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Steady finite-amplitude Rayleigh–Bénard convection of ethylene glycol–copper nanoliquid in a high-porosity medium made of 30% glass fiber-reinforced polycarbonate

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Abstract

In the paper, we make linear and nonlinear stability analyses of Rayleigh–Bénard convection in a Newtonian nanoliquid-saturated high-porosity medium. Single-phase model is used for nanoliquids, and values of thermophysical quantities concerning ethylene glycol–copper nanoliquid-saturated porous medium are calculated using mixture theory or phenomenological relations. The study is carried out for free-free, rigid-rigid and rigid-free isothermal boundaries. Boundary effects on onset of convection are shown to conform to classical predictions. The addition of copper nanoparticles to ethylene glycol is shown to lead to advanced onset of convection in the porous medium and thereby to a substantial increase in heat transport. Theoretical explanation is provided for the enhanced heat transfer situation in the medium. With suitable scaling in quantities, the result concerning heat transfer in ethylene glycol–copper nanoliquid-saturated porous medium is shown to be obtainable from those of ethylene glycol-saturated porous medium without copper nanoparticles. Nanoparticles serve the purpose of cooling and porous matrix retains the heat, thereby meaning that residence time of heat in the system can be regulated by using nanoparticles and porous matrix.

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Abbreviations

\(\alpha \) :

Thermal diffusivity of the nanoliquid in saturated porous medium (\({\rm m}^{2}\,{\rm s}^{-1}\))

\(\alpha _{1}\) :

Thermal diffusivity of the base liquid in saturated porous medium

\(\beta \) :

Thermal expansion coefficient of the nanoliquid in saturated porous medium (\({\rm K^{-1}}\))

\(\beta _{1}\) :

Thermal expansion coefficient of the base liquid in saturated porous medium

\(\chi \) :

Nanoparticle volume fraction

\(\Delta T\) :

Temperature difference

\(\Lambda \) :

Brinkman number or ratio of viscosities

\(\mu \) :

Viscosity of the nanoliquid

\(\mu ^{\prime}\) :

Viscosity of the nanoliquid in saturated porous medium (kg (m s)−1)

\(\nu \) :

Wave number (\({\rm m}^{-1}\))

\(\phi \) :

Porosity

\(\Psi \) :

Non-dimensional stream function

\(\psi \) :

Dimensional stream function

\(\rho \) :

Density of the nanoliquid in saturated porous medium (kg m–4)

\(\sigma ^{2}\) :

Porous parameter

\(\Theta \) :

Non-dimensional temperature

ABC :

Amplitudes of convection

\(C_\mathrm{p}\) :

Specific heat capacity of the nanoliquid in saturated porous medium at constant pressure (J (kg K)−1)

D :

Ozoe heat transfer diminishment parameter

E :

Ozoe heat transfer enhancement parameter

\(g=(0,0,-g)\) :

Acceleration due to gravity (\({\rm m\,s}^{-2}\))

K :

Permeability of the porous medium

k :

Thermal conductivity of the nanoliquid in saturated porous medium

\(k_{1}\) :

Thermal conductivity of the base liquid in saturated porous medium

\(k_\mathrm{l}\) :

Thermal conductivity of the base liquid

\(k_\mathrm{nl}\) :

Thermal conductivity of the nanoliquid (W (m K)−1)

\(k_\mathrm{np}\) :

Thermal conductivity of the nanoparticle

M :

Ratio of specific heats

\(\mathrm{Nu}\) :

Nusselt number of the nanoliquid in saturated porous medium

\(\mathrm{Nu}_{1}\) :

Nusselt number of the base liquid in saturated porous medium

\(\mathrm{Nu}_\mathrm{nl}\) :

Nusselt number of the nanoliquid

p :

Pressure

\(q=(u,0,w)\) :

Velocity vector (\({\rm m\,s^{-1}}\))

Ra:

Rayleigh number of the nanoliquid in saturated porous medium

T :

Dimensional temperature (K)

\(T_{0}\) :

Reference temperature

uw :

Horizontal and vertical velocity components

xX :

Dimensional and non-dimensional horizontal coordinates

zZ :

Dimensional and non-dimensional vertical coordinates

\(\mathrm{Nu}_\mathrm{l}\) :

