Skip to main content
SpringerLink
Log in

Mixed convective magnetonanofluid flow over a backward facing step and entropy generation using extended Darcy–Brinkman–Forchheimer model

  • Published:
Journal of Thermal Analysis and Calorimetry Aims and scope Submit manuscript

Abstract

This article focuses on a numerical study for the effect of porosity and internal heat generation/absorption parameters on the mixed convective flow of nanofluid over a backward facing step along with the entropy generation. The channel downstream bottom wall is isothermally heated while the remaining walls of the channel are thermally insulated. The dimensionless governing equations are discretized utilizing the Galerkin-based finite element method. The discrete systems of nonlinear algebraic equations are computed using the Newton method and the corresponding linearized systems are treated using the monolithic geometric multigrid solver. Effect of different physical parameters in the specified ranges such as \((10 \le Re \le 200)\), \((0.01 \le Ri \le 20)\), \((0\le Ha \le 100)\), \((-10 \le q \le 10)\), \((10^{-6} \le Da \le 10^{-3})\), \((0.2 \le \epsilon \le 0.8)\) and \((0^\circ \le \gamma \le 180^\circ )\) on the fluid flow is analyzed with the help of the streamlines, isotherm patterns, and various graphs. It is noticed that the average heat transfer increases with the porosity parameter and reduces with the internal heat generation parameter. Furthermore, the entropy generation produced by the heat transfer and fluid friction is found to amplify with the porosity parameter, whereas a decline is recorded in it due to the magnetic field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Abbreviations

\(C_\mathrm{p}\) :

Specific heat (\(\hbox {J kg}^{-1}\hbox {K}^{-1}\))

Be :

Bejan number

g :

Gravitational acceleration (\(\hbox {m s}^{-2}\))

Ha :

Hartmann number

Gr :

Grashof number

\(k_\mathrm{m}\) :

Effective thermal conductivity of porous medium (\(\hbox {W m}^{-1}\hbox {K}^{-1}\))

K :

Permeability parameter (\(\hbox {m}^2\))

k :

Thermal conductivity (\(\hbox {W m}^{-1}\hbox {K}^{-1}\))

H :

Step size (m)

\(Nu_{\text {avg}}\) :

Average Nusselt number

P :

Dimensionless pressure

p :

Pressure (\(\hbox {N m}^{-2}\))

Pr :

Prandtl number

Ri :

Richardson number

q :

Dimensionless heat generation/absorption parameter

Re :

Reynolds number

T :

Temperature (K)

\(S_\mathrm{HT}\) :

Dimensionless entropy generation due to heat transfer

\(S_\mathrm{FF}\) :

Dimensionless entropy generation due to fluid friction

\(S_\mathrm{MF}\) :

Dimensionless entropy generation due to magnetic field

u, v :

Dimensional velocity components (\(\hbox {m s}^{-1}\))

x, y :

Dimensional space coordinates (m)

\(u_0\) :

Velocity at the inlet

X, Y :

Dimensionless space coordinates

U, V :

Dimensionless velocity components

\(\alpha\) :

Thermal diffusivity (\(\hbox {m}^2\hbox { s}^{-1}\))

\(\gamma\) :

Magnetic field inclination angle

\(\rho\) :

Density (\(\hbox {kg m}^{-3}\))

\(\beta\) :

Expansion coefficient (\(\hbox {K}^{-1}\))

\(\mu\) :

Dynamic viscosity (\(\hbox {kg m}^{-1}\hbox {s}^{-1}\))

\(\sigma\) :

Electrical conductivity (\(\hbox {S m}^{-1}\))

\(\theta\) :

Dimensionless temperature

\(\phi\) :

Solid volume fraction

\(\nu\) :

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

\(\text {avg}\) :

Average

nf:

Nanofluid

f:

Fluid

c:

Cold

h:

Hot

P:

Nanoparticle

References

  1. Lin J, Armaly B, Chen T. Mixed convection in buoyancy-assisted vertical backward-facing step flows. Int J Heat Mass Transf. 1990;33:2121–32.

