7
CONDENSATION AND EVAPORATION
7.1 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 7.1 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. 7.2(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. 7.2(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
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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 7.1 Flowchart of the different modes of condensation.
(a) Filmwise condensation
Figure 7.2 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.
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Figure 7.3 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, 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 mistlike quality and is depicted in Fig. 7.3 (a). Two more examples of homogeneous condensation are shown in Figs. 7.3 (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. 7.3 (b)]. When vapor bubbles are introduced into the bulk liquid, as shown in Fig. 7.3 (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. 7.4 (a) shows the transverse distributions of temperature T and mass fractions ω1v and ω2v 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. 7.4 (b). Dew point line refects the point at which the binary vapor/gas begins to condense. When there is less ω1 and more ω2, 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
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T T∞ T∞ Tsat
T
Tw
Tδ
Tδ Tw
ωv ω1v∞ + ω2v∞ =1
ω2v∞ ω1v∞
y
Boiling point line
0 ω1δ ω1v∞ ω1vδ (b)
Variation of ω 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
1 ω1v
(a)
Figure 7.4 Temperature and mass fraction in the condensation of a binary miscible vapor mixture: (a) T and ω distribution in the condensate film and vapor boundary layer; (b) variation of T and ω 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. 7.4 (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 7.4(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
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Figure 7.5 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. 7.5 and is caused by the different surface tension forces of the two components in relation to the vapor and solid wall. Figures 7.6 and 7.7 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 Pure comp. 1 liquid region heavy w/ comp. 1
Dew point line
Temperature
Miscible liquid region heavy w/ comp. 2
Liquid-vapor region heavy w/ comp. 2 Boiling point line Immiscible liquid region
Dew point line Liquid-vapor region heavy w/ comp. 1 Miscible
0
mass fraction of component 1 (ω1)
Figure 7.6 Phase diagram of liquids with a miscibility gap.
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Pure comp. 2
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 7.7 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. Evaporation processes generally occur from liquid films, drops, and jets. Films may flow on a heated or adiabatic surface as a result of gravity or vapor shear. Drops may evaporate from a heated substrate, or they may be suspended in a gas mixture or immiscible fluid. Jets may be cylindrical in shape or elongated (ribbon-like). In film evaporators, a liquid film is evaporated in order to: (1) cool the surface on which it flows, (2) cool the liquid itself, or (3) increase the concentration of some component(s) in the liquid. Heat may be transferred to the film surface by conduction or convection. The design of film evaporators is most frequently constrained by the extent of the liquid superheat, since nucleate boiling must be prevented. Nucleate boiling can lead to product deterioration because elevated temperatures increase chemical reaction rates. Also, the accumulation of deposits on walls (fouling) as a result of nucleate boiling hinders heat transfer. To avoid nucleate boiling, liquid superheating should not exceed 3
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Pure comp. 1
595
Dew point line
Vapor region
to 40 K, depending on the liquid. This chapter deals with evaporation with no nucleate boiling. In addition to the capability for low liquid superheating, film evaporators may be designed to achieve short residence time (that is, contact time between the liquid and the evaporator wall), which further reduces the tendency for chemical reactions to occur. The tendency for fouling is also reduced in more rapidly flowing films. These features are ideal for heat-sensitive fluids such as those in food processing and polymer devolatilization (Alhusseini et al., 1998). A thin liquid film in an evaporator can be produced by several arrangements, two of which are falling and climbing films (See Fig. 7.8). In falling film evaporators, the liquid is evenly distributed around the opening of a vertical channel – usually a tube – and falls under gravitational force as its surface evaporates to a vapor or gas mixture. If the vapor is drawn to the bottom, the flow is cocurrent as shown in Fig. 7.8(a) and the shear stress of the vapor on the liquid film increases its speed downward. When the vapor is drawn to the top, the flow is countercurrent as shown in Fig. 7.8(b). A climbing film evaporator Fig. 7.8(c) is designed so that liquid is fed into the bottom, and vapor shear pushes the liquid film along the wall to the top. In all of these cases, the lack of a hydrostatic pressure drop eliminates the corresponding temperature drop along the evaporator length, thereby increasing the uniformity in temperature. A more complicated example, shown in Fig. 7.8 (d), is a two-phase closed thermosyphon where evaporation takes place in the evaporator section and liquid and vapor flow in opposite directions. The residence time of a liquid in a falling film evaporator may be increased with a countercurrent configuration. If this configuration is chosen, however, the likelihood of entrainment, flooding, or liquid blocking should be considered. Unlike the cocurrent configuration, countercurrent flow is more prone to
Figure 7.8 Examples of film evaporators.
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r θ rI ωv ωv,I ωv,∞
Tw t2 > t 1 TI t2 t1 T∞ rI
Figure 7.9 Evaporation from a liquid droplet on a heated wall.
entrainment, which occurs when liquid is carried upward by the vapor (see Section 7.3.8). Liquid blocking – when liquid bridges the channel and blocks the vapor path – can also be more likely in a countercurrent configuration. In addition to the film evaporation described above, evaporations from liquid droplets attached to a heated wall or surrounded by hot gas can also find their application in irrigation of crops, firefighting, and combustion. Figure 7.9 shows evaporation from a liquid droplet attached to a heated wall; its application can be found in surface spray cooling, where liquid droplets are sprayed onto a hot surface. Heat is conducted through the droplet to the interface, where an abrupt temperature drop takes place due to evaporation. The mass fraction of the vapor component in the gas mixture, ωv , is greatest next to the interface, and by diffusion the mass fraction decreases to its bulk level with increasing distance from the wall. The size of the liquid drop and the temperature distribution at two different times ( t 2 > t1 ) are shown in Fig. 7.9. As time goes on, the liquid droplet becomes smaller (as indicated by the dashed-line), while the temperature at the heated wall and interface remain unchanged. When a liquid droplet is surrounded by a hot gas mixture, evaporation takes place on the surface of the droplet, as shown in Figure 7.10. Such processes are used in bulk cooling as well as spray fuel injection in diesel engines. The time
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r θ rI ωv ωv,I ωv,∞ t2 > t 1 t2 t1 T∞
T∞ TI
rI
Figure 7.10 Evaporation from a liquid droplet suspended in vapor-gas mixture.
required for such a droplet to completely evaporate as a member of a dispersed phase will determine the overall heat transfer (cooling effect). The partial pressure, pv, and the mass fraction, ωv, of the vapor in the gas mixture are the greatest adjacent to the surface of the droplet, and diffusion causes the partial pressure and mass fraction to decrease with increasing distance from the interface. The temperature profile in the drop is dominated by transient conduction, but convection takes place outside the drop. Since the evaporating liquid is directly supplied by the liquid droplet, evaporation on the drop surface is dominated by heat transfer, not by mass diffusion. Section 7.2 presents the mechanisms of dropwise condensation, starting with the surface tension and capillary pressure, followed by a detailed discussion on thermal resistance in the condensation process, and heat transfer coefficient for dropwise condensation. Section 7.3 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 configurations. Section 7.3 closes with a
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discussion of the effect of noncondensable gases. This chapter is closed by discussion on evaporation from a vertical falling film on a heated surface and spray cooling in Section 7.4.
7.2 Dropwise Condensation
7.2.1 Surface Tension and Capillary Pressure
When a liquid is in contact with another medium, be it liquid, vapor, or solid, a force imbalance occurs at the boundaries between the different phases. For example, a liquid molecule surrounded by other liquid molecules will not experience any resultant force since it will be attracted in all directions equally. However, if the same liquid molecule is at or near a liquid-vapor interface, then the resultant molecular attraction of a liquid molecule on the surface would be in the direction of the liquid, since forces between the interacting gas and liquid molecules are less than the forces between the liquid molecules. It is basically due to the asymmetry of the force field acting on a molecule on the surface tending to pull it back to a higher density region or phase. If a liquid is bounded by its own vapor, then the force in the surface layer is directed into liquid because, in general, the liquid is denser than the vapor. As a result of this effect, the liquid will tend toward the shape of minimum area and behave like a rubber membrane under tension. In this context, if the surface area of the liquid is to be increased, then negative work must be done on the liquid against the liquid-toliquid molecular forces. Any increase in the surface area will require movement of molecules from the interior of the liquid out to the surface. The work or energy required to increase the surface area can be obtained from the following relation, which is also the definition of surface tension
σ = ∂A T , p
∂G
(7.1)
where G is the surface free energy, and A is the surface area. Equation (7.1) is valid for solid-liquid, solid-vapor, liquid-vapor, and liquid-liquid interfaces. Liquid-liquid interfaces are present between two immiscible liquids, such as oil and water. From this expression, the surface tension can be described as a fundamental quantity which characterizes the surface properties of a given liquid. Additionally, the surface tension is referred to as free energy per unit area or as force per unit length. Surface tension exists at all phase interfaces; i.e., solid, liquid and vapor. Therefore, the shape that the liquid assumes is determined by the combination of the interfacial forces of the three phases. The surface in which interfacial tension exists is not two-dimensional, but three-dimensional with very
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small thickness. In this very thin region, properties differ from the bordering bulk phases. It is worthwhile to discuss surface tension from the molecular perspective in order to understand the mechanism of surface tension for different substances. Surface tension can be considered as the summation of two parts: one part is due to dispersion force, and the other part is due to specific forces, like metallic or hydrogen bonding (Fowkes, 1965). Surface tension force in a nonpolar liquid is due only to the dispersion force; therefore, the surface tension for a nonpolar fluid is very low. For a hydrogen-bonded liquid, surface tension is slightly higher because the surface tension is due to both dispersion forces and hydrogen bonding. The surface tension for a liquid metal is highest because the surface tension is due to a combination of dispersion forces and metallic bonding, and metallic bonding is much stronger than hydrogen bonding. The surface tensions for different liquids are quantitatively demonstrated in Table 7.1.
Table 7.1 Surface tensions for different liquids at liquid-vapor interface Types of liquid Nonpolar liquid Hydrogen-bonded liquid (polar) Metallic liquid Liquid Helium Nitrogen Ammonia Water Mercury Silver Temperature (°C) -271 -153 -40 20 20 1100 Surface tension (mN/m) 0.26 0.20 35.4 72.9 484 878
It is generally necessary to specify two radii of curvature to describe an arbitrarily-curved surface, RI and RII, as shown in Fig. 7.11. The surface section is taken to be small enough that RI and RII are approximately constant. If the surface is now displaced outward by a small distance, the change in area is ΔA = ( x + dx )( y + dy ) − xy (7.2)
Figure 7.11 Arbitrarily-curved surface with two radii of curvature RI and RII.
