Transport in Porous Media

, Volume 98, Issue 1, pp 103–124 | Cite as

Thermosolutal Convection in a Rectangular Concentric Annulus: A Comprehensive Study

  • Sofen K. Jena
  • Swarup K. MahapatraEmail author
  • Amitava Sarkar


This investigation presents numerical treatment of governing equations pertaining to thermosolutal flow within an annulus and an application of a model describing the important physical phenomenon as found in muffle furnace. The inner side of the annulus is exposed to high temperature and high solute concentration while the outer side of the annulus is maintained at low temperature and low solute concentration. Darcy-Brinkman-Forchheimer model is used to study the flow, heat and solute transfer in a non-Darcian saturated porous media. The solution is obtained upon application of control volume integration. Modified MAC method is used for the numerical solution of governing equations. Gradient dependent consistent hybrid upwind scheme of second order (GDCHUSSO) is used for discretization of the convective terms. The parameters such as Rayleigh-Darcy number, Darcy number, buoyancy ratio and width ratio, that govern the flow phenomenon have been identified and their effects are critically examined. The fluid flow pattern in the annular space and the associated heat and mass transfer are conceptualized from the obtained isoconcentration, isotherm and flowline contour maps.


Annulus space Double diffusion Porous media Width ratio 

List of Symbols



Solute mass fraction (\(\text{ Kg}\,\text{ m}^{-3}\))


Specific heat capacity (\(\text{ J}\, \text{ Kg}^{-1}\, \text{ K}^{-1}\))


Mass diffusivity (\(\text{ m}^{2}\, \text{ s}^{-1}\))


Darcy number


Forchheimer term


Acceleration due to gravity (\(\text{ m}\, \text{ s}^{-2}\))


Inner length of the cavity (m)


Nodal index


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


Permeability of the medium (\(\text{ m}^{2}\))


Outer length of the cavity (m)


Lewis number


Local Nusselt number


Average Nusselt number


Unit normal vector


Pressure (\(\text{ N}\, \text{ m}^{-2}\))


Dimensionless pressure

\(\Pr \)

Prandtl number


Rayleigh number


Thermal Rayleigh-Darcy number


Solutal Rayleigh-Darcy number


Local Sherwood number


Average Sherwood number


Absolute temperature (K)


Time (s)


Dimensional velocity component (\(\text{ m}\, \text{ s}^{-1}\))


Dimensionless velocity


Dimensional coordinate (m)


Dimensionless coordinate

Greek symbols

\(\alpha \)

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

\(\beta _\mathrm{{T}}\)

Coefficient of thermal expansion (\(\text{ K}^{-1}\))

\(\beta _\mathrm{{C}}\)

Coefficient of compositional expansion (\(\text{ m}^{3}\text{ Kg}^{-1}\))

\(\chi \)

Heat capacity ratio

\(\varepsilon \)

Porosity of the medium

\(\mu \)

Dynamic viscosity (\(\text{ Kg}\, \text{ m}^{-1}\, \text{ s}^{-1}\))

\(\nu \)

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

\(\varTheta \)

Dimensionless temperature

\(\varPhi \)

Dimensionless solute mass fraction

\(\rho \)

Density (\(\text{ Kg}\, \text{ m}^{-3}\))

\(\tau \)

Dimensionless time

\(\mathfrak R \)

Width ratio

\(\wp \)

Buoyancy ratio







Lower, left











The authors wish to acknowledge Prof. Oleg Iliev (Fraunhofer ITWM, Germany), Prof. T. Sundararajan (IIT Madras), Prof. S.N. Panigrahi & Prof. P. Rath (IIT Bhubaneswar) for their advice and motivation.


