Nanofluids of the Future

  • Liqiu Wang
  • Michel Quintard
Part of the Advances in Transport Phenomena book series (ADVTRANS, volume 1)


Nanofluids are a new class of fluids engineered by dispersing nanometer-size structures (particles, fibers, tubes, droplets) in base fluids. The very essence of nanofluids research and development is to enhance fluid macroscale and megascale properties such as thermal conductivity through manipulating microscale physics (structures, properties and activities). Therefore, the success of nanofluid technology depends very much on how well we can address issues like effective means of microscale manipulation, interplays among physics at different scales, and optimization of microscale physics for the optimal megascale properties. In this chapter we review methodologies available to effectively tackle these central but difficult problems and identify the future research needs as well. The reviewed techniques include nanofluids synthesis through liquid-phase chemical reactions in continuous-flow microfluidic microreactors, scaling-up by the volume averaging, and constructal design with the constructal theory. The identified areas of future research contain microfluidic nanofluids, thermal waves, and constructal nanofluids. While our focus is on heat-conduction nanofluids, the methodologies are equally valid for the other types of nanofluids. The review could serve as a coherent, inspiring and realistic plan for future research and development of nanofluid technology.


Heat Transfer Heat Conduction Heat Mass Transfer Representative Elementary Volume Versus Versus Versus 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Choi, S.U.S., Zhang, Z.G., Keblinski, P.: Nanofluids. In: Nalwa, H.S. (ed.) Encyclopedia of Nanoscience and Nanotechnology, vol. 6, pp. 757–773. American Scientific Publishers (2004)Google Scholar
  2. 2.
    Peterson, G.P., Li, C.H.: Heat and mass transfer in fluids with nanoparticle suspensions. Adv. Heat Transfer 39, 257–376 (2006)Google Scholar
  3. 3.
    Das, S.K., Choi, S.U.S., Yu, W.H., Pradeep, T.: Nanofluids: Science and Technology. John Wiley & Sons, Chichester (2008)Google Scholar
  4. 4.
    Wen, D.S., Ding, Y.L., Williams, R.: Nanofluids turn up the heat. TCE 771, 32–34 (2005)Google Scholar
  5. 5.
    Pileni, M.P.: Magnetic fluids: fabrication, magnetic properties, and organization of nanocrystals. Adv. Funct. Mater. 11, 323–336 (2001)CrossRefGoogle Scholar
  6. 6.
    Wasan, D.T., Nikolov, A.D.: Spreading of nanofluids on solids. Nature 423, 156–159 (2003)CrossRefGoogle Scholar
  7. 7.
    Gorman, J.: Nanofluid flow: detergents may benefit from new insight. Sci. News 163, 292–293 (2003)CrossRefGoogle Scholar
  8. 8.
    Chen, H.S., Ding, Y.L., He, Y.R., Tan, C.Q.: Rheological behaviour of ethylene glycol based titania nanofluids. Chemical Physics Letters 444, 333–337 (2007)CrossRefGoogle Scholar
  9. 9.
    Zhang, L.L., Jiang, Y.H., Ding, Y.L., Povey, M., York, D.: Investigation into the antibacterial behaviour of suspensions of ZnO nanoparticles (ZnO nanofluids). Journal of Nanoparticle Research 9, 479–489 (2007)CrossRefGoogle Scholar
  10. 10.
    Pomogailo, A.D., Kestelman, V.N.: Metallopolymer Nanocomposites. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Dice, G.D., Mujumdar, S., Elezzabi, A.Y.: Plasmonically enhanced diffusive and subdiffusive metal nanoparticle-dye random laser. Appl. Phys. Lett. 86, 131105 (2005)CrossRefGoogle Scholar
  12. 12.
    Duan, X., Huang, Y., Cui, Y., Wang, J., Lieber, C.M.: Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices. Nature 409, 241–245 (2001)CrossRefGoogle Scholar
  13. 13.
    Duan, X., Huang, Y., Agarwal, R., Lieber, C.M.: Single-nanowire electrically driven lasers. Nature 421, 66–69 (2003)CrossRefGoogle Scholar
  14. 14.
