Heat and Mass Transfer

, Volume 49, Issue 11, pp 1659–1679 | Cite as

Hydrodynamic and thermal interactions of a cluster of solid particles in a pool of liquid of different Prandtl numbers using two-fluid model

  • Pallab S. Mahapatra
  • Nirmal K. MannaEmail author
  • Koushik Ghosh


The knowledge of thermal interaction between hot particles and liquid is essential for many engineering applications. The main focus of the present study is to understand the underlying phenomena of transient interaction between the hot particles and the liquid of varying Prandtl number under different parametric conditions. Analysis is carried out numerically using in-house multiphase code based on Eulerian two-fluid laminar model. The code is validated against existing results. The dispersion and penetration characteristics of the particles are observed to be a strong function of Prandtl number as well as volume fraction and particle diameter, with a stronger mushrooming observed for lower particle size or high Prandtl number liquid. The thermal interaction is observed to be between the particles and the narrow thermal envelope surrounding the particles. The particles cooling rate are observed to be several orders faster in a liquid with lower Prandtl number.


Prandtl Number Heat Transfer Rate Particle Temperature Liquid Sodium Thermal Interaction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Drag coefficient


Lift coefficient


Particle diameter (m)


Gravitational acceleration (m/s2)


Prandtl Number


Pressure (Pa)


Reynolds number


Time (s)


Temperature (K)


Normalized temperature


Radial velocity (m/s)


Axial velocity (m/s)

\( \vec{v} \)

Velocity vector (m/s)


Relative velocity (m/s)


Density (kg/m3)

\( \mathop {Q_{pl} }\limits^{.} \)

Heat transfer rate from particles to liquid (kg/s)


Viscosity (kg/m s)


Volume fraction







Liquid initial




Particle initial



Authors gratefully acknowledge several constructive suggestions made by Prof. Achintya Mukhopadhyay (Now on lien to Indian Institute of Technology, Madras) and Prof. Swarnendu Sen of Department of Mechanical Engineering, Jadavpur University. The financial support for this work from Bhabha Atomic Research Centre (BARC), India, and Council of Science and Industrial Research (CSIR), India is gratefully acknowledged. The authors also acknowledge the encouragement and suggestions of Deb Mukhopadhyay of BARC for carrying out the research.