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If dxdy ≈ 0 , then
ΔA = y dx + x dy
(7.3)
The work required to displace the surface is: δ W = σ ( x dy + y dx ) (7.4) Displacement acting on the area xy over the distance dz also creates a pressure difference Δp across the surface – capillary pressure (pcap). The work attributed to generating this pressure difference is (7.5) δ W = Δp xy dz = pcap xy dz From the geometry of Fig. 7.11, it follows that
x + dx x = RI + dz RI
(7.6)
or
dx = x dz RI y dz RII
(7.7)
Similarly,
dy =
(7.8)
For the surface to be in equilibrium across this differential change, the two expressions for the work must be equal: (7.9) i.e.,
σ
xy dz xy dz + = Δp xy dz RII RI
(7.10)
The pressure difference between two phases becomes
1 1 pcap = Δp = σ + = σ ( K1 + K 2 ) RI RII
(7.11)
where K1 and K2 are curvatures of the surface. This expression is called the Young-Laplace equation, and it is the fundamental equation for capillary pressure. It can be seen that when the two curvature radii are equal, in which case the curved surface is spherical, eq. (7.11) can be reduced to 2σ pcap = Δp = (7.12)
R
In addition to the surface tension at a liquid-vapor ( v ) interface discussed above, surface tensions can also exist at a solid-liquid interface ( s ) and solidvapor interface ( sv ); this can be demonstrated using a liquid-vapor-solid system as in Fig. 7.12. The contact line is the locus of points where the three phases intersect. The contact angle, θ , is the angle through the liquid between the tangent to the liquid-vapor interface and the tangent to the solid surface. The contact angle is defined for the equilibrium condition. In 1805, Young published
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Figure 7.12 Drop of liquid on a planar surface.
the basic equation for the contact angle on a smooth, insoluble, and homogeneous solid: σ − σ s cos θ = sv (7.13) σ v which follows from a balance of the horizontal force components (Faghri, 1995), as shown in Fig. 7.12. 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. When the contact angle, θ, 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
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surface alone. If the surface tension between liquid-vapor interface σ v is greater than , dropwise condensation occurs (Shafrin and Zisman, 1960). Critical surface tensions for selected solid surfaces are given in Table 7.2. It can be seen that the critical surface tensions for all solids listed in Table 7.2 are below the surface tension of water at 1 atm ( σ v = 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 7.2 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 (10-3 N/m)
31 46 10 6 31 33 39 18
7.2.2 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 7.13 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
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Figure 7.13 Schematic of the resistance to heat flow in the condensation process: (a) filmwise condensation; (b) dropwise condensation.
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 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 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: (7.14) ΔTtotal = Tvapor − Tw = ΔTvapor + ΔTδ + ΔTcap + ΔTdroplet 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,
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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. 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. The heat flux at the interface can be obtained by (Faghri, 1995) 2 M v pv vv 2α hv ′′ qδ = 1− (7.15) ( Tv − T ) T v 2π R T 2hv 2 − α v v u v where α is the accommodation coefficient. The corresponding heat transfer coefficient across the interface is obtained by 2 ′′ qδ M v pv vv 2α hv = hδ = 1− (7.16) 2π R T (Tv − T ) 2 − α Tv vv 2hv u v 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 hv = hδ = (7.17) 2π R T (Tv − T ) 2 − α Tv vv uv The temperature drop across the interface of a single liquid droplet – assuming it is hemispherical – is
ΔTδ = qd hδ (π D 2 / 2)
(7.18)
where π D 2 / 2 = Av is the surface area of the hemispherical liquid droplet.
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Resistance Due to the Capillary Depression of the Equilibrium Saturation Temperature Upon formation in their nucleation sites, droplets grow only if they form with a radius that exceeds the equilibrium radius. The minimum equilibrium radius of a newly formed droplet that will not spontaneously disappear is (Faghri and Zhang (2006): 2σ v (7.19) rmin =
RgT ln[ pv / psat (T )]
The relationship between saturation pressure and corresponding temperature can be obtained by Clapeyron-Clausius equation (Faghri, 1995):
dp hv p = dT RgT 2
(7.20)
which can be integrated to obtain:
ln
pv h = − v psat Rg
1 1 − Tw Tv
(7.21)
2 Substituting eq. (7.21) into eq. (7.19) and assuming T = Tw , and TvTw ≈ Tw , the expression for minimum equilibrium radius size becomes 2vσ Tv D rmin = min = (7.22)
2
hv (Tv − Tw )
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. (7.22) to get the following expression for the temperature drop across the capillary depression: 4v σ T ΔTcap = sat (7.23)
hv D
Combining eq. (7.23) with the minimum equilibrium droplet size expression, eq. (7.22), one obtains
ΔTcap =
(Tsat − Tw ) Dmin D
(7.24)
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
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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:
ΔTdroplet = qd ( D / 2) 4π k ( D / 2) 2
(7.25)
7.2.3 Heat Transfer Coefficient for Dropwise Condensation
By substituting the expressions for temperature drops through the interface, eq. (7.18); capillary depression, eq. (7.24); and liquid droplets, eq. (7.25), into eq. (7.14) and neglecting the temperature drop in the vapor phase, the temperature drop is obtained:
ΔTtotal = Tsat − Tw = 2qd hδ π D
2
+ (Tv − Tw )
qd Dmin + 2kπ D D
(7.26)
The heat flux through a single droplet can then be given by rearranging eq. (7.26), i.e., π D2 (1 − Dmin / D ) qd = (7.27) ΔTtotal
2
(1 / hδ
+ D / 4k )
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 qd 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 − Dmin / D ) ′′ q′′ = nD D 2 dD (7.28)
2
Dmin
(1 / hδ + D / 4k )
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 = nD D 2 dD (7.29)
2
Dmin
(1 / hδ + D / 4k )
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
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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 Rcons = 3π k
Dmax
Dmin
′′ nD D 2 dD [1 − f ( D)]
(7.30)
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
htotal 1 = + Rcons hdrops
−1
(7.31)
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 μm, 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 h= 255510 22 o C < Tsat < 100 o C Tsat > 100 o C
(7.32)
Example 7.1 On a clear winter night, dropwise condensation occurred on the inner surface of window glass at a temperature of 5 °C. The air 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
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dew point is the saturation temperature corresponding to pwater, i.e., Tsat = 22.1 °C. The latent heat of vaporization at this temperature is hv =2448.8 kJ/kg. Assuming eq. (7.32) 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
m ′′ =
q′′ h (Tsat − Tw ) 96276.4 × (22.1 − 5) = = = 0.672 kg/m 2 -s 3 hv hv 2448.8 × 10
7.3 Filmwise Condensation
7.3.1 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 7.14 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 / μ , 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 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
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Reδ ≈ 30
Reδ ≈ 1800
Figure 7.14 Flow regimes of film condensate on a vertical wall.
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.
7.3.2 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 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. 7.15, 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
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Figure 7.15 Physical model and coordinate system for condensation of a binary vapor mixture (Fujii, 1991; Reproduced with kind permission of Springer Science and Business Media).
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 , 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 , uδ , 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. 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 ( ρ ρv ). 8. The vapor mixture can be treated as an ideal gas, and the thermal diffusion is negligible.
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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:
∂u ∂v + =0 ∂x ∂y
u
(7.33) (7.34) (7.35)
∂u ∂u ∂ 2 u 1 dp + v = ν +g− 2 ∂x ∂y ρ dx ∂y
u
∂T ∂T ∂ 2T + v = α ∂x ∂y ∂y 2
For the vapor boundary layer, the continuity, momentum, energy, and species equations are:
∂uv ∂vv + =0 ∂x ∂y
uv uv ρ ∂uv ∂u ∂ 2 uv + vv v = ν v + g 1 − v∞ 2 ρv ∂x ∂y ∂y
(7.36) (7.37) (7.38) (7.39)
∂Tv ∂T ∂ 2Tv ∂ω ∂T + vv v = α v + Dc p12 1v v 2 ∂x ∂y ∂y ∂y ∂y
uv
∂ω1v ∂ω ∂ 2ω1v + vv 1v = D ∂x ∂y ∂y 2
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. (7.38) 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 p12 = c p1v − c p 2 v c p1vω1v + c p 2 vω2 v
=
c p1v − c p 2 v c pv
(7.40)
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This isobaric specific heat term is assumed to be constant even though it has a larger fluctuation than the other physical properties, because overall the second term on the right-hand side of eq. (7.38) is much smaller than the first. Other definitions of the terms in eqs. (7.33) – (7.39) are as follows, beginning with the mass fractions ω1v and ω2 v in the vapor phase: ρ ω1v = 1v (7.41) ρv ρ ω2 v = 2 v (7.42) ρv where ρ v = ρ1v + ρ 2 v (7.43) Therefore, it follows from eqs. (7.41) – (7.43) that a relationship between ω1v and ω2 v can be written as follows: ω1v + ω2 v = 1 (7.44) Note that only the continuity equation for component 1 of the vapor is given. However, it should be immediately recognized from eq. (7.44) 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:
Mω p1 = 1 + 1 2v p M 2ω1v
−1
(7.45)
−1
p2 M 2ω1v = 1 + p M1ω2 v
(7.46)
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: u = 0 , y = 0 (7.47) v = 0 , y = 0 (7.48) T = Tw , y = 0 (7.49) 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 → ∞ (7.50) Tv = Tv∞ , y → ∞ (7.51) ω1v = ω1v∞ , y → ∞ (7.52) 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: uδ = uvδ = uδ (7.53)
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∂u ∂uv μ = μv ∂y δ ∂y δ dδ dδ ′′ ′′ ρ u dx − v = ρ v uv dx − vv = m′′ = m1 + m2 δ δ Tδ = Tvδ = Tδ ∂T ∂Tv k = hv m′′ + kv ∂y δ ∂y δ ω1v = ω1vδ
(7.54) (7.55) (7.56) (7.57)
(7.58) where μ is the dynamic viscosity, k is the thermal conductivity and hv 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 (7.53), (7.56), and (7.58) are the boundary conditions that refer to the continuity of velocity, temperature, and mass fraction at the liquid-vapor interface, respectively. Equation (7.54) recognizes that the shear stresses in the liquid and vapor layers are equal at the liquid-vapor interface. Equation (7.57) 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 (7.55) is the overall condensation mass continuity at the interface. The mass flux of the ′′ noncondensable component 2, m2 x , at the interface in eq. (7.55), 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 ) (7.59)
∂y
′′ m2 = − ρ D21
∂ω2 ′′ ′′ + ω2 (m1 + m2 ) ∂y ∂c1 ′′ + x1nT ∂y ∂c2 ′′ + x 2 nT ∂y
(7.60)
and the molar fluxes of the vapor in the binary vapor system are [see eq. (1.101)]
′′ n1 = − D12 ′′ n2 = − D21
(7.61) (7.62)
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. (7.45) and (7.46). nT = n1 + n2 is the total molar flux of component 1 and 2. Equations (7.61) and (7.62) are often expressed in terms of partial pressure, i.e.,
′′ n1 = − ′′ n2 = −
p D ∂p1 ′′ + nT 1 Ru T ∂y p p D ∂p2 ′′ + nT 2 Ru T ∂y p
(7.63) (7.64)
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where Ru is the universal gas constant. Equations (7.61) – (7.64) 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:
ω1 =
′′ m1x 1x + m2 x ′′ ′′ m
(7.65)
If the noncondensable component cannot be dissolved into the liquid film, ω1 will be unity. Effects of noncondensable gas on film condensation will be discussed in Section 7.3.7.
