Skip to main content
SpringerLink
Log in

Analytical determination of the heat transfer coefficient for gas, liquid and liquid metal flows in the tube based on stochastic equations and equivalence of measures for continuum

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

The stochastic equations of continuum are used for determining the heat transfer coefficients. As a result, the formulas for Nusselt (Nu) number dependent on the turbulence intensity and scale instead of only on the Reynolds (Peclet) number are proposed for the classic flows of a nonisothermal fluid in a round smooth tube. It is shown that the new expressions for the classical heat transfer coefficient Nu, which depend only on the Reynolds number, should be obtained from these new general formulas if to use the well-known experimental data for the initial turbulence. It is found that the limitations of classical empirical and semiempirical formulas for heat transfer coefficients and their deviation from the experimental data depend on different parameters of initial fluctuations in the flow for different experiments in a wide range of Reynolds or Peclet numbers. Based on these new dependences, it is possible to explain that the differences between the experimental results for the fixed Reynolds or Peclet numbers are caused by the difference in values of flow fluctuations for each experiment instead of only due to the systematic error in the experiment processing. Accordingly, the obtained general dependences of Nu for a smooth round tube can serve as the basis for clarifying the experimental results and empirical formulas used for continuum flows in various power devices. Obtained results show that both for isothermal and for nonisothermal flows, the reason for the process of transition from a deterministic state into a turbulent one is determined by the physical law of equivalence of measures between them. Also the theory of stochastic equations and the law of equivalence of measures could determine mechanics which is basis in different phenomena of self-organization and chaos theory.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kolmogorov, A.N.: Curves in Hilbert space invariants with respect to a one-parameter group of motion. Dokl. Akad. Nauk 26(1), 6–9 (1940)

    Google Scholar 

  2. Kolmogorov, A.N.: A new metric invariant of transitive dynamic systems and automorphisms of the Lebesgue spaces. Dokl. Akad. Nauk SSSR 119(5), 861–864 (1958)

    MathSciNet  MATH  Google Scholar 

  3. Kolmogorov, A.N.: About the entropy per time unit as a metric invariant of automorphisms. Dokl. Akad. Nauk SSSR 124(4), 754–755 (1959)

    MathSciNet  MATH  Google Scholar 

  4. Kolmogorov, A.N.: Mathematical models of turbulent motion of an incompressible viscous fluid. Usp. Mat. Nauk 59(1(355)), 5–10 (2004)

    Article  MathSciNet  Google Scholar 

  5. Landau, L.D.: On the problem of a turbulence. Dokl. Akad. Nauk 44(8), 339–342 (1944)

    MathSciNet  Google Scholar 

  6. Gilmor, R.: Catastrophe Theory for Scientists and Engineers. Dover, New York (1993)

    Google Scholar 

  7. Haller, G.: Chaos Near Resonance. Springer, Berlin (1999). doi:10.1007/978-1-4612-1508-0

    Book  MATH  Google Scholar 

  8. Halmos, P.: Theory of Measures. D.Van Nostrand Company Inc, New York (1950)

    Book  Google Scholar 

  9. Klimontovich, Y.L.: What are stochastic filtering and stochastic resonance? Usp. Fiz. Nauk 42, 37–44 (1999). doi:10.1070/pu1999v042n01abeh000445

    Article  Google Scholar 

  10. Landahl, M.T., Mollo-Christensen, E.: Turbulence and Random Processes in Fluid Mechanics. Cambridge University Press, Cambridge (1986). doi:10.1002/aic.690340127

    MATH  Google Scholar 

  11. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963). doi:10.1175/1520-0469

    Article  ADS  Google Scholar 

  12. Dmitrenko, A.V.: Calculation of pressure pulsations for a turbulent heterogeneous medium. Dokl. Fiz. 52(7), 384–387 (2007). doi:10.1134/s1028335807120166

    Article  MATH  Google Scholar 

  13. Dmitrenko, A. V.: Calculation of the boundary layer of a two-phase medium. High Temp. 40(5), 706–715 (2002). http://link.springer.com/article/10.1023%2FA%3A1020436720213

  14. Priymak, V.G.: Splitting dynamics of coherent structures in a transitional round-pipe flow. Dokl. Fiz. 58(10), 457–463 (2013). doi:10.1134/s102833581310008x

    Article  Google Scholar 

  15. Dmitrenko, A. V.: Fundamentals of heat and mass transfer and hydrodynamic of single-phase and two-phase media. In: Criterion, Integral, Statistical and DNS Methods. Galleya Press, Moscow (2008). http://search.rsl.ru/ru/catalog/record/6633402

  16. Dmitrenko, A.V.: Heat and mass transfer and friction in injection to a supersonic region of the Laval nozzle. Heat Transf. Res. 31(6–8), 338–399 (2000)

    Google Scholar 

  17. Orzag, S.A., Kells, L.C.: Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 96(1), 159–205 (1980). doi:10.1017/s0022112080002066

    Article  ADS  MATH  Google Scholar 

  18. Feigenbaum, M.: The transition to aperiodic behavior in turbulent systems. Commun. Math. Phys. 77(1), 65–86 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Rabinovich, M.I.: Stochastic self-oscillations and turbulence. Usp. Fiz. Nauk 125, 123–168 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  20. Monin, A.S.: On the nature of turbulence. Usp. Fiz. Nauk 125, 97–122 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  21. Raymond, J.P.: Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations. J. Math. Pures. Appl. 87(6), 627–669 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fursikov, A.V.: Moment theory for Navier–Stokes equations with a random right-hand side. Izv. Ross. Akad. Nauk 56(6), 1273–1315 (1992)

    MathSciNet  Google Scholar 

  23. Newton, P.K.: The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere. J. Fluid Mech. 786, 1–4 (2016). January

    Article  ADS  MathSciNet  Google Scholar 

  24. Dmitrenko, A.V.: Equivalence of measures and stochastic equations for turbulent flows. Dokl. Fiz. 58(6), 228–235 (2013). doi:10.1134/s1028335813060098

    Article  MathSciNet  MATH  Google Scholar 

  25. Dmitrenko, A. V.: Equivalent measures and stochastic equations for determination of the turbulent velocity fields in incompressible isothermal medium. Abstracts of papers presented at International Conference on “Turbulence and Wave Processes”, Lomonosov Moscow State University, 26–28 November 2013, pp. 39–40 (2013). http://www.dubrovinlab.msu.ru/turbulencemdm100ru/

  26. Dmitrenko, A.V.: Analytical estimation of velocity and temperature fields in a circular tube on the basis of stochastic equations and equivalence of measures. J. Eng. Phys. Thermophys. 88(6), 1569–1576 (2015). doi:10.1007/s10891-015-1344-x

    Article  Google Scholar 

  27. Dmitrenko, A.V.: An estimation of turbulent vector fields, spectral and correlation functions depending on initial turbulence based on stochastic equations. The Landau fractal equation. Int. J. Fluid Mech. Res. 43(3), 82–91 (2016). doi:10.1615/InterJFluidMechRes.v43.i3.60

  28. Dmitrenko, A.V.: Stochastic equations for continuum and determination of hydraulic drag coefficients for smooth flat plate and smooth round tube with taking into account intensity and scale of turbulent flow. Contin. Mech. Thermodyn. 29(1), 1–9 (2016). doi:10.1007/s00161-016-0514-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Dmitrenko, A. V.: Natural connection between the deterministic (laminar) and chaotic (turbulent) motions in a continuous medium—the equivalence of measures. Scientific Discovery Diploma No. 458, Certificate on Scientific Discovery Registration No. 583, International Academy of Authors of Scientific Discoveries and Inventions, Russian Academy of Natural Sciences (2013). http://www.raen.info/activities/reg_o/?46

  30. Dmitrenko, A.V.: The Theory of Equivalent Measures and the Theory of Sets with Repetitive, Counting Fractal Elements. Stochastic Thermodynamics and Turbulence. The Correlator “Determinancy-Randomness”. Galleya Press, Moscow (2013). http://search.rsl.ru/ru/catalog/record/6633402

  31. Dmitrenko, A.V.: Some analytical results of the theory of equivalence measures and stochastic theory of turbulence for nonisothermal flows. Adv. Stud. Theor. Phys. 8(25), 1101–1111 (2014). doi:10.12988/astp.2014.49131

    Article  Google Scholar 

  32. Dmitrenko, A.V.: The theory of equivalence measures and stochastic theory of turbulence for non-isothermal flow on the flat plate. Int. J. Fluid Mech. Res. 43(2), 182–187 (2016). doi:10.1615/InterJFluidMechRes.v43.i2. http://www.dl.begellhouse.com/jousrnals

  33. Dmitrenko, A.V.: Determination of critical Reynolds numbers for non-isothermal flows with using stochastic theories of turbulence and equivalent measures. Heat Transf. Res. 47(1), 338–399 (2016). 2015, doi:10.1615/HeatTransRes.014191

  34. Davidson, P.A.: Turbulence. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  35. Pope, S.B.: Turbulent Flows. Cambridge: Cambridge University Press (2000). doi:10.1017/cbo9780511840531

  36. Hinze, J.O.: Turbulence, 2nd edn. McGraw-Hill, New York (1975)

    Google Scholar 

  37. Schlichting, H.: Boundary-Layer Theory, 6th edn. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  38. Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics, vol. 1 and 2. M.I.T. Press, Moscow (1971). doi:10.1002/aic.690180546

    Google Scholar 

  39. Miyazaki, E., Inoue, H., Kimoto, T., Yamashita, S., Inoue, S., Yamaoka, N.: Heat transfer and temperature fluctuation of lithium flowing under transverse magnetic field. J. Nucl. Sci. Technol. 23(7), 582–593 (1986)

    Article  Google Scholar 

  40. Sukoriansky, S., Branover, H., Klaiman, D., Greenspan, E.: Heat transfer enhancement possibilities and implications for liquid metal blanket design. In: Proceedings on 12th IEEE Symposium on Fusion Engineering. Monterey, CA (1987)

  41. Branover, H., Greenspan, E., Sukoriansky, S., Talmage, G.: Turbulence and feasibility of self-cooled liquid metal blankets for fusion reactors. Fusion Technol. 10, 822–829 (1986)

    Article  Google Scholar 

  42. Lyon, R.N.: Liquid metal heat transfer coefficients. Chem. Eng. Prog. 47(12), 87–93 (1951)

    Google Scholar 

  43. Ibragimov, M., Subbotin, V.I., Ushakov, P.A.: Investigation of heat transfer in the turbulent flow of liquid metals in tubes. At. Energiya 8(1), 54–56 (1960)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the program of increasing the competitive ability of National Research Nuclear University MEPhI (agreement with the Ministry of Education and Science of the Russian Federation of August 27, 2013, Project No. 02.a03.21.0005).

Author information

Authors and Affiliations

  1. Department of Thermal Physics, National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, Moscow, Russia, 115409

    Artur V. Dmitrenko

  2. Department of Power Engineering, Moscow State University of Railway Engineering (MIIT), 9 Obraztsov St., Moscow, Russia, 127994

    Artur V. Dmitrenko

Authors
  1. Artur V. Dmitrenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Artur V. Dmitrenko.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dmitrenko, A.V. Analytical determination of the heat transfer coefficient for gas, liquid and liquid metal flows in the tube based on stochastic equations and equivalence of measures for continuum. Continuum Mech. Thermodyn. 29, 1197–1205 (2017). https://doi.org/10.1007/s00161-017-0566-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-017-0566-x

Keywords

Search

Navigation