Skip to main content
SpringerLink
Log in

Reynolds Analogy

  • Chapter
  • First Online:
Bubble Systems

Part of the book series: Mathematical Engineering ((MATHENGIN))

  • 1403 Accesses

Abstract

We give a coherent treatment of the hydrodynamic theory of heat exchange based on the perceptions about the unity of the mechanisms of turbulent transfer of heat and momentum (the Reynolds analogy). A conclusion is made that for nonequilibrium two-phase flows the Reynolds analogy is the natural method of describing heat transfer processes, which is capable of applying the results of theoretical analysis to obtain relatively simple analytic solutions. Two extreme cases of motion of two-phase bubble mixture in pipes are singled out: the sliding bubble flow and the coring bubble flow. For each of these limiting regimes of bubble flow, it proved possible to construct, a closed hydrodynamic model of the flow, which is capable of determining both the velocity profiles and the magnitudes of the hydrodynamic drag. Approximate formulas describing the results of accurate analytic studies are obtained. The calculations results and the set of available experimental data were found to be in a good agreement. The relations obtained are used to derive equations for the Reynolds analogy for nonequilibrium flows of a two-phase mixture. A new similarity criterion is obtained taking into account the relative role of the convective and conductive heat transfer mechanisms for flows with “double disequilibrium” (superheated near-wall layer of liquid– subcooled liquid in the core). The potency of this approach is illustrated by solving a number of problems on physics of surface boiling. Besides, it proved possible to obtain analytic solutions of such problems, no introduction of empirical constants matched from experimental data begin made. A solution to the problem of maximal (over the growth—condensation cycle) diameter of parietal bubbles is put forward. A comparison with the set of available experimental data with liquid subcoolings 3–80 K, flow velocities 0.2–9.2 m/s, and densities of heat flux 0.38–8.53 MW/m2 showed a good agreement. The solution of the problem of pressure losses for flow boiling of subcooled liquid showed a good matching both for parameters that are characteristic of power units (velocities 0.5–2.0 m/s, heat fluxes up to 1.5 MW/m2, pressures up to 14 MPa) and for high-performance heat exchange systems (velocities up to 20 m/s, heat fluxes up to 40 MW/m2, pressures up to 14.7 MPa). A consistent use of the Reynolds analogy has enabled us to solve the problem of heat transfer and hydrodynamics of film boiling under forced motion conditions. The relation thus obtained shows a good match with the available experimental data for water (pressures 0.1–21.6 MPa, flow velocities up to 14.2 m/s, densities of heat flux up to 81.2 MW/m2), liquid helium and nitrogen. It is shown that for elevated velocities and subcoolings of liquid, in the regime of film boiling one may, without destruction of the heat-transfer surface, remove heat fluxes exceeding 100 MW/m2. A conclusion is made that the application of the Reynolds analogy holds the key to solving a number of other problems in physics of nucleate boiling.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We recall that in the classical courses in hydrodynamics, the boundary of viscous sublayer is usually assumed to be \( \delta_{*} = 11.7l_{*} \).

  2. 2.

    The laws of turbulent micro motions were considered in detail in Chap. 5.

  3. 3.

    Since the appearance of the paper by Avdeev (1982), several different explicit expressions for the friction coefficient were put forward; see, for example, Sonnad and Goudar (2007). However, it has a more involved structure and, in a some cases, a lower accuracy than formula (11.48).

  4. 4.

    As one could expect from physical considerations, these data give values of \( d_{\hbox{max} } \) that are 2–3 times smaller than those predicted by formula (11.66).

  5. 5.

    Up to a small supplement due to the variation over the channel length of the mean size of near-wall bubbles and the mean density of their packing.

  6. 6.

    There is a relatively narrow region of influence of the velocity of liquid forced motion on the heat transfer during boiling, which can be taken into account, for example, in accordance with the recommendations by Labuntsov and Yagov (1978).

  7. 7.

    Excluding the special case of boiling in deep vacuum.