  1. Asan, R.C.: Natural convection in an annulus between two isothermal concentric square ducts. Int. Commun. Heat Mass Transf. 27, 367–376 (2000)CrossRefGoogle Scholar
  2. Badruddin, I.A., Al-Rashed, A.A.A.A., Ahmed, N.J.S., Kamangar, S.: Investigation of heat transfer in square porous-annulus. Int. J. Heat Mass Transf. 55, 2184–2192 (2012)CrossRefGoogle Scholar
  3. Barletta, A., Storesletten, L.: Onset of convective rolls in a circular porous duct with external heat transfer to a thermally stratified environment. Int. J. Therm. Sci. 50, 1374–1384 (2011)CrossRefGoogle Scholar
  4. Barletta, A., Celli, M., Kuznetsov, A.V.: Transverse heterogeneity effects in the dissipation-induced instability of a horizontal porous layer. ASME J. Heat Transf. 133, 122601–1-8 (2011)Google Scholar
  5. Barletta, A., Celli, M., Kuznetsov, A.V.: Heterogeneity and onset of instability in Darcy’s flow with a prescribed horizontal temperature gradient. ASME J. Heat Transf. 134, 042602–1-8 (2012)Google Scholar
  6. Beghein, C., Haghighat, F., Allard, F.: Numerical study of double diffusive natural convection in a square cavity. Int. J. Heat Mass Transf. 35, 833–846 (1992)CrossRefGoogle Scholar
  7. Bejan, A.: Mass and heat transfer by natural convection in a vertical cavity. Int. J. Heat Fluid Flow 6, 149–159 (1985)CrossRefGoogle Scholar
  8. Bishop, E.H., Mack, L.R., Scanlan, J.A.: Heat transfer by natural convection between concentric spheres. Int. J. Heat Mass Transf. 9, 649–662 (1966)CrossRefGoogle Scholar
  9. Brandt, A., Dendy, J.E., Ruppel, H.: The multi-grid method for semi-implicit hydrodynamic codes. J. Comput. Phys. 34, 348–370 (1980)CrossRefGoogle Scholar
  10. Brinkman, H.C.: On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A 1, 81–86 (1947)CrossRefGoogle Scholar
  11. Chen, F., Chen, C.F.: Double-diffusive fingering convection in a porous medium. Int. J. Heat Mass Transf. 36, 793–807 (1993)CrossRefGoogle Scholar
  12. Chiu, C.P., Chen, R.C.: Transient natural convection heat transfer between concentric and vertically eccentric sphere. Int. J. Heat Mass Transf. 39, 1439–1452 (1996)CrossRefGoogle Scholar
  13. Cho, C.H., Chang, K.S., Park, K.H.: Numerical simulation of natural convection in concentric and eccentric horizontal cylindrical annuli. ASME J. Heat Transf. 104, 624–630 (1982)CrossRefGoogle Scholar
  14. Chorin, A.J.: Numerical methods for solving incompressible viscous flow problems. J. Comput. Phys 2, 12–26 (1967)CrossRefGoogle Scholar
  15. Darcy, H.P.G.: Les Fontaines Publiques de la ville de Dijon. Vector Dalmont, Paris (1856)Google Scholar
  16. Forchheimer, P.: Wasserbewegung durch Boden. Z. Ver. Deut. Ing. 45, 1736–1741, 1781–1788 (1901)Google Scholar
  17. Garg, V.K.: Natural convection between concentric spheres. Int. J. Heat Mass Transf. 35, 1935–1945 (1992)CrossRefGoogle Scholar
  18. Gentry, R.A., Martin, R.E., Daly, B.J.: An Eulerian differencing method for unsteady incompressible flow problems. J. Comput. Phys. 1, 87–118 (1966)CrossRefGoogle Scholar
  19. Hackbush, W.: Iterative Solution of Large Sparse System of Equations. Springer-Verlag, New York (1994)CrossRefGoogle Scholar
  20. Haldenwang, P.: Resolution tridimensionnelle des equations de Navier-Stokes par methodes spectrales Tchebycheff, These d’etat. Universite de Provence, Marseille (1984)Google Scholar
  21. Hirt, C.W., Cook, J.L.: Calculating three dimensional flows around structures and over rough terrain. J. Comput. Phys. 10, 324–340 (1972)CrossRefGoogle Scholar
  22. Ingham, D.B.: Heat transfer by natural convection between spheres and cylinders. Numer. Heat Transf. Part A 4, 53–67 (1981)Google Scholar
  23. Issa, R.I.: Solution of the implicitly discretized fluid flow equations by operator-splitting. J. Comput. Phys. 62, 40–65 (1985)CrossRefGoogle Scholar
  24. Kim, S.W., Benson, T.J.: Comparison of the SMAC, PISO and iterative time advancing schemes for unsteady flows. Comput. Fluids 21, 435–454 (1992)CrossRefGoogle Scholar
