**Condensation** heat transfer plays an important role in many engineering applications, notably electric power generation, process industries, refrigeration and air-conditioning. Many different physical phenomena are involved in the condensation process, their relative importance depending on the circumstances and application. (For examples, see Air Conditioning, Condensers, Desalination, and Refrigeration.)

When a liquid and its vapor are in contact, molecules pass from liquid to vapor and from vapor to liquid. **Condensation** occurs when the number of molecules entering the liquid phase exceeds that of leaving molecules. Under these circumstances, the temperature of the vapor in the immediate vicinity (a few mean free paths) of the vapor-liquid interface is higher than that of the liquid. The *interface temperature drop* increases with increasing condensation rate and with decreasing pressure but in most circumstances (an exception being the case of liquid metals), this is very small and equilibrium conditions at the interface can be assumed. A brief summary of interface matter transfer during condensation is given by Niknejad and Rose (1981).

The most common and best understood case of condensation heat transfer is that of film *condensation* of a pure quiescent vapor on a solid surface. The problem of calculating the heat transfer rate for a plane vertical surface and for a horizontal cylinder with uniform surface temperatures, and where the condensate flow is laminar and governed only by gravity and viscous forces, has been solved by Nusselt (1916). Nusselt’s main assumptions are that heat transfer across the condensate film is by pure conduction, the effect of vapor drag in supporting the falling condensate film is negligible, and the properties of the condensate may be taken to be uniform across the film, i.e., are essentially independent of temperature.

The well-known *Nusselt equations* are:

For the vertical plane surface, the mean value of Nusselt number is

and for the horizontal tube

where ρ is the liquid density, g is acceleration due to gravity, h_{lg} is the latent heat of evaporation, η is the liquid viscosity, λ is the liquid thermal conductivity, Δρ is the density difference between vapor and condensate, ΔT is the vapor-to-surface temperature difference; L is the plate height; and d, the tube diameter. For the case of the tube, the additional assumption that film thickness is small compared with the tube radius is needed. Since the theory predicts that the radial film thickness tends to infinity at the bottom of the tube, this assumption is evidently invalid for the lower part of the tube. The fact that the heat transfer rate is inaccurate where the film becomes thick is relatively unimportant because it is small and makes a minor contribution to the total heat transfer rate for the tube. Equations (1) and (2) have been well verified experimentally for condensation of pure (only one molecular constituent) vapor. In order to obtain the constant in Eq. (2), numerical integration is required twice. Nusselt’s slightly inaccurate value (0.8024 (2/3)^{1/4} = 0.725) is due to his use of planimetry. (See Condensation of Pure Vapor.)

As discussed by Rose (1988), more recent theoretical studies in which the effects of inertia and convection in the condensate film and vapor drag on the condensate surface are included have shown that these effects are unimportant. More recently, it has been shown by Memory and Rose (1991) that the effect of variable wall temperature, which occurs in practice during condensation on a horizontal tube, also has a negligible effect on the *mean* heat transfer rate. Thus, Eqs. (1) and (2) can be used with confidence for condensation of pure “stationary” vapors when the condensate flow is laminar.

In the case of significant vapor flow rate over the condensing surface, the effect of drag on the condensate becomes significant. In some circumstances, the effect of vapor drag overwhelms that of gravity. Since 1960, *vapor shear stress effects* have been studied extensively. Some of the more important contributions are described by Rose (1988).

The relative importance of vapor shear stress and gravity on the motion of the condensate film is measured by the dimensionless parameter F=ηh_{lg}gx/λΔTu^{2}_{∞}, where x is the relevant linear dimension (plate height or tube diameter) and u_{∞} is the free-stream vapor velocity. For downward vapor flow over a horizontal tube, an approximate analysis gives

Nu_{d} is the mean Nusselt number for the condensate film and is a Reynolds number using the vapor approach velocity and condensate properties. Equation (3)—which indicates that for F > 10, gravity dominates while for F < 0.1, vapor shear stress is controlling—agrees quite well with experimental data from several investigations using various condensing fluids. (See Tubes, Condensation on Outside in Crossflow.)