References

  • Abdelmessih, A.H., Hooper, F.C., Nangia, S.: Flow effects on bubble growth and collapse in surface boiling. Int. J. Heat Mass Transf. 15(1), 115–125 (1972)

    Article  Google Scholar 

  • Armand, A.A., Nevstrueva E.I.: Investigation of two phase flow in vertical tube. Izv. VTI (Proc. Dzerzhinskiy All-Union Power Eng. Inst.). 2, 1–8 (1950) (In Russian)

    Google Scholar 

  • Auton, T.R.: The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199–218 (1987)

    Article  MATH  Google Scholar 

  • Avdeev, A.A.: Reynolds analogy for undeveloped forced flow surface boiling. Therm. Eng. 3, 23–26 (1982)

    Google Scholar 

  • Avdeev, A.A.: Hydrodynamics of turbulent flows of bubble two-phase mixture. High Temp. 21(4), 707–715 (1983)

    Google Scholar 

  • Avdeev, A.A.: Heat transfer and hydraulic drag under film boiling of subcooled liquid in channels. Therm. Eng. 4, 39–42 (1986a)

    Google Scholar 

  • Avdeev, A.A.: Application of Reynolds analogy to the study of forced flow surface boiling. High Temp. 24(1), 111–119 (1986b)

    Google Scholar 

  • Avdeev, A.A.: The boundaries of the area of influence of surface boiling on steam quality and hydraulic resistance. Therm. Eng. 9, 44–47 (1988)

    Google Scholar 

  • Avdeev, A.A., Pehterev, V.P.: Hydraulic resistance when subcooled forced flow boiling. Therm. Eng. 6, 49–52 (1985)

    Google Scholar 

  • Avdeev, A.A., Pehterev, V.P.: Forced flow subcooled liquid boiling. High Temp. 24(5), 912–920 (1986)

    Google Scholar 

  • Avdeev, A.A., Pehterev, V.P.: Criterial equation for calculating the subcooled boiling flows. Therm. Eng. 2, 20–25 (1987)

    Google Scholar 

  • Bartolomey, G.G.: Experimental research and development of methods of calculation of heat transfer and hydrodynamics of forced flow subcooled boiling. Dr.-Ing. Hab. Thesis. Moscow, MEI (Moscow Power Energetic Univ.) (1989) (In Russian)

    Google Scholar 

  • Bartolomey, G.G., Kharitonov, Y.V., Kovrizhnykh, V.P., et al.: The investigation of hydraulic resistance during forced flow subcooled boiling in a uniformly heated tube. Therm. Eng. 7, 64–65 (1979)

    Google Scholar 

  • Bejan, A.: Convection Heat Transfer. Wiley, Hoboken (2013)

    Book  MATH  Google Scholar 

  • Bibeau, E.L., Salcudean, M.: A study of bubble ebullition in forced-convective subcooled nucleate boiling at low pressure. Int. J. Heat Mass Transf. 37(15), 2245–2259 (1994)

    Article  Google Scholar 

  • Colebrook, C.F., White, C.M.: Experiments with fluid friction in roughened pipes. Proc. Royal Society of London. Ser. A, Math. and Physical Sci. 367–381 (1937)

    Google Scholar 

  • Drew, D., Lathey, R.T.: The virtual mass and lift force on a sphere in a rotating and straining flow. Int. J. Multiph. Flow 13, 113–121 (1987)

    Article  MATH  Google Scholar 

  • Dreytser, G.A., Kalinin, E.K., Yarkho, S.A.: Intensification of Heat Transfer in Channels. Mashinostroenie, Moscow (1990). (in Russian)

    Google Scholar 

  • Filonenko, G.K.: Correlation for hydraulic resistance coefficient of smooth tubes. Izv. VTI. (Proc. Dzerzhinskiy All-Union Therm. Eng. Inst.). 162(10), (1948) (In Russian)

    Google Scholar 

  • Fukuyama, Y., Kuriyama, T., Hirata, N.: Boiling transition and the possibility of spontaneous nucleation under high subcooling and high mass flux density in a tube. In: Proceedings of the 8th International Heat Transfer Conference, vol. 5, pp. 2197–2202 (1986)

    Google Scholar 

  • Fung, K.K., Groenvald D.C.: A physical model of subcooled and low quality film boiling of water in vertical flow at atmospheric pressure. In: Proceedings of the 7th International Heat Transfer Conference, vol. 4, pp. 381–386 (1982)

