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Thermal Non-equilibrium in Porous Annulus: A New Finite Element Solution Technique

  • Research Article - Mechanical Engineering
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Abstract

Thermal non-equilibrium between solid and fluid phases of porous medium is an important condition that exists in many applications when the thermal conductivity ratio and interphase heat transfer coefficient between the solid and fluid phases have low values. The condition of thermal non-equilibrium requires that the mathematical model be comprised of two energy equations to simulate the temperature variations of solid and fluid phases. This necessitates the two energy equations related to solid and fluid phases to be solved. However, this brings an additional difficulty to simulate thermal non-equilibrium in porous annulus. The difficulty of simulating thermal non-equilibrium condition arises due to involvement of multiple partial differential equations that should be solved simultaneously. The aim of the present work is to propose a novel method that combines the two energy equations into a single equation at the solution stage. The adopted methodology depends on converting the partial differential equations into simpler form by utilising the finite element method. The current work comprehensively demonstrates the feasibility of proposed method along with its validation with respect to heat transfer prediction in porous annulus. The proposed method is found to have good accuracy to simulate the heat transfer in porous annulus.

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Abbreviations

A :

Element area

Ar:

Annulus aspect ratio

g :

Gravitational acceleration

h, H :

Dimensional and non-dimension interphase heat transfer coefficient

H t :

Cylinder height

k :

Thermal conductivity

K :

Permeability of porous medium (m2)

L :

Width of porous medium

\({{\overline{\text{Nu}}}}\) :

Average Nusselt number

Ra:

Rayleigh number

\(R_{d}\) :

Radiation parameter

\(T,\;\bar{T}\) :

Temperature

u, v :

r and z direction velocity

r, z :

Coordinates

α :

Thermal diffusivity

\(\beta_{\text{T}}\), \(\beta_{\text{r}}\) :

Thermal expansion and absorption coefficient, respectively

γ = ks/kp :

Conductivity ratio

ρ :

Density

v :

Kinematic viscosity

\(\sigma\) :

Stephan Boltzmann constant

\(\psi \,,\bar{\psi }\) :

Stream function

h:

Hot

c:

Cold

s:

Solid

f:

Fluid

t:

Total

References

  1. Merkin, J.H.: Free convection from a vertical cylinder embedded in a saturated porous medium. Acta Mech. 62, 19–28 (1986)

    MATH  Google Scholar 

  2. Rajamani, R.C.; Srinivas, C.; Nithiarasu, P.; Seetharamu, K.N.: Convective heat transfer in axisymmetric porous bodies. Int. J. Numer. Methods Heat Fluid Flow 5, 829–837 (1995)

    MATH  Google Scholar 

  3. Badruddin, I.A.; Zainal, Z.A.; Aswatha Narayana, P.A.; Seetharamu, K.N.: Natural convection through an annular vertical cylindrical porous medium. Int. J. Heat Exch. 7(2), 251–262 (2006)

    MATH  Google Scholar 

  4. Badruddin, I.A.; Zainal, Z.A.; Khan, Z.A.; Mallick, Z.: Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus. Int. J. Therm. Sci. 46(3), 221–227 (2007)

    Google Scholar 

  5. Badruddin, I.A.; Zainal, Z.A.; Narayana, P.A.; Seetharamu, K.N.: Heat transfer by radiation and natural convection through a vertical annulus embedded in porous medium. Int. Commun. Heat Mass Transfer 33(4), 500–507 (2006)

    MATH  Google Scholar 

  6. Saeid, N.H.; Mohamad, A.A.: Periodic free convection from a vertical plate in a saturated porous medium non-equilibrium model. Int. J. Heat Mass Transfer 48, 3855–3863 (2005)

    MATH  Google Scholar 

  7. Saeid, N.H.: Analysis of mixed convection in a vertical porous layer, using non-equilibrium model. Int. J. Heat Mass Transfer 47, 5619–5627 (2004)

    MATH  Google Scholar 

  8. Badruddin, I.A.; Abdullah, A.A.A.A.; Ahmed, N.J.S.; Kamangar, S.: Investigation of heat transfer in square porous-annulus. Int. J. Heat Mass Transf. 55(7–8), 2184–2192 (2012)

    Google Scholar 

  9. Badruddin, I.A.; Zainal, Z.A.; Narayana, P.A.; Seetharamu, K.N.: Numerical analysis of convection conduction and radiation using a non-equilibrium model in a square porous cavity. Int. J. Therm. Sci. 46(1), 20–29 (2007)

    Google Scholar 

  10. Ahmed, N.J.S.; Badruddin, I.A.; Kanesan, J.; Zainal, Z.A.; Ahamed, K.S.N.: Study of mixed convection in an annular vertical cylinder filled with saturated porous medium, using thermal non-equilibrium model. Int. J. Heat Mass Transf. 54(17–18), 3822–3825 (2011)

    MATH  Google Scholar 

  11. Chamkha, A.J.; Aly, A.M.; Mansour, M.A.: Similarity solution for unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and chemical reaction effects. Chem. Eng. Commun. 197(6), 846–858 (2010)

