Condensation within tube banks is subject to the combined effects of *vapor shear* and *falling condensate* from upper tubes. The latter is called *condensate inundation*. Tube orientation may be vertical or horizontal. However, horizontal orientation is more common when a pure vapor is to be condensed. Table 1 summarizes the factors that affect condensation heat transfer of pure vapors in a horizontal tube bank.

**Table 1. Factors that affect condensation of pure vapors in horizontal tube banks**

Item | Factor | |

Geometry | Plain Finned | |

Layout | In-line | |

Tube | Staggered | |

Number of tube rows | Horizontal | |

Vertical | ||

Pitch-to-diameter ratio | ||

Vapor | Velocity | |

Flow direction | ||

Condensate | Inundation rate | |

Mode of inundation |

Plain tubes are commonly used for condensation of steam and other fluids with high liquid thermal conductivity. Integral fin tubes and more advanced three-dimensional fin tubes are often used for condensation of organic fluids with low liquid thermal conductivity. Both in-line and staggered tube banks are commonly employed. For power condensers operated at low pressures, the maximum vapor velocity exceeding 100 m/s is realized at the inlet of tube bank. For refrigerant condensers, on the other hand, the vapor velocity at the inlet of condenser shell is designed to be less than about 6 m/s. The vapor flow may be downward, horizontal, upward and any other direction depending on the location in the tube bank. For the downward flow of vapor, the condensate inundation rate depends only on the condensation rate at the upper rows. For the other flow directions, it is not possible to estimate the inundation rate accurately.

The mode of condensate inundation depends on the inundation rate and vapor velocity. Discussions on the inundation modes are given by Marto (1984, 1988) and Collier and Thome (1994). The inundation modes are illustrated in Figure 1. At low vapor velocities, condensate drains in discrete drops (Figure 1a), then in condensate columns (Figure 1b), and then in a condensate sheet (Figure 1c) as the inundation rate increases. In a closely packed staggered tube bank, side drainage may occur depending on the condition (Figure 1d). The condensate impinging on the lower tube causes splashing, ripples and turbulence on the condensate film (Figure 1e). At high vapor velocities, the condensate leaving the tube is disintegrated into small drops and impinges on the other tubes (Figure 1f). A wide variety of condensate drainage mode shown in Figure 1 results in a wide breadth of experimental data regarding the effect of condensate inundation. Review of relevant literature are given by Marto (1984, 1988) and Collier and Thome (1994). Butterworth (1981) discusses condensate inundation without vapor shear. Cavallini et al. (1990) give a comparison of proposed correlations with available experimental data.

Figure 2 is a schematic representation of the results of various experimental measurements on the coordinates of
versus n [see Marto (1984)], where
is the mean heat transfer coefficient for a vertical row of n tubes and α_{1} is the heat transfer coefficient for the top tube. In Figure 2, a number of theoretical and empirical relationships are also presented. For a stagnant vapor, Nusselt (1916) extended his analysis of laminar film condensation on a single horizontal plain tube to include the effect of sheet mode drainage (See also Condensation and Condensation of Pure Vapors.) Jakob (1949) generalized the Nusselt analysis and derived the following equation:

The expression for α_{1} is given by Nusselt (1916) as follows:

where g is the gravitational acceleration, ρ_{L} is the liquid density, ρ_{G} is the vapor density, λ_{L} is the liquid thermal conductivity, h_{LG} is the specific enthalpy of evaporation, η_{L} is the liquid dynamic viscosity, D is the tube diameter and ΔT is the condensation temperature difference. Based on the observation of condensate drainage in operating condensers, Kern (1958) suggested a smaller dependence on the row number such that:

Based on the side drainage model, Eissenberg (1972) derived the following equation

Equation 4 predicts a much smaller dependence on the row number than Equations 1 and 3, as shown in Figure 2.