    Google Scholar 

  • Giarratano, P.J., Hess, R.C., Jones, M.C.: Forced Convection Heat Transfer to Subcritical Helium-I, pp. 404–416. Springer, US (1995)

    Google Scholar 

  • Griffith, P., Clark, J.A., Rohzenow, W.M.: Void volumes in subcooled boiling systems. ASME paper, 58-HT-19 (1958)

    Google Scholar 

  • Gunter, F.C.: Photographic study of surface boiling heat transfer to water with forced convection. Trans. ASME. J Heat Transf. 73(2), 115–123 (1951)

    Google Scholar 

  • Herring, E.A., Davis, M.R.: Structural development of gas-liquid mixture flows. J. Fluid Mech. 73, 97–123 (1976)

    Article  Google Scholar 

  • Ibragimov, M.H., Bobkov, V.P., Tychinskij, N.A.: Investigation of the behavior of the gas phase in a turbulent flow of gas-water mixture in channels. High Temp. 11(5), 1051–1061 (1973)

    Google Scholar 

  • Inoue, A., Aoki, S., Koga, T., et al.: Void-fraction, bubble and liquid velocity profiles of two-phase bubble flow in a vertical pipe. Trans. JSME. 42(360), 2521–2531 (1976) (In Japanese)

    Google Scholar 

  • Kapitsa, P.L. Kapitsa S.P.: Wave flow of thin layers of viscous liquid. ZhETF (J. Eksp. Theor. Phys.). 18(1), 3–28 (1948) (In Russian)

    Google Scholar 

  • Katsumata, I., Hirata, M.: Disappearance of DNB conditions for strong forced convection boiling heat transfer in a tube. Trans. JSME 43(375), 4257–4266 (1977)

    Article  Google Scholar 

  • Koshkin, V.K., Kalinin, E.K., Yarkho, S.A., et al.: Investigation of film boiling in a turbulent flow of subcooled liquid nitrogen. Heat Mass Transf. 2, 153–161 (1968) (In Russian)

    Google Scholar 

  • Koshkin, V.K., Kalinin, E.K., Dreytser, G.A., et al.: Nonstationary Heat Transfer. Mashinostoenie, Moscow (1985). (In Russian)

    Google Scholar 

  • Krepper, E., Rzehak, R., Lifante, C., et al.: CFD for subcooled flow boiling: coupling wall boiling and population balance models. Nucl. Eng. Des. 255, 330–346 (2013)

    Article  Google Scholar 

  • Kurilenko, A.A., Dymenko, S.K.: Heat transfer in the rod film boiling regime of hydrogen in pipes. IFZh (J. Eng. Phys. Thermophys.) 40(4), 586–591 (1981) (In Russian)

    Google Scholar 

  • Labuntsov, D.A.: An approximate theory of heat transfer in developed nucleate boiling. Izv. AN SSSR. Energetica i Transport (Proc. USSR Acad. Sci. Power Ind. Transp.). 1, 58–71 (1963) (in Russian)

    Google Scholar 

  • Labuntsov, D.A.: Questions of nucleate boiling heat transfer. Therm. Eng. 9, 14–19 (1972)

    Google Scholar 

  • Labuntsov, D.A., Yagov, V.V.: Mechanics of Simple Gas-Liquid Structures. Izd. MEI (Moscow Power Energetic Univ. Publ.), Moscow (1978) (In Russian)

    Google Scholar 

  • Lucas, D., Tomiyama, A.: On the role of the lateral lift force in poly-dispersed bubbly flow. Int. J Multiph. Flow. 37, 1178–1190 (2011)

    Article  Google Scholar 

  • Malnes, D.: Slip ratio and friction factors in the bubble flow regime in vertical tubes. Rep. of Kjeller Inst. fur Atomenergy, KR-110, (1966)

    Google Scholar 

  • Miropol’skiy, Z.L.: Heat transfer in film boiling of stem-water mixture in the steam generating tubes. Therm. Eng. 5, 49–52 (1963)

