Atomic Energy

, Volume 122, Issue 3, pp 156–171 | Cite as

Heat Exchange in Turbulent Flow. Part 1. Turbulent Prandtl Number

  • P. L. Kirillov

Knowledge of the characteristics of heat-exchange intensity in a turbulent coolant flow is one of the main problems of safety validation of nuclear power facilities. The maximum temperature of fuel elements and other structures is determined on the basis of the heat exchange coefficients. However, most computational relations of empirical or semi-empirical character have in many cases an error ±(15–20)%, which is unsatisfactory given the current requirements of the design of nuclear power facilities. The present review of the investigations of the turbulent boundary layer for different media performed in our country and abroad in 1950–2016 makes it possible to identify a different approach to the analysis of heat exchange intensity. The first part of the review examines the structural characteristics of the boundary layer and the evaluation of the turbulent Prandtl number \( {\mathrm{P}}_{{\mathrm{r}}_{\mathrm{T}}} \) for different media.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. Kays, “Turbulent Prandtl number – where we are?” Trans. ASME, J. Heat Transfer, 116, No. 2, 284–295 (1994).CrossRefGoogle Scholar
  2. 2.
    A. Reynolds, “The prediction of turbulent Prandtl and Schmidt number,” Int. J. Heat Mass Transfer, 18, No. 9, 1055–1069 (1975).ADSCrossRefGoogle Scholar
  3. 3.
    B. Launder, “Second-moment closure and its use in modelling turbulent industrial flow,” Int. J. Numer. Meth. in Fluids, 9, No. 8, 963–985 (1989).ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Y. Nagano and C. Kim, “A two-equation model for heat transport in wall turbulent shear flows,” Trans. ASME, J. Heat Transfer, 110, No. 3, 583–589 (1988).ADSCrossRefGoogle Scholar
  5. 5.
    S. A. Rogozhkin, A. A. Aksenov, S. V. Zhluktov, et al.,” Development of a model of turbulent heat transfer for liquid metal coolant and its verification,” Vychisl. Mekh. Splosh. Sred, 7, No. 3, 306–316 (2014).Google Scholar
  6. 6.
    V. I. Borodulin and Y. S. Kachanov, “Universal mechanism of generation of wall turbulence and determinate turbulence,” Mekh. Zhidk. Gaza. Vest. Nizhegorod. Univ. im. Lobachevskogo, No. 4 (3), 653–655 (2014).Google Scholar
  7. 7.
    Y. A. Bae, “A new formulation of variable-turbulent Prandtl number for heat transfer to supercritical fluids,” Int. J. Heat Mass Transfer, 92, 792–806 (2016).CrossRefGoogle Scholar
  8. 8.
    S. S. Kutateladze, Wall Turbulence, Nauka, Novosibirsk (1973).Google Scholar
  9. 9.
    M. D. Millionshchikov, Turbulent Flow in a Boundary Layer and in Pipes, Nauka, Moscow (1963).Google Scholar
  10. 10.
    K. P. Zybin and V. A. Sirota, “ Model stretched vortices and validation of the statistical properties of turbulence,” Usp. Fiz. Nauk, 185, No. 6, 593–611 (2015).CrossRefGoogle Scholar
  11. 11.
    R. Nedderman, “The measurements of velocities in the wall region of turbulent liquid pipe flow,” Chem. Eng. Sci., 16, No. 1–2, 120–126 (1961).CrossRefGoogle Scholar
  12. 12.
    K. Trinh, “Reflections on a penetration theory of turbulent heat transfer” (2010),
  13. 13.
    E. W. Repik and Y. P. Sosedko, Turbulent Boundary Layer. Methodology and Experimental Results, Fizmatlit, Moscow (2007).Google Scholar
  14. 14.
    A. Poincaré, Theory of Vortices [Russian translation], ITs Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk (2000).Google Scholar
  15. 15.
    D. Ruelle, Randomness and Chaos [Russian translation], ITs Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk (2001).Google Scholar
  16. 16.
