As shown in Figure 1, in the inlet region of a horizontal condenser tube the volume flow rate of superheated vapor is usually high, and the flow pattern is single-phase vapor flow. If the temperature of the vapor decreases below saturation temperature near the cooling tube wall, the vapor starts to condense on the surface of the tube. The condensed liquid flows as a liquid film on the tube wall co-currently with the vapor flow and the flow pattern becomes annular flow. In the vapor core, liquid also flows as droplets due to strong interfacial shear force if the velocity of vapor is high. In a condenser, the volume flow rate of vapor and liquid mixture continuously decreases as the mixture flows downstream, where the role of the gravitational force in the phase distribution becomes stronger. Then the flow pattern becomes wavy, separated, then becomes plug flow with vapor flowing on the top side of the tube cross section. Finally, all of the vapor changes into liquid, and single liquid flow appears near the end of the condenser.

In the case of a vertical tube condensing fluid usually flows downwards. The phase distribution is axisymmetric and the flow pattern is annular flow over almost whole region of the condenser tube and changes to slug flow just upstream of the end of the condensing section.

Local heat transfer coefficient α, at the cross-section where the quality is x, is given by Shah (1979) as follows:

where α_{L} is the heat transfer coefficient given when liquid flows as a single-phase flow with same liquid mass flow rate, and P_{s}/P_{cri} is the reduced pressure ratio of the saturation pressure P_{s}. This equation is useful being independent of the orientation of the pipe and the flow direction, although the accuracy is not high.

The heat transfer coefficients during condensation, however, depends strongly on the phase distribution, i.e., the flow pattern, especially when the existence of dried area on the condensing surface controls the heat transfer. Therefore, the prediction of flow pattern is important to estimate accurately the heat transfer coefficient.

In the case of horizontal flow, some correlations are available to estimate the heat transfer coefficient α expressed as Nusselt Number Nu (≡ α l/λ, where l is the characteristic length of the heat transfer area, and λ the thermal conductivity) for respective flow condition.

where Pr is the Prandtl Number of the vapor and f is the friction factor which is given by

The following formulas have been also widely used:

*Dittus-Boelter correlation*(4)*Colburn correlation*(5)

(Narrower range of Re and Pr than those in Dittus–Boelter’s correlation), where Re is the Reynolds Number (≡ ūD/ν_{v}, where ū
is the mean vapor velocity, D the tube diameter, and ν_{v} the kinematic viscosity of the vapor).

The Nusselt number decreases with the decrease in the quality, i.e., the increase of the condensate flow rate. Miropolsky (1974) proposed a similar type of correlation to that given by Equation (2), where the fluid properties are defined by the average calorimetric flow temperature.

Fujii et al. (1977) proposed the following correlation of the condensation heat transfer coefficient a in the case where the superheated vapor is condensing on its condensate:

where St is the Stanton Number, ū
is the mean velocity of vapor, Pr_{v} the Prandtl Number of vapor and f the Friction Factor given by the next equation,

Here ε is the void fraction, φ_{v} the Martinelli parameter for the two-phase pressure drop,
the mass velocity of vapor, D is the tube diameter and η_{v} the absolute viscosity of vapor. They suggested also that void fraction ε can be obtained by the next relation proposed by Fauske (1961),

Notice that the heat flux through the tube wall is expressed as

where D is the tube diameter, 2r_{i} the average diameter of the vapor-liquid interface, h_{LG} the latent heat of condensation,
the mass flux of the condensate and
the heat flux calculated by the heat transfer coefficient given by St described in Equation (6) above.

The heat transferred to the liquid comes only from the condensation of the vapor on the liquid layer in this region. Therefore, heat transfer can be estimated in the same way as mentioned in the previous section, if the heat flux at the interface is assumed to be zero.

A typical example of condensation in a closed tube is a heat pipe. As shown in Figure 2 vapor generated in a heating section, i.e., an evaporator, flows into a condenser where the vapor condenses. The condensate returns to the evaporator as a liquid film on a tube wall. Acordingly in this case, *reflux condensation* heat transfer takes place between up-going vapor flow and down-coming liquid film flow, as for example, in a vertical *two-phase thermosyphon*, i.e., a *wickless heat pipe*.

Laminar film condensation on a vertical plate was analyzed by Nusselt (1916). The local heat transfer coefficient α is given as

where ρ_{L} and ρ_{G}, are the liquid and gas densities, λ_{L} the liquid thermal conductivity, z the distance from the start of condensation, η_{L} the absolute liquid viscosity and ΔT the difference between the saturation temperature and the wall temperature, and
where h_{LG} is the latent heat of evaporation and c_{L} the specific heat of the liquid.

The local heat transfer in a closed tube is not as simple as Nusselt's physical model in the following respects [Fukano et al. (1990)]:

Nusselt assumed a smooth surface. Actually, however, the interface is not smooth as shown in Figure 3. Flow pattern changes from the dropwise through the smooth surface to the two-dimensional wave and finally the three dimensional wave — in this order — from the top of the condenser to the bottom.

The vapor–liquid interfacial shear stress is not zero contrary to the assumption made by Nusselt.

The boundary condition of the film thickness δ, i.e., δ = 0 at the start of condensation, is used in the Nusselt analysis. However, many fine droplets are observed to flow onto the top wall of the tube, so that the film thickness is not zero at the tube top.

Therefore, the Nusselt number is usually lower in the top region of the condenser than that given by the Nusselt analysis due to the larger film thickness. Also, it is higher in the bottom region due to the quick sweeping away of the condensate by large two or three dimensional waves resulting in the smaller film thickness with smaller thermal resistance.

If the average film thickness δ is estimated by the mass and the heat balances of the liquid film, the local heat transfer coefficient α is obtained by assuming that heat is transferred only by conduction through liquid film as follows:

where C means the fraction of the wave layer in which the thermal resistance is zero, and is given by the dimensionless shear stress follows,

The effect of the wave on the heat transfer does not appear until the interfacial shear stress exceeds a certain value, i. e., τ_{i}* > 0.08.

#### REFERENCES

Fauske, H. K. (1961) *Heat Transfer and Fluid Mechanics*, Stanford University Press.

Fujii, T. et al. (1977) *Refrigeration*, 52, 596.

Fukano, T. et al. (1990) *Proc. of 9th Int. Heat Transfer Conf*, 6.

Miropolsky, Z. L. (1974) *Proc. 5th Int. Heat Transfer Conf*, 3.

Nusselt, W. (1916) *2. ver. deut. Ing.*, 60, 541, 569.

Petukov, B. S. (1970) *Advances in Heat Transfer*, 6.

Shah, M. M. (1979) *Int. J. Heat Mass Trans.*, 22-4.