    Google Scholar 

  • Miropol’skiy, Z.L., Shneerova, R.I., Shitsman, M.E.: Effect of heat flux and liquid velocity on the flow resistance in the pipes. Trudy. TsKTI (Polzunov Central Boiler and Turbine Inst. Proc.). 59, 31–41 (1965) (In Russian)

    Google Scholar 

  • Nakoryakov, V.E., Burdyukov, A.P., Pokusaev, B.G. et al.: Investigation of turbulent flows of two-phase media. Ins. Izd. ITF (Inst. Thermal Phys. Publ.). Novosibirsk (1973) (In Russian)

    Google Scholar 

  • Nevstrueva, E.I., Tyutyaev, V.V.: The study of thermal-hydraulic characteristics of two-phase flow in a single steam generating channels and draft devices. Otchyot NPO “Energia” (Scientific and Production Association “Energia” Report), № OE-0151/76 (1976) (In Russian)

    Google Scholar 

  • Nakoryakov, V.E., Kashinskiy, O.N., Burdyukov, A.P., et al.: Local characteristics of upward gas-liquid flows. Int. J. Multiph. Flow 7(1), 63–81 (1981)

    Article  Google Scholar 

  • Ornatskiy, A.P.: Critical heat load and heat transfer at the forced movement of water in the pipes in the field of ultra-high pressures. Therm. Eng. 3, 66–69 (1963)

    Google Scholar 

  • Ornatskiy, A.P., Kichigin, A.M.: Investigation of the hydraulic resistance for the flow of water in the pipe of small diameter and large thermal loads. Therm. Eng. 8, 56–60 (1961)

    Google Scholar 

  • Prandtl, L.: Eine Beziehung zwischen Wärmeaustausch und Strömungswiderstand der Flüssigkeiten. Phys. Z. 11(2), 1072–1078 (1910)

    MATH  Google Scholar 

  • Peng, X.F., Wang, B.X.: Turbulent film boiling heat transfer for liquid flowing with high velocity through a horizontal flat duct. Int. J. Heat Mass Transf. 34(4/5), 1293–1299 (1991)

    Article  MathSciNet  Google Scholar 

  • Prasher, H., Beyer, M., Carl, H., et al.: Evolution of the structure of a gas-liquid two-phase flow in a large vertical pipe. Nucl. Eng. Des. 237, 1848–1861 (2007)

    Article  Google Scholar 

  • Reynolds, O.: On the extent and action of the heating surface for steam boilers. Proc. of the Lit. and Phil. Soc. of Manchester. 14, 7–12 (1875)

    Google Scholar 

  • Sato, Y., Sadatomi, M., Sekoguchi, K.: Momentum and heat transfer in two-phase bubble flow. II. Acomparision between experimental data and theoretical calculations. Int. J. Multiph. Flow. 7(2), 179–190 (1981)

    Google Scholar 

  • Sato, Y., Sekoguchi, K.: Liquid velocity distribution in vertical two-phase bubble flow. Int. J. Multiph. Flow 2(1), 79–95 (1975)

    Article  MATH  Google Scholar 

  • Schlihting, H.: Grenzschicht-Theorie. Verlag G. Braun, Karcluhe (1968)

    Google Scholar 

  • Sekoguchi, K., Nakazatomi, M., Sato, Y., et al.: Forced convective heat transfer in vertical air-water bubble flow. Bull. JSME 23(184), 1625–1631 (1980)

    Article  Google Scholar 

  • Sekoguchi, K., Sato, Y., Honda, T.: Two-phase bubble flow (1-st report, an experimental investigation of sparsely populated air bubbles in water stream in vertical duct). Trans. JSME 40(333), 1395–1403 (1974)

    Article  Google Scholar 

  • Serizawa, A.: A study of forced convective subcooled flow boiling. Two-phase Momentum Heat Mass Transf. Chem. Proc. Energy Eng. Syst. 1, 231–242 (1979)

    Google Scholar 

  • Serizawa, A., Kataoka, I., Michiyoshi, I.: Turbulence structure of air-water bubbly flow. II. Local properties. Int. J. Multiph. Flow 2(3), 235–246 (1975)

    Article  Google Scholar 

  • Sonnad, J.R., Goudar, C.T.: Explicit reformulation of the Colebrook-White equation for turbulent friction factor calculation. Ind. Eng. Chem. Res. 46, 2593–2600 (2007)