    CAS  Google Scholar 

  2. Erturk E. Numerical solutions of 2-D steady incompressible flow over a backward-facing step, part I: high Reynolds number solutions. Comput Fluids. 2008;37(6):633–55.

    CAS  Google Scholar 

  3. Ratha D, Sarkar A. Analysis of flow over backward facing step with transition. Int J Therm Sci. 2014;10:1–10.

    Google Scholar 

  4. Siavashi M, Ghasemi K, Yousofvand R, Derakhshan S. Computational analysis of swcnh nanofluid-based direct absorption solar collector with a metal sheet. Sol Energy. 2018;170:252–62.

    CAS  Google Scholar 

  5. Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S. A review of the applications of nanofluids in solar energy. Int J Heat Mass Transf. 2013;57(2):582–94.

    CAS  Google Scholar 

  6. Saldana J, Anand N. Numerical solution of mixed convective flow over a three-dimensional horizontal backward facing step. J Heat Transf. 2005;127:1027–36.

    Google Scholar 

  7. Selimefendigil F, Oztop H. Numerical analysis of laminar pulsating flow at a backward facing step with an upper wall mounted adiabatic thin fin. Comput Fluids. 2013;88:93–107.

    Google Scholar 

  8. Khanafer K, Al-Azmi B, Al-Shammari A, Pop I. Mixed convection analysis of laminar pulsating flow and heat transfer over a backward-facing step. Int J Heat Mass Transf. 2008;51:5785–93.

    Google Scholar 

  9. Iwai H, Nakabe K, Suzuki K. Flow and heat transfer characteristics of backward-facing step laminar flow in a rectangular duct. Int J Heat Mass Transf. 2000;43:457–71.

    Google Scholar 

  10. Nie J, Armaly B. Convection in laminar three-dimensional separated flow. Int J Heat Mass Transf. 2004;47:5407–16.

    Google Scholar 

  11. Selimefendigil F, Oztop H. Control of laminar pulsating flow and heat transfer in backward-facing step by using a square obstacle. J Heat Transf. 2014;136:081701.

    Google Scholar 

  12. Saldana J, Anand N. Flow over a three-dimensional horizontal forward-facing step. Numer Heat Transf. 2008;53:1–17.

    Google Scholar 

  13. Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. ASME Fluid Eng. 1995;231:99–105.

    CAS  Google Scholar 

  14. Oztop H, Nada A. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow. 2008;29:1326–36.

    Google Scholar 

  15. Selimefendigil F, Oztop H. Pulsating nanofluids jet impingement cooling of a heated horizontal surface. Int J Heat Mass Transf. 2014;69:54–65.

    CAS  Google Scholar 

  16. Roy G, Nguyen CT, Lajoie P-R. Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids. Superlattices Microstruct. 2004;35(3):497–511. https://doi.org/10.1016/j.spmi.2003.09.011.

    Article  CAS  Google Scholar 

  17. Nada A. Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step. Int J Heat Fluid Flow. 2008;29:242–9.

    Google Scholar 

  18. Saha L, Somadder M, Uddin KS. Mixed convection heat transfer in a lid driven cavity with wavy bottom surface. Am J Appl Math. 2013;5:92–101.

    Google Scholar 

  19. Hussain S, Mehmood K, Sagheer M. MHD mixed convection and entropy generation of water–alumina nanofluid flow in a double lid driven cavity with discrete heating. J Magn Magn Mater. 2016;419:140–55.

    CAS  Google Scholar 

  20. Hussain S, Ahmed S, Mehmood K, Sagheer M. Effects of inclination angle on mixed convective nanofluid flow in a double lid-driven cavity with discrete heat sources. Int J Heat Mass Transf. 2017;106:847–60.

