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The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium

Abstract

The paper develops a theory of double-diffusive nanofluid convection in porous media. This theory is applied to investigating the onset of nanofluid convection in a horizontal layer of a porous medium saturated by a nanofluid for the case when the base fluid of the nanofluid is itself a binary fluid such as salty water. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. Both non-oscillatory and oscillatory cases are investigated by using Galerkin method; the stability boundaries for these cases are approximated by simple and useful analytical expressions.

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Abbreviations

C :

Solute concentration

D B :

Brownian diffusion coefficient (m2/s)

D T :

Thermophoretic diffusion coefficient (m2/s)

H :

Dimensional layer depth (m)

k :

Thermal conductivity of the nanofluid (W/m K)

K :

Permeability (m2)

Le:

Thermo-solutal Lewis number, defined by Eq. (30)

Ln:

Thermo-nanofluid Lewis number, defined by Eq. (23)

N A :

Modified diffusivity ratio, defined by Eq. (28)

N B :

Modified particle-density increment, defined by Eq. (29)

N CT :

Soret parameter, defined by Eq. (32)

N TC :

Dufour parameter, defined by Eq. (31)

p*:

Pressure (Pa)

p :

Dimensionless pressure, \({p^{\ast}K/\mu \alpha_{\rm m}}\)

Ra:

Thermal Rayleigh-Darcy number, defined by Eq. (24)

Rm:

Basic-density Rayleigh number, defined by Eq. (26)

Rn:

Nanoparticle Rayleigh number, defined by Eq. (27)

Rs:

Solutal Rayleigh number, defined by Eq. (25)

t*:

Time (s)

t :

Dimensionless time, \({t^{\ast}\alpha_{\rm m} /H^{2}}\)

T*:

Nanofluid temperature (K)

T :

Dimensionless temperature, \({\frac{T^\ast-T^\ast_c}{T^\ast_h -T^\ast_c }}\)

\({T^*_{\rm c}}\) :

Temperature at the upper wall (K)

\({T^*_{\rm h}}\) :

Temperature at the lower wall (K)

(u, v, w):

Dimensionless velocity components, \({(u^\ast,v^\ast,w^\ast)H/\alpha_{\rm m}}\) (m/s)

v :

Nanofluid velocity (m/s)

(x, y, z):

Dimensionless Cartesian coordinates, (x*, y*, z*)/H; z is the vertically-upward coordinate

(x*, y*, z*):

Cartesian coordinates (m)

α m :

Effective thermal diffusivity of the porous medium (m/s2)

β C :

Solutal volumetric coefficient

β T :

Thermal volumetric coefficient (K−1)

ε :

Porosity

μ :

Viscosity of the fluid (N s/m2)

ρ :

Fluid density (kg/m3)

ρ p :

Nanoparticle mass density (kg/m3)

σ :

Thermal capacity ratio

ϕ*:

Nanoparticle volume fraction

ϕ :

Relative nanoparticle volume fraction, \({\frac{\phi^\ast-\phi^\ast_0}{\phi^\ast_1 -\phi^\ast_0}}\)

*:

Dimensional variable

′:

Perturbation variable

b:

Basic solution

f:

Fluid

p:

Particle

References

  1. Abbassi H., Aghanajafi C.: Evaluation of heat transfer augmentation in a nanofluid-cooled microchannel heat sink. J. Fusion Energy 25, 187–196 (2006)

  2. Anoop K.B., Kabelac S., Sundararajan T., Das S.K.: Rheological and flow characteristics of nanofluids: influence of electroviscous effects and particle agglomeration. J. Appl. Phys. 106, 034909 (2009)

  3. Buongiorno J.: Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006)

  4. Buongiorno, J., Hu, L.-W.: Nanofluid coolants for advanced nuclear power plants. Paper no. 5705, Proceedings of ICAPP ‘05, Seoul, May 15–19 (2005)

  5. Choi, S.: Enhancing thermal conductivity of fluids with nanoparticles. ASME FED-Vol. 231/ MD-Vol. 66, 99–105 (1995)

  6. Das S.K., Choi S.U.S.: A review of heat transfer in nanofluids. Adv. Heat Transf. 41, 81–197 (2009)

  7. Das S.K., Choi S.U.S., Yu W., Pradeep T.: Nanofluids: Science and Technology. Wiley, Hoboken, NY (2008)

  8. Feng Y., Yu B., Feng K., Xu P., Zou M.: Thermal conductivity of nanofluids and size distribution of nanoparticles by monte carlo simulations. J. Nanoparticle Res. 10, 1319–1328 (2008)

  9. Ganguly S., Sikdar S., Basu S.: Experimental investigation of the effective electrical conductivity of aluminum oxide nanofluids. Powder Technol. 196, 326–330 (2009)

