Applied Mathematics and Mechanics

, Volume 35, Issue 7, pp 831–848 | Cite as

Comparative numerical study of single and two-phase models of nanofluid heat transfer in wavy channel

  • M. M. Rashidi
  • A. Hosseini
  • I. PopEmail author
  • S. Kumar
  • N. Freidoonimehr


The main purpose of this study is to survey numerically comparison of two-phase and single phase of heat transfer and flow field of copper-water nanofluid in a wavy channel. The computational fluid dynamics (CFD) prediction is used for heat transfer and flow prediction of the single phase and three different two-phase models (mixture, volume of fluid (VOF), and Eulerian). The heat transfer coefficient, temperature, and velocity distributions are investigated. The results show that the differences between the temperature field in the single phase and two-phase models are greater than those in the hydrodynamic field. Also, it is found that the heat transfer coefficient predicted by the single phase model is enhanced by increasing the volume fraction of nanoparticles for all Reynolds numbers; while for the two-phase models, when the Reynolds number is low, increasing the volume fraction of nanoparticles will enhance the heat transfer coefficient in the front and the middle of the wavy channel, but gradually decrease along the wavy channel.

Key words

nanofluid two-phase model wavy channel semi implicit method for pressure linked equation (SIMPLE) method 



specific heat


skin friction coefficient


particle diameter


gravitational acceleration


height of channel


convective heat transfer coefficient




length of channel


Nusselt number \((\tfrac{{hD_H }} {{K_f }}) \)


Reynolds number \((\tfrac{{\rho _f U_{in} D_H }} {{\mu _f }}) \)


