BOOKCOMP, Inc. — John Wiley & Sons / Page 714 / 2nd Proofs / Heat Transfer Handbook / Bejan 714 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [714], (80) Lines: 2747 to 2793 ——— 4.988pt PgVar ——— Normal Page PgEnds: T E X [714], (80) REFERENCES Andreani, M., and Yadigaroglu, G. (1997). A 3-D Eulerian–Lagrangian Model of Dispersed Flow Boiling Including aMechanistic Description of the Droplet Spectrum Evolution, Parts 1 and 2, Int. J. Heat Mass Transfer, 40, 1753–1793. Baker, O. (1954). Design of Pipe Lines for Simultaneous Flow of Oil and Gas, Oil Gas J., July, p. 26. Barnea, D., and Taitel, Y. (1986). Flow Pattern Transition in Two-Phase Gas–Liquid Flows, in Encyclopedia of Fluid Mechanics, Gulf Publishing, Houston, TX, Vol. 3, pp. 403–474. Berenson, P. J. (1960). Transition Boiling Heat Transfer from a Horizontal Surface, MIT Heat Transfer Lab. Tech. Rep. 17. Bergles, A. E. (1996). The Encouragement and Accommodation of High Heat Fluxes, Proc. 2nd European Thermal Sciences and 14th UIT National Heat Transfer Conference 1996, Edizioni ETS, Piso, Vol. 1, pp. 3–11. Bromley, A. L. (1950). Heat Transfer in Stable Film Boiling, Chem. Eng. Prog., 46, 221-227. Carey, V. P. (1992). Liquid–Vapor Phase-Change Phenomena, Hemisphere Publishing (Taylor & Francis), New York. Celata, G. P. (1997). Modeling of the Critical Heat Flux in Subcooled Flow Boiling, presented at the Convective Flow and Pool Boiling Conference, Kloster Irsee, Germany, May 18–23. Chen, J. C. (1963). A Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow, ASME-63-HT-34, ASME, New York. Collier, J. G., and Thome, J. R. (1994). Convective Boiling and Condensation, 3rd ed., Oxford University Press, Oxford, U.K. Cooper, M. G. (1984). Heat Flow Rates in Saturated Nucleate Pool Boiling: A Wide Ranging Examination Using Reduced Properties, in Advances in Heat Transfer, Vol. 16, J. P. Hartnett and T. F. Irvine, eds., Academic Press, New York, pp. 157–239. Delhaye, J. M., Giot, M., and Riethmuller, M. L. (1981). Thermohydraulics of Two-Phase Sys- tems for Industrial Design and Nuclear Engineering, Hemisphere Publishing, Washington, DC. Dhir, V. K., and Liaw, S. P. (1987). Framework for a Unified Model for Nucleate and Transition Pool Boiling, in Radiation, Phase Change Heat Transfer and Thermal Systems, ASME- HTD-81, ASME, New York, pp. 51–58. Dougall, R. S., and Rohsenow, W. M. (1963). Film Boiling on the Inside of Vertical Tubes with Upward Flow of the Fluid at Low Vapor Qualities, Report 9079-26, Mechanical Engineering Department, Engineering Project Laboratory, MIT, Cambridge, MA, Sept. Fair, J. R. (1960). What You Need to Design Thermosyphon Reboilers, Pet. Refiner, 39(2), 105. Forester, H. K., and Zuber, N. (1955). Dynamics of Vapor Bubbles and Boiling Heat Transfer, AIChE J., 1(4), 531–535. Fritz, W. (1935). Berechnung des Maximal Volume von Dampfblasen, Phys. Z., 36, 379–388. Ganic, E. N., and Rohsenow, W. M. (1977). Dispersed Flow Heat Transfer, Int. J. Heat Mass Transfer, 20, 855–866. Ginoux, J. J. (1978). Two-Phase Flows and Heat Transfer with Application to Nuclear Reactor Design Problems, Hemisphere Publishing, Washington, DC. Gnielinski, V. (1976). New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow, Int. Chem. Eng., 16, 359–368. BOOKCOMP, Inc. — John Wiley & Sons / Page 715 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 715 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [715], (81) Lines: 2793 to 2840 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [715], (81) Gorenflo, D. (1993). Pool Boiling, in VDI-Heat Atlas (English version), VDI-Verlag, D ¨ ussel- dorf, Germany. Groeneveld, D. C. (1973). Post-Dryout