    Singh, A.K.: Thermal conductivity of nanofluids. Defence Science Journal 58, 600–607 (2008)Google Scholar
  15. 15.
    Li, C.H., Williams, W., Buongiorno, J., Hu, L.W., Peterson, G.P.: Transient and Steady-State Experimental Comparison Study of Effective Thermal Conductivity of Al2O3/Water Nanofluids. J. Heat Transfer 130, 040301/1–044503/4 (2008)Google Scholar
  16. 16.
    Wang, L.Q., Wei, X.H.: Nanofluids: Synthesis, Heat Conduction, and Extension. J. Heat Transfer 131, 033102/1–033102/7 (2009)Google Scholar
  17. 17.
    Jang, S.P., Choi, S.U.S.: Effects of Various Parameters on Nanofluid Thermal Conductivity. J. Heat Transfer 129, 617–623 (2007)CrossRefGoogle Scholar
  18. 18.
    Vadasz, P.: Heat Conduction in Nanofluid Suspensions. J. Heat Transfer 128, 465–477 (2006)CrossRefGoogle Scholar
  19. 19.
    Lee, S., Choi, S.U.S., Li, S., Eastman, J.A.: Measuring Thermal Conductivity of Fluids Containing Oxide Nanoparticles. J. Heat Transfer 121, 280–289 (1999)CrossRefGoogle Scholar
  20. 20.
    Wei, X.H., Zhu, H.T., Wang, L.Q.: CePO4 Nanofluids: Synthesis and Thermal Conductivity. J. Thermophysics Heat Transfer 23, 219–222 (2009)CrossRefGoogle Scholar
  21. 21.
    Das, S.K., Putra, N., Thiesen, P., Roetzel, W.: Temperature Dependence of Thermal Conductivity Enhancement for Nanofluids. J. Heat Transfer 125, 567–574 (2003)CrossRefGoogle Scholar
  22. 22.
    He, Y.R., Men, Y.B., Liu, X., Lu, H.L., Chen, H.S., Ding, Y.L.: Study on forced convective heat transfer of non-Newtonian nanofluids. J. Thermal Sci. 18, 20–26 (2009)CrossRefGoogle Scholar
  23. 23.
    Tzou, D.Y.: Thermal Instability of Nanofluids in Natural Convection. Int. J. Heat Mass Transfer 51, 2967–2979 (2008)zbMATHCrossRefGoogle Scholar
  24. 24.
    Buongiorno, J.: Convection Transport in Nanofluids. J. Heat Transfer 128, 240–250 (2006)CrossRefGoogle Scholar
  25. 25.
    Xuan, Y.M., Li, Q.: Investigation on Convective Heat Transfer and Flow Features of Nanofluids. J. Heat Transfer 125, 151–155 (2003)CrossRefGoogle Scholar
  26. 26.
    Milanova, D., Kumar, R.: Heat Transfer Behavior of Silica Nanoparticles Experiment in Pool Boiling. J. Heat Transfer 130, 042401/–042401/6 (2009)Google Scholar
  27. 27.
    Kim, S.J., McKrell, T., Buongiorno, J., Hu, L.W.: Alumina Nanoparticles Enhance the Flow Boiling Critical Heat Flux of Water at Low Pressure. J. Heat Transfer 130, 044501/1–044501/3 (2008)Google Scholar
  28. 28.
    Kim, S.J., McKrell, T., Buongiorno, J., Hu, L.W.: Experimental Study of Flow Critical Heat Flux in Alumina-Water, Zinc-Oxide-Water, and Diamond-Water Nanofluids. J. Heat Transfer 131, 043204/1–043204/7 (2009)Google Scholar
  29. 29.
    Kedzierski, M.A.: Effect of CuO Nanoparticle Concentration on R134a/Lubricant Pool-Boiling Heat Transfer. J. Heat Transfer 131, 043205/1–043205/7 (2009)Google Scholar
  30. 30.
    Wu, D.X., Zhu, H.T., Wang, L.Q., Liu, L.M.: Critical Issues in Nanofluids Preparation, Characterization and Thermal Conductivity. Current Nanoscience 5, 103–112 (2009)CrossRefGoogle Scholar
  31. 31.