When the vapor contains more than one molecular species, the problem is complicated by diffusion of species in the vapor. For example, for a two-constitutent vapor where only one constituent condenses (e.g., steam-air), the mixture is rich in the *noncondensing gas* near the interface where vapor molecules are removed. The tendency is for noncondensing gas molecules to diffuse away from the interface so that in the steadystate, the rate at which gas molecules arrive at the surface with the condensing vapor is equal to their diffusion rate away from the surface. Even in the absence of forced convection of the vapor-gas mixture, the density difference, which results from the composition difference between that of the bulk vapor and that of the vapor adjacent to the interface, leads to natural convection. The process by which the steadystate is reached is therefore one of diffusion in the presence of convection. The fact that the vapor-gas mixture adjacent to the condensate surface is rich in non-condensing gas causes the temperature at the interface to be lower than in the bulk. Assuming equilibrium at the interface, the temperature is equal to the saturation temperature corresponding to the partial pressure of the vapor, which may be significantly lower than in the more remote vapor. Composition (or partial pressure) and temperature boundary layer are set up in the vapor adjacent to the interface. This gives an effective heat transfer resistance since the temperature drop across the condensate film, and hence the heat transfer rate, is significantly reduced. Detailed boundary layer solutions of this problem for free and forced convection, notably by Koh, Sparrow, Fujii et al., have been given. Earlier works are discussed and approximate equations are given by Rose (1969) for the free convection problem, and Rose (1980) for the forced convection case. When two or more constitutents of the vapor condense together, the situation is similar to that described above since the more volatile constituent is more concentrated at the condensate surface. Extensive treatments of these problems for the case of the plane vertical condensing surface and laminar flow of vapor and condensate have been given by Fujii (1991). (See Condensation of Multicomponent Vapors.)

Approximate methods, based on the *analogy* between heat and diffusive mass transfer in the vapor, are widely used for multiconstituent problems. The essence of the method is that the differential equations expressing conservation of energy and molecular species can be arranged in identical form by appropriate nondimensionalization. Known results (theoretical or experimental) for heat transfer problems are used to infer results for the corresponding mass transfer diffusion problems. The method has wide utility but is approximate since the boundary condition on the normal vapor velocity at the surface is not the same for the heat and mass transfer problems. For the heat transfer problem, the normal velocity at the (solid) heat-exchanger wall is zero. In the case of condensation, where molecules pass through the vapor-liquid interface, the normal velocity is not zero. The validity of the analogy depends on the smallness of the “suction parameter” (–v_{o}/u)Re^{1/2}, where v_{o} is the normal outward (i.e., negative for condensation) vapor velocity at the condensate surface, u is the free-stream velocity parallel to the surface and Re is the free-stream vapor Reynolds number. The results are strictly correct only in the limiting case of zero condensation rate. A widely-used approximate *stagnant film model* extends the range of validity of the analogy. The heat-mass transfer analogy and the stagnant film model are discussed by Lee and Rose (1983), Butterworth (1983) and Webb (1990).

The foregoing refers exclusively to laminar flow conditions. For tall condensing surfaces, or under conditions of high vapor shear stress, transition to turbulent flow in the condensate film may occur. This brings to the problem unresolved difficulties associated with the general problem of Turbulence. For moderate or low vapor velocities, the “effective height” of the surface for a horizontal tube (≈d) is small and laminar flow of the condensate is expected. Moreover, since turbulent mixing enhances heat transfer across the condensate film, the Nusselt solution [Eq. (2)] is conservative and is widely used in design calculations. Various models for turbulent film condensation from an essentially stationary vapor exist in the literature and predict somewhat different results. For gravity-dominated flow, transition to turbulence has been found to occur at film Reynolds numbers, 4Γ/η, rather lower than 2 000, where Γ is the condensate flow rate per width of surface. In the presence of high-vapor shear stress, the problem is more complicated. Turbulent film condensation on a vertical surface under free-convection conditions can, in principle, be analyzed by an approach similar to that used for single-phase pipe flow. However, this problem is relatively unimportant since in practice, condensing surfaces are usually not sufficiently tall for turbulence to occur under purely free-convection conditions. Significant shear stress, due to vapor flow along the condensate surface, promotes the onset of turbulence. The analysis is then more complicated, particularly when the vapor flow is not in the same direction of gravity.

