Journal of Thermal Analysis and Calorimetry

, Volume 132, Issue 2, pp 1291–1306 | Cite as

Effects of partial slip on entropy generation and MHD combined convection in a lid-driven porous enclosure saturated with a Cu–water nanofluid

  • A. J. Chamkha
  • A. M. Rashad
  • T. Armaghani
  • M. A. Mansour
Article

Abstract

In this work, the influences of heat generation/absorption and nanofluid volume fraction on the entropy generation and MHD combined convection heat transfer in a porous enclosure filled with a Cu–water nanofluid are studied numerically with of partial slip effect. The finite volume technique is utilized to solve the dimensionless equations governing the problem. A comparison with already published studies is conducted, and the data are found to be in an excellent agreement. The minimization of entropy generation and the local heat transfer according to various values of the controlling parameters are reported in detail. The outcome indicates that an augmentation in the heat generation/absorption parameter decreases the Nusselt number. Also, when the volume fraction is raised, the Nusselt number and entropy generation are reduced. The impact of Hartmann number on heat transfer and the Richardson number on the entropy generation and the thermal rendering criteria are also presented and discussed.

Keywords

Heat generation/absorption Entropy generation Nanofluid Partial slip Nusselt number 

List of symbols

B

Dimensionless of heat source/sink length

\( B_{0} \)

Magnetic field strength (T)

\( Be \)

Bejan number

\( b \)

Length of heat source (m)

\( C_{\text{p}} \)

Specific heat at constant pressure \( ( {\text{J}}\;{\text{kg}}\;{\text{K}}^{ - 1} ) \)

D

Dimensionless heat source position

Da

Darcy number

d

Location of heat sink and source (m)

H

Length of cavity (m)

Ha

Hartmann number, \( B_{0} L\sqrt {\sigma_{\text{f}} /\rho_{\text{f}} \nu_{\text{f}} } \)

Gr

Grashof number, \( g\beta_{\text{f}} H^{3} \Delta T/\upsilon^{2}_{\text{f}} \)

g

Acceleration due to gravity (m s−2)

K

Permeability of porous medium (m2)

k

Thermal conductivity (W m−1 K−1)

Nu

Local Nusselt number

\( Nu_{\text{m}} \)

Average Nusselt number of heat source

\( p \)

Fluid pressure (Pa)

\( P \)

Dimensionless pressure, \( pH/\rho_{\text{nf}} \alpha_{\text{f}}^{2} \)

Pr

Prandtl number, \( \upsilon_{\text{f}} /\alpha_{\text{f}} \)

Re

Reynolds number, \( V_{0} H/\upsilon_{\text{f}} \)

S

Entropy generation (W K−1 m−3)

T

Temperature (K)

\( T_{\text{c}} \)

Cold wall temperature (K)

\( T_{\text{h}} \)

Heated wall temperature (K)

u,v

Velocity components in x, y directions (m s−1)

\( U,V \)

Dimensionless velocity components, u/V 0, v/V 0

\( x,y \)

Cartesian coordinates (m)

\( X,Y \)

Dimensionless coordinates, x/L, y/L

Greek symbols

\( \alpha \)

Thermal diffusivity, \( {\text{m}}^{2} \;{\text{s}}^{ - 1} ,{\text{k}}/\rho c_{\text{p}} \)

\( \beta \)

Thermal expansion coefficient, K−1

\( \phi \)

Solid volume fraction

σ

Effective electrical conductivity \( (\upmu {\text{S}}\;{\text{cm}}^{ - 1} ) \)

\( \theta \)

Dimensionless temperature, \( {{(T - T_{\text{c}} )} \mathord{\left/ {\vphantom {{(T - T_{\text{c}} )} {(T_{\text{h}} - T_{\text{c}} }}} \right. \kern-0pt} {(T_{\text{h}} - T_{\text{c}} }}) \)

\( \mu \)

Dynamic viscosity (N s m−2)