Nusselt number of the base liquid

h :

Distance between the plates (m)

0:

At reference value

1:

Liquid property in saturated porous medium

b:

Basic state

c:

Critical

l:

Base liquid

nl:

Nanoliquid

np:

Nanoparticle

s:

Solid

\(\prime\) :

Perturbed quantity

FF:

Free-free boundaries

RF:

Rigid-free boundaries

RR:

Rigid-rigid boundaries

References

  1. 1.

    https://www.matbase.com/material-categories/natural-and-synthetic-composites/polymer-matrix-composites-pmc/reinforced-polymers/material-properties-of-polycarbonate-30-percent-glass-fiber-reinforced-pc-gf30.html.

  2. 2.

    Abu-Nada E. Rayleigh–Bénard convection in nanofluids: effect of temperature dependent properties. Int J Therm Sci. 2011;50(9):1720–30.

  3. 3.

    Adler P. Porous media: geometry and transports. Boston: Elsevier; 2013.

  4. 4.

    Bergman TL, Incropera FP, Lavine AS, Dewitt DP. Fundamentals of heat and mass transfer. Sixth ed. New York: Wiley; 2011.

  5. 5.

    Bianco V, Manca O, Nardini S, Vafai K. Heat transfer enhancement with nanofluids. New York: CRC Press; 2015.

  6. 6.

    Bourantas GC, Skouras ED, Loukopoulos VC, Burganos VN. Heat transfer and natural convection of nanofluids in porous media. Eur J Mech B Fluids. 2014;43:45–56.

  7. 7.

    Brinkman HC. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20:571.

  8. 8.

    Buongiorno J. Convective transport in nanofluids. J Heat Transfer. 2006;128:240–50.

  9. 9.

    Chandrasekhar S. Hydrodynamic and hydromagnetic stability. London: Clarendon Press; 1961.

  10. 10.

    Chandrasekhar S, Reid WH. On the expansion of functions which satisfy four boundary conditions. Proc Natl Acad Sci USA. 1957;43:521–7.

  11. 11.

    Corcione M. Rayleigh–Bénard convection heat transfer in nanoparticle suspensions. Int J Heat Fluid Flow. 2011;32:65–77.

  12. 12.

    Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J Heat Transfer. 2003;125:567–74.

  13. 13.

    Dhananjay Y, Agrawal GS, Bhargava R. Rayleigh–Bénard convection in nanofluid. Int J Appl Math Mech. 2011;7:61–76.

  14. 14.

    Dullien FAL. Porous media: fluid transport and pore structure. Second ed. San Diego: Academic Press; 2012.

  15. 15.

    Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ. Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett. 2001;78:718–20.

  16. 16.

    Hamilton RL, Crosser OK. Thermal conductivity of heterogeneous two-component systems. Ind Eng Chem Fundam. 1962;1:187–91.

  17. 17.

    Ingham DB, Pop I. Transport phenomena in porous media. Oxford: Elsevier; 1998.

  18. 18.

    Kasaeian A, Azarian RD, Mahian O, Kolsi L, Chamkha AJ, Wongwises S, Pop I. Nanofluid flow and heat transfer in porous media: a review of the latest developments. Int J Heat Mass Transfer. 2017;107:778–91.

  19. 19.

    Kaviany M. Principles of heat transfer in porous media. New York: Springer; 2012.

  20. 20.

    Khanafer K, Vafai K. Applications of nanofluids in porous medium. J Therm Anal Calorim. 2019;135(2):1479–92.

  21. 21.

    Khanafer K, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf. 2003;46(19):3639–53.

  22. 22.

    Kim J, Kang YT, Choi CK. Analysis of convective instability and heat transfer characteristics of nanofluids. Phys Fluids (1994-present). 2004;16:2395–401.

  23. 23.

    Liu D, Yu L. Single-phase thermal transport of nanofluids in a minichannel. J Heat Transf. 2011;133:031009-1–-11.

  24. 24.

    Minkowycz WJ, Sparrow EM, Abraham JP. Nanoparticle heat transfer and fluid flow, vol. 4. Boca Raton: CRC Press; 2012.

  25. 25.

