0
Research Papers: Evaporation, Boiling, and Condensation

Using Generalized Dimensional Analysis to Obtain Reduced Effective Model Equations for Condensation in Slender Tubes With Rotational Symmetry

[+] Author and Article Information
Andrea Dziubek

Assistant Professor
Department of Engineering,
Science, and Mathematics,
SUNY Institute of Technology,
Utica, NY 13502
e-mail: dziubea@sunyit.edu

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received February 12, 2012; final manuscript received December 27, 2012; published online April 9, 2013. Assoc. Editor: Louis C. Chow.

J. Heat Transfer 135(5), 051501 (Apr 09, 2013) (11 pages) Paper No: HT-12-1053; doi: 10.1115/1.4023350 History: Received February 12, 2012; Revised December 27, 2012

In this paper, we study the continuum physics model equations for condensation (two phase flow problems) in vertical tubes with small diameter and obtain reduced model equations. We found that generalization of dimensional analysis to multiple spatial dimensions is an excellent tool for that purpose, so that a review of this method is also part of the paper. We obtain the nondimensional numbers of the problem and derive reduced bulk and interface equations. The problem is characterized by three length scales, tube radius R, tube length L, and initial film thickness H. For small ratio ɛ=H/L, we derive a single ordinary differential equation for the condensate film thickness as function of axial position with tube radius as parameter, which agrees well with commonly used (parametric) models from literature. Our model is based on the physical dimensions of the problem which gives a greater geometrical flexibility and a wider range of applicability. We also discuss the effect of surface tension and the limit of the model.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Carey, V. P., 1992, Liquid-Vapor Phase-Change Phenomena, Hemisphere Publishing Cooperation, New York.
Kandlikar, S. G., Shoji, M., and Dhir, V. K., 1999, Handbook of Phase Change: Boiling and Condensation, Taylor & Francis, London.
Bergles, A. E., 2011, “Stability and Enhancement of Boiling in Microchannels,” ASME 9th International Conference on Nanochannels, Microchannels, and Minichannels. [CrossRef]
Barbosa, J. R., 2011, “Recent Developments in Vapor Compression Technologies for Small Scale Refrigeration Applications,” ASME 9th International Conference on Nanochannels, Microchannels, and Minichannels. [CrossRef]
Collier, J. G., Thome, J. R., 1994, Convective Boiling and Condensation, 3rd ed., Clarendon Press, Oxford, UK.
Fiedler, S., and Auracher, H., 2004, “Experimental and Theoretical Investigation of Reflux Condensation in an Inclined Small Diameter Tube,” Int. J. Heat Mass Transfer, 47(19–20), pp. 4031–4043. [CrossRef]
Wang, B., and Du, X., 1999, “Heat Transfer Characteristics for Flow Condensation in Vertical Thin Tube,” J. Shanghai Jiaotong Univ., 33(8), pp. 970–973.
Wang, B., and Du, X., 2003, “Study on Transport Phenomena for Flow Film Condensation in Vertical Mini-Tube With Interfacial Waves,” Int. J. Heat Mass Transfer, 46(11), pp. 2095–2101. [CrossRef]
Feng, Z., and Serizawa, A., 1999, “Visualization of Two-Phase Flow Patterns in an Ultra-Small Tube,” Proceedings of the 18th Multiphase Flow Symposium of Japan, Osaka, Japan, pp. 33–36.
Zhao, T. S., and Liao, Q., 2002, “Theoretical Analysis of Film Condensation Heat Transfer Inside Vertical Mini Triangular Channels,” Int. J. Heat Mass Transfer, 45(13), pp. 2829–2842. [CrossRef]
Panday, P. K., 2003, “Two-Dimensional Turbulent Film Condensation of Vapours Flowing Inside a Vertical Tube and Between Parallel Plates: A Numerical Approach,” Int. J. Refrig., 26(4), pp. 492–503. [CrossRef]
Nebuloni, S., and Thome, J. R., 2010, “Numerical Modeling of Laminar Annular Film Condensation for Different Channel Shapes,” Int. J. Heat Mass Transfer, 53(13–14), pp. 2615–2627. [CrossRef]
Nebuloni, S., and Thome, J. R., 2012, “Numerical Modeling of the Conjugate Heat Transfer Problem for Annular Laminar Film Condensation in Microchannels,” ASME J. Heat Trans., 134(5), p. 051021. [CrossRef]
Narain, A., and Phan, L., 2007, “Nonlinear Stability of the Classical Nusselt Problem of Film Condensation and Wave Effects,” ASME J. Appl. Mech., 74(2), pp. 279–290. [CrossRef]