**Figure 2. Effect of vertical row number on the mean heat transfer coefficient of a bank of horizontal plain tubes. (See Marto (1984))**

For condensation without appreciable vapor shear, a number of other methods have also been proposed to account for the effect of condensate inundation. Two well-known empirical equations, which are due to Short and Brown (1951) and Grant and Osment (1968), are respectively written as

where Nu_{f} =
, Re_{f} = 2Г_{n}/η_{L} is the condensate drainage per unit length from the nth tube and y_{n} is the condensation rate per unit length on the nth tube. Butterworth (1981) have shown that Equations (3), (5) and (6) are in very close agreement despite their apparent difference. Marto (1988) and Collier and Thome (1994) recommend Equation (3) for design purposes. When the tube bank is inclined with respect to the horizontal, pendant drops on the tube are driven by gravity toward the lower tube end. As a result, the inundation rate decreases as the angle of inclination increases. This results in an increase in the mean heat transfer coefficient. Figure 3 shows experimental data for steam condensing in 7 tubes wide, n tubes deep (n = 1, 3, 6, and 12) in-line tube banks obtained by Shklover and Buevich (1978). In Figure 3, the ratio of the mean heat transfer coefficient for tube banks with and without tube inclination
is plotted as a function of the inclination angle φ. It is seen from Figure 3 that the effect of tube inclination is more significant for a deeper tube bank.

**Figure 3. Effect of the angle of inclination on the mean heat transfer coefficient of a bank of plain tubes: downward flow of steam. [Data of Shklover and Buevich (1978).]**

At medium to high vapor velocities, condensation in tube banks is considerably affected by the flow direction of vapor. Experimental data showing the effect of vapor flow direction are shown in Figure 4 and are due to Fujii (1981). In Figure 4, the results for downward, horizontal and upward flow of low pressure steam condensing in 5 × 15 (width × depth) in-line and staggered banks of horizontal plain tubes are plotted on the coordinates of
versus F, where Nu = αD/λ_{L}, Re_{L} =
, F =
, α is the mean heat transfer coefficient for a tube row perpendicular to the vapor flow direction,
is the vapor velocity based on the maximum flow cross section (i.e., the velocity calculated in the absence of any tube) just upstream of the tube row and ΔT is the mean condensation temperature difference of the tube row. The results for the downward and horizontal flow are almost identical and are correlated fairly well by the following equation:

The data for the upward flow are lower than those for the downward and horizontal flow. The difference is largest in the region 0.1 < F < 0.5 where the effects of vapor shear and gravity on the condensate flow are of the same order of magnitude. Most of the data for the upward flow in the region of 0.1 < F <1 are even lower than the prediction of the Nusselt (1916) equation, which is written in terms of and F as

For the downward flow of vapor, experimental data showing the combined effects of vapor shear and condensate inundation are presented in Figure 5 and are due to Honda et al. (1988). In Figure 5, the results for near atmospheric R-113 vapor condensing in 3 × 15 in-line and staggered banks of horizontal plain tubes are plotted on the coordinates of Nu_{f} versus Re_{f} with the vapor velocity at the tube bank inlet u_{G,in} as a parameter. The value of Ref is calculated assuming the gravity drained flow model. Two lines in Figure 5 show the Nusselt (1916) equation for a single tube, which is written in terms of Nu_{f} and Re_{f} as

and the following empirical equation for a nearly stagnant vapor:

Equation (10) is based on the experimental data for R-12 and R-21 condensing on vertical columns of horizontal plain tubes with D = 10 ~ 45 mm obtained by Kutateladze and Gogonin (1979) and Kutateladze et al. (1985). The data for moving vapor is close to Eq. (10) at the smallest u_{o},i_{n} and deviates toward a higher value as u_{G,in} increases. For the downward flow of vapor, a number of empirical relationships have been proposed to predict the combined effects of vapor shear and condensate inundation. A common feature of the proposed relationships is that the heat transfer coefficient is calculated from the superposition of two contributions; i.e., gravity drained condensation and vapor shear controlled condensation. Butterworth (1977) proposed the following

where α_{sh} is the heat transfer coefficient in the vapor-shear-controlled regime and α_{g} is the heat transfer coefficient in the gravity-controlled regime. The expression for α_{sh} is written as

where ε is the void fraction for the tube bank (i.e. free volume divided by total volume). The expression for α_{g} is given by Equation (2). Equation (12) is based on the Shekriladze and Gomelauri (1966) analysis for the effect of vapor shear on condensation on horizontal tubes. Fujii and Oda (1986) and Cavallini et al. (1988) also proposed empirical equations in which the inundation effect is expressed in terms of n. Based on the experimental data for atmospheric pressure steam obtained by Nobbs (1975), McNaught (1982) proposed the following empirical equation:

He proposed to model the process in the vapor shear controlled regime as the two-phase forced convection. The derived expression for ash is written as

where α_{L} is the liquid-phase forced convection coefficient across a tube bank and X_{tt} is the Lackhart–Martinelli parameter given by

The expression for α_{g} is given by

where a = 0.13 and 0.22 for the staggered and in-line tube banks, respectively, and α_{1} is given by Equation (2). Based on their own experimental results shown in Figure 5, Honda et al. (1988) derived the following equations:

**Staggered tube bank**where

(18)(19)P

_{t}is the transverse tube pitch, P_{1}is the longitudinal tube pitch, and U_{Gn}is the vapor velocity at the nth row based on the minimum flow cross section. For the first row, the leading coefficient on the right-hand side of Equation 19 should be replaced by 0.13.*In-line tube bank*where

The definitions of Nu_{g}, Ref_{f,sh}, Re_{G}, Re_{L} and u_{Gn} are the same as those for the staggered tube bank. For the first row, the leading coefficient on the right-hand side of Equation 21 should be replaced by 0.042. Cavallini et al. (1990) and Collier and Thome (1994) recommend Equations 17 and 20 for refrigerants. These equations generally give a conservative prediction for steam at low vapor velocities.

**Figure 4. Comparison of data for horizontal, downward and upward flow of steam in in-line and staggered tube banks. [Data of Fujii (1981).]**

**Figure 5. Combined effects of vapor shear and condensate inundation on the local heat transfer coefficient in banks of horizontal plain tubes: downward flow of R-l 13. [Data of Honda et al. (1988)].**

Various enhancement techniques have been proposed for shell-side condensation of organic fluids with low liquid thermal conductivity. A review of relevant literature is given by Webb (1994). A comparison of four kinds of enhancement techniques applied to the condensation of refrigerants in tube banks is shown in Figure 6 and is due to Yabe (1991). In Figure 6, Tube No. 1 is a horizontal *integral-fin tube* with fin density of 2000 fins per meter, Tube No. 2 is the *Thermoexcel-C*^{TM} tube with saw-tooth-shape fins, Tube No. 3 is a vertical *fluted tube* with flute pitch of 1 mm and Tube No. 4 is a vertical plain tube fitted with electrodes for *electrohydro-dynamic enhancement*. At present, the highest heat transfer performance is provided by the horizontal integral-fin tube with optimized fin geometry. A theoretical model of film condensation in banks of horizontal integral-fin tubes is proposed by Honda et al. (1989). (See also Augmentation of Heat Transfer, Two Phase.)

**Figure 6. Comparison of four enhancement techniques for condensation of refrigerants in tube banks. [See Yabe (1991).]**

#### REFERENCES

Butterwortha, D. (1977) Developments in the Design of Shell-and-Tube Condensers, ASME Paper No. 77-WA/HT-24, Atlanta.

Butterworth, D. (1981) Inundation without vapor shear, *Power Condenser Heat Transfer Technology*, P. J. Marto and R. H. Nunn Eds., Hemisphere Publishing Corporation, New York.

Cavallini, A., Frizzerin, S. and Rosetto, L. (1988) Refrigerant 113 Vapor Condensation on a Horizontal Tube Bundle, Inst. di Fisica Tecnica dell' University di Padova, Rept. No. 133.

Cavallini, A., Longo, G. and Rosetto, L. (1990) Condensation of pure saturated vapor on the outside of tube bundles, *Phase-Interface Phenomena in Multiphase Flow*, G. F. Hewitt, F. Mayinger and J. R. Riznic Eds., Hemisphere Publishing Corporation, New York.