Heat Transfer at Reactor Operating Conditions, Report AECL-4513, ANS Topical Meeting on Water Reactor Safety, Salt Lake City, UT. Groeneveld, D. C., and Delorme, G. G. J. (1976). Prediction of the Thermal Non-Equilibrium in the Post-dryout Regime, Nucl. Eng. Des., 36, 17–26. Gungor, K. E., and Winterton, R. H. S. (1986). A General Correlation for Flow Boiling in Tubes and Annuli, Int. J. Heat Mass Transfer, 29, 351–358. Gungor, K. E., and Winterton, R. H. S. (1987). Simplified General Correlation for Saturated Flow Boiling and Comparisons of Correlations with Data,Chem. Eng. Res. Des., 65, 148 156. Han, C. Y., and Griffith, P. (1965). The Mechanism of Heat Transfer in Nucleate Pool Boiling, 1: Bubble Initiation, Growth and Departure, Int. J. Heat Mass Transfer, 8, 887. Hewitt, G. F., and Roberts, D. N. (1969). Studies of Two-Phase Flow Patterns by Simultaneous X-ray and Flash Photography, AERE-M 2159, Her Majesty’s Stationery Office, London. Hsu, Y. Y. (1962). On the Size Range of Active Nucleation Cavities on a Heating Surface, J. Heat Transfer, 84, 207–213. Hsu, Y. Y., and Graham, R. W. (1976). Transport Processes in Boiling and Two-Phase Systems, Hemisphere Publishing, Washington, DC. Jung, D. S., McLinden, M., Radermacher, R., and Didion, D. (1989). A Study of Flow Boiling Heat Transfer with Refrigerant Mixtures, Int. J. Heat Mass Transfer, 32(9), 1751–1764. Kandlikar, S. G. (1990). A General Correlation of Saturated Two-Phase Flow Boiling Heat Transfer inside Horizontal and Vertical Tubes, J. Heat Transfer, 112, 219–228. Kattan, N., Thome, J. R., and Favrat, D. (1998a). Flow Boiling in Horizontal Tubes, 1: Devel- opment of a Diabatic Two-Phase Flow Pattern Map, J. Heat Transfer, 120, 140–147. Kattan, N., Thome, J. R., and Favrat, D. (1998b). Flow Boiling in Horizontal Tubes, 2: New Heat Transfer Data for Five Refrigerants, J. Heat Transfer, 120, 148–155. Kattan, N., Thome, J. R., and Favrat, D. (1998c). Flow Boiling in Horizontal Tubes, 3: De- velopment of a New Heat Transfer Model Based on Flow Patterns, J. Heat Transfer, 120, 156–165. Katto, Y. (1994). Critical Heat Flux, Int. J. Multiphase Flow, 20, 53–90. Katto, Y. (1996). Critical Heat Flux Mechanisms, in Convective Flow Boiling, J. C. Chen, ed., Taylor & Francis, Washington, DC, pp. 29–44. Kelvin, Lord (1871). Philos. Mag., 42(4), 448. Keshock, E. G., and Siegel, R. (1964). Forces Acting on Bubbles in Nucleate Boiling under Normal and Reduced Gravity Conditions, NASA-TN-D-2299. Klimenko, V. V. (1988). A Generalized Correlation for Two-Phase Forced Flow Heat Transfer, Int. J. Heat Mass Transfer, 31(3), 541–552. Kutateladze, S. S. (1948). On the Transition to Film Boiling under Natural Convection, Kotlo- turbostroenie, 3, 10. Lienhard, J. E., Sun, K. H., and Dix, G. E. (1976). Low Flow Film Boiling Heat Transfer on Vertical Surfaces, II: Empirical Formulations and Application to BWR-LOCA Analysis, presented at U.S. National Heat Transfer Conference, St. Louis, MO. Lienhard, J. H. (1981). A Heat Transfer Textbook, Prentice-Hall, Englewood Cliffs, NJ, p. 411. Lienhard, J. H., and Dhir, V. K. (1973a). Extended Hydrodynamic Theory of the Peak and Minimum Pool Boiling Heat Fluxes, NASA-CR-2270, July. BOOKCOMP, Inc. — John Wiley & Sons / Page 716 / 2nd Proofs / Heat Transfer Handbook / Bejan 716 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [716], (82) Lines: 2840 to 2880 ——— 4.0pt PgVar ——— Normal Page PgEnds: T E X [716], (82) Lienhard, J. H., and Dhir, V. K. (1973b). Hydrodynamic Prediction of Peak Pool Boiling Heat Fluxes from Finite Bodies, J. Heat Transfer, 95, 152–158. Mikic, B. B., Rohsenow, W. M., and