\( \nu \)

Kinematic viscosity \( ( {\text{m}}^{2} \;{\text{s}}^{ - 1} ) \)

\( \rho \)

Density (kg m−3)

Subscripts

c

Cold

0

Reference

f

Pure fluid

h

Hot

m

Average

nf

Nanofluid

p

Nanoparticle

References

  1. 1.
    Khanafer KM, Al-Amiri AM, Pop I. Numerical simulation of unsteady mixed convection in a driven cavity using an externally excited sliding lid. Eur J Mech B/Fluids. 2007;26:669–87.CrossRefGoogle Scholar
  2. 2.
    Rahman MDM, Alim MA, Saha S, Chowdhury MK. A numerical study of mixed convection in a square cavity with a heat conducting square cylinder at different locations. J Mech Eng. Instit Eng Bangl ME. 2008;39:78–85.Google Scholar
  3. 3.
    Moshizi SA, Malvandi A. Different modes of nanoparticle migration at mixed convection of Al2O3–water nanofluid inside a vertical microannulus in the presence of heat generation/absorption. J Therm Anal Calorim. 2016;126:1947–62.CrossRefGoogle Scholar
  4. 4.
    Garoosi F, Rohani B, Rashidi MM. Two phase simulation of natural convection and mixed convection of the nanofluid in a square cavity with internal and external heating. Powder Technol. 2015;275:304–21.CrossRefGoogle Scholar
  5. 5.
    Mashaei PR, Shahryari M, Madani S. Numerical hydrothermal analysis of water–Al2O3 nanofluid forced convection in a narrow annulus filled by porous medium considering variable properties. J Therm Anal Calorim. 2016;126:891–904.CrossRefGoogle Scholar
  6. 6.
    Hady FM, Ibrahim FS, Abdel Gaied SM, Eid MR. Effect of heat generation/absorption natural convective boundary layer flow from a vertical cone embedded in a porous medium filled with a non-Newtonian nanofluid. Int Commun Heat Mass Transfer. 2011;38:1414–20.CrossRefGoogle Scholar
  7. 7.
    Choi SUS. Enhancing thermal conductivity of fluid with nanoparticles. Dev Appl Non-Newtonian Flows. 1995;66:99–105.Google Scholar
  8. 8.
    Ahmad S, Pop I. Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Int Commun Heat Mass Transf. 2010;37(8):987–91.CrossRefGoogle Scholar
  9. 9.
    Tiwari RK, Das MK. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transf. 2007;50:2002–18.CrossRefGoogle Scholar
  10. 10.
    Cimpean DS, Pop I. Fully developed mixed convection flow of a nanofluid through an inclined channel filled with a porous medium. Int J Heat Mass Transf. 2012;55:907–14.CrossRefGoogle Scholar
  11. 11.
    Gorla RSR, Chamkha AJ, Rashad AM. Mixed convective boundary layer flow over a vertical wedge embedded in a porous medium saturated with a nanofluid-natural convection dominated regime. Nanoscale Res Lett. 2011;6:207–14.CrossRefGoogle Scholar
  12. 12.
    Ghalambaz M, Noghrehabadi A. Effects of heat generation/absorption on natural convection of nanofluids over the vertical plate embedded in a porous medium using drift flux model. J Comput Appl Res Mech Eng. 2014;3:113–24.Google Scholar
  13. 13.
    Matin MH, Ghanbari B. Effects of Brownian motion and thermophoresis on the mixed convection of nanofluid in a porous channel including flow reversal. Transp Porous Media. 2014;101:115–36.CrossRefGoogle Scholar
  14. 14.
    Srinivasacharya D, Kumar PV. Mixed convection along an inclined wavy surface in a nanofluid saturated porous medium with wall heat flux. J Nanofluids. 2016;5:120–9.CrossRefGoogle Scholar
  15. 15.