Turbulence is more often encountered for condensation inside a tube. In this case, the problem is generally complicated by the presence of significant vapor shear stress on the condensate film since even when all of the vapor is condensed in the tube, the shear stress for a portion of the tube towards the inlet end is generally significant owing to the high vapor velocity resulting from the small tube cross-section. Condensation inside tubes is beset with all the problems and uncertainties of Two-Phase Flow. The only case which can be analyzed wholly satisfactorily is that of downward vapor flow in a vertical tube with a laminar condensate film on the wall (stratified flow). In this case, the problem is the same as that for external flow, except that account must be taken of the progressive reduction of vapor flow rate, and hence shear stress, due to condensation. Approaches used in other cases are outlined by Butterworth (1983). (See Tubes, Condensation in.)

In many practical applications, condensation occurs in bundles or banks of horizontal tubes (shell-side condensation). In these cases, there is the additional complication of *inundation* (condensate from higher or upstream tubes falling or impinging on lower or downstream tubes). This leads to thicker condensate films on the inundated tubes. At the same time, the condensate film on inundated tubes is disturbed and the heat transfer coefficient may be enhanced. Nusselt’s (1916) approach for a vertical, in-line column of horizontal tubes assumes that condensate drains to lower tubes in the form of a continuous laminar film. This leads to a simple expression for the average heat transfer coefficient for a column of N tubes:

where α_{1}, for the uppermost tube, is given by Eq. (2). In view of the more probable mode of *drainage* or inundation with condensate film disturbance due to splashing from droplets, columns or unstable broken films of liquid, it is not surprising that Eq. (4) has been found to be overconservative. Many experimental studies of condensation on tube banks have been made. The data are widely-scattered owing primarily to the effects of noncondensing gases, turbulence and vapor velocity. Various correlations have been proposed and approximate methods used in practiced are discussed by Butterworth (1983). (See Tube Banks, Condensation Heat Transfer in.)

Numerous techniques for *enhancement of film condensation heat transfer* have been proposed [see, for instance, Webb (1981)]. Notable amongst these, for shell-side condensation, are low (fin height small in relation to tube diameter) integral-fin tubes. In this case, it is found that for horizontal tubes, the enhancement of the heat transfer coefficient can significantly exceed the increase in surface area due to the presence of the fins. The reasons for this are: 1) the vertical or near-vertical fin flanks have small heights so that the heat transfer coefficients are large [see Eqs. (1) and (2)] surface tension effects give rise to an additional mechanism for draining condensate from parts of the surface. The latter arises from the pressure gradient set up in the presence of a condensate surface of varying curvature, e.g., for the two-dimensional case:

where P is pressure, s is distance measured along the surface, σ is surface tension and r is the local radius of curvature of the condensate film. At the same time, surface tension has a detrimental effect on the heat transfer coefficient due to capillary retention of condensate between the fins and the consequent “blanketing” of heat transfer surface on the lower part of the surface. The extent of condensate retention can be calculated from:

as formulated by Honda et al. (1983). In Eq. (6), φ is the angle measured from the top of the tube to the position where the interfin tube space is filled with retained condensate; β is the angle between the fin flank and radial plane; b is the distance between adjacent fins measured at the fin tip; and d_{o} is the tube diameter over the fins.

In an early theoretical solution of the problem of condensation on low-finned tubes by Beatty and Katz (1948), the vertical fin-flanks and horizontal interfin tube spaces were treated on the basis of the Nusselt theory and effects of surface tension were ignored. This simple approach proved quite successful in practice for relatively low-surface tension fluids. This is partly because condensate retention in this case is small, and partly because the beneficial and detrimental effects of surface tension tend, to some extent, to nullify each other. More detailed and complicated models have been proposed, notably by Honda and Nozu (1987). These require numerical solution and are less readily applied than the simple analytical result of Beatty and Katz. The various models are discussed by Marto (1988). A recent semi-empirical approach, which includes surface tension effects, has been given by Rose (1994). The result, in the form of an equation for the “enhancement ratio”, is in good agreement with experimental data from seven investigations using four condensing fluids and 41 tube/fin geometries. (See also Augmentation of Heat Transfer, Two-Phase Systems.)