    CAS  Google Scholar 

  21. Mehmood K, Hussain S, Sagheer M. Mixed convection in alumina–water nanofluid filled lid-driven square cavity with an isothermally heated square blockage inside with magnetic field effect: Introduction. Int J Heat Mass Transf. 2017;109:397–409.

    CAS  Google Scholar 

  22. Mehmood K, Hussain S, Sagheer M. Numerical simulation of MHD mixed convection in alumina–water nanofluid filled square porous cavity using KKL model: effects of non-linear thermal radiation and inclined magnetic field. J Mol Liq. 2017;238:485–98.

    CAS  Google Scholar 

  23. Mahian O, Kolsi L, Amani M, Estellé P, Ahmadi G, Kleinstreuer C, Marshall JS, Siavashi M, Taylor RA, Niazmand H, Wongwises S, Hayat T, Kolanjiyil A, Kasaeian A, Pop I. Recent advances in modeling and simulation of nanofluid flows-part I: fundamentals and theory. Phys Rep. 2019;790:1–48.

    CAS  Google Scholar 

  24. Mahian O, Kolsi L, Amani M, Estellé P, Ahmadi G, Kleinstreuer C, Marshall JS, Taylor RA, Abu-Nada E, Rashidi S, Niazmand H, Wongwises S, Hayat T, Kasaeian A, Pop I. Recent advances in modeling and simulation of nanofluid flows-part II: applications. Phys Rep. 2019;791:1–59.

    CAS  Google Scholar 

  25. Abbasi H, Nassrallah S. MHD flow and heat transfer in a backward-facing step. Int Commun Heat Mass Transf. 2007;34:231–7.

    Google Scholar 

  26. Yousofvand R, Derakhshan S, Ghasemi K, Siavashi M. MHD transverse mixed convection and entropy generation study of electromagnetic pump including a nanofluid using 3D LBM simulation. Int J Mech Sci. 2017;133:73–90.

    Google Scholar 

  27. Mehrez Z, Cafsi A, Belghith A, Quere P. MHD effects on heat transfer and entropy generation of nanofluid flow in an open cavity. J Magn Magn Mater. 2015;364:214–24.

    Google Scholar 

  28. Pirmohammadi M, Ghassemi M. Effect of magnetic field on convection heat transfer inside a tilted square enclosure. Int Commun Heat Mass Transf. 2009;36:776–80.

    CAS  Google Scholar 

  29. Kherbeet A, Mohammed H, Munisamy K, Salman B. The effect of step height of microscale backward-facing step on mixed convection nanofluid flow and heat transfer characteristics. Int J Heat Mass Transf. 2014;68:554–66.

    CAS  Google Scholar 

  30. Aswadi A, Mohammed H, Shuaib N. Laminar forced convection flow over a backward facing step using nanofluids. Int Commun Heat Mass Transf. 2010;37:950–7.

    Google Scholar 

  31. Nasrin R, Alim A, Chamkha J. Modeling of mixed convective heat transfer utilizing nanofluid in a double lid-driven chamber with internal heat generation. Int J Numer Methods Heat Fluid Flow. 2014;24:36–57.

    Google Scholar 

  32. Xu ZG, Zhao CY. Enhanced boiling heat transfer by gradient porous metals in saturated pure water and surfactant solutions. Appl Therm Eng. 2016;100:68–77.

    CAS  Google Scholar 

  33. Musto M, Bianco N, Rotondo G, Toscano F, Pezzella G. A simplified methodology to simulate a heat exchanger in an aircraft’s oil cooler by means of a porous media model. Appl Therm Eng. 2016;94:836–45.

    Google Scholar 

  34. Fu J, Tang Y, Li J, Ma Y, Chen W, Li H. Four kinds of the two-equation turbulence model’s research on flow field simulation performance of DPF’s porous media and swirl-type regeneration burner. Appl Therm Eng. 2016;93:397–404.

    Google Scholar 

  35. Siavashi M, Blunt MJ, Raisee M, Pourafshary P. Three-dimensional streamline-based simulation of non-isothermal two-phase flow in heterogeneous porous media. Comput Fluids. 2014;103:116–31.