7.3.3 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. 7.16. 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.
Figure 7.16 Overview of the control volume under consideration in the Nusselt analysis.
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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. 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. (7.33). The momentum equation in the liquid phase is eq. (7.34), 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
dp dpv = = ρv g dx dx
(7.66)
Substituting eq. (7.66) into eq. (7.34), the momentum equation becomes
ρ u
∂u ∂u ∂2u + v = μ 2 + g( ρ − ρ v ) ∂y ∂y ∂x
(7.67)
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 ∂y
2
=
g
μ
( ρ v − ρ )
(7.68)
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:
u( x , y) =
( ρ − ρ v ) g y2 yδ − 2 μ
(7.69)
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.
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The mass flow rate per unit width of surface, Γ, 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 (7.70) Γ = ρ udy = 0 3μ 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:
(7.71) 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. 7.16 is given as
dq′ = k ΔT
q′′ =
k (Tsat − Tw )
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′ = hv dΓ (7.73) where dΓ is found by differentiating the expression for mass flow rate per unit surface, eq. (7.70). It is found to be ρ ( ρ − ρ v ) gδ 2 (7.74) dΓ = dδ μ This expression and the expression for heat flow dq′ across the control volume are then substituted into the energy equation, eq. (7.73), and the following is found: k μ ΔT dδ (7.75) = δ3 dx ρ ( ρ − ρ v ) ghv Finally, δ can be found by integrating the above and using the boundary condition that δ = 0 at x = 0:
δ =
4k μ x ΔT ρ ( ρ − ρ v ) ghv
1/ 4
δ
dx
(7.72)
(7.76)
The local heat transfer coefficient hx is found to be
k q′′ 3/ ρ ( ρ − ρ v ) ghv hx = = = k 4 Tsat − Tw δ 4 μ x ΔT
1/4
(7.77)
The local Nusselt number, Nux, of the laminar film condensation process on a vertical plate is found to be
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h x ρ ( ρ − ρ v ) ghv x 3 Nux = x = 4k μ ΔT k
1/4
(7.78)
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
h= 1 L
L
0
hx ( x ) dx
(7.79)
Substituting the expression for hx , eq. (7.77), into eq. (7.79) and integrating, the following expressions are obtained:
ρ ( ρ − ρ v ) ghv L3 hL Nu = = 0.943 μ k ΔT k
1/4
(7.80) also be
The local and average heat transfer coefficient can nondimensionalized in term of Reynolds number Reδ , defined as
Reδ = 4Γ
(7.81) μ Substituting eq. (7.70) into eq. (7.81), the Reynolds number for laminar film condensation becomes 4 ρ ( ρ − ρ ) gδ 3 (7.82) Reδ = 2 v 3μ Substituting eqs. (7.81) and (7.76) into eqs. (7.78) and (7.80), the following nondimensional correlations are obtained:
hx k
2 μ ρ ( ρ − ρ v ) g 2 μ ρ ( ρ − ρ v ) g 1/3 − = 1.1Reδ 1/3
(7.83) (7.84)
h k
1/3
− = 1.47 Reδ 1/3
where the bracket term on the left-hand side, 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′ k ΔT dΓ d = = hv + dx dx dx δ
δ
0
ρ c p u(Tsat − T ) dy
(7.85)
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. (7.69) and using a linear temperature profile,
Tsat − T y = 1− δ Tsat − Tw
(7.86)
to evaluate eq. (7.85), the energy balance can be written as
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k ΔT
where
δ
′ = hv
dΓ dx
(7.87)
3 c p (Tsat − Tw ) ′ hv = hv 1 + 8 hv
(7.88)
Equation (7.87) is identical to the energy balance in eq. (7.72) and (7.73) ′ except that hv has been replaced by hv . 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:
c p (Tsat − Tw ) ′ hv = hv 1 + 0.68 hv
(7.89)
Finally, if it is desirable to include the effects of vapor superheat in the above analysis, the latent heat obtained by eqs. (7.88) or (7.89) can be further modified by adding c pv (Tv − Tsat ) . 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 g cos θ . It follows from eqs. (7.77) and (7.80) that hx ∝ (cos θ )1/4 and h ∝ (cos θ )1/4 .
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. (7.82). While experimental observations of the condensate indicate that the laminar flow regime usually becomes wavy in the general
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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 Ar1/5 (7.90) where Archimedes number is defined as ρ σ 3/2 Ar = 2 1/2 (7.91) μ g ( ρ − ρ v )3/2 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:
h= k
2 (ν
/ g)
1/3
Reδ − 5.2
1.22
Reδ
, 30 ≤ Reδ ≤ 1800
(7.92)
To use eq. (7.92) 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. (7.82). The mass flow rate of the condensate can be determined by
Γ= q′ hL (Tsat − Tw ) = ′ ′ hv hv
(7.93)
Substituting eq. (7.93) into eq. (7.81), one obtains
Reδ = 4hL (Tsat − Tw ) ′ hvν
′ Reδ hvν 4 L (Tsat − Tw )
(7.94)
Rearranging eq. (7.94) yields
h=
(7.95)
Combining eq. (7.92) and (7.95) yields the Reynolds number for film condensation with waves as follows:
3.7 Lk (Tsat − Tw ) g Reδ = 4.81 + 2 ′ μ hv ν
1/3 0.82
(7.96)
After the Reynolds number is determined by eq. (7.96), the heat transfer coefficient can be determined by eq. (7.92).
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
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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:
hx k
2 μ ρ ( ρ − ρ v ) g 1/3 0.25 = 0.023Reδ Pr−0.5 , Pr ≥ 10
(7.97)
Chun and Seban (1971) obtained the following experimental correlation for the local turbulent heat transfer coefficient for evaporation of water from a vertical wall, which is applicable for local heat transfer coefficient for condensation:
hx k
2 ν g
1/3
= 3.8 × 10−3 Re0.4 Pr−0.65 , Re > 5800 Pr−1.06
(7.98)
Butterworth (1983) obtained the average heat transfer coefficient for film condensation that covers laminar, wavy laminar, and turbulent flow by combining eqs. (7.84), (7.92), and (7.97) as follows
h= k
2 (ν
/ g)
1/3
0.75 8750 + 58 Pr−0.5 (Reδ − 253)
Reδ
, 1 < Reδ ≥ 7200
(7.99)
The Reynolds number, Reδ , is needed in order to use eq. (7.99) to determine the heat transfer coefficient for turbulent film condensation. Equation (7.95) was obtained by energy balance and it is valid for all film condensation regimes. Combining eqs. (7.99) and (7.95), the Reynolds number for turbulent flow is obtained:
0.069 Lk Pr 0.5 (T − T ) g sat w Reδ = 2 ν ′ μ hv
1/3
− 151Pr0.5 + 253
4 /3
(7.100)
which can be used together with eq. (7.99) to determine the heat transfer coefficient for turbulent film condensation.
Example 7.2 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 ˚C. What are the average heat transfer and condensation rates?
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Solution: The saturation temperature of steam at 1 atm is Tsat =100 ˚C. The vapor density at this temperature is ρ v = 0.5974 kg/m3 , and the latent heat of vaporization is hv = 2251.2kJ/kg . The liquid properties evaluated at T f = (Tsat + Tw ) / 2 = 90 ˚C are
ρ = 965.3 kg/m3 ,
c p = 4.206 kJ/kg-K ,
μ = 0.315 × 10−3 kg/m-s,
k = 0.675 W/m-K , and ν = μ / ρ = 0.326 × 10−6 m 2 /s. .
The revised latent heat of vaporization is
c p (Tsat − Tw ) ' hv = hv 1 + 0.68 hv 4.206 × (100 − 80) = 2251.2 × 1 + 0.68 = 2308.4 kJ/kg 2251.2
Assuming the film condensation is laminar (as will be verified later), the heat transfer coefficient can be obtained from eq. (7.80), i.e.,
ρ ( ρ − ρ v ) g k3hv h = 0.943 μ ΔT L
1/4
965.3 × (965.3 − 0.5974) × 9.8 × 0.6753 × 2308.4 ×103 = 0.943 × 0.315 × 10−3 (100 − 80) × 1 = 5340.2W/m 2 -K
1/4
The heat transfer rate is then
q = hLb(Tsat − Tw ) = 5340.2 × 1× 1.5 × (100 − 80) = 1.602 × 105 W
The condensation rate is
m=
q 1.602 × 105 = = 0.0694 kg/s ′ hv 2308.4 × 103
The assumption of laminar film condensation is now checked by obtaining the Reynolds number defined in eq. (7.81), i.e.,
μ 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. (7.96), i.e.,
3.7 Lk (Tsat − Tw ) g Reδ = 4.81 + 2 ν ′ μ hv
1/3 0.82
Reδ =
4Γ
=
4m 4 × 0.0694 = = 588 μ b 0.315 × 10−3 × 1.5
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1/3 3.7 × 1× 0.675 × (100 − 80) 9.8 = 4.81 + 0.315 × 10−3 × 2308.4 ×103 (0.326 × 10−6 ) 2 = 730.9
0.82
which confirms that the film condensation is in the wavy regime. The heat transfer coefficient is obtained from eq. (7.92), i.e.,
h= = k
2 (ν
/ g)
1/3
Reδ − 5.2 730.9
1/3
1.22
Reδ = 7160 W/m 2 -K
0.675 [(0.326 × 10 ) / 9.8]
−6 2
730.9
1.22
− 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
m=
q 2.15 × 105 = = 0.0931 kg/s ′ hv 2308.4 × 103
which is much higher than the condensation rate obtained by assuming laminar film condensation.
