Abstract
The present study reports results of numerical investigation of natural convection in a spherical annulus filled with gas saturated porous medium. Boussinesq approximation has been relaxed considering variation of density, viscosity, and thermal conductivity of the gas with temperature. The momentum and energy equation along with boundary condition and property variation equations are solved by employing successive accelerated replacement scheme. There exists a reference parameter (q = 0.4) at which if thermo physical properties are evaluated, the average Nusselt number values for variable properties and constant properties are nearly same. Also, the effect of variable properties is negligible for temperature difference ratio, \(\theta ^{*}<0.1\).
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Abbreviations
- c:
-
Specific heat of fluid (J/kgK)
- \(\hbox {C}_{\rho } ,\hbox {C}_{\mu } ,\hbox {C}_\mathrm{k}\) :
-
Constants for property variation (–)
- D:
-
Width of porous annulus (m)
- g:
-
Acceleration due to gravity \((\hbox {m/s}^{2})\)
- K:
-
Permeability of porous medium \((\hbox {m}^{2})\)
- k:
-
Thermal conductivity of porous medium (W/mK)
- Nu:
-
Local Nusselt number (–)
- \(\overline{\hbox {Nu}}\) :
-
Average Nusselt number (–)
- \(\hbox {n}_{\rho } ,\hbox {n}_{\mu } ,\hbox {n}_{k}\) :
-
Exponents for property variation (–)
- p:
-
Fluid pressure \((\hbox {N/m}^{2})\)
- P:
-
Dimensionless pressure (–)
- q:
-
Reference parameter (–)
- r:
-
Radial co-ordinate (m)
- R:
-
Dimensionless radial co-ordinate (–)
- Ra:
-
Modified Rayleigh number (–)
- rr:
-
Radius ratio (–)
- T:
-
Temperature (K)
- \(\hbox {u},\hbox {v}\) :
-
Velocity in r and \(\emptyset \) direction (m/s)
- \(\hbox {U},\hbox {V}\) :
-
Dimensionless velocity in r and \(\emptyset \) directions (–)
- \(\alpha \) :
-
Thermal diffusivity \((\hbox {m}^{2}/\hbox {s})\)
- \(\beta \) :
-
Coefficient of thermal expansion of the fluid \((\hbox {K}^{-1})\)
- \(\varepsilon \) :
-
Error tolerance limit (–)
- \(\varphi \) :
-
Porosity of porous media (–)
- \(\emptyset \) :
-
Polar co-ordinate (rad)
- \(\mu \) :
-
Dynamic viscosity of fluid \((\hbox {Ns/m}^{2})\)
- \(\rho \) :
-
Fluid density \((\hbox {kg/m}^{3})\)
- \(\bar{\rho }\) :
-
Dimensionless fluid density (–)
- \(\Psi \) :
-
Dimensionless stream function (–)
- \(\theta \) :
-
Dimensionless temperature (–)
- \(\theta ^{*}\) :
-
Temperature difference ratio (–)
- \(\omega \) :
-
Acceleration factor (–)
- i:
-
Inner wall
- o:
-
Outer wall
- s:
-
Solid
- f:
-
Fluid
- q:
-
Reference
References
Baytas, A.C., Grosan, T., Pop, I.: Free convection in spherical annular sectors filled with a porous medium. Transp. Porous Media 49, 191–207 (2002)
Bejan, A., Kraus, A.D.: Heat Transfer Handbook. Wiley, New York (2003)
Bejan, A., Dincer, I., Lorente, S., Miguel, A.F., Reis, A.H.: Porous and Complex Flow Structures in Modern Technologies. Springer, New York (2004)
Burns, P.J., Tien, C.L.: Natural convection in porous media bounded by concentric spheres and horizontal cylinders. Int. J. Heat Mass Transf. 6, 929–939 (1979)
Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media. Elsevier, Oxford (2005)
Kothandaraman, C.P., Subramanyan, S.: Heat and mass Transfer Data Book. New Age International Pvt. Ltd, New Delhi (2003)
Marpu, D.R., Satyamurty, V.V.: Influence of variable fluid density on free convection in rectangular porous media. Trans. ASME J. Energy Res. Technol. 4, 214–220 (1989)
Marpu, D.R., Satyamurty, V.V.: Investigation on the validity of Boussinesq approximation on free convectionin vertical porous annulus. Warme Stoffubertrag 3, 141–147 (1991)
Mey, S., Merker, G.P.: Free convection in a shallow cavity with variable properties- 2 Porous media. Int. J. Heat Mass Transf. 9, 1833–1837 (1987)
Neild, D.A.: Onset of convection in a porous layer saturated by an ideal gas. Int. J. Heat Mass Transf. 10, 1605–1606 (1982)
Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)
Peirotti, M.B., Giavedoni, M.D., Deiber, J.A.: Natural convective heat transfer in a rectangular porous cavity with variable fluid properties–validity of the Boussinesq approximation. Int. J. Heat Mass Transf. 12, 2571–2581 (1987)
Pop, I., Ingham, D.B., Cheng, P.: Transient free convection between two concentric spheres filled with a porous medium. J. Thermophys. Heat Transf. 4, 724–727 (1993)
Pop, I., Ingham, D.B.: Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001)
Saatdjian, N.: Natural convection in a porous layer saturated with a compressible ideal gas. Int. J. Heat Mass Transf. 12, 1681–1682 (1980)
Sangita, Sinha, M.K., Sharma, R.V.: Natural convection in a spherical porous annulus: the Brinkman extended Darcy flow model. Transp. Porous Media 100, 321–335 (2013)
Vafai, K.: Handbook of Porous Media, 2nd edn. Taylor and Francis, New York (2005)
Vadasz, P. (ed.): Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008)
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Sangita, Sinha, M.K. & Sharma, R.V. Influence of Property Variation on Natural Convection in a Gas Saturated Spherical Porous Annulus. Transp Porous Med 104, 521–535 (2014). https://doi.org/10.1007/s11242-014-0346-z
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DOI: https://doi.org/10.1007/s11242-014-0346-z