    A. Pyadishyus and A. Shlanchyauskas, Turbulent Transport in the Near-Wall Layers, Moklas, Vilnius (1987), pp. 53–62.Google Scholar
  17. 17.
    S. Nychas, H. Hershey, and R. Brodkye, “A visual study of turbulent shear flows,” Fluid Mech.,61, No. 3, 513–540 (1973).ADSCrossRefGoogle Scholar
  18. 18.
    F. G. Galimzyanov and R. F. Galimzyanov, The Theory of Internal Turbulent Motion, Expert, Ufa (1999).Google Scholar
  19. 19.
    J. Blom and D. Vries, “The magnitude of the turbulent Prandtl number,” in: Heat and Mass Transfer: Proc. 3rd All-Union Heat and Mass Transfer Meeting, A. V. Lykov and B. M. Smol’skii (eds.), Energiya, Moscow (1961), Vol. 1, pp. 147–154.Google Scholar
  20. 20.
    J. Blom, An Experimental Determination of the Turbulent Prandtl Number in a Developing Temperature Boundary Layer: Thesis Doct. Techn. Sci., Eindhoven Techn. High School (1970), https:/
  21. 21.
    W. Corcoran, F. Page, W. Schlinger, and B. Sage, “Temperature gradient in turbulent gas streams,” Industr. Eng. Chem., 44, 410–430 (1952).CrossRefGoogle Scholar
  22. 22.
    C. Sleicher, “Experimental velocity and temperature profiles for air in turbulent pipe flow,” Trans. ASME, 80, 693–704 (1958).Google Scholar
  23. 23.
    B. Sage and E. Venezian, “Temperature gradients in turbulent gas streams: effect of viscous dissipation on the evaluation of total conductivity,” J. Am. Inst. Chem. Engin. (AIChE), 7, 688–692 (1961).CrossRefGoogle Scholar
  24. 24.
    H. Ludwieg, “Bestimmung das Verhaltnisses der Austauschkoeffizienten für Wärme und Impuls bei Turbulenten Grenzschichten,” Z. Flugwiss., 4, 73–81 (1956).Google Scholar
  25. 25.
    D. Johnson, “Velocity and temperature fluctuation measurements in a turbulent boundary layer downstream of a stepwise discontinuity in wall temperature,” Trans. ASME, J. Appl. Mech., 26, 325–336 (1959).zbMATHGoogle Scholar
  26. 26.
    R. Johnk and T. Hanratty, “Temperature profiles of air in a pipe. P. I and II,” Chem. Eng. Sci., 17, 867–892 (1962).CrossRefGoogle Scholar
  27. 27.
    S. Isakoff and T. Drew, Heat and Momentum Transfer in Turbulent Flow of Mercury: General Discussion on Heat Transfer, Inst. Mech. Engineers and ASME (1951), pp. 405– 449.Google Scholar
  28. 28.
    H. Brown, B. Amstead, and B. Short, “The transfer of heat and momentum in a turbulent stream of mercury,” Trans. ASME, 79, 279–285 (1957).Google Scholar
  29. 29.
    A. Sesonke, S. Schrock, and E. Buyoco, “Eddy diffusivity ratios for mercury flowing in a tube,” Chem. Eng. Progr. Symp. Ser., 61, No. 57, 101–107 (1965).Google Scholar
  30. 30.
    R. Gowen and J. Smith, “Turbulent heat transfer from smooth and rough surfaces.,” Int. J. Heat Mass Transfer, 11, No. 11, 1657–1673 (1968).CrossRefGoogle Scholar
  31. 31.
    N. M. Galin, “On the tensor of the coefficients of turbulent thermal conductivity,” Teplofiz. Vys. Temp., 13, No. 5, 984–988 (1975).ADSGoogle Scholar
  32. 32.