surface geometry function


inlet temperature


wall temperature


velocity component in X-direction






velocity component in Y -direction

Greek Letters


thermal expansion coefficient


volume fraction


kinematic viscosity


dynamic viscosity





base fluid








solid particle



Chinese Library Classification


2010 Mathematics Subject Classification

82D80 80A20 


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  1. [1]
    Burns, J. C. and Parks, T. A review of steam soak operations in California. Journal of Fluid Mechanics, 29, 405–416 (1967)CrossRefGoogle Scholar
  2. [2]
    Asako, Y. and Faghri, M. Finite volume solutions for laminar flow an heat transfer in a corrugated duct. Journal of Heat Transfer, 109, 627–634 (1987)CrossRefGoogle Scholar
  3. [3]
    Rush, T. A., Newell, T. A., and Jacobi, A. M. An experimental study of flow and heat transfer in sinusoidal wavy passages. International Journal of Heat and Mass Transfer, 42, 1541–1553 (1999)CrossRefGoogle Scholar
  4. [4]
    Wang, C. C., Jang, J. Y., and Chiou, N. F. A heat transfer and friction correlation for wavy fin-and-tube heat exchangers. International Journal of Heat and Mass Transfer, 42, 1919–1924 (1999)CrossRefGoogle Scholar
  5. [5]
    Wang, C. C. and Chen, C. K. Forced convection in a wavy-wall channel. International Journal of Heat and Mass Transfer, 45, 2587–2595 (2002)CrossRefzbMATHGoogle Scholar
  6. [6]
    Fabbri, G. Heat transfer optimization in corrugated wall channels. International Journal of Heat and Mass Transfer, 43, 4299–4310 (2000)CrossRefzbMATHGoogle Scholar
  7. [7]
    Comini, G., Nonino, C., and Savino, S. Effect of aspect ratio on convection enhancement in wavy channels. Numerical Heat Transfer, Part A, 44, 21–37 (2003)CrossRefGoogle Scholar
  8. [8]
    Nilpueng, K. and Wongwises, S. Flow pattern and pressure drop of vertical upward gas-liquid flow in sinusoidal wavy channels. Experimental Thermal and Fluid Science, 30, 523–534 (2006)CrossRefGoogle Scholar
  9. [9]
    Chang, S. W., William Lees, A., and Chou, T. Heat transfer and pressure drop in furrowed channels with transverse and skewed sinusoidal wavy walls. International Journal of Heat and Mass Transfer, 52, 4592–4603 (2009)CrossRefGoogle Scholar
  10. [10]
    Assato, M. and de Lemos, M. J. S. Turbulent flow in wavy channels simulated with nonlinear models and a new implicit formulation. Numerical Heat Transfer, Part A, 56, 301–324 (2009)CrossRefGoogle Scholar
  11. [11]
    Stone, K. and Vanka, S. P. Numerical study of developing flow and heat transfer in a wavy passage. Journal of Fluids Engineering, 121, 713–719 (1999)CrossRefGoogle Scholar
  12. [12]
    Choi, U. S. Enhancing thermal conductivity of fluids with nanoparticles. ASME FED, 231, 99–103 (1995)Google Scholar
  13. [13]
    Maxwell, J. C. A Treatise on Electricity and Magnetism, 2nd ed., Cambridge Oxford University Press, Oxford (1904)Google Scholar
  14. [14]
    Maxwell, J. C. Electricity and Magnetism, Clarendon Press, Oxford (1873)Google Scholar
  15. [15]
    Lee, S., Choi, S. U. S., Li, S., and Eastman, J. A. Measuring thermal conductivity of fluids containing oxide nanoparticles. Journal of Heat Transfer, 121, 280–289 (1999)CrossRefGoogle Scholar
  16. [16]
    Masuda, H., Ebata, A., Teramae, K., and Hishinuma, N. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles (dispersions of-Al2O3, SiO2, and TiO2 ultra-fine particles) (in Japanese). Netsu Bussei, 4, 227–233 (1993)CrossRefGoogle Scholar
  17. [17]
    Eastman, J. A., Choi, S. U. S., Li, S., Yu, W., and Thompson, L. J. Anomalously increase effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Applied Physics Letters, 78(6), 718–720 (2001)CrossRefGoogle Scholar
  18. [18]
    Xuan, Y. and Li, Q. Heat transfer enhancement of nanofluids. International Journal of Heat and Fluid Flow, 21, 58–64 (2000)CrossRefGoogle Scholar
  19. [19]
    Pak, B. C. and Cho, Y. I. Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Experimental Heat Transfer, 11(2), 151–170 (1998)CrossRefGoogle Scholar
  20. [20]
    Wang, X., Xu, X., and Choi, S. U. S. Thermal conductivity of nanoparticles-fluid mixture. Journal of Thermophysics and Heat Transfer, 13(4), 474–480 (1999)CrossRefGoogle Scholar
  21. [21]
    Williams, W., Buongiorno, J., and Hu, L. W. Experimental investigation of turbulent convective heat transfer and pressure loss of alumina/water and zirconia/water nanoparticle colloids (nanofluids) in horizontal tubes. ASME Journal of Heat Transfer, 130(1), 42412–42419 (2008)CrossRefGoogle Scholar
  22. [22]
    Xuan, Y. and Li, Q. Investigation on convective heat transfer and flow features of nanofluids. Journal of Heat Transfer, 125(1), 151–155 (2003)CrossRefGoogle Scholar
  23. [23]
    Heris, S. Z., Etemad, S. G., and Esfahany, M. N. Experimental investigation of Oxide nanofluids laminar flow convective heat transfer. International Communications in Heat and Mass Transfer, 33, 529–535 (2006)CrossRefGoogle Scholar
  24. [24]
    Heris, S. Z., Esfahany, M. N., and Etemad, S. G. Experimental investigation of convective heat transfer of Al2O3/water nanofluid in circular tube. International Journal of Heat and Fluid Flow, 28, 203–210 (2007)CrossRefGoogle Scholar
  25. [25]
    Demir, H., Dalkilic, A. S., Kürekci, N. A., Duangthongsuk, W., and Wongwise, S. Numerical investigation on the single phase forced convection heat transfer characteristics of TiO2 nanofluids in a doubletube counter flow heat exchanger. International Communications in Heat and Mass Transfer, 38, 218–228 (2011)CrossRefGoogle Scholar
  26. [26]
    Ahmed, M. A., Shuaib, N. H., Yusoff, M. Z., and Al-Falahi, A. H. Numerical investigations of flow and heat transfer enhancement in a corrugated channel using nanofluid. International Communications in Heat and Mass Transfer, 38, 1368–1375 (2011)CrossRefGoogle Scholar
  27. [27]
    Santra, A. K., Sen, S., and Chakraborty, N. Study of heat transfer due to laminar flow of copperwater nanofluid through two isothermally heated parallel plates. International Journal of Thermal Science, 48, 391–400 (2009)CrossRefGoogle Scholar
  28. [28]
    Heidary, H. and Kermani, M. J. Effect of nano-particles on forced convection in sinusoidal-wall channel. International Communications in Heat and Mass Transfer, 37, 1520–1527 (2010)CrossRefGoogle Scholar
  29. [29]
    Mirmasoumi, S. and Behzadmehr, A. Numerical study of laminar mixed convection of a nanofluid in a horizontal tube using two-phase mixture model. Applied Thermal Engineering, 28, 717–727 (2008)CrossRefGoogle Scholar
  30. [30]
    Behzadmehr, A., Avval, M. S., and Galanis, N. Prediction of turbulent forced convection of a nanofluid in a tube with uniform heat flux using a two phase approach. International Journal of Heat and Fluid Flow, 28, 211–219 (2007)CrossRefGoogle Scholar
  31. [31]
    Akbari, M., Galanis, N., and Behzadmehr, A. Comparative analysis of single and two-phase models for CFD studies of nanofluid. International Journal of Thermal Sciences and Heat Transfer, 50, 1343–1354 (2011)CrossRefGoogle Scholar
  32. [32]
    Lotfi, R., Saboohi, Y., and Rashidi, A. M. Numerical study of forced convective heat transfer of nanofluids: comparison of different approaches. International Communications in Heat and Mass Transfer, 37, 74–78 (2010)CrossRefGoogle Scholar
  33. [33]
    Buongiorno, J. Convective transport in nanofluid. Journal of Heat Transfer, 128, 240–250 (2006)CrossRefGoogle Scholar
  34. [34]
    Rashidi, M. M., Abelman, S., and Freidoonimehr, N. Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid. International Journal of Heat and Mass Transfer, 62, 515–525 (2013)CrossRefGoogle Scholar
  35. [35]
    Rashidi, M. M. and Erfani, E. The modified differential transform method for investigating nano boundary-layers over stretching surfaces. International Journal of Numerical Methods for Heat & Fluid Flow, 21, 864–883 (2011)CrossRefMathSciNetGoogle Scholar
  36. [36]
    Rashidi, M. M., Freidoonimehr, N., Hosseini, A., Anwar Bég, O., and Hung, T. K. Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration. Meccanica, 49, 469–482 (2014)CrossRefGoogle Scholar
  37. [37]
    Durst, F., Ray, S., Unsal, B., and Bayoumi, O. A. The development lengths of laminar pipe and channel flows. ASME Journal of Fluid Engineering, 127, 1154–1160 (2005)CrossRefGoogle Scholar
  38. [38]
    Khanafer, K. and Vafai, K. A critical synthesis of thermophysical characteristics of nanofluids. International Journal of Heat and Mass Transfer, 54, 4410–4428 (2011)CrossRefzbMATHGoogle Scholar
  39. [39]
    Khanafer, K., Vafai, K., and Lightstone, M. Buoyancy-driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids. International Journal of Heat and Mass Transfer, 46, 3639–3653 (2003)CrossRefzbMATHGoogle Scholar
  40. [40]
    Maïgaa, S. B., Palma, S. J., and Nguyena, C. T. Heat transfer enhancement by using nanofluids in forced convection flows. International Journal of Heat and Fluid Flow, 26, 530–546 (2005)CrossRefGoogle Scholar
  41. [41]
    Yu, W. and Choi, S. U. S. The role of interfacial layers in the enhanced thermal of nanofluids: a renovated Maxwell mode. Journal of Nanoparticle Research, 5(1–2), 167–171 (2003)CrossRefGoogle Scholar
  42. [42]
    Manninen, M., Taivassalo, V., and Kallio, S. On the Mixture Model for Multiphase Flow, VTT Publications, Tekniikantie (1996)Google Scholar
  43. [43]
    Schiller, L. and Naumann, A. A drag coefficient correlation. Zeitschrift Vereines Deutscher Ingenieure, 77, 318–325 (1935)Google Scholar
  44. [44]
    Drew, D. A. and Lahey, R. T. Particulate Two-phase Flow, Butterworth-Heinemann, Boston (1993)Google Scholar
  45. [45]
    Ranz, W. E. and Marshall, W. R. Evaporation from drops. Chemical Engineering Progress, 48, 141–146 (1952)Google Scholar
  46. [46]
    Patankar, S. V. Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Francis Group, New York (1990)Google Scholar
  47. [47]
    Haghshenas Fard, M., Esfahany, M. N., and Talaie, M. R. Numerical study of convective heat transfer of nanofluids in a circular tube two-phase model versus single-phase model. International Communications in Heat and Mass Transfer, 37, 91–97 (2010)CrossRefGoogle Scholar
  48. [48]
    Kalteh, M., Abbassi, A., Saffar-Aval, M., and Harting, J. Eulerian-Eulerian two-phase numerical simulation of nanofluid laminar forced convection in a microchannel. International Journal of Heat and Fluid Flow, 32, 107–116 (2011)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • M. M. Rashidi
    • 1
  • A. Hosseini
    • 1
  • I. Pop
    • 2
    Email author
  • S. Kumar
    • 3
  • N. Freidoonimehr
    • 4
  1. 1.Department of Mechanical Engineering, Faculty of EngineeringBu-Ali Sina UniversityHamedanIran
  2. 2.Department of MathematicsBabeçBolyai UniversityCluj-NapocaRomania
  3. 3.Department of MathematicsNational Institute of TechnologyJamshedpurIndia
  4. 4.Young Researchers & Elite Club, Hamedan BranchIslamic Azad UniversityHamedanIran

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