  25. Koennings, S., Tessmar, J., Blunk, T., Göpferich, A.: Confocal microscopy for the elucidation of mass transport mechanisms involved in protein release from lipid-based matrices. Pharm. Res. 24, 1325–1335 (2007)CrossRefGoogle Scholar
  26. Mack, L.R., Hardee, H.C.: Natural convection between concentric spheres at low rayleigh numbers. Int. J. Heat Mass Transf. 11, 387–396 (1968)Google Scholar
  27. Muralidhar, K., Varghese, M., Pillai, K.M.: Application of an operator splitting algorithm for advection-diffusion problems. Numer. Heat Transf. Part B 23, 99–114 (1993)CrossRefGoogle Scholar
  28. Muralidhar, K., Sundararajan, T.: Computational Fluid Flow and Heat Transfer, 2nd edn. Narosa Publishing House Pvt. Ltd, New Delhi (2004)Google Scholar
  29. Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)CrossRefGoogle Scholar
  30. Nishimura, T., Wakamatsu, M., Morega, A.M.: Oscillatory double diffusive convection in a rectangular enclosure with combined horizontal temperature and concentration gradients. Int. J. Heat Mass Transf. 41, 1601–1611 (1998)CrossRefGoogle Scholar
  31. Pinder, G.F., Gray, W.G.: Essentials of Multiphase Flow and Transport in Porous Media. Wiley, Hoboken (2008)CrossRefGoogle Scholar
  32. Raithby, G.D., Torrance, K.E.: Upstream-weighted differencing schemes and their applications to elliptic problems involving fluid flow. Comput. Fluids 2, 191–206 (1974)CrossRefGoogle Scholar
  33. Roache, P.J.: Computational Fluid Dynamics. Hermosa Albuquerque (revised printing), New Mexico (1985)Google Scholar
  34. Rueda, F.J., Schladow, S.G., Clark, J.F.: Mechanisms of contaminant transport in a multi-basin lake. Ecol. Appl. 18, A72–A88 (2008)CrossRefGoogle Scholar
  35. Sankar, M., Bhuvaneswari, M., Sivasankaran, S., Younghae, Do: Buoyancy induced convection in a porous cavity with partially thermally active sidewalls. Int. J. of Heat Mass Transf. 54, 5173–5182 (2011)Google Scholar
  36. Sezai, I., Mohamad, A.A.: Three-dimensional double-diffusive convection in a porous cubic enclosure due to opposing gradients of temperature and concentration. J. Fluid Mech. 400, 333–353 (1999)CrossRefGoogle Scholar
  37. Sezai, I., Mohamad, A.A.: Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradient. Phys. Fluids 12, 2210–2223 (2000)CrossRefGoogle Scholar
  38. Sing, S.N., Chen, J.: Numerical solution for free convection between concentric spheres at moderate Grashof numbers. Numer. Heat Transf. Part A 3, 441–459 (1980)Google Scholar
  39. Stein, D., van der Heyden, F.H.J., Koopmans, W.J.A., Dekker, C.: Pressure-driven transport of confined DNA polymers in fluidic channels. Proc. Natl. Acad. Sci. 103, 15853–15858 (2006)CrossRefGoogle Scholar
  40. Trevisan, O.V., Bejan, A.: Mass and heat transfer by natural convection in a vertical slot filled with porous medium. Int. J. Heat Mass Transf. 29, 403–415 (1986)CrossRefGoogle Scholar
  41. Trevisan, O.V., Bejan, A.: Combined heat and mass transfer by natural convection in a vertical enclosure. ASME J. Heat Transf. 109, 104–112 (1987)CrossRefGoogle Scholar
  42. Venka, S.P., Chen, B.C.-J., Sha, W.T.: A semi-implicit calculation procedure for flow described in body-fitted coordinate systems. Numer. Heat Transf. A 3, 1–19 (1980)Google Scholar
  43. Wright, J.L., Douglass, R.W.: Natural convection in narrow-gap spherical annuli. Int. J. Heat Mass Transf. 29, 725–739 (1986)CrossRefGoogle Scholar
  44. Yin, S.H., Powe, R.E., Scanlan, J.A., Bishop, E.H.: Natural convection flow patterns in spherical annuli. Int. J. Heat Mass Transf. 16, 1785–1795 (1973)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Sofen K. Jena
    • 1
    • 3
  • Swarup K. Mahapatra
    • 2
    Email author
  • Amitava Sarkar
    • 3
  1. 1.Department of Flows and Materials SimulationFraunhofer Institute for Industrial Mathematics (ITWM)KaiserslauternGermany
  2. 2.School of Mechanical SciencesIndian Institute of Technology Bhubaneswar BhubaneswarIndia
  3. 3.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia

Personalised recommendations