    A. Quarmby and R. Quirk, “Axisymmetric and non-axisymmetric turbulent diffusion in a plain circular tube at high Schmidt number,” Int. J. Heat Mass Transfer, 17, No. 1, 143–147 (1974).CrossRefGoogle Scholar
  33. 33.
    M. Kh. Ibragimov. V. I, Subbotin, and G. S. Taranov, “Determination of the correlation between the velocity and temperature fluctuations in turbulent air flow in a pipe,” Dokl. AN SSSR, 183, No. 5, 1032–1035 (1968).Google Scholar
  34. 34.
    N. M. Galin and P. L. Kirillov, Heat and Mass Transfer (in nuclear power), Energoatomizdat, Moscow (1987), pp. 152–156.Google Scholar
  35. 35.
    M. Kh. Ibragimov, V. I. Merkulov, and V. I. Subbotin, “Statistical characteristics of wall temperature pulsations of a heat exchanger in high heat fluxes,” in: Liquid Metals, Atomizdat, Moscow (1967), pp. 71–81.Google Scholar
  36. 36.
    A. A. Shlanchyauskas and M.-R. M. Drizhyus, “The temperature profiles near a wall in a turbulent layer of different liquids,” Tr. AN Litovskoi SSR, Ser. B. Vilnius, 1, No. 64, 189–203 (1971).Google Scholar
  37. 37.
    A. F. Polyakov, “Wall effect on temperature pulsations in a viscous sublayer.” Teplofiz. Vys. Temp., 12 , No. 2, 328–337 (1974).ADSGoogle Scholar
  38. 38.
    V. P. Bobkov, Yu. I. Gribanov, M. Kh. Ibragimov, et al., “Measurement of the intensity of temperature fluctuations in turbulent flow of mercury in a tube,” Teplofiz. Vys. Temp., 3, No. 5, 708–716 (1965).Google Scholar
  39. 39.
    B. S. Petukhov, L. G. Genin, S. A. Kovalev, S. L Solov’ev, “Foundations of a semi-empirical theory of turbulence,” in: Heat Transfer in Nuclear Power Plants, Izd. MEI, Msocow (2003), pp. 181–216.Google Scholar
  40. 40.
    B. Blackwell, W. Kays, and R. Moffat, The Turbulent Boundary Layer on a Porous Plate: an Experimental Study of the Heat Transfer Behaviour with Adverse Pressure Gradients, Rep. HMT-16, Stanford Univ., Dept. Mech. Eng., Thermosci. Div., Stanford (1972).Google Scholar
  41. 41.
    N. M. Galin, “On the diffusion coefficients in anisotropic turbulence,” Izv. Akad. Nauk SSSR, Ser. Energet. Transp., No. 6, 130–139 (1980).Google Scholar
  42. 42.
    A. Quarmby and R. Quirk, “Measurements of the radial and tangential eddy diffusivities of heat and mass in turbulent flow in plain tube,” Int. J. Heat Mass Transfer, 15, No. 11, 2309–2327 (1972).CrossRefGoogle Scholar
  43. 43.
    C. Tien, “On Jenkins model of eddy diffusivities for momentum and heat,” Trans. ASME, J. Heat Transfer, 83, No. 3, 389–390 (1961).CrossRefGoogle Scholar
  44. 44.
    K. D. Voskresenskii and E. S. Turilina, “An approximate calculation of the heat emission of molten metals,” in: Heat Transfer and Thermal Modeling, AN SSSR, Moscow (1959).Google Scholar
  45. 45.
    A. V. Gulevitch, A. V. Zrodnikov, V. Y. Pupko, A. A. Shimanskii, “Mathematical model of turbulent motion taking into account molecular viscosity and thermal conductivity,” in: Application of Perturbation Theory in Engineering Problems of Nuclear Power, Energoatomizdat, Moscow (1993), pp. 177–192.Google Scholar
  46. 46.
    V. Yakhot, S. Orszag, and A. Yakhot, “Heat transfer in turbulent fluids. 1. Pipe flows,” Int. J. Heat Mass Transfer, 30, No. 1, 15–22 (1987).ADSCrossRefzbMATHGoogle Scholar
  47. 47.