  10. Ghazvini M., Akhavan-Behabadi M.A., Esmaeili M.: The effect of viscous dissipation on laminar nanofluid flow in a microchannel heat sink. IME J. Mech. Eng. Sci. 223, 2697–2706 (2009)

  11. Ghazvini M., Shokouhmand H.: Investigation of a nanofluid-cooled microchannel heat sink using fin and porous media approaches. Energy Convers. Manage. 50, 2373–2380 (2009)

  12. Hwang K.S., Jang S.P., Choi S.U.S.: Flow and convective heat transfer characteristics of water-based Al2O3 nanofluids in fully developed laminar flow regime. Int. J. Heat Mass Transf. 52, 193–199 (2009)

  13. Jain S., Patel H.E., Das S.K.: Brownian dynamic simulation for the prediction of effective thermal conductivity of nanofluid. J. Nanoparticle Res. 11, 767–773 (2009)

  14. Kim S.Y., Koo J.M., Kuznetsov A.V.: Effect of anisotropy in permeability and effective thermal conductivity on thermal performance of an aluminum foam heat sink. Numer. Heat Transf. A 40, 21–36 (2001)

  15. Kim S.Y., Kuznetsov A.V.: Optimization of pin-fin heat sinks using anisotropic local thermal nonequilibrium porous model in a jet impinging channel. Numer. Heat Transf. A 44, 771–787 (2003)

  16. Kuznetsov A.V., Nield D.A.: Thermal instability in a porous medium saturated by a nanofluid: Brinkman model. Transp. Porous Med. 81, 409–422 (2010a)

  17. Kuznetsov A.V., Nield D.A.: Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Transp. Porous Med 83, 425–436 (2010b)

  18. Kuznetsov, A.V., Nield, D.A.: The effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid: Brinkman model. J. Porous Med 14, to appear (2011)

  19. Lee S., Choi S.U.S., Li S., Eastman J.A.: Measuring thermal conductivity of fluids containing oxide nanoparticles. ASME J. Heat Transf. 121, 280–289 (1999)

  20. Masuda H., Ebata A., Teramae K., Hishinuma N.: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7, 227–233 (1993)

  21. Merabia S., Shenogin S., Joly L., Keblinski P., Barrat J.: Heat transfer from nanoparticles: a corresponding state analysis. Proc. Nat. Acad. Sci. 106, 15113–15118 (2009)

  22. Nelson I.C., Banerjee D., Ponnappan R.: Flow loop experiments using polyalphaolefin nanofluids. J. Thermophys. Heat Transf. 23, 752–761 (2009)

  23. Nield D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4, 533–560 (1968)

  24. Nield D.A., Kuznetsov A.V.: Thermal instability in a porous medium layer saturated by a nanofluid. Int. J. Heat Mass Transf. 52, 5796–5801 (2009)

  25. Nield, D.A., Kuznetsov, A.V.: The onset of double-diffusive convection in a nanofluid layer. ASME J. Heat Transf. submitted (2010)

  26. Pearlstein A.J., Harris R.M., Terrones G.: The onset of convective instability in a triple diffusive layer. J. Fluid Mech. 202, 443–465 (1989)

  27. Rea U., McKrell T., Hu L., Buongiorno J.: Laminar convective heat transfer and viscous pressure loss of alumina-water and zirconia-water nanofluids. Int. J. Heat Mass Transf. 52, 2042–2048 (2009)

  28. Salloum M., Ma R.H., Weeks D., Zhu L.: Controlling nanoparticle delivery in magnetic nanoparticle hyperthermia for cancer treatment: experimental study in agarose gel. Int. J. Hyperthermia 24, 337–345 (2008a)

  29. Salloum M., Ma R., Zhu L.: An in-vivo experimental study of temperature elevations in animal tissue during magnetic nanoparticle hyperthermia. Int. J. Hyperthermia 24, 589–601 (2008b)

  30. Salloum M., Ma R., Zhu L.: Enhancement in treatment planning for magnetic nanoparticle hyperthermia: optimization of the heat absorption pattern. Int. J. Hyperthermia 25, 309–321 (2009)

  31. Tsai T., Chein R.: Performance analysis of nanofluid-cooled microchannel heat sinks. Int. J. Heat Fluid Flow 28, 1013–1026 (2007)

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Correspondence to A. V. Kuznetsov.

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Kuznetsov, A.V., Nield, D.A. The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium. Transp Porous Med 85, 941–951 (2010). https://doi.org/10.1007/s11242-010-9600-1

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Keywords

  • Nanofluid convection
  • Porous media
  • Brownian motion
  • Thermophoresis
  • Natural convection
  • Horizontal layer