    Choi, S.U.S.: Nanofluids: From Vision to Reality Through Research. J. Heat Transfer 131, 033106/1–033106/9 (2009)Google Scholar
  32. 32.
    Eastman, J.A., Phillpot, S.R., Choi, S.U.S., Keblinski, P.: Thermal transport in nanofluids. Annu. Rev. Mater. Res. 34, 219–246 (2004)CrossRefGoogle Scholar
  33. 33.
    Phelan, P.E., Bhattacharya, P., Prasher, R.S.: Nanofluids for heat transfer applications. Annu. Rev. Heat Transfer 14, 255–275 (2005)Google Scholar
  34. 34.
    Sobhan, C.B., Peterson, G.P.: Microscale and Nanoscale Heat Transfer: Fundamentals and Engineering Applications. CRC Press, Boca Raton (2008)Google Scholar
  35. 35.
    Wang, L.Q., Xu, M.T., Wei, X.H.: Multiscale theorems. Adv. Chemical Engineering 34, 175–468 (2008)CrossRefGoogle Scholar
  36. 36.
    Wang, L.Q.: Flows through porous media: a theoretical development at macroscale. Transport in Porous Media 39, 1–24 (2000)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Choi, S.U.S., Eastman, J.A.: Enhanced heat transfer using nanofluids. United States Patent, US 6221275 B1 (2001)Google Scholar
  38. 38.
    Eastman, J.A., Choi, S.U.S., Li, S., Yu, W., Thompson, L.J.: Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett. 78, 718–720 (2001)CrossRefGoogle Scholar
  39. 39.
    Chang, H., Tsung, T.T., Chen, L.C., Yang, Y.C., Lin, H.M., Lin, C.K., Jwo, C.S.: Nanoparticle suspension preparation using the arc spray nanoparticle synthesis system combined with ultrasonic vibration and rotating electrode. Int. J. Adv. Manufacturing Tech. 26, 552–558 (2005)CrossRefGoogle Scholar
  40. 40.
    Lo, C.H., Tsung, T.T., Chen, L.C., Su, C.H., Lin, H.M.: Fabrication of copper oxide nanofluid using submerged arc nanoparticle synthesis system (SANSS). J. Nanoparticle Research 7, 313–320 (2005)CrossRefGoogle Scholar
  41. 41.
    Lo, C.H., Tsung, T.T., Chen, L.C.: Shaped-controlled synthesis of Cu-based nanofluid using submerged arc nanoparticle synthesis system (SANSS). J. Crystal Growth 277, 636–642 (2005)CrossRefGoogle Scholar
  42. 42.
    Romano, J.M., Parker, J.C., Ford, Q.B.: Application opportunities for nanoparticles made from condensation of physical vapors. Adv. Powder Metallurgy Particulate Materials 2, 12–13 (1997)Google Scholar
  43. 43.
    Zhu, H.T., Lin, Y.S., Yin, Y.S.: A novel one-step chemical method for preparation of copper nanofluids. J. Colloid Interface Sci. 277, 100–103 (2004)CrossRefGoogle Scholar
  44. 44.
    Zhu, H.T., Zhang, C.Y., Liu, S.Q., Tang, Y.M., Yin, Y.S.: Effects of nanoparticle clustering and alignment on thermal conductivities of Fe3O4 aqueous nanofluids. Appl. Phys. Lett. 89, 023123 (2006)Google Scholar
  45. 45.
    Zhu, H.T., Zhang, C.Y., Tang, Y.M.: Novel synthesis and thermal conductivity of CuO nanofluids. J. Phys. Chem. C 111, 1646–1650 (2007)CrossRefGoogle Scholar
  46. 46.
    Wei, X.H., Kong, T.T., Zhu, H.T., Wang, L.Q.: CuS/Cu2S nanofluids: synthesis and thermal conductivity. Int. J. Heat Mass Transfer (in press, 2009)Google Scholar
  47. 47.
    Wei, X.H., Zhu, H.T., Kong, T.T., Wang, L.Q.: Synthesis and Thermal Conductivity of Cu2O Nanofluids. Int. J. Heat Mass Transfer 52, 4371–4374 (2009)CrossRefGoogle Scholar
  48. 48.
