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Analysis of Entropy Generation and Energy Transport of Cu-Water Nanoliquid in a Tilted Vertical Porous Annulus

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Abstract

The physical structure in several industrial applications which includes cooling of electronic equipment, heat exchangers, nuclear reactors, and solar collectors, could aptly represent the cylindrical annular porous geometry. The prior knowledge of buoyant flow and thermal transport rates in this geometry provides the vital information to the design engineers. In this article, we analyze the convective nanoliquid flow and associated thermal dissipation as well as entropy generation rates in an inclined annular enclosure filled with nanoliquid saturated porous medium. The vertical surfaces of inner and outer cylinders are maintained at uniform, but different temperatures and horizontal boundaries are kept insulated. The momentum equations are modeled utilizing the Darcy law, the coupled partial differential equations are numerically solved adopting the time splitting and line over relaxation techniques. For the numerical simulations, a vast range of parameters such as the Darcy Rayleigh number (10 ≤ \( Ra_{D}\) ≤ 103), annulus inclination angle (0° ≤ γ ≤ 60°), aspect ratio (0.5 ≤ Ar ≤ 2) and nanoparticle volume fraction (0 ≤ ϕ ≤ 0.05) are considered. The contributions of heat transfer and fluid friction entropies to global entropy production in the geometry are determined through the Bejan number. The numerical results reveal that the convective flow, heat transfer and entropy generation rates could be controlled with the aid of cavity inclination angle. It is found that the shallow annular enclosure gives better thermal performance with minimum entropy generation regardless of \( Ra_{D}\), γ and ϕ. Further, the results are in excellent agreement with standard benchmark simulations. The predicted results could provide some vital information to enhance the system efficiency.

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Data Availability statement

The datasets generated and analysed during the current investigation are available within the manuscript.

Abbreviations

Ar :

Aspect ratio

D :

Width of the annulus (m)

g :

Acceleration due to gravity (m/s2)

H :

Height of the annulus (m)

k :

Thermal conductivity (W/m K−1)

K :

Permeability (m2)

Ra D :

Darcy Rayleigh number

T :

Dimensionless temperature

(R,Z):

Radial, axial coordinates

(U,W):

Velocity components

α :

Thermal diffusivity (m2/s)

β :

Thermal expansion coefficient (K−1)

γ :

Inclination angle (°)

θ :

Dimensional temperature (K)

ν :

Kinematic viscosity (m2/s)

ϕ :

Nanoparticle volume fraction

f :

Base fluid

nf :

Nanofluid

p :

Nanoparticle

References

  1. Prasad, V.: Numerical study of natural convection in a vertical porous annulus with constant heat flux on inner wall. Int. J. Heat Mass Transf. 29, 841–853 (1986)

    Google Scholar 

  2. di Schio, E.R., Celli, M., Pop, I.: Buoyant flow in a vertical fluid saturated porous annulus: The Brinkman model. Int. J. Heat Mass Transf. 54(7–8), 1665–1670 (2011)

    MATH  Google Scholar 

  3. Sankar, M., Park, Y., Lopez, J.M., Do, Y.: Numerical study of natural convection in a vertical porous annulus with discrete heating. Int. J. Heat Mass Transf. 54(7–8), 1493–1505 (2011)

    MATH  Google Scholar 

  4. Sankar, M., Park, J., Kim, D., Do, Y.: Numerical study of natural convection in a vertical porous annulus with an internal heat source: effect of discrete heating. Numer. Heat Transf. Part A 63(9), 687–712 (2013)

    Google Scholar 

  5. Gourari, S., Mebarek-Oudina, F., Hussein, A.K., Kolsi, L., Hassen, W., Younis, O.: Numerical study of natural convection between two coaxial cylinders. Int. J. Heat Technol. 37(3), 779–786 (2019)

    Google Scholar 

  6. Choi, S.U.S., Eastman, J.A.: Enhancing thermal conductivity of fluids with nanoparticles. ASME FED 231, 99–105 (1995)

    Google Scholar 

  7. Abouali, O., Falahatpisheh, A.: Numerical investigation of natural convection of Al2O3 nanofluid in vertical annuli. Heat Mass Transf. 46, 15–23 (2009)

    Google Scholar 

  8. Mebarek-Oudina, F., Bessaih, R., Mahanthesh, B., Chamkha, A.J., Raza, J.: Magneto-thermal-convection stability in an inclined cylindrical annulus filled with a molten metal. Int. J. Numer. Meth. Heat Fluid Flow 31(4), 1172–1189 (2020)

    Google Scholar 

  9. Reddy, N.K., Sankar, M.: Buoyant convective transport of nanofluids in a non-uniformly heated annulus. J. Phys. Conf. Ser. 1597, 012055 (2020)

    Google Scholar 

  10. Reddy, N.K., Swamy, H.A.K., Sankar, M.: Buoyant convective flow of different hybrid nanoliquids in a non-uniformly heated annulus. Eur. Phys. J. Spec. Top. 230(5), 1213–1225 (2021)

    Google Scholar 

  11. Sankar, M., Reddy, N.K., Do, Y.: Conjugate buoyant convective transport of nanofluids in an enclosed annular geometry. Sci. Rep. 11, 17122 (2021)

    Google Scholar 

  12. Chen, S., Liu, Z., Bao, S., Zheng, C.: Natural convection and entropy generation in a vertically concentric annular space. Int. J. Therm. Sci. 49(12), 2439–2452 (2010)

    Google Scholar 

  13. Baytas, A.C.: Entropy generation for natural convection in an inclined porous cavity. Int. J. Heat Mass Transf. 43(12), 2089–2099 (2000)

    MATH  Google Scholar 

  14. Bouabid, M., Magherbi, M., Hidouri, N., Brahim, A.B.: Entropy generation at natural convection in an inclined rectangular cavity. Entropy 13(5), 1020–1033 (2011)

    MATH  Google Scholar 

  15. Basak, T., Singh, A.K., Sruthi, T.P.A., Roy, S.: Finite element simulations on heat flow visualization and entropy generation during natural convection in inclined square cavities. Int. Commun. Heat Mass Transf. 51, 1–8 (2014)

    Google Scholar 

  16. Bhargava, R., Sharma, S., Bhargava, P., Anwar Beg, O., Kadir, A.: Finite element simulation of nonlinear convective heat and mass transfer in a micropolar fluid-filled enclosure with Rayleigh number effects. Int. J. Appl. Comput. Math. 3, 1347–1379 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Alsabery, A.I., Chamkha, A.J., Saleh, H., Hashim, I.: Natural convection flow of nanofluid in an inclined square enclosure partially filled with a porous medium. Sci. Rep. 7, 2357 (2017)

    Google Scholar 

  18. Rashad, A.M., Armaghani, T., Chamkha, A.J., Mansour, M.A.: Entropy generation and MHD natural convection of a nanofluid in an inclined square porous cavity: effects of a heat sink and source size and location. Chin. J. Phys. 56(1), 193–211 (2018)

    Google Scholar 

  19. Baghsaz, S., Rezanejad, S., Moghimi, M.: Numerical investigation of transient natural convection and entropy generation analysis in a porous cavity filled with nanofluid considering nanoparticle sedimentation. J. Mol. Liq. 279, 327–341 (2019)

    Google Scholar 

  20. Alnaqi, A.A., Aghakhani, S., Pordanjani, A.H., Bakhtiari, R., Asadi, A.: and Minh-Duc, Effects of magnetic field on the convective heat transfer rate and entropy generation of a nanofluid in an inclined square cavity equipped with a conductor fin: considering the radiation effect. Int. J. Heat Mass Transf. 133, 256–267 (2019)

    Google Scholar 

  21. Selimefendigil, F., Oztop, H.F.: Effects of conductive curved partition and magnetic field on natural convection and entropy generation in an inclined cavity filled with nanofluid. Physica A 540, 123004 (2020)

    MathSciNet  MATH  Google Scholar 

  22. Li, Z., Hussein, A.K., Younis, O., Afrand, M., Feng, S.: Natural convection and entropy generation of a nanofluid around a circular baffle inside an inclined square cavity under thermal radiation and magnetic field effects. Int. Commun. Heat Mass Transf. 116, 104650 (2020)

    Google Scholar 

  23. Li, Y., Firouzi, M., Karimipour, A., Afrand, M.: Effect of an inclined partition with constant thermal conductivity on natural convection and entropy generation of a nanofluid under magnetic field inside an inclined enclosure: applicable for electronic cooling. Adv. Powder Technol. 31(2), 645–657 (2020)

    Google Scholar 

  24. Sankar, M., Swamy, H.A.K., Do, Y., Altmeyer, S.: Thermal effects of nonuniform heating in a nanofluid-filled annulus: Buoyant transport versus entropy generation. Heat Transf (2021). https://doi.org/10.1002/htj.22342

    Article  Google Scholar 

  25. Jino, L., Kumar, A.V.: Cu-Water nanofluid MHD quadratic natural convection on square porous cavity. Int. J. Appl. Comput. Math. 7, 164 (2021)

    MathSciNet  Google Scholar 

  26. Sun, Q., Pop, I.: Free convection in a triangle cavity filled with a porous medium saturated with nanofluids with flush mounted heater on the wall. Int. J. Therm. Sci. 50, 2141–2153 (2011)

    Google Scholar 

  27. Mansour, M.A., Ahmed, S.E.: A numerical study on natural convection in porous media-filled an inclined triangular enclosure with heat sources using nanofluid in the presence of heat generation effect. Eng. Sci. Technol. Int. J. 18(3), 485–495 (2015)

    Google Scholar 

  28. Jagadeesha, R.D., Prasanna, B.M.R., Sankar, M.: Double diffusive convection in an inclined parallelogrammic porous enclosure. Procedia Eng. 127, 1346–1353 (2015)

    Google Scholar 

  29. Ghalambaz, M., Sheremet, M.A., Pop, I.: Free convection in a parallelogrammic porous cavity filled with a nanofluid using Tiwari and Das nanofluid model. PLoS ONE 10(5), e0126486 (2015)

    Google Scholar 

  30. Cho, C., Chiu, C., Lai, C.: Natural convection and entropy generation of Al2O3-water nanofluid in an inclined wavy-wall cavity. Int. J. Heat Mass Transf. 97, 511–520 (2016)

    Google Scholar 

  31. Ali, M.M., Abdul Alim, Md., Akhter, R., Ahmed, S.S.: MHD natural convection flow of CuO/water nanofluid in a differentially heated hexagonal enclosure with a tilted square block. Int. J. Appl. Comput. Math. 3, 1047–1069 (2017)

    MathSciNet  Google Scholar 

  32. Dutta, S., Biswas, A.K., Pati, S.: Natural convection heat transfer and entropy generation inside porous quadrantal enclosure with nonisothermal heating at the bottom wall. Numer. Heat Transf. Part A 73(4), 222–240 (2018)

    Google Scholar 

  33. Armaghani, T., Rashad, A.M., Vahidifar, O., Mishra, S.R., Chamkha, A.J.: Effect of discrete heat source location on heat transfer and entropy generation of nanofluid in an open inclined L-shaped cavity. Int. J. Numer. Meth. Heat Fluid Flow 29(4), 1363–1377 (2019)

    Google Scholar 

  34. Singh, K., Pandey, A.K., Kumar, M.: Entropy generation impact on flow of micropolar fluid via an inclined channel with non-uniform heat source and variable fluid properties. Int. J. Appl. Comput. Math. 6, 85 (2020)

    MathSciNet  MATH  Google Scholar 

  35. Oztop, H.F., Al-Salem, K.: A review on entropy generation in natural and mixed convection heat transfer for energy systems. Renew. Sustain. Energy Rev. 16, 911–920 (2012)

    Google Scholar 

  36. Mahian, O., Kianifar, A., Kleinstreuer, C., Al-Nimr, M.A., Pop, I., Sahin, A.Z., Wongwises, S.: A review of entropy generation in nanofluid flow. Int. J. Heat Mass Transf. 65, 514–532 (2013)

    Google Scholar 

  37. Biswal, P., Basak, T.: Entropy generation vs energy efficiency for natural convection based energy flow in enclosures and various applications: a review. Renew. Sustain. Energy Rev. 80, 1412–1457 (2017)

    Google Scholar 

  38. Huminic, G., Huminic, A.: Entropy generation of nanofluid and hybrid nanofluid flow in thermal systems: a review. J. Mol. Liq. 302, 112533 (2020)

    MATH  Google Scholar 

  39. Upreti, H., Pandey, A.K., Kumar, M.: Assessment of entropy generation and heat transfer in three-dimensional hybrid nanofluids flow due to convective surface and base fluids. J. Porous Media 24, 35–50 (2021)

    Google Scholar 

  40. Upreti, H., Pandey, A.K., Kumar, M., Makinde, O.D.: Ohmic heating and non-uniform heat source/sink roles on 3D Darcy-Forchheimer flow of CNTs nanofluids over a stretching surface. Arab. J. Sci. Eng. 45, 7705–7717 (2020)