7.3.4 Effects of Vapor Motion
Laminar Condensate Flow
It was assumed in Section 7.3.3 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. 7.13, the shear stress at the y = δ location of the interfacial control volume will now have a finite real value. The boundary layer momentum equation for liquid is once again given as
ρ u
∂u dp ∂u ∂2u + v = − + μ 2 + ρ g dx ∂y ∂y ∂x
(7.101)
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.,
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dp dp = ρv g + dx dx v
(7.102)
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, ρ v , denoted as
* ρv = ρv + g dx v Substituting eq. (7.103) into eqs. (7.102) and (7.101) produces
1 dp
(7.103)
ρ u
∂u ∂u ∂2u * + v = μ 2 + g( ρ − ρ v ) ∂y ∂y ∂x
(7.104)
The left-hand side of eq. (7.104) 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 ∂y
2
=
g
μ
* ( ρ v − ρ )
(7.105)
Integrating eq. (7.105) 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 τδ y u( x , y) = (7.106) yδ − + 2 μ μ 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 − ρ v ) gδ 3 τ δ ρδ 2 (7.107) Γ = ρ udy = + 0 3μ 2 μ 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. (7.107) 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:
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(7.108) δ where Tsat and Tw are the saturation temperature and wall temperature respectively (ΔT = Tsat – Tw). Rewriting eq. (7.108) in terms of heat transfer rate for the control volume shown in Fig. 7.16, the heat flow is given as
dx (7.109) δ Since no subcooling of the liquid film exists, the latent heat effects of condensation dominate the process. Thus dq′ = hv dΓ (7.110) where dΓ 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 ρδ dΓ = + (7.111) dδ μ μ dq′ = k ΔT
q′′ =
k ( Tsat − Tw )
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: k μ ΔT dδ (7.112) = * dx ρ ρ − ρ v ghvδ 3 + τ δ ρ hvδ 2
(
)
Finally, δ can be found by integrating the above and using the boundary condition that δ = 0 at x = 0.
δ4 +
* 3 ρ − ρ v
(
4τ δ δ 3
)
4k μ x ΔT = * g ρ ρ − ρ v ghv
(
)
(7.113)
This equation can be nondimensionalized by using the following dimensionless numbers (Rohsenow et al., 1956): δ (7.114) δ* =
LF x 4c p ΔT x* = LF Pr hv
* τδ = * LF ρ − ρ v g
(7.115) (7.116)
(
τδ
)
where
2 μ LF = ρ ρ − ρ* g v
1/3
(
)
(7.117)
is the characteristic length of the non-dimensional problem. Equation (7.113), which relates film thickness to the vertical location on the plate, can be rewritten as follows:
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x * = (δ * ) 4 +
4 *3 * δ τδ 3
()
(7.118)
It directly follows that the mean Nusselt number and Reynolds number for laminar flow with finite vapor shear can be expressed as follows:
h Nu = L k
1/3 * * 2 2 δ * τδ μ 4δ + = * 3 x* x* ρ ( ρ − ρ v ) g 2* 4Γ 4 * 3 Reδ = =δ + 2 δ * τδ μ 3
()
3
()
2
(7.119) (7.120)
()
()
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. 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 (7.121) x where the modified local Nusselt number is defined as
Nu* x
2 h μ = x k ρ ( ρ − ρ v ) g 1/3
(7.122)
and the dimensionless interfacial shear stress is ρτ δ + τδ = 2/3 ρ ( ρ − ρ v ) μ g
(7.123)
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 (7.124) where hgrav is the heat transfer coefficient for gravity-dominated film condensation – determined with eqs. (7.78) or (7.83) – and hshear is the heat transfer coefficient for shear-dominated film condensation, eq. (7.121). 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 7.22).
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Turbulent Condensate Flow
The velocity and pressure in turbulent flow experience large fluctuations and they can be expressed as (7.125) u = u + u′ v = v + v′ (7.126) p = p + p′ (7.127) 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 ∂u 1 ∂p 1 ∂ ∂u +v =− + − ρ u′v′ μ ρ ∂x ρ ∂y ∂y ∂x ∂y
(7.128)
where the bar notation denotes the mean value averaged over time and the prime notation denotes velocity fluctuations. Introducing the following notation:
− ρ u′v′ = ρε ∂u ∂y
(7.129)
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. (7.128) 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:
τ app = μ
∂u − ρ u′v′ ∂y
(7.130)
Substituting eq. (7.129) into eq. (7.130) gives
τ app = μ
∂u ∂u ∂u + ρε = ρ (ν + ε ) ∂y ∂y ∂y
(7.131)
which is the shear stress expression for turbulent flow consisting of both laminar and turbulent portions. If the following is defined ε (7.132) ε + = 1+ ν eq. (7.131) can be simplified to
τ app = ρε +ν
∂u ∂y
(7.133)
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. (7.133) 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 7.17 was adapted from Rohsenow et al. (1956) and shows the variation of the average film condensation heat transfer coefficient with Reynolds number
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Reδ,tr
* τ δ = 0, 2.5, 5, 10, 20, 50
Transition points
Reδ
Figure 7.17 Variation of the mean film condensation heat transfer coefficient with Reynolds * number and τ δ as predicted by Rohsenow et al. (1956).
* 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. 7.17 were presented by Rohsenow et al. (1956) as the following (marked on the plot as solid circles):
ρ ρ * * Reδ ,tr = 1800 − 246 1 − v τ δ + 0.667 1 + v τ δ ρ ρ
1/3
()
3
(7.134)
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
h=
1/2 0.065 Pr
(τ δ )
* 1/2
g k 2 v
1/3
(7.135)
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).
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7.3.5 Turbulent Film Condensation
We will now consider turbulent flow for a specific application – turbulent condensate flow in a circular tube, as shown in Fig. 7.18. 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. 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. 7.18 is considered. For the case of countercurrent flow, a force balance results in
dp p π r 2 + ρ v gVv + ρ gV = p + Δx π r 2 + τ ( 2π r Δx ) dx
()
(7.136)
where the volume of the vapor and the liquid portion of the control volume are as follows: 2 (7.137) Vv = ( R − δ ) πΔx
V = ( R − y ) π Δx − Vv
2
(7.138)
Figure 7.18 Physical model of the condensation phenomena in contact with flowing vapor.
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Substituting eqs. (7.137) and (7.138) into eq. (7.136) and dividing through by
Δx to solve for the shear stress, the following is obtained:
τ=
(R −δ ) R−y dp ρ g − − ( ρ − ρ v ) g dx 2 2( R − y)
2
(7.139)
The shear stress at the liquid-vapor interface can be found by letting y = δ in eq. (7.139), i.e., R −δ dp τδ = (7.140) ρv g −
2 dx
Substituting eq. (7.140) into eq. (7.139), the shear stress at any radius, τ , can be related to the shear stress at the liquid film surface, τ δ , by
τ=
2 R(δ − y) − δ 2 + y 2 R−y τ δ + ( ρ − ρ v ) g R −δ 2( R − y)
(7.141)
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. (7.140) reduces to
R dp (7.142) ρv g − dx 2 If it is further assumed that ρ ρv , eq. (7.141) would reduce to
τδ =
(7.143) which can be written into a generalized form that includes the case of cocurrent flow, i.e., τ = ±τ δ + ρ g (δ − y ) (7.144) 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 + ρ gδ (7.145) The velocity profile in the liquid film can be found from the following differential equation when all axial terms and the curvature are neglected:
d du ( v + ε m ) + g = 0 dy dy
τ = τ δ + ρ g (δ − y )
(7.146)
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 (7.147)
−μ ∂u = ±τ δ , ∂y y =δ
(7.148)
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Integrating eq. (7.146) twice with respect to y and applying boundary conditions specified by eqs. (7.147) and (7.148), the velocity profile in the liquid film is obtained: y [ g(δ − y) ± τ / ρ ] δ (7.149) u= dy 0 v + ε m The liquid Reynolds number is obtained by the following expression:
(7.150) 0 μ μ To generalize the problem statement, the following nondimensional variables are defined:
Reδ =
4Γ
= 4 ρ
δ
udy
δ+ =
δuf ν
ν 2 , δ* = δ g
−1/3
, y+ =
yu f
ν
* , τδ =
τ δ (ν g) −2 /3 ρ
xu f Du f ε u + u+ = u , x + = , ε m = 1 + m , D+ = ν v v f
(7.151)
where u f is the fractional velocity, defined as
τ uf = w ρ
1/ 2
(7.152)
Applying these nondimensional variables to eqs. (7.145), (7.149), and (7.150), their nondimensional forms are obtained as follows: 2/3 + u3 τ δ u f ( v g ) − ( v g ) δ + = 0 (7.153) f
u=
+
y+
(1 − gν y
+
/ u3 f
0
+ εm
) dy
+
(7.154) (7.155)
Reδ = 4
δ
0
u + dy +
+ It should be noted that if τ δ = 0 and u3 = ν gδ + then eq. (7.153) would reduce to f
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 / μ ) ′′ ρ c p + m (7.156) = qw = μ hv
Pr
Prt dy dx
where Prt is the turbulent Prandtl number, which will be discussed thoroughly towards the end of this subsection. Equation (7.156) can be nondimensionalized to obtain
d (Γ / μ ) dx +
= NT
δ+
0
−1 Pr Prt + + Pr − 1 + ε m dy Prt
−1
(7.157)
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where
NT =
(Tsat − Tw ) c p
hv Pr
(7.158)
Finally, the local heat transfer coefficient can be found directly from eq. (7.156):
′′ qw = hx = Tsat − Tw
δ
0
−1 νε ρ c + m dy p Pr Prt
−1
(7.159)
Nondimensionalizing eq. (7.159) as a Nusselt number, the following is obtained:
hδ Nux = x = δ + k
δ+
0
−1 Pr Prt + − 1 + ε m dy + Pr Prt
−1
(7.160)
An average heat transfer coefficient is desirable in many practical + applications. A modified Nusselt number Nux is related to Nux by (7.161) δ The average modified Nusselt number is then found from the following relation:
= Nux
+ + Nux =
hx v k g
1/3
uf
(ν g )−1/3
Nu + =
x+ Nu + d + L 0
1
(7.162)
The dimensionless shear stress at the interface can be written as f ρ + d (Γ / μ ) * + + + τ δ = (ν g)−2/3 u 2 (uv + u,δ ) E v (uv + u,δ ) + f dx + 2 ρ
(7.163)
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.,
f [1 + 0.045( Mg+ − 5.9)] Reδ ≤ 75 fE = −0.2 + f [1 + 0.045 Re v ( Mg − 5.9)] Reδ > 75
(7.164)
where
1/2 0.6 ν ρ τ δ 0.78 Reδ ν v ρv τ c + Mg = 1/2 0.7 ν ρ τ δ 0.50 Reδ ν v ρv τ c
Reδ ≤ 75
(7.165)
Reδ > 75
The characteristic stress is given by
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τδ 1 2 τ w = + (7.166) τ c 3 3 τδ 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: 2 2 y+ τ 1 1 τ + +2 τ ε m = + 1 + 0.64y 1 − exp − + exp −1.66 1 − 2 2 τw τ w A τ w (7.167) where A+ = 25.1 and τ / τ w = 1 − y + ( gν ) / u3 . f
−1
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:
Prt = 1 − exp(− y + / A + ) 1 − exp − y+ Pr / B+
(
)
(7.168)
where
B+ =
c ( log
i i =1
5
10
Pr )
i −1
(7.169)
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. (7.153). The dimensionless eddy diffusivity,
+ ε m , is obtained from eq. (7.153). Equations (7.154) and (7.155) 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. (7.163). The process is repeated until the Re values between two consecutive iterations differ by less than 0.5%. The local convective heat transfer coefficient can be obtained from eqs. (7.161) and (7.162). The above procedure can be repeated for different x until heat transfer coefficients are obtained at all locations.