    Wang, L.Q., Liu, F.: Forced convection in slightly curved microchannels. Int. J. Heat Mass Transfer 50, 881–896 (2007)zbMATHCrossRefGoogle Scholar
  49. 49.
    Wang, L.Q., Yang, T.L.: Multiplicity and stability of convection in curved ducts: review and progress. Adv. Heat Transfer 38, 203–255 (2004)Google Scholar
  50. 50.
    Wang, L.Q., Cheng, K.C.: Flow transitions and combined free and forced convective heat transfer in rotating curved channels: the case of positive rotation. Phys. Fluids 8, 1553–1573 (1996)zbMATHCrossRefGoogle Scholar
  51. 51.
    Sudarsan, A.P., Ugaz, V.M.: Fluid mixing in planar spiral microchannels. Lab on a Chip 6, 74–82 (2006)CrossRefGoogle Scholar
  52. 52.
    Hong, C.C., Choi, J.W., Ahn, C.H.: A novel in-plane passive microfluidic mixer with modified Tesla structures. Lab on a Chip 4, 109–113 (2004)CrossRefGoogle Scholar
  53. 53.
    Sudarsan, A.P., Ugaz, V.M.: Multivortex micromixing. Proceedings of the National Academy of Sciences of the United States of America 103, 7228–7233 (2006)CrossRefGoogle Scholar
  54. 54.
    Alleborn, N., Nandakumar, K., Raszillier, H., Durst, F.: Further contributions on the two-dimensional flow in a sudden expansion. J. Fluid Mech. 330, 169–188 (1997)zbMATHCrossRefGoogle Scholar
  55. 55.
    Nguyen, N.T.: Micromixers: Fundamentals, Design and Fabrication. William-Andrew (2008)Google Scholar
  56. 56.
    Tice, J.D., Song, H., Lyon, A.D., Ismagilov, R.F.: Formation of droplets and mixing in multiphase microfluidics at low values of the reynolds and the capillary numbers. Langmuir 19, 9127–9133 (2003)CrossRefGoogle Scholar
  57. 57.
    Günther, A., Jhunjhunwala, M., Thalmann, M., Schmidt, M.A., Jensen, K.F.: Micromixing of miscible liquids in segmented gas−liquid flow. Langmuir 21, 1547–1555 (2005)CrossRefGoogle Scholar
  58. 58.
    Hosokawa, K., Fujii, T., Endo, I.: Handling of picoliter liquid samples in a poly (dimethylsiloxane)-based microfluidic device. Anal. Chem. 71, 4781–4785 (1999)CrossRefGoogle Scholar
  59. 59.
    Handique, K., Burns, M.A.: Mathematical modeling of drop mixing in a slit-type microchannel. J. Micromech. Microeng. 11, 548–554 (2001)CrossRefGoogle Scholar
  60. 60.
    Kashid, M.N., Gerlach, I., Goetz, S., Franzke, J., Acker, J.F., Platte, F., Agar, D.W., Turek, S.: Internal circulation within the liquid slugs of a liquid-liquid slug-flow capillary microreactor. Ind. Eng. Chem. Res. 44, 5003–5010 (2005)CrossRefGoogle Scholar
  61. 61.
    Grigoriev, R.O.: Chaotic mixing in thermocapillary-driven microdroplets. Phys. Fluids 17, 033601 (2005)Google Scholar
  62. 62.
    Muradoglu, M., Stone, H.A.: Mixing in a drop moving through a serpentine channel: A computational study. Phys. Fluids 17, 073305 (2005)Google Scholar
  63. 63.
    Garstecki, P., Fischbach, M.A., Whitesides, G.M.: Design for mixing using bubbles in branched microfluidic channels. Appl. Phys. Lett. 86, 244108 (2005)CrossRefGoogle Scholar
  64. 64.
    Salman, W., Angeli, P., Gavriilidis, A.: Sample pulse broadening in Taylor flow microchannels for screening applications. Chem. Eng. Technol. 28, 509–514 (2005)CrossRefGoogle Scholar
  65. 65.
    Garstecki, P., Fuerstman, M.J., Fischbach, M.A., Sia, S.K., Whitesides, G.M.: Mixing with bubbles: a practical technology for use with portable microfluidic devices. Lab Chip 6, 207–212 (2006)CrossRefGoogle Scholar
  66. 66.
    Fan, J., Zhang, Y.X., Wang, L.Q.: Bubble formation in microfluidic T-junctions (submitted, 2009)Google Scholar
  67. 67.
    Tan, Y.C., Lee, A.: Micro/naodroplets in microfluidic devices. In: Bhushan, B. (ed.) Springer Handbook of Nanotechnology, pp. 571–587. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  68. 68.
    Wang, L.Q., Zhang, Y.X., Cheng, L.: Magic microfluidic T-junctions: valving and bubbling. Chaos, Solitons & Fractals 39, 1530–1537 (2009)CrossRefGoogle Scholar
  69. 69.
    Shah, R.K., Shum, H.C., Rowat, A.C., Lee, D.Y., Agresti, J.J., Utada, A.S., Chu, L.Y., Kim, J.W., Fernandez-Nieves, A., Martinez, C.J., Weitz, D.A.: Designer emulsions using microfluidics. Materials Today 11, 18–27 (2008)CrossRefGoogle Scholar
  70. 70.
    Wei, X.H., Wang, L.Q.: Microfluidic Cu2O nanofluids (submitted, 2009)Google Scholar
  71. 71.
    Xuan, Y.M., Li, Q., Zhang, X., Hu, W.: Aggregation structure and thermal conductivity of nanofluids. AICHE Journal 49, 1038–1043 (2003)CrossRefGoogle Scholar
  72. 72.
    Koo, J., Kleinstreuer, C.: A new thermal conductivity model for nanofluids. J. Nanoparticle Research 6, 577–588 (2004)CrossRefGoogle Scholar
  73. 73.
    Jang, S.P., Choi, S.U.S.: Role of Brownian motion in the enhanced thermal conductivity of nanofluids. Appl. Phys. Lett. 84, 4316–4318 (2004)CrossRefGoogle Scholar
  74. 74.
    Bhattacharya, P., Saha, S.K., Yadav, A., Phelan, P.E., Prasher, R.S.: Brownian dynamics simulation to determine the effect thermal conductivity of nanofluids. J. Appl. Phys. 95, 6492–6494 (2004)CrossRefGoogle Scholar
  75. 75.
    Prasher, R., Bhattacharya, P., Phelan, P.E.: Thermal conductivity of nanoscale colloidal solutions (nanofluids). Phys. Rev. Lett. 94, 025901 (2005)Google Scholar
  76. 76.
    Prasher, R., Bhattacharya, P., Phelan, P.E.: Brownian-motion-based convective-conductive model for the effective thermal conductivity of nanofluids. J. Heat Transfer 128, 588–595 (2006)CrossRefGoogle Scholar
  77. 77.
    Yu, W., Choi, S.U.S.: The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Maxwell model. J. Nanoparticle Research 5, 167–171 (2003)CrossRefGoogle Scholar
  78. 78.
    Yu, W., Choi, S.U.S.: The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Hamilton-Crosser model. J. Nanoparticle Research 6, 355–361 (2004)CrossRefGoogle Scholar
  79. 79.
    Xue, L., Keblinski, P., Phillpot, S.R., Choi, S.U.S., Eastman, J.A.: Effect of liquid layering at the liquid-solid interface on thermal transport. Int. J. Heat Mass Transfer 47, 4277–4284 (2004)zbMATHCrossRefGoogle Scholar
  80. 80.
    Xie, H., Fujii, M., Zhang, X.: Effect of interfacial nanolayer on the effective thermal conductivity of nanoparticle-fluid mixture. Int. J. Heat Mass Transfer 48, 2926–2932 (2005)CrossRefGoogle Scholar
  81. 81.
    Ren, Y., Xie, H., Cai, A.: Effective thermal conductivity of nanofluids containing spherical nanoparticles. J. Phys. D 38, 3958–3961 (2005)CrossRefGoogle Scholar
  82. 82.
    Leong, K.C., Yang, C., Murshed, S.M.S.: A model for the thermal conductivity of nanofluids: the effect of interfacial layer. J. Nanopart. Res. 8, 245–254 (2006)CrossRefGoogle Scholar
  83. 83.
    Wang, B.X., Zhou, L.P., Peng, X.F.: A fractal model for predicting the effective thermal conductivity of liquid with suspension of nanoparticles. Int. J. Heat Mass Transfer 46, 2665–2672 (2003)zbMATHCrossRefGoogle Scholar
  84. 84.
    Prasher, R., Phelan, P.E., Bhattacharya, P.: Effect of aggregation kinetics on the thermal conductivity of nanoscale colloidal solutions (nanofluid). Nano Letters 6, 1529–1534 (2006)CrossRefGoogle Scholar
  85. 85.
    Rusconi, R., Rodari, E., Piazza, R.: Optical measurements of the thermal properties of nanofluids. Appl. Phys. Lett. 89, 261916 (2006)CrossRefGoogle Scholar
  86. 86.
    Putnam, S.A., Cahill, D.G., Braun, P.V., Ge, Z.B., Shimmin, R.G.: Thermal conductivity of nanoparticle suspensions. J. Appl. Phys. 99, 084308Google Scholar
  87. 87.
    Eapen, J., Williams, W.C., Buongiorno, J., Hu, L.W., Yip, S.: Mean-field versus microconvection effects in nanofluid thermal conduction. Phys. Rev. Lett. 99, 095901 (2007)Google Scholar
  88. 88.
    Das, S.K., Choi, S.U.S., Patel, H.E.: Heat transfer in nanofluids: a review. Heat Transfer Engng. 27, 3–19 (2006)CrossRefGoogle Scholar
  89. 89.
    Keblinski, P., Prasher, R., Eapen, J.: Thermal conductance of nanofluids: is the controversy over? J. Nanopart. Res. 10, 1089–1097 (2008)CrossRefGoogle Scholar
  90. 90.
    Murshed, S.M.S.: Correction and comment on thermal conductance of nanofluids: is the controversy over? J. Nanopart. Res. 11, 511–512 (2009)CrossRefGoogle Scholar
  91. 91.
    Wang, L.Q., Zhou, X.S., Wei, X.H.: Heat Conduction: Mathematical Models and Analytical Solutions. Springer, Heidelberg (2008)Google Scholar
  92. 92.
    Whitaker, S.: The Method of Volume Averaging. Kluwer Academic, Dordrecht (1999)Google Scholar
  93. 93.
    Wang, L.Q.: Generalized Fourier law. Int. J. Heat Mass Transfer 37, 2627–2634 (1994)zbMATHCrossRefGoogle Scholar
  94. 94.
    Auriault, J.L.: Heterogeneous medium: is an equivalent macroscopic description possible? Int. J. Engng. Sci. 29, 785–795 (1991)zbMATHCrossRefGoogle Scholar
  95. 95.
    Quintard, M., Whitaker, S.: One- and two-equation models for transient diffusion processes in two-phase systems. Adv. in Heat Transfer 23, 369–464 (1993)Google Scholar
  96. 96.
    Ochoa-Tapia, J.A., Whitaker, S.: Heat transfer at the boundary between a porous medium and a homogeneous fluid. Int. J. Heat Mass Transfer 40, 2691–27076 (1997)zbMATHCrossRefGoogle Scholar
  97. 97.
    Ochoa-Tapia, J.A., Whitaker, S.: Heat transfer at the boundary between a porous medium and a homogeneous fluid: The one-equation model. J. Porous Media 1, 31–46 (1998)zbMATHGoogle Scholar
  98. 98.
    Howes, F.A., Whitaker, S.: The spatial averaging theorem revisited. Chem. Eng. Sci. 40, 1387–1392 (1985)CrossRefGoogle Scholar
  99. 99.
    Gray, W.G., Leijnse, A., Kolar, R.L., Blain, C.A.: Mathematical Tolls for Changing Spatial Scales in the Analysis of Physical Systems. CRC Press, Boca Raton (1993)Google Scholar
  100. 100.
    Carbonell, R.G., Whitaker, S.: Heat and mass transfer in porous media. In: Bear, J., Corapcioglu, M.Y. (eds.) Fundamentals of Transport Phenomena in Porous Media, pp. 123–198. Martinus Nijhoff (1984)Google Scholar
  101. 101.