    Moradi A, Toghraie D, Meghdadi IAH, Hosseinian A. An experimental study on mwcnt-water nanofluids flow and heat transfer in double-pipe heat exchanger using porous media. J Therm Anal Calorim. 2019;132:1–11.

  26. 26.

    Nagata M. Bifurcations at the Eckhaus points in two-dimensional Rayleigh–Bénard convection. Phys Rev E. 1995;52:6141–5.

  27. 27.

    Nield DA, Bejan A. Convection in porous media. 3rd ed. New York: Springer; 2006.

  28. 28.

    Nield DA, Kuznetsov AV. Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transfer. 2009;52:5796–801.

  29. 29.

    Nield DA, Kuznetsov AV. The onset of convection in a horizontal nanofluid layer of finite depth. Eur J Mech B Fluids. 2010;29(3):217–23.

  30. 30.

    Platten JK, Legros JC. Convection in liquids. Berlin: Springer; 2012.

  31. 31.

    Savithiri S, Pattamatta A, Das SK. Rayleigh–Bénard convection in water-based alumina nanofluid: a numerical study. Numer Heat Transf Part A Appl. 2017;71:202–14.

  32. 32.

    Shenoy A, Sheremet MA, Pop I. Convective flow and heat transfer from wavy surfaces: viscous fluids, porous media, and nanofluids. Boca Raton: CRC Press; 2016.

  33. 33.

    Sheremet MA, Cimpean DS, Pop I. Free convection in a partially heated wavy porous cavity filled with a nanofluid under the effects of brownian diffusion and thermophoresis. Appl Therm Eng. 2017;113:413–8.

  34. 34.

    Sheremet MA, Groşan T, Pop I. Steady-state free convection in right-angle porous trapezoidal cavity filled by a nanofluid: buongiornos mathematical model. Eur J Mech B Fluids. 2015;53:241–50.

  35. 35.

    Sheremet MA, Pop I. Free convection in a porous horizontal cylindrical annulus with a nanofluid using buongiornos model. Comput Fluids. 2015;118:182–90.

  36. 36.

    Siddheshwar PG, Kanchana C, Kakimoto Y, Nakayama A. Steady finite-amplitude Rayleigh–Bénard convection in nanoliquids using a two-phase model-theoretical answer to the phenomenon of enhanced heat transfer. ASME J Heat Transf. 2017;139:012402-1–8.

  37. 37.

    Siddheshwar PG, Meenakshi N. Amplitude equation and heat transport for Rayleigh–Bénard convection in Newtonian liquids with nanoparticles. Int J Appl Comput Math. 2015;2:1–22.

  38. 38.

    Siddheshwar PG, Ramachandramurthy V, Uma D. Rayleigh–Bénard and Marangoni magnetoconvection in Newtonian liquid with thermorheological effects. Int J Eng Sci. 2011;49(10):1078–94.

  39. 39.

    Siddheshwar PG, Sekhar GN, Jayalatha G. Effect of time-periodic vertical oscillations of the rayleigh–bénard system on nonlinear convection in viscoelastic liquids. J Non Newton Fluid Mech. 2010;165(19–20):1412–8.

  40. 40.

    Straughan B. Stability and wave motion in porous media, vol. 165. New York: Springer; 2008.

  41. 41.

    Tzou DY. Thermal instability of nanofluids in natural convection. Int J Heat Mass Transfer. 2008;51:2967–79.

  42. 42.

    Xuan Y, Roetzel W. Conceptions for heat transfer correlation of nanofluids. Int J Heat Mass Transfer. 2000;43:3701–7.

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Acknowledgements

One of the authors, T N Sakshath, is thankful to the Department of Backward Classes Welfare, Government of Karnataka, for the financial support and also to the Bangalore University for supporting his research.

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Correspondence to P. G. Siddheshwar.

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Siddheshwar, P.G., Sakshath, T.N. Steady finite-amplitude Rayleigh–Bénard convection of ethylene glycol–copper nanoliquid in a high-porosity medium made of 30% glass fiber-reinforced polycarbonate. J Therm Anal Calorim (2020). https://doi.org/10.1007/s10973-019-09214-4

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Keywords

  • Nanoliquid
  • Rayleigh–Bénard convection
  • Porous medium
  • Linear
  • Nonlinear
  • Stability
  • Single-phase