  1. 1.
    Hall RW, Fletcher DF (1995) Validation of CHYMES: simulant studies. Nucl Eng Des 155:97–114CrossRefGoogle Scholar
  2. 2.
    Fletcher DF, Witt PJ (1996) Numerical studies of multiphase mixing with application to small scale experiments. Nucl Eng Des 166:135–145CrossRefGoogle Scholar
  3. 3.
    Angelini S, Yuen WW, Theofanous TG (1995) Premixing-related behavior of steam explosions. Nucl Eng Des 155:115–157CrossRefGoogle Scholar
  4. 4.
    Angelini S, Theofanous TG, Yuen WW (1997) The mixing of particle clouds plunging into water. Nucl Eng Des 177:285–301CrossRefGoogle Scholar
  5. 5.
    Meyer L, Schumacher G (1996) Queos, a simulation-experiment of the premixing phase of a steam explosion with hot spheres in water base case experiments. FZKA Report 5612, Forschungszentrum KarlsruheGoogle Scholar
  6. 6.
    Annunziato A, Yerkess A, Addabbo C (1999) FARO and KROTOS Code simulation and analysis at JRC Ispra. Nucl Eng Des 189:359–378CrossRefGoogle Scholar
  7. 7.
    Berthoud G, Crecy Fd, Duplat F, Meignen R, Valette M Premixing of corium into water during a fuel–coolant interaction: the models used in the 3 fields version of the MC3D code and two examples of validation on BILLEAU and FARO experiments. In: JAERI-Conf 97-011, Tokai-mura, Japan, May 19–21 1997. Proceeding of the OECD/CSNI specialist meeting on fuel coolant interactionGoogle Scholar
  8. 8.
    Pohlner G, Vujic Z, Bürger M, Lohnert G (2006) Simulation of melt jet breakup and debris bed formation in water pools with IKEJET/IKEMIX. Nucl Eng Des 206(19–21):2026–2048CrossRefGoogle Scholar
  9. 9.
    Melissari B, Argyropoulos SA (2005) Development of a heat transfer dimensionless correlation for spheres immersed in a wide range of Prandtl number fluids. Int J Heat Mass Transf 48:4333–4341CrossRefGoogle Scholar
  10. 10.
    Srinivasan V, Moon K-M, Greif D, Wang DM, M-h Kim (2010) Numerical simulation of immersion quench cooling process using an Eulerian multi-fluid approach. Appl Therm Eng 30:499–509CrossRefGoogle Scholar
  11. 11.
    Sahu AK, Chhabra RP, Eswaran V (2009) Effects of Reynolds and Prandtl numbers on heat transfer from a square cylinder in the unsteady flow regime. Int J Heat Mass Transf 52:839–850CrossRefGoogle Scholar
  12. 12.
    Wallis GB (1969) One-dimensional two-phase flow. McGraw- Hill, New YorkGoogle Scholar
  13. 13.
    Ishii M, Hibiki T (1975) Thermo fluid dynamics of two-phase flow. Eyrolles, PariszbMATHGoogle Scholar
  14. 14.
    Gidaspow D (1994) Multiphase flow and fluidization. Academic Press, New YorkzbMATHGoogle Scholar
  15. 15.
    Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39:201–225CrossRefGoogle Scholar
  16. 16.
    Peng D, Merriman B, Osher S, Zhao H, Kang M (1999) A PDE-based fast local level set method. J Comput Phys 155:410–438MathSciNetCrossRefGoogle Scholar
  17. 17.
    Unverdi SO, Tryggvason G (1992) A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys 100:25–37CrossRefGoogle Scholar
  18. 18.
    Lakehal D (2002) On the modelling of multiphase turbulent flows for environmental and hydrodynamic applications. Int J Multiph Flow 28:823–863CrossRefGoogle Scholar
  19. 19.
    Chahed J, Roig V, Masbernat L (2003) Eulerian–Eulerian two-fluid model for turbulent gas–liquid bubbly flows. Int J Multiph Flow 29:23–49CrossRefGoogle Scholar
  20. 20.
    Yeoh GH, Tu JY (2006) Two-fluid and population balance models for subcooled boiling flow. Appl Math Model 30:1370–1391CrossRefGoogle Scholar
  21. 21.
    Hudson J, Harris D (2006) A high resolution scheme for Eulerian gas–solid two-phase isentropic flow. J Comput Phys 216:494–525MathSciNetCrossRefGoogle Scholar
  22. 22.
    Antal SP Jr, Lahey RT, Flaherty JE (1991) Analysis of phase distribution in fully developed laminar bubbly two-phase flow. Int J Multiph Flow 17:635–652CrossRefGoogle Scholar
  23. 23.
    Yusuf R, Halvorsen B, Melaaen MC (2011) Eulerian–Eulerian simulation of heat transfer between a gas–solid fluidized bed and an immersed tube-bank with horizontal tubes. Chem Eng Sci 66:1550–1564CrossRefGoogle Scholar
  24. 24.
    Wachem BGMv, Almstedt AE (2003) Methods for multiphase computational fluid dynamics. Chem Eng J 96:81–98CrossRefGoogle Scholar
  25. 25.
    Kalteh M, Abbassi A, Saffar-Avval M, Harting J (2011) Eulerian–Eulerian two-phase numerical simulation of nanofluid laminar forced convection in a microchannel. Int J Heat Fluid Flow 32:107–116CrossRefGoogle Scholar
  26. 26.
    Fard MH, Esfahany MN, Talaie MR (2010) Numerical study of convective heat transfer of nanofluids in a circular tube two-phase model versus single-phase model. Int Commun Heat Mass Transfer 37:91–97CrossRefGoogle Scholar
  27. 27.
    Clift R, Grace JR, Weber ME (1978) Bubbles, drops and particles. Academic press, New YorkGoogle Scholar
  28. 28.
    Ishii M, Chawla TC (1979) Local drag laws in dispersed two phase flow. NUREG/CR-1230:79–105Google Scholar
  29. 29.
    Khan AR, Richardson JF (1987) The resistance to motion of a solid sphere in a fluid. Chem Eng Commun 62:135–150CrossRefGoogle Scholar
  30. 30.
    Wallis GB (1974) The terminal speed of single drops or bubbles in an infinite medium. Int J Multiph Flow 1:491–511CrossRefGoogle Scholar
  31. 31.
    Schiller L, Naumann Z (1935) A drag coefficient correlation. Z Ver Deutsch Ing 77:318–320Google Scholar
  32. 32.
    Drew D, Lahey RT (1979) Application of general constitutive principles to the derivation of multidimensional two phase flow equations. Int J Multiph Flow 5:243–264CrossRefGoogle Scholar
  33. 33.
    Leskovar M, Mavko B (2002) Simulation of the isothermal QUEOS steam explosion premixing experiment Q08. J Mech Eng 48:449–458Google Scholar
  34. 34.
    Ranz WE, Marshall WR (1952) Evaporation from drops: part I and II. Chem Eng Prog 48(141–146):173–180Google Scholar
  35. 35.
    Witte L (1968) An experimental investigation of forced convection heat transfer from a sphere to liquid sodium. ASME J Heat Transf 90:9–12CrossRefGoogle Scholar
  36. 36.
    Whitaker S (1972) Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed beds and tube bundles. AIChE J 18(2):361–371CrossRefGoogle Scholar
  37. 37.
    Argyropoulos SA, Mikrovas AC (1996) An experimental investigation on natural and forced convection in liquid metals. Int J Heat Mass Transf 39(3):547–561CrossRefGoogle Scholar
  38. 38.
    Patankar SV (1980) Numerical heat transfer and fluid flow. Taylor & FrancisGoogle Scholar
  39. 39.
    Mahapatra PS, Ghosh K, Manna NK, Mukhopadhyay A, Sen S (2011) Study of dispersion and cooling of a cluster of solid spherical particles in quiescent liquid of different Prandtl numbers. Paper presented at the 21st National & 10TH ISHMT-ASME heat and mass transfer conference, IIT Madras, India, December 27–30Google Scholar
  40. 40.
    Leskovar M (2000) Comparison of multiphase mixing simulations performed on a staggered and collocated grid. Paper presented at the international conference nuclear energy in Central Europe 2000, SloveniaGoogle Scholar
  41. 41.
    Theofanous TG, Yuen WW, Angelini S (1999) The verification basis of the PM-ALPHA code. Nucl Eng Des 189:59–102CrossRefGoogle Scholar
  42. 42.
    Ishii M, Zuber N (1979) Drag coefficient and relative velocity in bubbly, droplet or particulate flow. AIChE J 25:843–855CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Pallab S. Mahapatra
    • 1
  • Nirmal K. Manna
    • 1
    Email author
  • Koushik Ghosh
    • 1
  1. 1.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia

Personalised recommendations