    Google Scholar 

  36. Ghasemizadeh R, Yu X, Butscher C, Hellweger F, Padilla I, Alshawabkeh A. Equivalent porous media (EPM) simulation of groundwater hydraulics and contaminant transport in karst aquifers. PLOS ONE. 2015;10(9):1–21. https://doi.org/10.1371/journal.pone.0138954.

    Article  CAS  Google Scholar 

  37. Alrwashdeh SS, Markötter H, Haußmann J, Arlt T, Klages M, Scholta J, Banhart J, Manke I. Investigation of water transport dynamics in polymer electrolyte membrane fuel cells based on high porous micro porous layers. Energy. 2016;102:161–5.

    CAS  Google Scholar 

  38. Khaled ARA, Vafai K. The role of porous media in modeling flow and heat transfer in biological tissues. Int J Heat Mass Transf. 2003;46:4989–5003.

    Google Scholar 

  39. Cascetta M, Cau G, Puddu P, Serra F. A comparison between CFD simulation and experimental investigation of a packed-bed thermal energy storage system. Appl Therm Eng. 2016;98:1263–72.

    CAS  Google Scholar 

  40. Qin L, Han J, Chen W, Yao X, Tadaaki S, Kim H. Enhanced combustion efficiency and reduced pollutant emission in a fluidized bed combustor by using porous alumina bed materials. Appl Therm Eng. 2016;94:813–8.

    CAS  Google Scholar 

  41. Tang GH, Bi C, Zhao Y, Tao W. Thermal transport in nano-porous insulation of aerogel: factors, models and outlook. Energy. 2015;90:701–21.

    CAS  Google Scholar 

  42. Emami RY, Siavashi M, Moghaddam GS. The effect of inclination angle and hot wall configuration on Cu–water nanofluid natural convection inside a porous square cavity. Adva Powder Technol. 2018;29(3):519–36.

    Google Scholar 

  43. Siavashi M, Yousofvand R, Rezanejad S. Nanofluid and porous fins effect on natural convection and entropy generation of flow inside a cavity. Adv Powder Technol. 2018;29(1):142–56.

    CAS  Google Scholar 

  44. Siavashi M, Rostami A. Two-phase simulation of non-Newtonian nanofluid natural convection in a circular annulus partially or completely filled with porous media. Int J Mech Sci. 2017;133:689–703.

    Google Scholar 

  45. Siavashi M, Bahrami HRT, Aminian E. Optimization of heat transfer enhancement and pumping power of a heat exchanger tube using nanofluid with gradient and multi-layered porous foams. Appl Therm Eng. 2018;138:465–74.

    CAS  Google Scholar 

  46. Maghsoudi P, Siavashi M. Application of nanofluid and optimization of pore size arrangement of heterogeneous porous media to enhance mixed convection inside a two-sided lid-driven cavity. J Therm Anal Calorim. 2019;135:947–61.

    CAS  Google Scholar 

  47. Asiaei S, Zadehkafi A, Siavashi M. Multi-layered porous foam effects on heat transfer and entropy generation of nanofluid mixed convection inside a two-sided lid-driven enclosure with internal heating. Transp Porous Media. 2019;126:223–47.

    CAS  Google Scholar 

  48. Siavashi M, Miri Joibary SM. Numerical performance analysis of a counter-flow double-pipe heat exchanger with using nanofluid and both sides partly filled with porous media. J Therm Anal Calorim. 2019;135:1595–610.

    CAS  Google Scholar 

  49. Koo J, Kleinstreuer C. Viscous dissipation effects in microtubes and microchannels. Int J Heat Mass Transf. 2004;47:3159–69.