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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.
7.3.6 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 cos θ :
ρ ( ρ − ρ v ) g cos θ hv L3 ′ hL L Nu = = 0.943 k μ k (Tsat − Tw )
1/4
(7.170)
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.,
D3hv g ( ρ − ρ v ) hD D Nu = = 0.729 k kν (Tsat − Tw )
1/4
(7.171)
which can be obtained by applying Nusselt analysis (see Problem 7.11). For laminar film condensation on a sphere, the average heat transfer coefficient can also be obtained by Nusselt analysis (see Problem 7.14).
D3hv g ( ρ − ρ v ) ′ hD D Nu = = 0.815 k kν (Tsat − Tw )
1/4
(7.172)
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:
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D 3hv g ( ρ − ρ v ) ′ h D ,n D Nu = = 0.729 k nkν (Tsat − Tw )
1/4
(7.173)
which indicates that the average heat transfer coefficient for all n cylinders, hD ,n , is related to the average heat transfer coefficient for the first tube of this array, hD , by
h D ,n = hD n1/4
(7.174)
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 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
ρ ( ρ − ρ v ) ghv L3 ′ hL Nu = = 1.079 k μ k ( Tsat − Tw )
1/5
(7.175)
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
ρ ( ρ − ρ v ) ghv D 3 ′ hD Nu D = = 1.368 k μ k ( Tsat − Tw )
1/5
(7.176)
Example 7.3 Saturated acetone at Tsat = 60 ˚C 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 ˚C. The vapor properties are evaluated at saturation temperature Tsat = 60 ˚C. The vapor density is
ρ v = 2.37 kg/m3 , and the latent heat of vaporization is hv = 517 kJ/kg .
The
liquid
properties
are
ρ = 756.0 kg/m3 ,
c p = 2255 J/kg-K ,
μ = 0.248 × 10−3 kg/m-s, k = 0.172 W/m-K , and ν = 0.328 ×10−6 m 2 /s. Find the heat transfer coefficient and the rate of condensation per unit length of the tube. Solution: The revised latent heat of vaporization is
c p (Tsat − Tw ) ′ hv = hv 1 + 0.68 hv 2.255 × (60 − 40) = 517 × 1 + 0.68 = 547.7 kJ/kg 517
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The heat transfer coefficient can be obtained from eq. (7.171), i.e.,
( ρ − ρ v ) g k3hv ′ h = 0.729 ν ΔTD
1/4
(756.0 − 2.37) × 9.8 × 0.1723 × 547.7 × 103 = 0.729 × 0.328 ×10−6 (60 − 40) × 0.03 = 2331.3 W/m 2 -K
1/4
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
m′ =
q′ 4394.4 = = 0.00802 kg/s-m ′ hv 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 ′ ghv μ D hD 1/2 = 0.64 Re D 1 + 1 + 1.69 2 Nu = k u∞ k ( Tsat − Tw ) 1/2
(7.177)
where ReD= u∞D/ν is based on the viscosity of liquid. Equation (7.177) 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 Pr ρ μ hL 1.508 = 0.872 Re1/2 + v v Nu = L k (1 + Ja / Pr )3/2 Ja ρ μ 1/3
(7.178)
where L is the length of the flat plate along the vapor flow direction, ReL = u∞L/ν is based on the viscosity of liquid, and Ja = c p (Tsat − Tw ) / hv is the Jakob number. Equation (7.178) is valid for Ja / Pr = 0.01 1 .
ρ μ / ρ v μv = 10 500
and
7.3.7 Effects of Noncondensable Gas
The physical model and governing equations for binary vapor (condensable vapor and noncondensable gas) have been discussed in Section 7.3.2. It can be seen from Fig. 7.15 that heat and mass transfer in the vapor phase must be
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y
Vapor
ωv ,∞ , pv ,∞
T∞
Tδ ′′ mv
δv
Tsat ( pv ,∞ )
liquid
δ
ωv ,δ
pv ,δ
x
Figure 7.19 Mass transfer in the equivalent laminar film.
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 = δ, 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 interface. At any point in space and time the summation of the partial pressures of this binary system must equal the constant total pressure. (7.179) pv + pg = p 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 7.3.2. 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, ωv, varies linearly from ωv,δ at the liquid-vapor interface
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to ωv,∞ at y = δ + δ v (see Fig. 7.19). 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. (7.61)]
′′ nv = − Dvg
∂cv ′′ + nv x v ∂y
Dvg
(7.180)
which can be rearranged to obtain
′′ nv = −
∂cv 1 − cv / cT ∂y
(7.181)
where cT is total molar concentration of vapor and noncondensable gas. Integrating eq. (7.181) over the equivalent laminar layer and considering ′′ nv and cT are constants, one obtains
′′ nv
δ
δ +δ v
dy = − Dvg cT
cv ,∞
cv ,δ
1 dcv cT − cv
(7.182)
Equation (7.182) can be rearranged to yield
′′ nv =
cT Dvg
δv
cT − cv,δ ln c −c v,∞ T
cT − cv,δ = cT hm ,G ln c −c v,∞ T
(7.183)
where (7.184) δv is the 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.,
hm ,G =
hm ,G =
Dvg
1
hG
ρ vg c p,vg
(7.185)
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. (7.183) can also be expressed in terms of mass flux, mv , and partial pressure of the condensable vapor, pv, in the mixture (see Problem 7.27)
′′ mv = ρ vg hm ,G ln
p − pv,δ p − pv,∞
(7.186)
The energy balance across a differential control volume at the liquid-vapor interface, as shown in Fig. 7.20, is (Stephan, 1992) ′′ (7.187) h (Tδ − Tw ) = mv hv + hG (Tvg − Tδ ) 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.
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Condensate liquid
Vapor
′′ mv hv
h (Tδ − Tw )
hG (Tvg − Tδ )
Interface Figure 7.20 Energy balance at the liquid-vapor interface for film condensation on a vertical plate including the effects of non-condensable gases.
Substituting eqs. (7.185) and (7.186) into the energy balance equation (7.187) 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):
Tδ − Tw = hG h
hv p − pv,δ ln + ξ Tvg − Tδ c p,vg p − pv,res
(
)
(7.188)
In the case of small inert gas content, the above equation will reduce to
Tδ − Tw = hG h
h p − pv,δ v ln c p, vg p − pv,∞
(7.189)
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: (7.190) (T − Tw ) δ sat However, if a noncondensable gas is present, the temperature drop across the film would be lower and the heat flux would be as follows:
q′′ = k
′′ where qvg
(7.191) (T − Tw ) δδ is the heat flux across the liquid film in the presence of a
′′ qvg =
k
noncondensable gas. The ratio of the heat fluxes obtained by eqs. (7.191) and (7.190) is
′′ qvg
q′′
=
Tδ − Tw ≤1 Tsat − Tw
(7.192)
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Substituting the expression for temperature difference across a liquid film in the presence of a noncondensable gas – eq. (7.189) – into the above ratio, the following is obtained:
′′ qvg
q′′
=
(Tsat
′′ qvg
q′′
p − pv,δ hG hv ln − Tw ) h cvg,∞ p − pv,∞
(7.193)
or, substituting mv from eq. (7.186):
=
′′ mv hv (Tsat − Tw )h
(7.194)
It can be seen from the above expression that for large (Tsat – Tw), the velocity or mass flow rate mv , 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 7.21 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.
′′ qvg q ′′
Tvg
Mass fraction of noncondensable gas
Figure 7.21 Falling film condensation of steam with non-condensable gas (Collier and Thome, 1996; Reprinted with permission from Oxford University Press).
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Example 7.4 A mixture of 20% steam and 80% air at 100 ˚C 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
ρ vg = 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 the liquid-vapor interface? If the temperature of the tube is 80 ˚C, 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 mass of water and air are Mv = 18.02 kg/kmol and Mg = 28.96 kg/kmol. The partial pressure of the steam can be obtained from eq. (7.45), i.e.,
pv,∞ M v (1 − ωv,∞ ) = p 1 + M gω v , ∞
−1
18.02 × (1 − 0.9) = 1.013 × 105 × 1 + 28.96 × 0.9
−1
= 0.9475 ×105 kPa
The Reynolds number of the mixture is ρ vg u∞ D 0.944 × 30 × 0.1 = = 34536 Re D = μvg 8.2 × 10−6 The Schmidt number of the mixture is μvg 8.2 × 10−6 = = 0.239 Sc = ρ vg Dvg 0.944 × 3.64 × 10−5 The empirical correlation for forced convective heat transfer across a cylinder is (see Table 1.9)
Nu D = 0.3 + 0.62 Re1/2 Pr1/3 D [1 + (0.4 / Pr) 2/3 ]1/4 0.62 Re1/2 Sc1/3 D [1 + (0.4 / Sc) 2/3 ]1/4 Re 5/8 D × 1 + 282000 Re 5/8 D 1 + 282000
4 /5
Analogy between mass and heat transfer gives us
4 /5
Sh D = 0.3 +
5/8 0.62 × 345361/2 × 0.2391/3 34536 = 0.3 + 1 + [1 + (0.4 / 0.239) 2/3 ]1/4 282000
4 /5
= 108.1
The mass transfer coefficient is therefore
hm ,G = Sh D Dvg D
=
108.1× 3.64 × 10−5 = 0.03935 m/s 0.1
The partial pressure of the vapor at the liquid-vapor interface, pv,δ , can be obtained from eq. (7.186)
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mv ′′ pv,δ = p − ( p − pv,∞ ) exp ρ vg hm,G
0.02 = 1.013 × 105 − (1.013 × 105 − 0.9475 × 105 ) exp 0.944 × 0.03935 = 0.900 × 105 Pa
The interfacial temperature, Tδ, is the saturation temperature corresponding to the above partial pressure. Assuming that the vapor behaviors like ideal gas, the Clapeyron-Clausius equation (2.168) can be used to obtain
ln 1 1 (7.195) − p Tδ Tsat = 2251.2 kJ/kg and Rg = 0.4615 kJ/kg-K for water at 1 atm. pv,δ =− hv Rg
−1
where
hv
Equation (7.195) can be rearranged to obtain
Rg pv,δ 1 ln − Tδ = p Tsat hv
1 0.4615 0.900 × 105 = − 373.15 2251.2 ln 1.013 × 105
−1
= 369.80K = 96.65 C
The ratio of heat fluxes with and without noncondensable gas can be obtained from eq. (7.192)
′′ qvg
q′′
=
Tδ − Tw 96.65 − 80.0 = = 0.8325 Tsat − Tw 100.0 − 80.0
In other words, the heat transfer is decreased by 16.75% due to the presence of noncondensable gas.