Pan, Y., 2001, “Condensation Characteristics Inside a Vertical Tube Considering the Presence of Mass Transfer, Vapor Velocity and Interfacial Shear,” Int. J. Heat Mass Transfer, 44(23), pp. 4475–4482. [CrossRef]
Benjamin, T. B., 1957, “Wave Formation in a Laminar Flow Down an Inclined Plane,” J. Fluid Mech., 2(6), pp. 554–573. [CrossRef]
Yih, C. S, 1963, “Stability of Liquid Flow Down an Inclined Plane,” Phys. Fluids, 6(3), pp. 321–334. [CrossRef]
Lin, S. P., 1975, “Stability of Liquid Flow Down a Heated Inclined Plane,” Lett. Heat Mass Transfer, 2(5), pp. 361–369. [CrossRef]
Marschall, E., and Lee, C. Y., 1973, “Stability of Condensate Flow Down a Vertical Wall,” Int. J. Heat Mass Transfer, 16(1), pp. 41–48. [CrossRef]
Ünsal, M., and Thomas, W. C, 1978, “Linearized Stability Analysis of Film Condensation,” ASME J. Heat Trans., 100(4), pp. 629–634. [CrossRef]
Spindler, B., 1982, “Linear Stability of Liquid Films With Interfacial Phase Change,” Int. J. Heat Mass Transfer, 25(2), pp. 161–173. [CrossRef]
Burelbach, J. P., Bankoff, S. G., and Davis, S. H., 1988, “Nonlinear Stability of Evaporating/Condensing Liquid Films,” J. Fluid Mech., 195, pp. 463–494. [CrossRef]
Joo, S. W., Davis, S. H., and Bankoff, S. G., 1991, “Long-Wave Instabilities of Heated Falling Films: Two-Dimensional Theory of Uniform Layers,” J. Fluid Mech., 230, pp. 117–146. [CrossRef]
Hwang, C. W., and Weng, C. I., 1987, “Finite-Amplitude Stability Analysis of Liquid Films Down a Vertical Wall With and Without Interfacial Phase Change,” Int. J. Multiphase Flow., 13(6), pp. 803–814. [CrossRef]
Fieg, G. P., and Roetzel, W., 1994, “Calculation of Laminar Film Condensation In/On Inclined Elliptical Tubes,” Int. J. Heat Mass Transfer, 37(4), pp. 619–624. [CrossRef]
Mosaad, M., 1999, “Combined Free and Forced Convection Laminar Film Condensation on an Inclined Circular Tube With Isothermal Surface,” Int. J. Heat Mass Transfer, 42(21), pp. 4017–4025. [CrossRef]
Wang, B., and Du, X., 2000, “Study on Laminar Film-Wise Condensation for Vapor Flow in an Inclined Small/Mini-Diameter Tube,” Int. J. Heat Mass Transfer, 43(10), pp. 1859–1868. [CrossRef]
Alekseenko, S. V., Nakoryakov, V. E., and Pokusaev, B. G, 1994, Wave Flow of Liquid Films, Begel House, New York.
Yoshimura, P. N., Nosoko, T., and Nagata, T., 1996, “Enhancement of Mass Transfer Into a Falling Laminar Liquid Film by Two-Dimensional Surface Waves—Some Experimental Observations and Modeling,” Chem. Eng. Sci., 51(8), pp. 1231–1240. [CrossRef]
Dziubek, A., 2012, “Equations for Two-Phase Flows: A Primer,” Meccanica, 47(8), pp. 1819–1836. [CrossRef]
Neemann, K., and Schade, H., 2001, “Dimensionsanalyse, Grundlagen und Anwendungen,” TUB Herman-Föttinger-Institut für Strömungsmechanik, TU Berlin, Germany.
Schlichting, H., 1979, Boundary Layer Theory, McGraw-Hill, New York.
Nusselt, W., “Die Oberflächenkondensation des Wasserdampfes,” Z. Ver. Dtsch. Ing., 60(27), pp. 541–575.
Bejan, A, 2004, Convection Heat Transfer, 3rd ed., Wiley, New York, p. 462ff.
Dalkilic, A. S., and Wongwises, S., 2009, “Intensive Literature Review of Condensation Inside Smooth and Enhanced Tubes,” Int. J. Heat Mass Transfer, 52(15–16), pp. 3409–3426. [CrossRef]
Wei, X., Fang, X., and Shi, R., 2012, “A Comparative Study of Heat Transfer Coefficients for Film Condensation,” Energy Sci. Technol., 3(1), pp. 1–9.
Shah, M. M., 2009, “An Improved and Extended General Correlation for Heat Transfer During Condensation in Plain Tubes,” HVAC&R Res., 15(5), pp. 889–913. [CrossRef]
Chen, S. L., Gerner, F. M., and Tien, C. L., 1987, “General Film Condensation Correlations,” Exp. Heat Transfer, 1(2), pp. 93–107. [CrossRef]
Dullin, H. R., Gottwald, G. A., and Holm, D. D., 2003, “Camassa-Holm, Korteweg-de Vries-5 and Other Asymptotically Equivalent Equations for Shallow Water Waves,” Fluid Dyn. Res., 33(1–2), pp. 73–95. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Vertical tube with condensate and vapor

Grahic Jump Location
Fig. 3

Water: Nusselt (84), ODE without (83), and with (82) surface tension

Grahic Jump Location
Fig. 4

Water: Nusselt (84), ODE without (83), and with (82) surface tension for d = 60 mm and d = 7 mm

Grahic Jump Location
Fig. 5

R134a: Nusselt (84), ODE without (83), and with (82) surface tension

Grahic Jump Location
Fig. 6

Local Nusselt number of Water and R134a

Grahic Jump Location
Fig. 2

Wavy condensate film

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In