Griffith, P. (1970). On Bubble Growth Rates, Int. J. Heat Mass Transfer, 13, 657–665. Mostinski, I. L. (1963). Application of the Rule of Corresponding States for Calculation of Heat Transfer and Critical Heat Flux, Teploenergetika, 4, 66. Nukiyama, S. (1934). The Maximum and Minimum Values of Heat Q Transmitted from Metal to Boiling Water under Atmospheric Pressure, J. Jpn. Soc. Mech. Eng., 37, 367–374 (in Japanese). Trans. in Int. J. Heat Mass Transfer, 9, 1419–1433 (1966). Palen, J. W. (1983). Kettle and Internal Reboilers, in Heat Exchanger Design Handbook, Vol. 3, E. U. Schl ¨ under, ed., Hemisphere Publishing, New York, pp. 3.6.1.1–3.6.5.6. Plesset, M. S., and Zwick, S. A. (1954). The Growth of Vapor Bubbles in Superheated Liquids, J. Appl. Phys., 25, 493–500. Rayleigh, Lord (1917). On the Pressure Developed in a Liquid during the Collapse of a Spherical Cavity, Philos. Mag., 34, 94–98; Scientific Papers, Vol. 6, Cambridge University Press, Cambridge, 1920, p. 504. Reid, R. C., Prausnitz, J. M., and Poling, B. E. (1987). The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York. Rohsenow, W. M. (1962). A Method of Correlating Heat Transfer Data for Surface Boiling of Liquids, Trans. ASME, 74, 969–975. Rohsenow, W. M. (1973). Boiling, in Handbook of Heat Transfer, W. M. Rohsenow and J. P. Hartnett, eds., McGraw-Hill, New York, Sec. 13. Rouhani, Z., and Axelsson, E. (1970). Calculation of Volume Void Fraction in the Subcooled and Quality Region, Int. J. Heat Mass Transfer, 13, 383–393. Schl ¨ under, E. U. (1983). Heat Transfer in Nucleate Boiling of Mixtures, Int. Chem. Eng., 23(4), 589–599. Shah, M. M. (1982). Chart Correlation for Saturated Boiling Heat Transfer Equations and Further Study, ASHRAE Trans., 88, pt. 1, 185–196. Shakir, S., and Thome, J. R. (1986). Boiling Nucleation of Mixtures on Smooth and Enhanced Surfaces, Proc. 8th International Heat Transfer Conference, San Francisco, Vol. 4, pp. 2081–2086. Steiner, D. (1993). Heat Transfer to Boiling Saturated Liquids, in VDI-W ¨ armeatlas (VDI Heat Atlas), for Verein Deutscher Ingenieure, VDI-Gessellschaft Verfahrenstechnik und Chemie- ingenieurwesen, D ¨ usseldorf, Germany. Transl. J. W. Fullarton. Steiner, D., and Taborek, J. (1992). Flow Boiling Heat Transfer in Vertical Tubes Correlated by an Asymptotic Model, Heat Transfer Eng., 13(2), 43–69. Stephan, K., and Abdelsalam, M. (1980). Heat Transfer Correlations for Natural Convection Boiling, Int. J. Heat Mass Transfer, 23, 73–87. Taitel, Y., and Dukler, A. E. (1976). A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas–Liquid Flow, AIChE J., 22(2), 43–55. Thome, J. R. (1989). Prediction of the Mixture Effect on Boiling in Vertical Thermosyphon Reboilers, Heat Transfer Eng., 12(2), 29–38. Thome, J. R. (1990). Enhanced Boiling Heat Transfer, Hemisphere Publishing (Taylor & Francis), Washington, DC. Thome, J. R. (1995a). Flow Boiling in Horizontal Tubes: A Critical Assessment of Current BOOKCOMP, Inc. — John Wiley & Sons / Page 717 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 717 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [717], (83) Lines: 2880 to 2919 ——— * 40.87401pt PgVar ——— Normal Page * PgEnds: PageBreak [717], (83) Methodologies, Proc. International Symposium of Two-Phase Flow Modelling and Exper- imentation, Rome, Vol. 1, pp. 41–52. Thome, J. R. (1995b). Comprehensive Thermodynamic Approach to Modelling Refrigerant– Lubricating Oil Mixtures, Int. J. HVAC&R Res., 1(2), 110–126. Thome, J. R. (1999). Flow Boiling inside Microfin Tubes: Recent Results and Design Methods, in Heat Transfer Enhancement of Heat Exchangers, S. Kakac¸ et al., eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, Series E: Applied Sciences, Vol. 355, pp. 467–486. Thome, J. R., and Shock, R. A. W. (1984). Boiling of Multicomponent Liquid Mixtures, in Advances in Heat Transfer, J. P. Harnett and T. F. Irvine, eds., Academic Press, New York, Vol. 16, pp. 59–156. Tong, L. S. (1965). Boiling Heat Transfer and Two-Phase Flow, Wiley, New York. Vachon, R. I., Nix, G. H., and Tanger, G. E. (1967). Evaluation of Constants for the Rohsenow Pool Boiling Correlation, 67-HT-33, U.S. National Heat Transfer Conference. van Stralen, S. J. D. (1966). The Mechanism of Nucleate Boiling in Pure Liquids and in a Binary Mixture, Parts I and II, Int. J. Heat Mass Transfer, 9, 995–1046. van Stralen, S. J. D., and Cole, R. (1979). Boiling Phenomena, Hemisphere Publishing, Wash- ington, DC. Wallis, G. B. (1969). One Dimensional Two-Phase Flow, McGraw-Hill, New York. Wallis, G. B., and Collier, J. G. (1968). Two-Phase Flow and Heat Transfer, Vol. 3, pp. 33–46, Notes for a Summer Shortcourse, July 15–26, Dartmouth College, Hanover, NH. Webb, R. L. (1994). Principles of Enhanced Heat Transfer, Wiley, New York. Webb, R. L. (1999). Prediction of Condensation and Evaporation in Micro-fin and Micro- channel Tubes, in Heat Transfer Enhancement of Heat Exchangers, S. Kakac¸ et al., eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, Series E: Applied Sciences, Vol. 355, pp. 529–550. Weisman, J. (1992). The Current Status of Theoretically Based Approaches to the Prediction of the Critical Heat Flux in Flow Boiling, Nucl. Technol., 99, July, 1–21. Whalley, P. (1987). Boiling, Condensation and Gas–Liquid Flow, Oxford University Press, Oxford. Witte, L. C., and Lienhard, J. W. (1982). On the Existence of Two “Transition” Boiling Curves, Int. J. Heat Mass Transfer, 25, 771–779. Zuber, N. (1959). Hydrodynamic Aspects of Boiling Heat Transfer, AEC Rep. AECU-4439, Physics and Mathematics, Atomic Energy Commission, Washington, DC. Z ¨ urcher, O., Thome, J. R., and Favrat, D. (1997). Prediction of Two-Phase Flow Patterns for Evaporation of Refrigerant R-407C inside Horizontal Tubes, Paper IX-1, Convective Flow and Pool Boiling Conference, Engineering Foundation, Irsee, Germany. Z ¨ urcher, O., Thome, J. R., and Favrat, D. (1998). Intube Flow Boiling of R-407C and R- 407C/Oil Mixtures, II: Plain Tube Results and Predictions, Int. J. HVAC&R Res., 4(4), 373–399. Z ¨ urcher, O., Thome, J. R., and Favrat, D. (1999). Evaporation of Ammonia in a Smooth Horizontal Tube: Heat Transfer Measurements and Predictions, J. Heat Transfer, 121, 89– 101. BOOKCOMP, Inc. — John Wiley & Sons / Page 719 / 2nd Proofs / Heat Transfer Handbook / Bejan 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [First Page] [719], (1) Lines: 0 to 98 ——— -0.81792pt PgVar ——— Normal Page PgEnds: T E X [719], (1) CHAPTER 10 Condensation M. A. KEDZIERSKI Building and Fire Research Laboratory National Institute of Standards and Technology Gaithersburg, Maryland J. C. CHATO Department of Mechanical and Industrial Engineering University of Illinois–Urbana-Champaign Urbana, Illinois T. J. RABAS Consultant Downers Grove, Illinois 10.1 Introduction 10.2 Vapor space film condensation 10.2.1 Nusselt’s analysis of a vertical flat plate 10.3 Film condensation on low fins 10.3.1 Introduction 10.3.2 Surface tension pressure gradient 10.3.3 Specified interfaces 10.3.4 Bond number 10.4 Film condensation on single horizontal finned tubes 10.4.1 Introduction 10.4.2 Trapezoidal fin tubes 10.4.3 Sawtooth fin condensing tubes 10.5 Electrohydrodynamic enhancement 10.5.1 Introduction 10.5.2 Vapor space EHD condensation 10.5.3 