    CAS  Google Scholar 

  50. Maxwell JC. A treatise on electricity and magnetism, vol. II. Cambridge: Oxford University Press; 1873.

    Google Scholar 

  51. Koo J, Kleinstreuer C. Laminar nanofluid flow in microheat-sinks. Int J Heat Mass Transf. 2005;48:2652–61.

    CAS  Google Scholar 

  52. Brinkman HC. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20:571–81.

    CAS  Google Scholar 

  53. Vafai K. Handbook of porous media. 2nd ed. New York: Taylor & Francis; 2005.

    Google Scholar 

  54. Nield DA, Bejan A. Convection in porous media. 4th ed. Berlin: Springer; 2013. p. 31–2.

    Google Scholar 

  55. Grott M, Knollenberg J, Krause C. Apollo lunar heat flow experiment revisited: a critical reassessment of the in situ thermal conductivity determination. J Geophys Res Planets. 2010;115:1–11. https://doi.org/10.1029/2010JE003612.

    Article  Google Scholar 

  56. Reddy KS, Sreedhar D. Thermal conductivity of natural fiber, glass fiber & CNTs reinforced epoxy composites. Int J Curr Eng Technol. 2016;6(4):1196–98. https://doi.org/10.14741/Ijcet/22774106/6.4.2016.18.

    Article  Google Scholar 

  57. Ghasemi K, Siavashi M. MHD nanofluid free convection and entropy generation in porous enclosures with different conductivity ratios. J Magn Magn Mater. 2017;442:474–90.

    CAS  Google Scholar 

  58. Siavashi M, Bordbar V, Rahnama P. Heat transfer and entropy generation study of non-Darcy double-diffusive natural convection in inclined porous enclosures with different source configurations. Appl Therm Eng. 2017;110:1462–75.

    CAS  Google Scholar 

  59. Chamkha AJ, Rashad AM, Mansour MA, Armaghani T, Ghalambaz M. Effects of heat sink and source and entropy generation on mhd mixed convection of a cu–water nanofluid in a lid-driven square porous enclosure with partial slip. Phys Fluids. 2017;29(5):052001.

    Google Scholar 

  60. Hussain S, Schieweck F, Turek S. Efficient Newton multigrid solution techniques for higher order space time Galerkin discretizations of incompressible flow. Appl Numer Math. 2014;83:51–71.

    Google Scholar 

  61. Alshuraiaan BS. Mixed convection flow and heat transfer over different geometries of backward-facing step. J Eng Res. 2013;1:211–33.

    CAS  Google Scholar 

  62. Acharya S, Dixit G, Hou Q. Laminar mixed convection in a vertical channel with a backstep: a benchmark study. ASME HTD. 1993;258:11–20.

    Google Scholar 

  63. Khanafer K, Al-Azmi B, Shammari A, Pop L. Mixed convection analysis of laminar pulsating flow and heat transfer over a backward-facing step. Int J Heat Mass Transf. 2008;51:5785–93.

    Google Scholar 

Download references

Acknowledgements

Calculations have been carried out on the LiDOng cluster at Technische Universität, Dortmund, Germany. The support by the LiDOng team at the ITMC at TU Dortmund is gratefully acknowledged. We would like to thank the LiDOng cluster team for their help and support. We also used FeatFlow (www.featflow.de) solver package and would like to acknowledge the support by the FeatFlow team.

Author information

Authors and Affiliations

  1. Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan

    Shafqat Hussain, Khalid Mehmood, Muhammad Sagheer & Asifa Ashraf

  2. Institut für Angewandte Mathematik (LS III), Technische Universität, Dortmund, Germany

    Shafqat Hussain

Authors
  1. Shafqat Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khalid Mehmood
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Sagheer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Asifa Ashraf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shafqat Hussain.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hussain, S., Mehmood, K., Sagheer, M. et al. Mixed convective magnetonanofluid flow over a backward facing step and entropy generation using extended Darcy–Brinkman–Forchheimer model. J Therm Anal Calorim 138, 3183–3203 (2019). https://doi.org/10.1007/s10973-019-08347-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-019-08347-w

Keywords

Search

Navigation