7.4 Falling Film Evaporation on a Heated Wall and Spray Cooling
As mentioned previously, falling film evaporators are used in both mass transfer and heat transfer applications. In general, the film’s mass flow rate decreases as evaporation occurs. If the film is being concentrated (as in fruit juice concentration), evaporation is also accompanied by an increase in viscosity. With the decrease in mass flow rate and the corresponding increase in viscosity, the Reynolds number decreases and the flow tends to progress from turbulent (if that ever existed) to wavy laminar, and then to laminar flow.
7.4.1 Classical Nusselt Evaporation
The simplest evaporation heat transfer coefficient calculation uses the Nusselt treatment of falling film evaporation, as shown in Fig. 7.22. A flat plate with constant temperature Tw is placed vertically and a liquid film flows downward due to gravity. Evaporation takes place on the surface of the liquid
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Γ0, Re0 y, v
Vapor (v)
Liquid ( )
Tw
Tv g
δ
ΓL, ReL
Figure 7.22 Schematic of Nusselt evaporation.
x, u
film because the temperature of the vertical plate is above the saturation temperature of the vapor corresponding to the partial pressure of the vapor in the mixture. Since the heat must be transferred from the heat source (vertical wall) to the liquid-vapor interface, where evaporation takes place, this is an example of heterogeneous evaporation. The following assumptions are made: 1. The vapor is quiescent. 2. Inertia in the laminar liquid film is negligible. 3. Heat transfer through the liquid film is by conduction only, i.e., convective terms in the energy equation are negligible. 4. Constant wall temperature. 5. The process occurs at steady state. 6. A nonslip condition exists at the wall. Proceeding from the assumptions of negligible inertia and constant fluid properties, the momentum balance is
μ
d 2 u dy 2 = − ( ρ − ρv ) g
(7.196)
Integrating eq. (7.196) twice, with respect to y and using the boundary conditions of non-slip at the wall (u = 0 at y = 0) and zero shear at the liquidvapor interface ( ∂u / ∂y = 0 at y = δ), the following velocity profile is obtained:
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u =
( ρ − ρ v ) g
μ
y2 δ y − 2
(7.197)
The mass flow rate per unit width is defined by integrating the velocity profile from the wall to the interface.
Γ=
δ
0
ρ udy
(7.198)
The liquid velocity profile, eq. (7.197), is substituted into eq. (7.198) and the result is integrated from the wall to the interface. The result is then rearranged to obtain liquid film thickness:
3μ Γ δ = ρ ( ρ − ρ v ) g
13
(7.199)
Since heat transfer through the film is by conduction only, the heat transfer coefficient is hx = k / δ . Considering eq. (7.199), the local heat transfer coefficient becomes
ρ ( ρ − ρv ) g hx = k 3μ Γ
13
(7.200)
Note that the film thickness δ drops out of the equation, and the heat transfer coefficient h is now a function of the mass flow rate per unit width, Γ . The local energy balance across the liquid film is
hv dΓ = −hx ( Tw − Tv ) dx
13
(7.201)
Combining eqs. (7.200) and (7.201), one obtains,
Γ1 3dΓ = − k ρ ( ρ − ρ v ) g 3μ hv
(Tw − Tv ) dx
13
(7.202)
Integrating from the inlet to some arbitrary length L,
ΓL
Γo
Γ1 3dΓ = −
L
0
k ρ ( ρ − ρ v ) g 3μ hv
(Tw − Tv ) dx
(7.203)
A relation for Γ at outlet is obtained in terms of the inlet mass flow rate, Γ 0 , for the case of constant wall and vapor temperatures:
4 Γ 4 3 = Γo 3 − L
4 k ρ ( ρ − ρ v ) g 3 hv 3μ Re = 4Γ
13
(Tw − Tv ) L
(7.204)
Now the mass flow rate per unit width is related to the Reynolds number by (7.205) μ The local heat transfer coefficient from eq. (7.200) can be expressed in terms of Reynolds number as follows:
hx k
2 μ ρ ( ρ − ρ v ) g 13
4 = 3
13
1 Re1 3
= 1.10 Re −1/3
(7.206)
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Equation (7.206) for local heat transfer coefficient is valid for both laminar condensation and evaporation at both constant wall temperature and constant heat flux at the wall. To calculate the average heat transfer coefficient, h , the energy flux over the entire length 0 < x < L is balanced with the mass flux from the interface: _ h ( Γ − Γ L ) μ hv ( Re0 − Re L ) (7.207) h = v 0 =
L ( Tw − Tv ) 4 L ( Tw − Tv )
Equation (7.207) is valid for laminar, wavy laminar, and turbulent with constant wall temperature. Equation (7.201), which reflects the local energy balance across the liquid film, can be written in terms of Reynolds number as
4 ( Tw − Tv ) d Re =− dx μ hv hx
(7.208)
Integrating eq. (7.208) in the interval of (0, L), one obtains
Re L
Re0
4 ( Tw − Tv ) L d Re =− μ hv hx
(7.209)
Combining eqs. (7.207) and (7.209), the average heat transfer coefficient is expressed as
h=−
_
( Re0 − Re L )
Re L Re0
1 d Re hx
(7.210)
Equation (7.210) for h is valid for laminar, wavy laminar, and turbulent flow. Substituting eq. (7.206) into eq. (7.210), an expression for the average heat transfer coefficient for laminar flow in terms of Reynolds number is obtained:
h k
_ 2 μ ρ ( ρ − ρ v ) g 13
4 = 3
43
( Re
( Reo − Re L )
43 43 o − Re L
)
13
(7.211)
In order to use eq. (7.211) to obtain the average heat transfer coefficient, it is necessary to know Reynolds number at x=L, Re L . It can be obtained by rewriting the relation for Γ, eq. (7.204), in terms of Reynolds number, as follows:
4 4 Re4 3 = Reo 3 − 4 L 3
43
k ( Tw − Tv ) L ρ ( ρ − ρ v ) g 4 3μ μ 3hv
(7.212)
The above analysis is valid for falling film evaporation on a vertical wall maintained at a constant temperature. Falling film evaporation from a vertical wall with constant heat flux is discussed in Example 7.5.
Example 7.5 A liquid film flows downward and evaporates on a vertical plate subjected to a constant heat flux. Determine the average heat transfer coefficient.
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Solution: The physical model of the problem under consideration is similar to that shown in Fig. 7.28 except that the thermal boundary condition is changed. The local heat transfer coefficient for falling film evaporation can still be calculated by eq. (7.200) or (7.206), because no assumption about boundary conditions were used to obtain them. For the case of constant heat flux, the energy balance accounts for the heat addition required for evaporation, i.e.,
hv dΓ ′′ = −qw dx
(7.213)
′′ Since qw is not a function of x, eq. (7.213) can be integrated in the interval of (0, L) to yield Γ = Γ0 − ′′ qw L hv ′′ 4qw L μ hv
(7.214)
Equation (7.214) can be rewritten in terms of Reynolds number as
Re L = Re0 −
(7.215)
which can be used to obtain the Reynolds number at x=L. The average heat transfer coefficient can be obtained by _ h ( Γ − Γ L ) μ hv ( Re0 − Re L ) (7.216) h = v 0 =
L ( Tw − Tv ) 4 L ( Tw − Tv )
which is similar to eq. (7.207) except that the wall temperature is the average value along the wall (0<x<L), i.e.,
Tw =
L
0
Tw dx
(7.217)
For the case with constant heat flux, the energy balance across the liquid film in terms of Reynolds number can still be represented by eq. (7.208). Integrating eq. (7.208) in the interval of (0, L) and considering the fact that Tw is not a constant, one obtains
Re L
Re0
4 ( Tw − Tv ) L d Re =− μ hv hx
(7.218)
Combining eqs. (7.216) and (7.218) to eliminate (Tw − Tv ) yields the same equation as eq. (7.210), which eventually leads to eq. (7.211). Therefore, eq. (7.211) is also valid for the case with constant heat flux heating, while the Reynolds number at x = L must be calculated by eq. (7.215).