    Article  Google Scholar 

  • Styrikovich, M.A., Shitsman, M.E., Miropol’skiy, Z.L.: Some data on temperature regime of vertical heating pipe in the near-critical pressures. Therm. Eng. 12, 32–36 (1955)

    Google Scholar 

  • Tolubinsky, V.I., Kostanchuk, D.M.: Vapour bubbles growth rate and heat transfer intensity at subcooled water boiling. In: Proceedings of the 4th International Heat Transfer Conference, vol. 5, paper B 2.8 (1970)

    Google Scholar 

  • Tolubinsky, V.I., Vasil’ev, A.A., Mitin, A.A.: Water film boiling heat transfer in the tube in the near critical pressure region. In: Proceedings of the 7th International Heat Transfer Conference, vol. 4, pp. 427–430 (1982)

    Google Scholar 

  • Tomiyama, A., Tamai, H., Zun, I., et al.: Transverse migration of single bubbles in a simple shear flows. Chem. Eng. Sci. 57, 1849–1858 (2002)

    Article  Google Scholar 

  • Treshyov, G.G.: Experimental study of the mechanism of surface boiling. In: Heat Transfer with High Heat Loads and other Special Conditions. Gosenergoizdat, Moscow, pp. 51–68 (1959) (In Russian)

    Google Scholar 

  • Treshyov, G.G.: The number of nucleation sites in surface boiling. In: The Convective Heat Transfer in Two-phase and Single-phase Flows. Gosenergoizdat, Moscow, pp. 118–129 (1961) (In Russian)

    Google Scholar 

  • Unal, H.C.: Maximum bubble diameter, Maximal bubble diameter, maximal bubble-growth time and bubble growth-rate during the subcooled nucleate flow boiling of water up to 17.7 MN/m2. Int. J. Heat Mass Transf. 19, 643–649 (1976)

    Article  Google Scholar 

  • Unal., H.C.: Void fraction and incipient point of boiling during the subcooled nucleate flow boiling of water. Int. J. Heat Mass Transf. 20(4), 409–419 (1977)

    Google Scholar 

  • Viannay, S., Karian, J.: Study of the accelerated cooling of a very hot wall with a forced flow of subcooled liquid in film boiling regime. In: Proceedings of the 7th International Heat Transfer Conference, vol. 4, pp. 431–435 (1982)

    Google Scholar 

  • Yagov, V.V., Dedov, A.V.: Heat transfer under conditions of film boiling in turbulent flow of subcooled liquid. Therm. Eng. 56(3), 201–209 (2009)

    Article  Google Scholar 

  • Yang, L., Guo, A., Zhang, W.: Computational fluid dynamics simulation of subcooled flow boiling in vertical rectangular 2-mm narrow channel. Adv. Mech. Eng. 7(7), 1–12 (2015)

    Google Scholar 

  • Zaruba, A., Lucas, D., Prasser, H., et al.: Bubble-wall interaction in a vertical gas-liquid flow: bouncing, sliding and bubble deformations. Chem. Eng. Sci. 62, 1591–1605 (2007)

    Article  Google Scholar 

  • Zeitoun, O., Shoukri, M.: Bubble behavior and mean diameter in subcooled flow boiling. Trans ASME J. Heat Transf. 118(1), 110–116 (1996)

    Article  Google Scholar 

  • Zeygarnik, Y.A., Kirillova, I.V., Klimov, A.I., et al.: Some of the results of measurements of the hydraulic resistance in subcooled boiling of water. High Temp. 21(2), 303–308 (1983)

    Google Scholar 

  • Zun, I.: The transverse migration of bubbles influenced by walls in vertical bubbly flow. Int. J. Multiph. Flow 6, 583–588 (1980)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dmitrovskoe shosse 46-1-6, 127238, Moscow, Russia

    Alexander A. Avdeev

Authors
  1. Alexander A. Avdeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander A. Avdeev .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Avdeev, A.A. (2016). Reynolds Analogy. In: Bubble Systems. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-29288-5_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-29288-5_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29286-1

  • Online ISBN: 978-3-319-29288-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation