Exact solutions for the Atangana-Baleanu time-fractional model of a Brinkman-type nanofluid in a rotating frame: Applications in solar collectors

  • Aamina
  • Farhad AliEmail author
  • Ilyas Khan
  • Nadeem Ahmad Sheikh
  • Madeha Gohar
Regular Article


Nanofluids are the next generation fluids that exhibit thermal properties superior to those of conventional fluids. Nanofluids play a vital role in various thermal applications such as automotive industries, heat exchangers, solar power generation, etc. Therefore, a generalized Brinkman-type fluid model has been developed to predict the heat transport properties of a flat-plate solar collector using a nanofluid in a rotating frame under the influence of transverse magnetic field B0 and two cases are discussed. i) B0 being fixed to the fluid (K = 0 ; ii) B0 being fixed to the plate (K = 0). Thermal radiation and concentration are also taken into account. Furthermore, the classical model is converted to a generalized model using the Atangana-Baleanu (AB) fractional derivative and then the exact solutions are obtained via the Laplace transform method. A parametric study of all the governing parameters is carried out and some other important results are illustrated in tabular form. A comparison of several nano-sized solid particles, i.e. SWCNT, MWCNT, CuO, Al2O3 and TiO2 has been done in the current investigation and it is concluded that adding SWCNT in the working fluid (water) can augment the heat transfer rate up to 36.61%, which consequently upgrades the working ability of flat-plate solar collectors by enhancing their absorption power of solar radiation.


