid film in the cylindrical coordinate system. Analyze heat transfer in the liquid film using the Nusselt type solution procedure. Figure P8.6 8.26. The reflux condensation in a small-diameter tube is shown in Fig. P8.7. It is assumed that the shear stress on the surface of the liquid film is negligible, and all other assumptions for the Nusselt analysis are applicable. Derive the differential equation for the liquid film thickness distribution and specify the corresponding boundary conditions. You do not need to consider the liquid layer at the lower portion of the crosssection. 8.27. The mass flux at the liquid vapor-interface for condensation of condensable vapor and noncondensable gas can be obtained from eq. (8.208). Show that the mass flux can also be obtained from eq. (8.211) if the mixture of condensable vapor and noncondensable gas can be treated as ideal gas. 8.28. Develop the film thickness and heat transfer coefficient, eqs. (8.324) and (8.325), for thin condensate layers on a rotating disk, where inertia and convective effects are neglected. 676 Transport Phenomena in Multiphase System φ φ φ θ (a) configuration Figure P8.7 (b) liquid film inside the tube 8.29. A horizontal elliptical tube with a major axis 2a and minor axis 2b is embedded in a porous medium filled with a dry saturated vapor, as shown in Fig P8.8. A dry saturated vapor flows downward with uniform vertical velocity U∞. Since the temperature of the elliptical tube, Tw, is below the saturation temperature of the vapor, Tsat, condensation takes place at the surface of the elliptical tube. Specify the governing equation and corresponding boundary conditions for this problem. U∞ g Figure P8.8 Chapter 8 Condensation 677