7.4.2 Laminar Falling Film with Surface Waves
Wavy flows of thin liquid films have higher heat transfer coefficients than smooth thin films. This effect is due to the former’s greater interfacial surface area and mixing action. For practical purposes, Chun and Seban (1971) suggested
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that the following correlation based on wavy laminar film condensation is valid for wavy laminar falling film evaporation:
Re hx = 0.876 4
0.11
hNusselt
(7.219)
An empirical correlation for the local heat transfer coefficient for laminarwavy flow is obtained by combining eq. (7.219) with eq. (7.206),
hx k
2 μ ρ ( ρ − ρ v ) g 13
= 0.828 Re−0.22 ,
Re > Rewavy
(7.220)
Chun and Seban (1971) suggested the following empirical correlation to predict the onset of wavy laminar flow: Re ≥ Rewavy = 2.43Ka −1/11 (7.221) where Ka is Kapitza number, a ratio of surface tension to viscous force: 4 μ g Ka = (7.222) ( ρ − ρ v ) σ 3 The average heat transfer coefficient is often relevant to the practical application and can be obtained by using eq. (7.210), which is also valid for laminar flow with waves. Substituting eq. (7.220) into eq. (7.210), one obtains an empirical correlation of the average heat transfer coefficient as follows:
h k
_ 2 μ ρ ( ρ − ρ v ) g 13
=
( Re
( Reo − Re L )
1.22 1.22 o − Re L
)
(7.223)
Substituting eq. (7.207) into eq. (7.223), an equation correlating Re L , the Reynolds number at x = L, is obtained:
Re1.22 L
= kL Re1.22 − 4 0
(Tw − Tv ) ρ ( ρ − ρv ) g
μ hv
2 μ
13
(7.224)
Example 7.6 A uniformly-distributed liquid water film at 0.01 kg/s is introduced at the top of a tube with an inner diameter of 4 cm and a height of 2 m. The inner surface temperature of the tube is uniformly 105 °C and the saturation temperature of the steam is 100 °C. Find the evaporation rate and the average heat transfer coefficient. Solution: The properties of the liquid water and the steam can be approximated as those at the saturation temperature: k = 0.68 W/m- o C,
ρ = 958.77 kg/m3 , μ = 2.79 ×10−4 N-s/m 2 , σ = 58.9 × 10−3 N/m, ρ v =
0.596 kg/m3 , hv = 2251.2 kJ/kg. It is assumed that the liquid film
thickness is much smaller than the radius of the tube. Therefore, the falling film analysis for evaporation over a vertical flat plate can be used. The Kapitza number is obtained from eq. (7.222):
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Ka =
4 μ g (2.79 × 10−4 ) 4 × 9.8 = = 3.032 × 10−13 ( ρ − ρv ) σ 3 (958.77 − 0.596) × (58.9 ×10−3 )3
Therefore, the Reynolds number for wavy film is
Re wavy = 2.43Ka −1/11 = 2.43 × (3.03 × 10−13 ) −1/11 = 33.4
The inlet mass flow rate per unit perimeter is
Γ0 =
The Reynolds number at the inlet is
Re0 = 4Γ 0
=
m0 0.01 = = 7.95 × 10−2 kg/s-m π D π × 0.04
4 × 7.95 × 10−2
= 1139.8 > Re
wavy μ 2.79 × 10−4 which is also under the transition Reynolds number for turbulent, Returb = 5840 Pr −1.05 = 1862 [see eq. (7.225)]. This means that the liquid film is in the laminar wavy regime. The Reynolds number at outlet can be obtained from eq. (7.224), i.e.,
Re1.22 = Re1.22 − 4 L 0
= 1139.81.22 − 4 ×
k L ( Tw − Tv ) ρ ( ρ − ρ v ) g 2 μ hv μ
13
958.77 × ( 958.77 − 0.596 ) × 9.8 × 3 −4 2.79 × 10 × 2251.2 × 10 (2.79 × 10−4 ) 2
0.68 × 2 × (105 − 100)
13
Thus
= 3257.5
Re L = 757.6
which is still in the wavy regime so the entire surface is wavy. The mass flow rate of the liquid at the outlet is
mL = m0
Re L 757.6 = 0.01× = 6.64 × 10−3 Re0 1139.8
The evaporation rate is therefore
mv = m0 − mL = 0.01 − 6.64 ×10−3 = 3.36 ×10−3 kg / m
The heat transfer coefficient can be obtained from eq. (7.223), i.e.,
ρ (ρ − ρ )g h = k 2 v μ
_ 13
( Re
( Reo − Re L )
1.22 1.22 o − Re L
)
× 1139.8 − 757.6 1139.81.22 − 757.61.22
958.77 × ( 958.77 − 0.596 ) × 9.8 =0.68 × (2.79 × 10−4 ) 2 =6017 W/m 2 - o C
13
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7.4.3 Turbulent Falling Film
Stephan (1992) presented an empirical relationship between Reynolds number and Prandtl number that could be used to determine whether a falling film is completely turbulent: Re ≥ Returb = 5840 Pr −1.05 (7.225) To predict local heat transfer in turbulent falling film flow on a wall heated under constant heat flux, the following empirical correlation, recommended by Chun and Seban (1971), can be used:
h k
2 μ ρ ( ρ − ρ v ) g 13
= 3.8 × 10−3 Re0.4 Pr 0.65
(7.226)
The average heat transfer coefficient can be obtained by substituting eq. (7.226) into eq. (7.210), i.e.,
h k
_ 2 μ ρ ( ρ − ρv ) g 13
= 2.28 × 10−3
(
( Reo − Re L )
Re0.6 − Re0.6 o L
)
Pr 0.65
(7.227)
As was demonstrated in Example 7.5, eq. (7.227) is valid for cases with either constant wall temperature or constant heat flux. However, the methods for determining the Reynolds number at x = L will vary for different thermal boundary conditions on the wall. For cases with constant wall temperature, the Reynolds number at x = L, Re L , is obtained by substituting eq. (7.207) into eq. (7.227), i.e.,
Re0.6 = Re0.6 − 9.12 × 10−3 0 L k L ( Tw − Tv ) ρ ( ρ − ρ v ) g 2 μ hv μ
13
(7.228)
For the case with constant heat flux, eq. (7.215) can be used to determine the
Reynolds number at x = L, Re L .
While gravitational force drives the falling film evaporation discussed here, liquid flow in a liquid film can also be driven by centrifugal force. Rahman and Faghri (1992) analyzed the processes of heating and evaporation in a thin liquid film adjacent to a horizontal disk rotating about a vertical axis at a constant angular velocity. The fluid emanates axisymmetrically from a source at the center of the disk and then is carried downstream by inertial and centrifugal forces. Closed-form analytical solutions were derived for fully developed flow and heat transfer. Simplified analyses were also presented for developing heat transfer in a fully-developed flow field. Rice et al. (2005) provided a detailed analysis for evaporation from a thin liquid film on a rotating disk including conjugate effects.
7.4.4 Surface Spray Cooling
When liquid drops are sprayed onto a hot surface, heat conduction causes the liquid to superheat and evaporate into a gas (see Figure 7.9). The cooling capacity of the process depends on the time required for an average-sized drop to
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cool. If the shape of the droplet can be assumed to be hemispherical during the evaporation process, the energy balance at the interface can be written as π /2 ∂T (r , θ ) dr I ρ hv 2π rI2 I = k 2π rI2 cos θ dθ − h 2π rI2 ( T∞ − Tsat ) (7.229)
dt
0
0
∂r
where rI is the radius of the droplet. Equation (7.229) can be simplified as π /2 ∂T (r , θ ) dr I (7.230) ρ hv I = k cos θ dθ − h ( T∞ − Tsat )
dt
∂r
Analytical solution of eq. (7.230) is very difficult because the temperature distribution in the liquid droplet is two-dimensional in nature. A scale analysis similar to that in Lock (1994) is presented here to estimate the time required for a droplet with an initial radius of Ri to evaporate completely. The scale of the radius of the droplet, rI, is Ri. Thus the scale of the temperature gradient at the interface is ∂T (rI , θ ) Tw − Tsat (7.231)
∂r Ri
The scale analysis of eq. (7.230) yields
ρ hv
Ri T − Tsat k w + h(T∞ − Tsat ) tf Ri
(7.232)
which can be rearranged as ρ hv Ri2
k tf
(Tw − Tsat ) +
hRi (T∞ − Tsat ) k
(7.233)
Since the liquid droplet is very small, hRi / k is very small, so the second term on the right-hand side of eq. (7.232) is much smaller than the first. Equation (7.232) can be simplified by neglecting the second term on the right-hand side, i.e.,
ρ hv
Ri2 (Tw − Tsat ) k t f
(7.234)
The physical significance of eq. (7.234) is that the latent heat of evaporation is balanced primarily by conduction in the liquid droplet, while the effect of convection on the surface of the droplet is negligible. The time that it takes to completely evaporate the droplet with an initial radius of Ri is therefore estimated by ρ hv Ri2 (7.235) tf
k ( Tw − Tsat ) ′′ For the cases where the substrate heat flux beneath the droplet, qw , is known, eq. (7.234) should be rewritten in terms of heat flux. The scale of the heat flux at the heating surface is ′′ qw k ( Tw − Tsat ) Ri
(7.236)
Combining eqs. (7.235) and (7.236) yields
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tf
ρ hv Ri
′′ qw
(7.237)
which indicates that a smaller drop will provide a larger cooling effect. While eqs. (7.235) and (7.237) provide the order of magnitude of the time required to completely evaporate the droplet, quantitative estimation of the evaporation time is often also desirable. A simple approximate analysis will be done below by estimating the conduction in the liquid droplet by the following correlation: (7.238) δ where A and δ are the average cross-sectional area of heat conduction and the average path length of the conduction, respectively. For a hemispherical droplet, the contact area between the droplet and the heated wall is Aw = π rI2 and the
qd = k A Tw − Tsat
interfacial area of the droplet is AI = 2π rI2 . Thus, we can take the average conduction area as
A= 1 3 ( Aw + AI ) = π rI2 2 2
(7.239)
The average path length for conduction is V (2 / 3)π rI3 4 = rI δ= = A (3 / 2)π rI2 9 Therefore, the conduction in the liquid droplet becomes
qd =
(7.240)
27 π k rI (Tw − Tsat ) 8
(7.241)
Replacing the first term on the right-hand side of eq. (7.230) with eq. (7.241), and dropping the second term on the right-hand side of eq. (7.230) (see the above scale analysis), the energy balance for the hemispherical droplet becomes
ρ hv 2π rI2
drI 27 = π k rI (Tw − Tsat ) 8 dt
(7.242)
which can be rearranged as
rI drI 27 k (Tw − Tsat ) = ρ hv dt 16 rI = Ri , t = 0
(7.243)
which is subject to the following initial conditions: (7.244) Integrating eq. (7.243) and considering its initial condition, eq. (7.244), one obtains the transient radius of the liquid droplet:
rI = Ri2 − 27 k (Tw − Tsat )t 8 ρ hv
(7.245)
The time required for the droplet to evaporate completely can be obtained by letting rI in eq. (7.245) equal zero, i.e.,
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tf =
8ρ hv Ri2 27k (Tw − Tsat )
(7.246)
which agrees with the results obtained by scale analysis, eq. (7.235). The above analysis is otherwise simplistic because it assumes that the shape of the droplet is hemispherical. In fact, the shape of the droplet depends on the velocity with which it impacts the wall as well as the wettability of the liquid droplet on the wall. If the contact area between the droplet and the wall is larger (due to higher impacting velocity or good wettability), the time for conduction through the liquid is shorter and the life of the drop will be shorter as well.
Example 7.7 A hemispherical liquid water droplet with an initial radius of 20 μm is attached to a heated wall and exposed to pure water vapor at 1 atm. The temperature of the heated wall is 180 °C. Estimate the time required for the liquid droplet to evaporate completely. Solution: The saturation temperature corresponding to 1 atm is Tsat = 100 °C. The properties of the liquid water can be approximately evaluated at the saturation temperature: k = 0.68 W/m- o C, ρ = 958.77 kg/m3 ,
hv = 2251.2 kJ/kg.
The time required for the liquid droplet to evaporate completely can be obtained from eq. (7.246), i.e., 8ρ hv Ri2
tf = = 27k (Tw − Tsat ) 8 × 958.77 × 2251.2 × 103 × (20 ×10−6 ) 2 = 0.0047 s = 4.7 ms 27 × 0.68 × (180 − 100)
References
Alhusseini, K.A., Tuzla, K., and Chen, J.C., 1998, “Falling Film Evaporation of Single Component Liquids,” ASME Journal of Heat Transfer, Vol. 41, pp. 16231632. Bejan, A., 1991, “Film Condensation on a Upward Facing Plate with Free Edges,” International Journal of Heat and Mass Transfer, Vol. 34, pp. 578-582. Brauer, H., 1956, “Stromung und Warmeubergang bei Reiselfilmen,” VDI Forschung, Vol. 22, pp. 1-40. Butterworth, D., 1981, “Simplified Methods for Condensation on a Vertical Surface with Vapor Shear,” UKAEA Rept. AERE-R9683. Butterworth, D., 1983, “Film Condensation of Pure Vapor,” Heat Exchanger Handbook, Chapter 2.6.2, Hemisphere, Washington, DC.