Collier, J. G. and Thome, J. R. (1994) *Convective Boiling and Condensation*, 3rd Edn., Clarendon Press, Oxford.

Eissenberg, D. M. (1972) An Investigation of the Variables Affecting Steam Condensation on the Outside of a Horizontal Tube Bundle, Ph. D. Thesis, Univ. of Tennessee, Knoxville.

Fujii, T. (1981) Vapor shear and condensate inundation: An overview, *Power Condenser Heat Transfer Technology*, P. J. Marto and R. H. Nunn Eds., Hemisphere Publishing Corporation, New York.

Fujii, T. and Oda, K. (1986) Correlation equation of heat transfer coefficient for condensate inundation in tube bundles, *Trans. JSME*, 52, 822-826.

Grant, I. D. R. and Osment, B. D. J. (1968) The effect of condensate drainage on condenser performance, NEL Report No. 350, National Engineering Laboratory, East Kilbride.

Honda, H., Uchima, B., Nozu, S., Nakata, H. and Fujii, T. (1988) Condensation of downward flowing R-l 13 vapor on bundles of horizontal smooth tubes, *Trans. JSME*, 54, 1453-1460, and *Heat Transfer-Jap. Res.*, 18, 6, 31–52.

Honda, H., Nozu, S. and Takeda, Y. (1989) A theoretical model of film condensation in a bundle of horizontal low finned tubes, *ASME Journal of Heat Transfer*, 111, 525–532.

Jakob, M. (1949) *Heat Transfer*, Vol. 1, John Wiley and Sons, Inc., New York.

Kern, D. Q. (1958) Mathematical development of loading in horizontal condensers, *AIChE J.*, 4, 157–160.

Kutateladze, S. S. and Gogonin, I. I. (1979) Heat transfer in film condensation of slowly moving vapor, *Int. J. Heat Mass Trans.*, 22, 1593–1599.

Kutateladze, S. S., Gogonin, I. I. and Sosunov, V. I. (1985) The influence of condensate flow rate on heat transfer in film condensation of stationary vapor on horizontal tube banks, *Int. J. Heat Mass Trans.*, 28, 1011–1018.

Marto, P. J. (1984) Heat transfer and two-phase flow during shell-side condensation, *Heat Transfer Engineering*, 5, 31–61.

Marto, P. J. (1988) Fundamentals of condensation, *Two-Phase Flow Heat Exchangers.* S. Kakac, A. E. Bergles and E. O. Fernandes Eds., Kluwer Academic Publishers.

McNaught, J. M. (1982) Two-phase forced convection heat transfer during condensation on horizontal tube bundles, *Proc. 7th Int. Heat Transfer Conf.*, Munich, 5, 125–131.

Nobbs, D. W. (1975) The Effect of Downward Vapor Velocity and Inundation on the Condensation Rates on Horizontal Tubes and Tube Banks, Ph. D. Thesis, Univ. of Bristol, Bristol.

Nusselt, W. (1916) Die Oberflächenkondensation des Wasserdampfes, *Zeit. Ver. Deut. Ing.*, 60, 541-546, 569–575.

Shekriladze, I. G. and Gomelauri, V. I. (1966) Theoretical study of laminar film condensation of flowing vapor, *Int. J. Heat Mass Trans.*, 9, 581–591. DOI: 10.1016/0017-9310(66)90092-5

Shklover, G. G. and Buevich, A. V. (1978) Investigation of steam condensation in an inclined bundle of tubes, *Thermal Eng.*, 49–52.

Short, B. E. and Brown, H. E. (1951) Condensation of vapor on vertical banks of horizontal tubes, *Proc. Inst. Mech. Eng.*, General Discussion on Heat Transfer, 27–31, 1951.

Webb, R. L. (1994) *Principles of Enhanced Heal Transfer*, John Wiley and Sons, Inc., New York.

Yabe, A. (1991) Active Heat Transfer Enhancement by Applying Electric Fields, Proc. 3rd *ASME/JSME Thermal Eng. Joint Conf*, 3, xv-xxiii.