In-tube EHD condensation 10.6 Condensation in smooth tubes 10.6.1 Introduction 10.6.2 Flow regimes in horizontal tubes Flow regimes in horizontal two-phase flow Effects of mass flux and quality Effects of fluid properties and tube diameter Potential role of surface tension Flow regime mapping Comparison of flow regime maps 719 BOOKCOMP, Inc. — John Wiley & Sons / Page 720 / 2nd Proofs / Heat Transfer Handbook / Bejan 720 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [720], (2) Lines: 98 to 166 ——— -5.03pt PgVar ——— Normal Page PgEnds: T E X [720], (2) 10.6.3 Heat transfer in horizontal tubes Effects of mass flux and quality Effects of tube diameter Effects of fluid properties Effects of temperature difference Gravity-driven condensation Shear-driven annular flow condensation Comparison of heat transfer correlations 10.6.4 Pressure drop 10.6.5 Effects of oil 10.6.6 Condensation of zeotropes 10.6.7 Inclined and vertical tubes 10.7 Enhanced in-tube condensation 10.7.1 Microfin tubes 10.7.2 Microfin tube pressure drop 10.7.3 Twisted-tape inserts 10.8 Film condensation on tube bundles 10.8.1 X-shell condensers (shell-side condensation) Tube-side flow and temperature maldistribution Condenser sizing methods Noncondensable gas management and proper venting techniques 10.8.2 In-tube condensers Nonuniform outside inlet flow and temperature distributions Noncondensable gas pockets 10.9 Condensation in plate heat exchangers 10.9.1 Introduction 10.9.2 Steam condensation heat transfer 10.9.3 Effect of inclination on heat transfer performance 10.9.4 Effect of inclination on pressure drop Appendix A Nomenclature References 10.1 INTRODUCTION Condensation is the process by which a vapor is converted to its liquid state. Because of the large internal energy difference between the liquid and vapor states, a signifi- cant amount of heat can be released during the condensation process. For this reason, the condensation process is used in many thermal systems. In general, a vapor will condense to liquid when it is cooled sufficiently or comes in contact with something (e.g., a solid or another fluid) that is below its equilibrium temperature. This chapter is concerned primarily with convective condensation (con- densation of a flowing vapor in a passage) and vapor space condensation (condensa- tion of stagnate vapor onto a surface). Film condensation occurs when the condensate BOOKCOMP, Inc. — John Wiley & Sons / Page 721 / 2nd Proofs / Heat Transfer Handbook / Bejan VAPOR SPACE FILM CONDENSATION 721 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [721], (3) Lines: 166 to 178 ——— 1.627pt PgVar ——— Normal Page PgEnds: T E X [721], (3) completely wets the surface in a continuous liquid film and can be associated with either convective or vapor space condensation. Dropwise condensation, usually as- sociated with vapor space condensation, occurs when the condensate “beads up” on the surface into drops of liquid as a consequence of the liquid’s lack of affinity for the surface. Heat transfer coefficients for dropwise condensation can be one to two or- ders of magnitude greater than those for film condensation. Unfortunately, dropwise condensation is not easily sustained in practice. 10.2 VAPOR SPACE FILM CONDENSATION 10.2.1 Nusselt’s Analysis of a Vertical Flat Plate Nusselt (1916) published a solution for steady-state laminar film condensation on a vertical flat plate. This pioneering work laid the foundation on which those working in the field of condensation still build their research. The cross section of the liquid film as analyzed by Nusselt is shown in Fig. 10.1. The vapor condenses at its saturation temperature (T sat ) due to a cooler wall temperature (T w ) of the vertical plate. The thickness of the condensate film (δ) increases along the length (s) due to mass transfer to the liquid–vapor interface. The film is drained by the influence of gravity alone in the downward s direction. Consequently, the film velocities in the y and z directions can be neglected compared to the velocity in the s direction (u). Moreover, the T w T sat dm . im mi ss ϩ d ds ds . im ss . d= dT dy | (ds)sq kЉ w w Ϫ y=0 z y s ds T(y) u(y) du dy y=␦ =0 g Energy balance Saturated vapor Liquid-vapor interface Liquid film Wall surface ␦(s) ⌫(s) Figure 10.1 Nusselt condensation on vertical flat plate. BOOKCOMP, Inc. — John Wiley & Sons / Page 722 / 2nd Proofs / Heat Transfer Handbook / Bejan 722 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [722], (4) Lines: 178 to 239 ——— 0.30623pt PgVar ——— Normal Page * PgEnds: Eject [722], (4) velocity and temperature changes in the y direction are much larger than those in the s and z directions. Accordingly, both momentum changes and convection in the s direction are negligible. As a result, if there is no shear stress at the liquid–vapor interface, the s-moment equation becomes µ l d 2 u dy 2 = dP ds − ρ l g (10.1) Equation (10.1) shows that the viscous forces are balanced by the sum of the gravity force (ρ l g) and pressure gradient in the s direction (dP /ds). The momentum equation can be integrated while using the no-slip condition at the wall and no shear at the liquid–vapor interface to yield the velocity profile of the film: u = y µ l y 2 − δ dP ds − ρ l g (10.2) The liquid mass flow rate of the film per unit width (Γ)is Γ = δ 0 ρ l u s dy = δ 3 3ν l dP ds − ρ l g (10.3) Differentiating eq. (10.3) gives dΓ ds = 1 3ν 1 d ds δ 3 dP ds − ρ l g (10.4) An energy balance on an incremental element of the film is shown in Fig. 10.1. If the convection of heat along the s direction is neglected, the heat balance becomes λ dΓ =−k l dT dy ds (10.5) where λ,k l , and T are the latent heat, thermal conductivity, and temperature of the film, respectively. By applying the Nusselt assumptions to the energy equation, it reduces to dT dy = T sat − T w δ (10.6) where T sat is the saturation temperature of the condensate and T w is the local wall temperature. Rearranging the incremental energy balance and substituting the temperature gra- dient gives the gradient of the mass flow rate: dΓ ds = −k l (T sat − T w ) λδ (10.7) Equating the foregoing two expressions for the mass flow rate gradient yields the ordinary differential equation that models Nusselt condensation: BOOKCOMP, Inc. — John Wiley & Sons / Page 723 / 2nd Proofs / Heat Transfer Handbook / Bejan FILM CONDENSATION ON LOW FINS 723 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [723], (5) Lines: 239 to 279 ——— 5.88013pt PgVar ——— Normal Page PgEnds: T E X [723], (5) d ds δ 3 dP ds − ρ l g = −3ν l k l (T sat − T w ) λδ (10.8) By applying the Nusselt assumption to the s-momentum equation for the vapor phase, one obtains dP/ds = ρ g g; and using the expression δ(dδ 3 /ds) = 3 4 (dδ 4 /ds), the solution to eq. (10.8) becomes δ = 4k l µ l (T sat − T w )s ρ l g(ρ l − ρ g )λ 1/4 (10.9) For film condensation, the convection of heat along length s can be neglected com- pared to conduction across the film. For these conditions, the temperature gradient of the film is approximately linear, and the heat transfer coefficient is k l /δ. Con- sequently, the condensation heat transfer coefficient for a vertical flat plate for no interfacial shear is h = ρ l g(ρ l − ρ g )k 3 l λ 4µ l (T sat − T w )s 1/4 (10.10) The average condensation heat transfer coefficient over s = 0tos = L is h = 0.943 ρ l g(ρ l − ρ g )k 3 l λ µ l (T sat − T w )L 1/4 (10.11) Equation (10.11) gives the heat transfer coefficient for laminar film condensation on a vertical flat plate for low pressures (ρ v ρ g ) and c p,1 (T sat − T w )/λ < 1. The average condensation heat transfer coefficient for a horizontal tube of outer diameter D o is obtained by replacing g with g sin φ and integrating around the tube with respect to the cylindrical coordinate angle φ from 0 to 180°: h = 0.729 ρ l g(ρ l − ρ g )k 3 l λ µ l (T sat − T w )D o 1/4 (10.12) 10.3 FILM CONDENSATION ON LOW FINS 10.3.1 Introduction The first application of a finned surface to condense vapor was probably done with the intent to enhance the heat transfer via additional surface area for a given projected area. Today, low fins (<1.5 mm) are specially designed to enhance condensation significantly by inducing surface tension drainage forces that rid the fins of insulating condensate. Gravity forces are always present, but the influence of gravity on the BOOKCOMP, Inc. — John Wiley & Sons / Page 724 / 2nd Proofs / Heat Transfer Handbook / Bejan 724 CONDENSATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [724], (6) Lines: 279 to 303 ——— -2.85397pt PgVar ——— Long Page PgEnds: T E X [724], (6) condensate drainage from a fin can be minimal for short or low fins. As described in the remainder of this section, the minimum fin height to encourage surface tension drainage depends on the surface tension of the fluid and shape of the fin. 10.3.2 Surface Tension Pressure Gradient Laplace (1966) has shown that if a liquid–vapor interface is curved, a pressure dif- ference across the interface must be present to establish mechanical equilibrium of the interface. The equilibrium condition is described by the difference between the pressure of the liquid (P l ) and the pressure of the vapor (P g ) by the two radii of cur- vature (r) of the interface and the surface tension of the fluid liquid–vapor interface (σ). This may be represented as P 1 − P g = σ 1 r 1 + 1 r 2 (10.13) The radii of curvature are defined as positive from the liquid side of the interface. Figure 10.2 shows r 1 as the curvature in the Y–X plane; r 2 is the curvature in the X–Z plane. Consider, for example, a still pond. Because the surface of a pond is flat or of infinite radius, eq. (10.13) predicts that the pressure of the water at the liquid side of the liquid–vapor interface is equal to the pressure of the surrounding air. Similarly, eq. (10.13) shows that the pressure of a curved liquid film for positive radii r 1 and r 2 must be greater than the vapor that surrounds it. A greater difference between the pressure of the liquid and vapor occurs for large surface tension and small radii of curvature (large curvature). The liquid–vapor interface of acondensate film must have a curvature (κ i ) decrease along the fin length to exhibit surface tension drainage. The curvature decrease of Figure 10.2 Convex fin and condensate film. . Boiling Heat Transfer with Refrigerant Mixtures, Int. J. Heat Mass Transfer, 32(9), 1751–1764. Kandlikar, S. G. (1990). A General Correlation of Saturated Two-Phase Flow Boiling Heat Transfer. New Heat Transfer Data for Five Refrigerants, J. Heat Transfer, 120, 148–155. Kattan, N., Thome, J. R., and Favrat, D. (1998c). Flow Boiling in Horizontal Tubes, 3: De- velopment of a New Heat Transfer. (1962). A Method of Correlating Heat Transfer Data for Surface Boiling of Liquids, Trans. ASME, 74, 969–975. Rohsenow, W. M. (1 973) . Boiling, in Handbook of Heat Transfer, W. M. Rohsenow and J.