    Google Scholar 

  12. Guedda, Mohammed; Ouahsine, Abdellatif: Similarity solutions of MHD flows in a saturated porous medium. European Journal of Mechanics-B/Fluids 33, 87–94 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Khuri, S.A.: Similarity solutions of the mixed convection boundary-layer flow in a porous medium. Int. J. Comput. Methods 4(04), 621–631 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Loganathan, P.; Puvi-Arasu, P.; Kandasamy, R.: Local non-similarity solution to impact of chemical reaction on MHD mixed convection heat and mass transfer flow over porous wedge in the presence of suction/injection. Applied Mathematics and Mechanics 31(12), 1517–1526 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Manjunatha, Siddapura S.; Gireesha, Bijjanal J.; Eshwarappa, Kunabevu M.; Bagewadi, Channabasappa S.: Similarity solutions for boundary layer flow of a dusty fluid through a porous medium over a stretching surface with internal heat generation/absorption. J. Porous Media 16(6), 501–514 (2013)

    Google Scholar 

  16. Merkin, J.H.: Free convective boundary-layer flow in a heat-generating porous medium: similarity solutions. The Quarterly Journal of Mechanics and Applied Mathematics 61(2), 205–218 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Achemlal, D.; Sriti, M.: Finite differences analysis on nonuniform mesh for free convection in a porous medium and comparison with similarity approach. Computational Thermal Sciences: An International Journal 7(3), 231–243 (2015)

    Google Scholar 

  18. del Teso, F.: Finite difference method for a fractional porous medium equation. Calcolo 51(4), 615–638 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Oka, Fusao; Yashima, Atsushi; Shibata, Toru; Kato, Mitsuru; Uzuoka, Ryosuke: FEM-FDM coupled liquefaction analysis of a porous soil using an elasto-plastic model. Appl. Sci. Res. 52(3), 209–245 (1994)

    MATH  Google Scholar 

  20. Liu, Wei; Yuan, Yirang: Finite difference schemes for two-dimensional miscible displacement flow in porous media on composite triangular grids. Comput. Math Appl. 55(3), 470–484 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Verma, A.K.; Singh, T.N.; Chauhan, N.K.; Sarkar, K.: A hybrid FEM-ANN approach for slope instability prediction. J. Inst. Eng. Ser. A 97(3), 171–180 (2016)

    Google Scholar 

  22. Verma, A.K.; Gautam, P.; Singh, T.N.; Bajpai, R.K.: Discrete element modelling of conceptual deep geological repository for high-level nuclear waste disposal. Arab. J. Geosci. 8(10), 8027–8038 (2015)

    Google Scholar 

  23. Lewis, R.W.; Garner, R.W.: A finite element solution of coupled electrokinetic and hydrodynamic flow in porous media. Int. J. Numerical Methods Engineering 5(1), 52–56 (1972)

    MATH  Google Scholar 

  24. Li, Xindong; Rui, Hongxing: A MCC finite element approximation of incompressible miscible displacement in porous media. Comput. Math Appl. 70(5), 750–764 (2015)

    MathSciNet  Google Scholar 

  25. Sajid, M.; Mahmood, R.; Hayat, T.: Finite element solution for flow of a third grade fluid past a horizontal porous plate with partial slip. Comput. Math Appl. 56(5), 1236–1244 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Balla, C.S.; Naikoti, K.: Soret and Dufour effects on free convective heat and solute transfer in fluid saturated inclined porous cavity. Engineering Science and Technology, an International Journal 18, 543–554 (2015)

    Google Scholar 

  27. Wansophark, N.; Malatip, A.; Dechaumphai, P.: Streamline upwind finite element method for conjugate heat transfer problems. Acta. Mech. Sin. 21(5), 436–443 (2005)

    MATH  Google Scholar 

  28. Ahmed, N.J.S.; Badruddin, I.A.; Zainal, Z.A.; Khaleed, H.M.T.; Kanesan, J.: Heat transfer in a conical cylinder with porous medium. Int. J. Heat Mass Transf. 52(13), 3070–3078 (2009)

    MATH  Google Scholar 

  29. Azeem, A.; Badruddin, I.A.; Idris, M.Y.I.; Nik-Ghazali, N.; Ahmed, N.J.S.; Al-Rashed, A.A.A.A.: Conjugate heat and mass transfer in square porous cavity. Indian J. Pure Appl. Phys. 54(12), 777–786 (2016)

    Google Scholar 

  30. Badruddin, I.A.; Azeem, K.; Idris, M.Y.I.; Nik-Ghazali, N.; Salman Ahmed, N.J.; Abdullah, A.A.A.: Simplified finite element algorithm to solve conjugate heat and mass transfer in porous medium. Int. J. Numer. Meth. Heat Fluid Flow 27(11), 2481–2507 (2017)

    Google Scholar 

  31. Badruddin, I.A.; Salman Ahmed, N.J.; Al-Rashed, A.A.A.A.; Nik-Ghazali, N.; Jameel, M.; Kamangar, S.; Yunus Khan, T.M.: Conjugate heat transfer in an annulus with porous medium fixed between solids. Transp. Porous Media 109(3), 589–608 (2015)

    MathSciNet  Google Scholar 

  32. Badruddin, I.A.: Investigation of heat transfer in irregular porous cavity subjected to various boundary conditions. Int. J. Numer. Meth. Heat Fluid Flow 29(1), 418–447 (2019)

    Google Scholar 

  33. Munjiza, Antonio: The Combined Finite-Discrete Element Method. Wiley, Hoboken (2004)

    MATH  Google Scholar 

  34. Verma, A.K.; Singh, T.N.: Assessment of tunnel instability—a numerical approach. Arab. J. Geosci. 3(2), 181–192 (2010)

    Google Scholar 

  35. Badruddin, I.A.: Heat and mass transfer with Soret/Dufour effect in irregular porous cavity. J. Thermophys. Heat Transfer 33(3), 647–662 (2019). https://doi.org/10.2514/1.T5666

    Article  Google Scholar 

  36. Badruddin, I.A.; Salman Ahmed, N.J.; Anqi, A.E.; Kamangar, S.: Conjugate heat and mass transfer in a vertical porous cylinder. J. Thermophys. Heat Transfer 33(2), 548–558 (2019). https://doi.org/10.2514/1.T5488

    Article  Google Scholar 

  37. Nandalur, A.A.; Kamangar, S.; Badruddin, I.A.: Heat transfer in a porous cavity in presence of square solid block. Int. J. Numer. Meth. Heat Fluid Flow 29(2), 640–656 (2019). https://doi.org/10.1108/HFF-05-2017-0193

    Article  Google Scholar 

  38. Azeem; Badruddin, I.A.; Nik-Ghazali, N.; Idris, M.Y.I.; Anqi, A.E.; Salman Ahmed, N.J.; Kamangar, S.; Al-Rashed, A.A.A.A.: Effect of size and location of solid on conjugate heat transfer in porous cavity. Indian J. Pure Appl. Phys. 56(10), 792–801 (2018)

    Google Scholar 

  39. Badruddin, I.A.; Zainal, Z.A.; Narayana, P.A.; Seetharamu, K.N.: Thermal non-equilibrium modeling of heat transfer through vertical annulus embedded with porous medium. Int. J. Heat Mass Transfer. 49(25–26), 4955–4965 (2006)

    MATH  Google Scholar 

  40. Raptis, A.: Radiation and free convection flow through a porous medium. Int. Commun. Heat Mass Transfer 25(2), 289–295 (1998)

    Google Scholar 

  41. Koriko, O.K.; Animasaun, I.L.: Scrutinization of thermal stratification, nonlinear thermal radiation and quartic autocatalytic chemical reaction effects on the flow of three-dimensional Eyring-Powell alumina-water nanofluid. Multidiscipline Modeling in Materials and Structures 14(2), 261–283 (2018)

    Google Scholar 

  42. Sivaraj, R.; Animasaun, I.L.; Olabiyi, A.S.; Saleem, S.; Sandeep, N.: Gyrotactic microorganisms and thermoelectric effects on the dynamics of 29 nm CuO-water nanofluid over an upper horizontal surface of paraboloid of revolution. Multidiscipline Modelling in Materials and Structures 14(4), 695–721 (2018)

    Google Scholar 

  43. Mahanthesh, B.; Gireesha, B.J.; Animasaun, I.L.: Exploration of non-linear thermal radiation and suspended nanoparticles effects on mixed convection boundary layer flow of nanoliquids on a melting vertical surface. Journal of Nanofluids 7, 1–11 (2018)

    Google Scholar 

  44. Lewis, R.W.; Morgan, K.; Thomas, H.R.; Seetharamu, K.N.: The Finite Element Method in Heat Transfer Analysis. Wiley, Chichester (1996)

    MATH  Google Scholar 

  45. Moaveni, S.: Finite Element Analysis Theory and Application with ANSYS, 3rd edn. Prentice-Hall, Inc, Upper Saddle River, NJ (2007)

    Google Scholar 

  46. Reddy, J.N.: An Introduction to The Finite Element Method, 3rd edn. McGraw-Hill, New York (2005)

    Google Scholar 

  47. Lewis, R.W.; Nithiarasu, P.; Seetharamu, K.N.: The Finite Element Method in Heat and Fluid Flow. Wiley, Chichester (2004)

    Google Scholar 

  48. Segerland, L.J.: Applied Finite Element Analysis. Wiley, New York (1982)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, College of Engineering, King Khalid University, P.O. Box 394, Abha, 61421, Kingdom of Saudi Arabia

    Irfan Anjum Badruddin

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  1. Irfan Anjum Badruddin
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Badruddin, I.A. Thermal Non-equilibrium in Porous Annulus: A New Finite Element Solution Technique. Arab J Sci Eng 45, 1279–1292 (2020). https://doi.org/10.1007/s13369-019-04298-4

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