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372 CHAPTER 11  Boiling and condensation

leads to pressure fluctuations, which can result in transient flow reversal if upstream compressibility is present (Gedupudi et al., 2011; Jain et al., 2019). Another characteristic is the possibility of occurrence of intermittent dryout caused by the cyclic passage of liquid slugs and long confined bubbles (Jain et al., 2019). Enhancing the two-phase heat transfer coefficient and the critical heat flux and reducing the flow instabilities and pressure drop are the research challenges in this area (Kandlikar et al., 2006; Karayiannis and Mahmoud, 2017).

11.5  Condensation

The condensation process is the reverse of the boiling process and involves the change of a vapor phase to a liquid phase. Just as liquid superheat is required to induce the nucleation of bubbles in boiling, vapor subcooling is required to induce the nucleation of droplets in condensation. If the condensate forms a continuous film, then it is called film-wise condensation (or simply, film condensation), which occurs on wetted surfaces. In drop-wise condensation (or simply, drop condensation), the vapor condenses into small liquid droplets of different sizes that coalesce and fall down the cooled surface, and it occurs on non-wetted surfaces. Drop condensation, as seen from the above definition, offers less thermal resistance, and this results in heat transfer coefficients as much as 5 to 10 times the values in film condensation. Though drop condensation would be preferred to film condensation, it is difficult to achieve or maintain drop condensation. Much of the recent research focuses on promoting drop condensation using microstructured surfaces, which is beyond the scope of this textbook. Additionally, there is a lack of reliable theories of drop condensation.

11.6  Film condensation on a vertical plate

Nusselt (1916) made the following simplifying assumptions to solve the film condensation problem analytically, which is often considered a classic treatise on the subject of condensation:

1.The condensate flow in the film is laminar.

2.The plate temperature, Tw, is uniform and less than the vapor saturation temperature, Tsat.

3.The fluid properties are constant.

4.Negligible shear stress at the liquid-vapor interface.

5.Momentum changes in the film are negligible.

6.The heat transfer across the film is by conduction only, and the condensate temperature profile is linear.

Fig. 11.7 shows the typical growth of condensate film as well as velocity and temperature profiles. The analysis is made per unit width of the plate in the direction

11.6 Film condensation on a vertical plate 373

FIGURE 11.7

Film condensation on a vertical plate.

perpendicular to the plane of the paper. Consider a force balance on a differential element of the film at any axial location, x, shown in Fig. 11.7.

Substituting Eq. (11.52) in Eq. (11.51),

11.7 Condensation on horizontal tubes 377

11.7  Condensation on horizontal tubes

In surface condensers, condensation occurs on a bank of horizontal tubes, as shown in Fig. 11.8.

Nusselt derived the following mean heat transfer coefficient relationship for laminar film condensation on the outside surface of a single horizontal tube:

For a vertical column of N horizontal tubes, Jakob (1936) proposed the following mean heat transfer coefficient correlation:

FIGURE 11.8

Film condensation on a bank of horizontal tubes.

378 CHAPTER 11  Boiling and condensation

The above equations clearly show that the Nth tube will have a much lower heat transfer coefficient than the first, which is intuitively apparent.

Note that the Nusselt theory neglects the shear at the liquid-vapor interface. The interfacial shear becomes significant for condensation with flowing vapor and hence is an important factor that influences the heat transfer coefficient for condensation in tubes. Readers can refer to advanced texts (Collier and Thome, 1994; Stephan, 1992) for condensation in vertical and horizontal tubes.

11.8  Two-phase pressure drop

Pressure drop is an important consideration in the design of heat transfer equipment and more so for two-phase flows. Two-phase drop involves the pressure drop due to wall and interfacial shear stresses, the acceleration pressure drop for the case of evaporating flows, and the pressure drop due to gravitational head. The homogenous model and the separated flow model are usually used to estimate the two-phase pressure drop. The separated flow model, which is more complex, makes better prediction than the homogenous model but is beyond the scope of the present study. The interested reader can refer to advanced texts, such as Collier and Thome (1994) and Stephan (1992).

The homogenous model is based on the following assumptions:

(i)the vapor and liquid velocities are equal (ug = uf = u )

(ii)the thermodynamic equilibrium exists between the phases, and

(iii)an appropriately defined single-phase friction factor can be used for two-phase flow.

The total pressure gradient is the sum of the frictional, acceleration, and gravitational head pressure gradients,

where G, v, and fTP are the total mass flux (the product of density and velocity), specific volume, and the two-phase Fanning friction factor, respectively. Note that the Fanning friction factor is expressed as τ w / (ρu2 / 2), where τ w is the wall shear stress. The Darcy friction factor is four times the Fanning friction factor.

11.8 Two-phase pressure drop 379

The acceleration pressure gradient is given by

Assuming vfg ≈ vg and neglecting the compressibility of the liquid phase,

where θ is the angle made by the inclined pipe with the horizontal. Substituting Eqs. (11.82), (11.85) and (11.86) in Eq. (11.81),

The Fanning friction factor ( fTP ) can be evaluated using a mean two-phase viscosity, .

McAdams et al. (1942) suggested the following relationship for the mean twophase viscosity.

Cicchitti et al. (1960) suggested the relationship

For turbulent flow, the two-phase Fanning friction factor can be evaluated using the Blasius equation