The foregoing relates to the case when the condensate wets the condensing surface and forms a continuous film. When the surface is not wetted, a quite different mode, namely Dropwise Condensation, may occur. In this case, minute droplets form at nucleation sites on the surface and growth takes place by condensation and coalescence with neighbors until drops reach a size at which they are removed from the surface by gravity or vapor shear stress. Moving drops sweep up stationary drops in their path, making available new area for condensation. The maximum-to-minimum drop size is around 10^{6}. To date, dropwise condensation has only been obtained with a few high-surface tension fluids (notably water). A nonwetting agent or “promoter” is required to promote dropwise condensation on metal surfaces. Heat transfer coefficients for dropwise condensation are much higher than those for film condensation under the same conditions. For steam at atmospheric pressure, the factor is around 20. Not surprisingly, this has stimulated research work over the past 60 years with the aim of finding an effective promoter. Although good promoters (e.g., dioctadecyl disulfide) are available, which form stable monomolecular layers on copper or copper-containing surfaces, and give dropwise condensation for hundreds or thousands of hours under clean laboratory conditions, effective promoters for use under industrial conditions have yet to be found. Recent surveys of dropwise condensation heat transfer have been given by Tanasawa (1991) and Rose (1994).

In some applications, such as desalination and geothermal power plant, use is made of direct contact condensation. This is a term applied to processes wherein vapor condenses on subcooled liquid drops, sprays, jets, films or in a liquid pool. Various types of equipment and the relevant mechanisms for direct contact condensation are given by Jacobs (1985). (See also Direct Contact Heat Transfer.)

#### REFERENCES

Beatty, K. O. and Katz, D. L. (1948) Condensation of Vapors on the Outside of Finned Tubes. *Chem. Eng. Prog.* 44, 55-70.

Butterworth, D. (1983) *Film Condensation of Pure Vapor*. Heat Exchanger Design Handbook, Ed. E. U. Schlunder, Hemisphere Publishing Corporation, New York.

Fujii, T. (1991) *Theory of Laminar Film Condensation*. Springer-Verlag, New York.

Honda, H., Nozu, S. and Mitsumori, K. (1983) Augmentation of Condensation on Horizontal Finned Tubes by Attaching a Porous Drainage Plate. Proc. ASME-JSME Thermal Engng. Joint Conf., 3, 289-296.

Honda, H. and Nozu, S. (1987) A Prediction Method for Heat Transfer during Film Condensation on Horizontal Integral Fin Tubes. *J. Heat Transfer*, 109, 218-225.

Jacobs, H. R. (1985) *Direct Contact Condensers*. Heat Exchanger Design Handbook, Ed. E. U. Schlunder, Hemisphere Publishing Corporation, New York.

Lee, W. C. and Rose, J. W. (1983) Comparison of Calculation Methods for Non-Condensing Gas Effects in Condensation on a Horizontal Tube. *IChemE Symp*. Ser. No. 75, 342-355.

Marto, P. J. (1988). An Evaluation of Film Condensation on Horizontal Integral-Fin Tubes. *J. Heat Transfer*, 110, 1287-1305.

Memory, S. B. and Rose, J. W. (1991) Free Convection Laminar Film Condensation on a Horizontal Tube with Variable Wall Temperature. *Int. J. Heat Mass Transfer*, 34, 2775-2778. DOI: 10.1016/S0017-9310(96)00374-2

Niknejad, J. and Rose, J. W. (1981) Interphase Matter Transfer—an Experimental Study of Condensation of Mercury. *Proc. R. Soc. Lond.* A 378, 305-327.

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Rose, J. W. (1980) Approximate Equations for Forced-Convection Condensation in the Presence of a Non-Condensing Gas on a Flat Plate and Horizontal Tube, *Int. J. Heat Mass Transfer*, 23, 539-546. DOI: 10.1016/0017-9310(80)90095-2

Rose, J. W. (1988) Fundamentals of Condensation Heat Transfer: Laminar Film Condensation. JSME *Int. Journal*, Series II, 31 (3), 357-375.

Rose, J. W. (1994) An Approximate Equation for the Vapor-Side Heat Transfer Coefficient for Condensation on Low-finned Tubes. *Int. J. Heat Mass Transfer*, 37, 865-875. DOI: 10.1016/0017-9310(94)90122-8

Rose, J. W. (1994) Dropwise Condensation. *Heat Exchanger Design Update*, 2.6.5, 1–11, Begell House.

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Webb, R. L. (1981) *The Use of Enhanced Surface Geometries in Condensers: An Overview*. Power Condenser Heat Transfer Technology, Eds. P. J. Marto and R. H. Nunn, Hemisphere Publishing Corporation, New York.