    Quintard, M., Kaviany, M., Whitaker, S.: Two-medium treatment of heat transfer in porous media: numerical results for effective parameters. Adv. Water Resour. 20, 77–94 (1997)CrossRefGoogle Scholar
  102. 102.
    Quintard, M., Whitaker, S.: Theoretical Analysis of Transport in Porous Media. In: Vafai, K. (ed.) Handbook of Heat Transfer in Porous Media, pp. 1–52. Marcel Dekker, New York (2000)Google Scholar
  103. 103.
    Quintard, M., Whitaker, S.: Local thermal equilibrium for transient heat conduction: Theory and comparison with numerical experiments. Int. J. Heat Mass Transfer 38, 2779–2796 (1995)zbMATHCrossRefGoogle Scholar
  104. 104.
    Tzou, D.Y.: Macro-to Microscale Heat Transfer: The Lagging Behavior. Taylor & Francis, Abington (1997)Google Scholar
  105. 105.
    Wang, L.Q., Wei, X.H.: Equivalence between dual-phase-lagging and two-phase-system heat conduction processes. Int. J. Heat Mass Transfer 51, 1751–1756 (2008)zbMATHCrossRefGoogle Scholar
  106. 106.
    Wang, L.Q., Zhou, X.S.: Dual-phase-lagging Heat-Conduction Equations. Shandong University Press (2000)Google Scholar
  107. 107.
    Wang, L.Q., Zhou, X.S.: Dual-phase-lagging Heat-Conduction Equations: Problems and Solutions. Shandong University Press (2001)Google Scholar
  108. 108.
    Wang, L.Q., Xu, M.T., Zhou, X.S.: Well-posedness and solution structure of dual-phase-lagging heat conduction. Int. J. Heat Mass Transfer 44, 1659–1669 (2001)zbMATHCrossRefGoogle Scholar
  109. 109.
    Xu, M.T., Wang, L.Q.: Thermal oscillation and resonance in dual-phase-lagging heat conduction. Int. J. Heat Mass Transfer 45, 1055–1061 (2002)zbMATHCrossRefGoogle Scholar
  110. 110.
    Wang, L.Q., Xu, M.T., Wei, X.H.: Dual-phase-lagging and porous-medium heat conduction processes. In: Vadasz, P. (ed.) Emerging Topics in Heat and Mass Transfer in Porous Media - from Bioengineering and Microelectronics to Nanotechnology, pp. 1–37. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  111. 111.
    Fan, J., Wang, L.Q.: Microstructural effects on macroscale thermal properties in nanofluids (submitted, 2009)Google Scholar
  112. 112.
    Bejan, A., Lorente, S.: Design with Constructal Theory. Wiley, Chichester (2008)CrossRefGoogle Scholar
  113. 113.
    Reis, A.H.: Constructal theory: from engineering to physics, and how flow systems develop shape and structure. App. Mech. Rev. 59, 269–282 (2006)CrossRefGoogle Scholar
  114. 114.
    Bejan, A., Lorente, S.: Constructal theory of configuration generation in nature and engineering. J. App. Phys. 100, 041301/1–041301/27 (2006)Google Scholar
  115. 115.
    Bai, C., Wang, L.Q.: Constructal design of particle volume fraction in nanofluids. J. Heat Transfer 131, 112402/1–112402/7 (2009)Google Scholar
  116. 116.
    Wang, L.Q.: An approach for thermodynamic reasoning. Int. J. Modern Phys. B 10, 2531–2551 (1996)CrossRefGoogle Scholar
  117. 117.
    Rocha, L.A., Lorente, S., Bejan, A.: Constructal design for cooling a disc-shaped area by conduction. Int. J. Heat Mass Transfer 45, 1643–1652 (2002)zbMATHCrossRefGoogle Scholar
  118. 118.
    Bejan, A.: Constructal-theory network of conducting paths for cooling a heat generating volume. Int. J. Heat Mass Transfer 40, 799–816 (1997)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Liqiu Wang
    • 1
  • Michel Quintard
    • 2
  1. 1.Department of Mechanical Engineeringthe University of Hong KongHong Kong
  2. 2.Institut de Mécanique des FluidesC.N.R.S.ToulouseFrance

Personalised recommendations