  1. 1.
    S.K. Verma, A.K. Tiwari, Energy Convers. Manag. 100, 324 (2015)CrossRefGoogle Scholar
  2. 2.
    Energyquest, Energy Quest (2007) www.energyquestcagov/story/chapter14html
  3. 3.
    Renewables 2016 Global Status Report, REN21 (2016) ISBN 978-3-9818107-0-7Google Scholar
  4. 4.
    2016 Snapshot of Global Photovoltaic Markets (International Energy Agency, 2017)Google Scholar
  5. 5.
    A. Ibrahim, M.Y. Othman, M.H. Ruslan, S. Mat, K. Sopian, Renew. Sustain. Energy Rev. 15, 352 (2011)CrossRefGoogle Scholar
  6. 6.
    F.L. Lansing, V. Clarke, R. Reynolds, Energy 4, 685 (1979)CrossRefGoogle Scholar
  7. 7.
    A. Kolb, E.R.F. Winter, R. Viskanta, Sol. Energy 65, 91 (1999)CrossRefGoogle Scholar
  8. 8.
    H.C. Brinkman, Appl. Sci. Res. 1, 27 (1949)CrossRefGoogle Scholar
  9. 9.
    H.C. Brinkman, Appl. Sci. Res. 1, 81 (1949)CrossRefGoogle Scholar
  10. 10.
    H. Darcy, Determination of the laws of flow of water through sand, in Physical Hydrology, edited by R.A. Freeze, W. Back (Hutchinson Ross, 1983)Google Scholar
  11. 11.
    W. Yu, H. Xie, J. Nanomater. 2012, 435873 (2012)Google Scholar
  12. 12.
    S.U. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, No. ANL/MSD/CP--84938, CONF-951135--29 (Argonne National Lab., IL, 1995)Google Scholar
  13. 13.
    R. Prasher, P.E. Phelan, P. Bhattacharya, Nano Lett. 6, 1529 (2006)CrossRefGoogle Scholar
  14. 14.
    H. Tyagi, P. Phelan, R. Prasher, The predicted efficiency of a nanofluid-based direct absorption solar receiver, in ASME 2007 energy sustainability conference (American Society of Mechanical Engineers, 2007) pp. 729--736Google Scholar
  15. 15.
    M. Saqib, I. Khan, S. Shafie, Chaos, Solitons Fractals 116, 79 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    U. Khan, N. Ahmed, S.T. Mohyud-Din, Appl. Therm. Eng. 113, 1107 (2017)CrossRefGoogle Scholar
  17. 17.
    H. Xie, H. Lee, W. Youn, M. Choi, J. Appl. Phys. 94, 4967 (2003)CrossRefGoogle Scholar
  18. 18.
    E. Natarajan, R. Sathish, Int. J. Adv. Manufact. Technol. (2009)
  19. 19.
    O. Mahian, A. Kianifar, S.A. Kalogirou, I. Pop, S. Wongwises, Int. J. Heat Mass Transfer 57, 582 (2013)CrossRefGoogle Scholar
  20. 20.
    T. Yousefi, F. Veisy, E. Shojaeizadeh, S. Zinadini, Exp. Therm. Fluid Sci. 39, 207 (2012)CrossRefGoogle Scholar
  21. 21.
    J.A. Khan, M. Mustafa, T. Hayat, M. Sheikholeslami, A. Alsaedi, PLoS ONE 10, e0116603 (2015)CrossRefGoogle Scholar
  22. 22.
    M. Karami, M.A. Bahabadi, S. Delfani, A. Ghozatloo, Sol. Energy Mater. Sol. Cells 121, 114 (2014)CrossRefGoogle Scholar
  23. 23.
    M.A. Sabiha, R. Saidur, S. Hassani, Z. Said, S. Mekhilef, Energy Convers. Manag. 105, 1377 (2015)CrossRefGoogle Scholar
  24. 24.
    N.A. Shah, I. Khan, Eur. Phys. J. C 76, 362 (2016)CrossRefGoogle Scholar
  25. 25.
    K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Vol. 111 (Elsevier, 1974)Google Scholar
  26. 26.
    M. Caputo, Geophys. J. Int. 13, 529 (1967)CrossRefGoogle Scholar
  27. 27.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  28. 28.
    A. Atangana, Eur. Phys. J. Plus 131, 373 (2016)CrossRefGoogle Scholar
  29. 29.
    A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    B.S.T. Alkahtani, Chaos, Solitons Fractals 89, 547 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    I. Koca, A. Atangana, Therm. Sci. 21, 2299 (2017)CrossRefGoogle Scholar
  32. 32.
    O.J.J. Algahtani, Chaos, Solitons Fractals 89, 552 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    N.A. Sheikh, F. Ali, I. Khan, M. Gohar, M. Saqib, Eur. Phys. J. Plus 132, 540 (2017)CrossRefGoogle Scholar
  34. 34.
    A.A. Tateishi, H.V. Ribeiro, E.K. Lenzi, Front. Phys. 5, 52 (2017)CrossRefGoogle Scholar
  35. 35.
    Q.Z. Xue, Physica B 368, 302 (2005)CrossRefGoogle Scholar
  36. 36.
    S. Aman, I. Khan, Z. Ismail, M.Z. Salleh, A.S. Alshomrani, M.S. Alghamdi, AIP Adv. 7, 015036 (2017)CrossRefGoogle Scholar
  37. 37.
    F. Ali, N.A. Sheikh, I. Khan, M. Saqib, J. Magn. & Magn. Mater. 423, 327 (2017)CrossRefGoogle Scholar
  38. 38.
    Z. Said, R. Saidur, N.A. Rahim, M.A. Alim, Energy Build. 78, 1 (2014)CrossRefGoogle Scholar
  39. 39.
    Y. He, S. Wang, J. Ma, F. Tian, Y. Ren, Nanosci. Nanotechnol. Lett. 3, 494 (2011)CrossRefGoogle Scholar
  40. 40.
    B.R. Kumar, T.S. Kumar, A.G. Kumar, Front. Heat Mass Transf. 6, 12 (2015)CrossRefGoogle Scholar
  41. 41.
    F. Ali, B. Aamina, I. Khan, N.A. Sheikh, M. Saqib, Int. J. Heat Technol. 4, 893 (2017)Google Scholar
  42. 42.
    C. Kleinstreuer, H. Chiang, Heat Transf. Eng. 11, 45 (1990)CrossRefGoogle Scholar
  43. 43.
    F. Ali, Aamina, I. Khan, N.A. Sheikh, M. Gohar, I. Tlili, Sci. Rep. 8, 15285 (2018)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Aamina
    • 1
  • Farhad Ali
    • 2
    • 3
    Email author
  • Ilyas Khan
    • 4
  • Nadeem Ahmad Sheikh
    • 1
  • Madeha Gohar
    • 1
  1. 1.Department of MathematicsCity University of Science and Information TechnologyKhyber PakhtunkhwaPakistan
  2. 2.Computational Analysis Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Department of Mathematics, College of Science Al-ZulfiMajmaah UniversityAl-MajmaahSaudi Arabia

Personalised recommendations