    E. Skupinski, J. Tortel, and L. Vautrey, “Determination des coefficients de convection d’un alliage sodium-potassium dans un tube circulaire,” Int. J. Heat Mass Transfer, 8, No. 6, p. 937–951 (1965).CrossRefGoogle Scholar
  48. 48.
    H. Buhr, A. Carr, and R. Balzhiser, “Temperature profiles in liquid metals and the effect of superimposed free convection in turbulent flow,” Int. J. Heat Mass Transfer, 11, No. 4, 641– 654 (1968).CrossRefGoogle Scholar
  49. 49.
    C. Sleicher, A. Awad, and K. Notter, “Temperature and eddy diffusivity profiles in NaK.” Int. J. Heat Mass Transfer, 16, No. 8, 1565–1575 (1973).CrossRefGoogle Scholar
  50. 50.
    D. Hollingsworth, W. Kays, and R. Moffat, Measurement and Prediction of the Turbulent Thermal Boundary Layer in Water on Flat and Concave Surfaces, Rep. HMT-41, Stanford Univ., Dept. Mech. Eng., Thermosci. Div., Stanford (1989).Google Scholar
  51. 51.
    M. Hishida, Y. Nagano, and M. Tagawa, “Transport process of heat and momentum in the wall region of turbulent pipe flow,” in: Proc. 8th Int. Heat Transfer Conf., San Francisco, USA, Aug. 17–22, 1986, Vol. 3, pp. 925–930.Google Scholar
  52. 52.
    B. Kader and A. Jaglom, “Heat and mass transfer laws for fully turbulent wall flows,” Int. J. Heat Mass Transfer, 15, No. 12, 2329–2351 (1972).CrossRefGoogle Scholar
  53. 53.
    B. S. Petukhov, “Turbulence in the Theory Heat Exchange,” in: Heat- Mass-Transfer-VI: Problem Reports at the 6th All-Union. Conf. on Heat and Mass Transfer, Minsk (1981), Pt. 1, pp. 21–51.Google Scholar
  54. 54.
    W. Kays, M. Crawford, and B. Weigand, Convective Heat and Mass Transfer, McGraw Hill, Boston (2005), 4th ed.Google Scholar
  55. 55.
    A. Malhotra and S. Kang, “Turbulent Prandtl number in circular pipes,” Int. J. Heat Mass Transfer, 27, No. 11, 2158–2161 (1984).CrossRefGoogle Scholar
  56. 56.
    F. Chen, X. Huai, J. Cai, et al., “Investigation on the applicability of turbulent-Prandtl-number models for liquid bismuth eutectic,” Nucl. Eng. Design, 257, 128–133 (2013).CrossRefGoogle Scholar
  57. 57.
    X. Cheng and Tak Nam-il, ” Investigation on turbulent heat transfer to liquid-bismuth eutectic flows in tubes,” Nucl. Eng. Design, 236, 385–393 (2006).CrossRefGoogle Scholar
  58. 58.
    S. Aoki, “A consideration on the heat transfer in liquid metal,” Bull. Tokyo Inst. Technol., 54, 63–73 (1963).Google Scholar
  59. 59.
    M. Jischa and H. Rieke, “About the prediction of turbulent Prandtl and Schmidt numbers from modeled transport equations,” Int. J. Heat Mass Transfer, 22, 1547–1555 (1979).Google Scholar
  60. 60.
    B. Hasan, “Turbulent Prandtl number and its use in prediction of heat transfer coefficient for liquids,” Nahrain Univ., College of Eng. J. (NUCEJ), 10, No. 1, 53–64 (2007),

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • P. L. Kirillov
    • 1
  1. 1.State Science Center of the Russian Federation – Leipunskii Institute for Physics and Power Engineering (GNTs RF – FEI)ObninskRussia

Personalised recommendations