    Jafarian B, Hajipour M, Khademi R. Conjugate heat transfer of MHD non-Darcy mixed convection flow of a nanofluid over a vertical slender hollow cylinder embedded in porous media. Transp Phenom Nano and Micro Scales. 2016;4:1–10.Google Scholar
  16. 16.
    Esfe MH, Akbari M, Karimipour A, Afrand M, Mahian O, Wongwises S. Mixed-convection flow and heat transfer in an inclined cavity equipped to a hot obstacle using nanofluids considering temperature-dependent properties. Int J Heat Mass Transf. 2015;85:656–66.CrossRefGoogle Scholar
  17. 17.
    Rashidi I, Mahian O, Lorenzini G, Biserni C, Wongwises S. Natural convection of Al2O3/water nanofluid in a square cavity: effects of heterogeneous heating. Int J Heat Mass Transf. 2014;74:391–402.CrossRefGoogle Scholar
  18. 18.
    Heris SZ, Pour MB, Mahian O, Wongwises S. A comparative experimental study on the natural convection heat transfer of different metal oxide nanopowders suspended in turbine oil inside an inclined cavity. Int J Heat Mass Transf. 2014;73:231–8.CrossRefGoogle Scholar
  19. 19.
    Ho CJ, Chen D, Yan W, Mahian O. Buoyancy-driven flow of nanofluids in a cavity considering the Ludwig-Soret effect and sedimentation: numerical study and experimental validation. Int J Heat Mass Transf. 2014;77:684–94.CrossRefGoogle Scholar
  20. 20.
    Estelle´ P, Mahia O, Mare´ T, Öztop HF. Natural convection of CNT water-based nanofluids in a differentially heated square cavity. J Therm Anal Calorim. 2017;128:1765–70.CrossRefGoogle Scholar
  21. 21.
    Mahian O, Kianifar A, Heris SZ, Wongwises S. Natural convection of silica nanofluids in square and triangular enclosures: theoretical and experimental study. Int J Heat Mass Transf. 2016;99:792–804.CrossRefGoogle Scholar
  22. 22.
    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 Transf. 2017;107:778–91.CrossRefGoogle Scholar
  23. 23.
    Bejan A. A study of entropy generation in fundamental convective heat transfer. J Heat Transf. 1979;101:718–25.CrossRefGoogle Scholar
  24. 24.
    Bejan A. Second-law analysis in heat and thermal design. Adv Heat Transf. 1982;15:1–58.CrossRefGoogle Scholar
  25. 25.
    Bejan A. Entropy generation minimization. Boca Raton: CRC Press; 1996.Google Scholar
  26. 26.
    Mahian O, Kianifar A, Kleinstreuer C, Al-Nimr MA, Pop I, Sahin AZ, Wongwises S. A review of entropy generation in nanofluid flow. Int J Heat Mass Transf. 2013;65:514–32.CrossRefGoogle Scholar
  27. 27.
    Armaghani T, Kasaeipoor A, Alavi N, Rashidi MM. Numerical investigation of water-alumina nanofluid natural convection heat transfer and entropy generation in a baffled L-shaped cavity. J Mol Liq. 2016;223:243–51.CrossRefGoogle Scholar
  28. 28.
    Chamkha AJ, Ismael MA, Kasaeipoor A, Armaghani T. Entropy generation and natural convection of CuO–water nanofluid in C-Shaped cavity under magnetic field. Entropy. 2016;18:50–60.CrossRefGoogle Scholar
  29. 29.
    Sheikholeslami M, Ellahi R, Ashorynejad HR, Domairry G, Hayat T. Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. J Comput Theor Nanosci. 2014;11:486–96.CrossRefGoogle Scholar
  30. 30.
    Ting TW, Hung YM, Guo N. Entropy generation of viscous dissipative nanofluid flow in thermal non-equilibrium porous media embedded in microchannels. Int J Heat Mass Transf. 2015;81:862–77.CrossRefGoogle Scholar
  31. 31.