    Google Scholar 

  41. Pandey, A.K., Kumar, M.: Natural convection and thermal radiation influence on nanofluid flow over a stretching cylinder in a porous medium with viscous dissipation. Alex. Eng. J. 56, 55–62 (2017)

    Google Scholar 

  42. Joshi, N., Upreti, H., Pandey, A.K., Kumar, M.: Heat and mass transfer assessment of magnetic hybrid nanofluid flow via bidirectional porous surface with volumetric heat generation. Int. J. Appl. Comput. Math. 7, 64 (2021)

    MathSciNet  Google Scholar 

  43. Singh, K., Pandey, A.K., Kumar, M.: Melting heat transfer assessment on magnetic nanofluid flow past a porous stretching cylinder. J. Egypt. Math. Soc. 29, 1 (2021)

    MathSciNet  MATH  Google Scholar 

  44. Singh, K., Pandey, A.K., Kumar, M.: Slip flow of micropolar fluid through a permeable wedge due to the effects of chemical reaction and heat source/sink with Hall and ion-slip currents: an analytic approach. Propuls Power Res. 9, 289–303 (2020)

    Google Scholar 

  45. Haq, I., Shahzad, M., Khan, W.A., Irfan, M., Mustafa, S., Ali, M., Sultan, F.: Characteristics of chemical processes and heat source/sink with wedge geometry. Case Stud. Therm. Eng. 14, 100432 (2019)

    Google Scholar 

  46. Ali, M., Hamza, S., Aldila, D., Sultan, F., Mustafa, S., Shahzad, M.: Evaluation of steady-state to identify the fast-slow completion-route in the multi-route reaction mechanism. Appl. Nanosci. 10, 3405–3410 (2020)

    Google Scholar 

  47. Ali, M., Shahzad, M., Sultan, F., Khan, W.A., Rashid, S.: Exploring the features of stratification phenomena for 3D flow of cross nanofluid considering activation energy. Int. Commun. Heat Mass Transf. 116, 104674 (2020)

    Google Scholar 

  48. Sultan, F., Ali, M., Mustafa, S., Shahzad, M., Iqbal, A.: The impact of the rate coefficient over the reaction mechanism. Appl. Nanosci. 10, 5375–5381 (2020)

    Google Scholar 

  49. Shahzad, M., Shah, S.I.A., Sultan, F., Ali, M.: The C-matrix augmentation in a multi-route reaction mechanism. Appl. Nanosci. 10, 5383–5390 (2020)

    Google Scholar 

  50. Sultan, F., Mustafa, S., Khan, W.A., Shahzad, M., Ali, M., Adnan, W., Rehman, S.: A numerical treatment on rheology of mixed convective Carreau nanofluid with variable viscosity and thermal conductivity. Appl. Nanosci. 10, 3295–3303 (2020)

    Google Scholar 

  51. Mejri, I., Mahmoudi, A., Abbassi, M.A., Omri, A.: Magnetic field effect on entropy generation in a nanofluid-filled enclosure with sinusoidal heating on both side walls. Powder Technol. 266, 340–353 (2014)

    Google Scholar 

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Acknowledgements

The authors thank the management and R&D centre, Presidency University for support and encouragement. HAKS and NKR sincerely acknowledge the university fellowship. The constructive suggestions from the anonymous referees are sincerely appreciated.

Funding

This research work has been financially supported by VGST, GoK under the Grant Number KSTePS/VGST-KFIST (L1)/2017.

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Authors and Affiliations

  1. Department of Mathematics, School of Engineering, Presidency University, Bengaluru, 560064, India

    H. A. Kumara Swamy & N. Keerthi Reddy

  2. Department of General Requirements, University of Technology and Applied Sciences, 516, Ibri, Oman

    M. Sankar

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Contributions

The research problem and the FORTRAN code was designed by MS. Numerical simulations and plotting of graphs was carried out by HAKS. Analysis of results and paper preparation was done by HAKS and NKR. MS proofread the paper and managed overall preparation.

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Correspondence to M. Sankar.

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Swamy, H.A.K., Sankar, M. & Reddy, N.K. Analysis of Entropy Generation and Energy Transport of Cu-Water Nanoliquid in a Tilted Vertical Porous Annulus. Int. J. Appl. Comput. Math 8, 10 (2022). https://doi.org/10.1007/s40819-021-01207-y

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