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Carey, V. P., 1992, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corp., Washington, D. C. Carpenter, F.S., and Colburn, A.P., 1951, “The Effect of Vapor Velocity on Condensation Inside Tubes,” Proceedings of General Discussion of Heat Transfer, Institute of Mechanical Engineers and American Society of Mechanical Engineers, pp. 20-26. Chun, K.R. and Seban, R. A., 1971, “Heat Transfer to Evaporating Liquid Films,” ASME Journal of Heat Transfer, Vol. 93, pp. 391-396. Collier, J.G., and Thome, J.R., 1996, Convection Boiling and Condensation, 3rd ed., Oxford University Press, Oxford. Eucken, A., 1937, Naturwissenschaften, Vol. 25, pp. 209. Faghri, A., 1986, “Turbulent Film Condensation in a Tube with Cocurrent and Countercurrent Vapor Flow,” AIAA Paper No. 86-1354. Faghri, A., 1995, Heat Pipe Science & Technology, Taylor & Francis, Washington, D.C. Faghri, A., Chen, M. M., and Morgan, M., 1989, “Heat Transfer Characteristics in Two-Phase Closed Conventional and Concentric Annular Thermosyphons,” ASME Journal of Heat Transfer, Vol. 111, No. 3, pp. 611-618. Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA. Fowkes, F.M., 1965, “Attractive Forces at Interfaces,” Chemistry and Physics of Interfaces, American Chemical Society, Washington, DC. Fujii, T., 1991, Theory of Laminar Film Condensation, Springer-Verlag, New York, New York. 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. 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. Hewitt, G.F., Bott, T.R., Shires, G.L., 1994, Process Heat Transfer, CRC Press, Boca Raton, FL
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Kutateladze, S.S., 1982, “Semi-Empirical Theory of Film Condensation of Pure Vapors,” International Journal of Heat 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. Lock, G.S.H., 1994, Latent Heat Transfer, Oxford Science Publications, Oxford University, Oxford, UK. 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. 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. Nusselt, W., 1916, “Die Oberflächenkondensation des Wasserdampfes,” Z. Vereins deutscher Ininuere, Vol. 60, pp. 541-575. Rahman, M.M., and Faghri, A., 1992, “Analysis of Heating and Evaporation from a Liquid Film Adjacent to a Horizontal Rotating Disk,” International Journal of Heat and Mass Transfer, Vol. 35, pp. 2644-2655. Rice, J., Faghri, A., and Cetegen, B.M., 2005, “Analysis of a Free Surface Flow for a Controlled Liquid Impinging Jet over a Rotating Disk Including Conjugate Effects, with and without Evaporation,” International Journal of Heat and Mass Transfer, Vol. 48, pp. 5192-5204. 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. Stephan, K., 1992, Heat Transfer in Condensation and Boiling, Springer-Verlag, Berlin.
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Szablewski, W., 1968, “Turbulent Parallelstromunjen,” Zeitshr. Ang. Math Mech., Vol. 48, p. 35.
Problems
7.1. 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. 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. (7.55)]
dδ dδ ρ u dx − v = ρ v uv dx − vv , y = δ I I
7.2.
7.3.
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? 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 (see Fig. P7.1). 100 W is transferred from the plate to the air by convective cooling. What is the condensation rate assuming dropwise condensation? q =100 W
7.4.
Air @ 1 atm
Saturated Steam @ 2 atm
Condensate u∞=2 m/s T∞=50 °C
Figure P7.1
7.5.
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%.
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Calculate the heat transfer rate for a single droplet of condensate 1 mm in diameter. Discuss which thermal resistances are most significant. 7.6. 7.7. Many assumptions are made in Nusselt analysis. Discuss the reasonability of these assumptions and their expected effects on heat transfer. 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. 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. 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?
7.8.
7.9.
7.10. 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 7.7. Find the corresponding average heat transfer coefficient for this configuration. The imposed pressure gradient that drives the flow is 2000 N/m3. 7.11. The effect of vapor flow on film condensation was considered in Section 7.3.4 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. 7.12. 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. 7.13. 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
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analysis to get the average heat transfer coefficient and compare your result with eq. (7.171). 7.14. 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.
Figure P7.2
7.15. A saturated vapor condenses on the outside surface of a constanttemperature cone (see Fig. P7.2). 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). 7.16. 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. (7.172). 7.17. 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.
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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. 7.18. The process of condensation inside a vertical tube is complex. Build a reasonable model; write governing equations and boundary conditions for the model. 7.19. 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. (7.85). If the temperature profile in the liquid can be assumed to be linear like eq. (7.86), show that the Nusselt analysis is still valid provided that the latent ′ heat, hv , is replaced by hv as defined in eq. (7.88).
uv0, Tv0 Adiabatic plate
y Vapor boundary layer Liquid film Cold plate (Tw) Figure P7.3
δ
x
7.20. A pure vapor flows into a horizontal channel formed by two parallel plates (see Fig. P7.3) 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 ρ v (uv − uδ ) 2 + mδ (uv − uδ ) , where the fraction coefficient
f = C Ren (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. 7.21. 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
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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? 7.22. The condenser section of a rotating heat pipe is shown in Fig. P7.4. 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.
Figure P7.4
7.23. 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 P7.5
7.24. 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
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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. P7.5. 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.
Figure P7.6
7.25. Under an ideal situation, no liquid accumulates at the bottom of the thermosyphon (see Fig. P7.6). 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. 7.26. The reflux condensation in a small-diameter tube is shown in Fig. P7.7. 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
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distribution and specify the corresponding boundary conditions. You do not need to consider the liquid layer at the lower portion of the crosssection.
φ φ φ
θ
(a) configuration
Figure P7.7
(b) liquid film inside the tube
Figure P7.8
7.27. The mass flux at the liquid vapor-interface for condensation of condensable vapor and noncondensable gas can be obtained from eq. (7.183). Show that the mass flux can also be obtained from eq. (7.186) if the mixture of condensable vapor and noncondensable gas can be treated as ideal gas. 7.28. Falling water film evaporation occurs on the inner sides of two parallel plates with a length of 1 m, and they are separated by a distance of 2b = 3 cm . The inlet and outlet liquid film temperatures are 50 ˚C and 45
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˚C, respectively. Find the mass flow rate of the liquid film per unit width,
Γ.
7.29. Air enters the bottom of a vertical channel formed by two parallel walls (see Fig. P7.8). A liquid water film is present on the left wall, which is maintained at a constant temperature. The right wall is adiabatic and has no liquid film on it. The liquid film on the left wall is so thin that its thickness can be neglected. The left wall can be considered as a surface where the concentration of the vapor equals the saturation concentration of the vapor at its partial pressure. Assuming that the air flow in the vertical channel is laminar and the air inlet velocity is uniform, find the governing equations and boundary conditions to describe this conjugate heat and mass transfer process. 7.30. Air flows into a tube with radius R and constant wall temperature Tw. Evaporation occurs on a thin liquid film on the inner wall of the tube. The thickness of the liquid film can be neglected in comparison with the radius of the tube. The inner wall can be considered as a surface where the concentration of the vapor equals the saturation concentration of the vapor at its partial pressure. The air flow at the inlet of the tube is fully developed, but the temperature and concentration distributions at the inlet are uniform. Specify the governing equations and corresponding boundary conditions of the problem. 7.31. Solve the energy and species equations of the above problem to obtain the temperature and concentration distributions in the tube. The Schmidt number, Sc, can be assumed to be equal to one. Obtain the local convective heat transfer coefficient for the system. 7.32. A two-phase thermosyphon, shown in Fig. P7.9, is needed to deliver a steady-state heat rate of Q = 1kW at an operating temperature of Tv = 333 K. An acceptable wall superheat is ΔT = 5 K. The inside diameter of the circular channel is D = 0.1 m, and the working fluid is water. Use classical Nusselt analysis to determine how long the evaporator should be to avoid dryout. 7.33. In a falling liquid film evaporation experiment, the temperature of the heating wall is 106 °C and the vapor pressure is at one atmosphere. The inlet mass flow rate of the liquid film per unit width is Γ 0 = 2.24 × 10−2 kg/s-m . Is the film flow laminar, laminar with waves, or turbulent? If the height of the vertical plate is 0.1 m, determine the average heat transfer coefficient. Please also check the overall energy balance by using eq. (7.207).
662 Advanced Heat and Mass Transfer
Amir Faghri, Yuwen Zhang, and John Howell
Copyright © 2010 Global Digital Press
Figure P7.9
7.34. A uniformly-distributed liquid water film at 0.06 kg/s is introduced at the top of a tube with an inner diameter of 4 cm and height of 3 m. The inner surface temperature of the tube is uniformly 105 °C, and the saturation temperature of the steam is 100 °C. Is the liquid film at the inlet and outlet laminar, laminar with waves, or turbulent? Find the evaporation rate and the average heat transfer coefficient. 7.35. Falling film evaporation on a vertical wall is analyzed in great detail in Section 7.6. How would you apply the solutions of falling film evaporation on a vertical heated wall to predict heat transfer of falling film evaporation on an inclined heating wall? The inclination angle between the heating wall and the vertical direction is θ . 7.36. You are asked to select the most weight-efficient evaporant for a particular evaporator. The temperature at which the evaporator will operate has not been set. Go through Appendix B.4 and rank each working fluid based on its heat of vaporization ( hv ). Also, for each working fluid, note what the operating temperature of the evaporator would be if the operating pressure was 1 atm. Which eva ρ ρ v orants stand out in the different temperature ranges?
Chapter 7 Condensation and Evaporation
663
Amir Faghri, Yuwen Zhang, and John Howell
Copyright © 2010 Global Digital Press
7.37. Consider evaporation for a constant temperature wall rotating disk with controlled liquid jet impingement at its center (see Fig. P7.10). Rotational forces cause the film to spread across the disk and then off the outer edge. The flow field is similar to a laminar, steady falling film, except that rotation replaces gravity as the driving force and the system must be solved in cylindrical coordinates. Obtain the heat transfer coefficient for the thin liquid film evaporation on the rotating disk by neglecting inertia and convective terms.
z
r Φ
ω
Figure P7.10
664 Advanced Heat and Mass Transfer
Amir Faghri, Yuwen Zhang, and John Howell
Copyright © 2010 Global Digital Press