    Bhatti MM, Abbas T, Rashidi MM. Numerical study of entropy generation with nonlinear thermal radiation on magnetohydrodynamics non-Newtonian nanofluid through a porous shrinking sheet. J Magnet. 2016;21:468–75.CrossRefGoogle Scholar
  32. 32.
     Das S, Chakraborty S, Jana RN, Makinde OD. Entropy analysis of unsteady magneto-nanofluid flow past accelerating stretching sheet with convective boundary condition. Applied Mathematics and Mechanics. 36;12:1593–1610.Google Scholar
  33. 33.
    Ismael MA, Armaghani T, Chamkha AJ. Conjugate heat transfer and entropy generation in a cavity filled with a nanofluid-saturated porous media and heated by a triangular solid. J Taiwan Inst Chem Eng. 2016;59:138–51.CrossRefGoogle Scholar
  34. 34.
    Armaghani T, Ismael MA, Chamkha AJ. Analysis of entropy generation and natural convection in an inclined partially porous layered cavity filled with a nanofluid. Canadian J of Physics. to be published.Google Scholar
  35. 35.
    Torabi M, Karimi N, Zhang K, Peterson GP. Generation of entropy and forced convection of heat in a conduit partially filled with porous media-Local thermal non-equilibrium and exothermicity effects. Appl Therm Eng. 2016;106:518–36.CrossRefGoogle Scholar
  36. 36.
    Khanafer KM, Chamkha AJ. Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Int J Heat Mass Transf. 1999;42:2465–81.CrossRefGoogle Scholar
  37. 37.
    Iwatsu R, Hyun JM, Kuwahara K. Mixed convection in a driven cavity with a stable vertical temperature gradient. Int J Heat Mass Transf. 1993;36:1601–8.CrossRefGoogle Scholar
  38. 38.
    Khanafer KM, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf. 2003;46:3639–53.CrossRefGoogle Scholar
  39. 39.
    Abu-Nada E, Chamkha AJ. Effect of nanofluid variable properties on natural convection in enclosures filled with an CuO–EG–water nanofluid. Int J Thermal Sci. 2010;49:2339–52.CrossRefGoogle Scholar
  40. 40.
    Maxwell JA. Treatise on electricity and magnetism. 2nd ed. Cambridge: Oxford University Press; 1904.Google Scholar
  41. 41.
    Brinkman HC. The viscosity of concentrated suspensions and solution. J Chem Phys. 1952;20:571–81.CrossRefGoogle Scholar
  42. 42.
    Chamkha AJ, Ismael MA. Conjugate heat transfer in a porous cavity filled with nanofluids and heated by a triangular thick wall. Int J Therm Sci. 2013;67:135–51.CrossRefGoogle Scholar
  43. 43.
    Mahmud S, Fraser RA. Magnetohydrodynamic free convection and entropy generation in a square porous cavity. Int J Heat Mass Transf. 2004;47:3245–56.CrossRefGoogle Scholar
  44. 44.
    Abolbashari MH, Freidoonimehr N, Nazari F, Rashidi MM. Analytical modeling of entropy generation for Casson nano-fluid flow induced by a stretching surface. Adv Powder Technol. 2015;26:542–52.CrossRefGoogle Scholar
  45. 45.
    Patankar SV. Numerical heat transfer and fluid flow. hemisphere, New York. 1980.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentPrince Sultan Endowment for Energy and Environment Prince Mohammad Bin Fahd UniversityAl-KhobarKingdom of Saudi Arabia
  2. 2.Rak Research and Innovation CenterAmerican University of Ras Al KhaimahRas Al KhaimahUnited Arab Emirates
  3. 3.Department of Mathematics, Faculty of ScienceAswan UniversityAswânEgypt
  4. 4.Department of Engineering, Mahdishahr BranchIslamic Azad UniversityMahdishahrIran
  5. 5.Department of Mathematics, Faculty of ScienceAssuit UniversityAssuitEgypt

Personalised recommendations