The effect of conducting bounding horizontal plates on species separation in porous cavity saturated by a binary mixture

Abdelkader Mojtabi, Bafétigué Ouattara, D. Andrew S. Rees, Marie Catherine Charrier-Mojtabi

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Abstract

In this paper, an analytical and numerical study of species separation in binary mixtures taking account of the presence of bounding plates for the cell is presented. A rectangular horizontal porous cavity saturated by a binary mixture and heated from below is considered. This cavity is bounded by thin plates of uniform thickness, the outer surfaces of which are subjected to a constant heat flux. The transition from the equilibrium solution to the convective one, either stationary or oscillatory, was previously studied by Ouattara et al. (2012). Thus in the first part of this paper, the critical parameters associated with the onset of long wavelength disturbances, obtained analytically, are recalled. Then the hypothesis of parallel flow is used to determine an analytical solution which describes the unicellular flow which may appear in the case of a large aspect ratio cell for a given range of separation ratio, ψ Rayleigh number, Ra, Lewis number, Le, the ratio of the plate to the porous layer thickness, δ and their thermal conductivity ratio, d. The analytical results are corroborated by direct numerical simulations. We verify that if d goes to infinity, the walls become infinitely conductive and we find the results obtained by Charrier-Mojtabi et al. (2007). When d tends to 0, the walls become infinitely thin the results obtained by Yacine et al. (2016) are recovered. A linear stability analysis of the unicellular flow is also presented. The eigenvalue problem resulting from the temporal stability analysis is solved by a Tau spectral method. The optimal Rayleigh number leading to an optimal value of the separation horizontal gradient is determined for different values of physical parameters. We show that the species separation depends sensitively on the ratio of the plate to the porous layer thickness, and the ratio of their thermal conductivities. Furthermore, we have shown that in the stationary state and for a given value of the thermal conductivity ratio (d = 29), the maximum separation is almost equal for walls of the same thickness than the one of porous cavity or for the case of porous cell delimited by the infinitely thin walls.

Original languageEnglish
Pages (from-to)479-488
Number of pages10
JournalInternational Journal of Heat and Mass Transfer
Volume126
Issue numberPart A
Early online date18 May 2018
DOIs
Publication statusPublished - 1 Nov 2018

Fingerprint

Binary mixtures
binary mixtures
conduction
cavities
Thermal conductivity
thermal conductivity
Rayleigh number
cells
Lewis numbers
thin walls
Linear stability analysis
parallel flow
Parallel flow
spectral methods
thin plates
Direct numerical simulation
direct numerical simulation
infinity
Heat flux
aspect ratio

Keywords

  • Convection
  • Porous medium
  • Soret effect
  • Species separation
  • Stability
  • Thermogravitational-diffusion

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanical Engineering
  • Fluid Flow and Transfer Processes

Cite this

The effect of conducting bounding horizontal plates on species separation in porous cavity saturated by a binary mixture. / Mojtabi, Abdelkader; Ouattara, Bafétigué; Rees, D. Andrew S.; Charrier-Mojtabi, Marie Catherine.

In: International Journal of Heat and Mass Transfer, Vol. 126, No. Part A, 01.11.2018, p. 479-488.

Research output: Contribution to journalArticle

Mojtabi, Abdelkader ; Ouattara, Bafétigué ; Rees, D. Andrew S. ; Charrier-Mojtabi, Marie Catherine. / The effect of conducting bounding horizontal plates on species separation in porous cavity saturated by a binary mixture. In: International Journal of Heat and Mass Transfer. 2018 ; Vol. 126, No. Part A. pp. 479-488.
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AB - In this paper, an analytical and numerical study of species separation in binary mixtures taking account of the presence of bounding plates for the cell is presented. A rectangular horizontal porous cavity saturated by a binary mixture and heated from below is considered. This cavity is bounded by thin plates of uniform thickness, the outer surfaces of which are subjected to a constant heat flux. The transition from the equilibrium solution to the convective one, either stationary or oscillatory, was previously studied by Ouattara et al. (2012). Thus in the first part of this paper, the critical parameters associated with the onset of long wavelength disturbances, obtained analytically, are recalled. Then the hypothesis of parallel flow is used to determine an analytical solution which describes the unicellular flow which may appear in the case of a large aspect ratio cell for a given range of separation ratio, ψ Rayleigh number, Ra, Lewis number, Le, the ratio of the plate to the porous layer thickness, δ and their thermal conductivity ratio, d. The analytical results are corroborated by direct numerical simulations. We verify that if d goes to infinity, the walls become infinitely conductive and we find the results obtained by Charrier-Mojtabi et al. (2007). When d tends to 0, the walls become infinitely thin the results obtained by Yacine et al. (2016) are recovered. A linear stability analysis of the unicellular flow is also presented. The eigenvalue problem resulting from the temporal stability analysis is solved by a Tau spectral method. The optimal Rayleigh number leading to an optimal value of the separation horizontal gradient is determined for different values of physical parameters. We show that the species separation depends sensitively on the ratio of the plate to the porous layer thickness, and the ratio of their thermal conductivities. Furthermore, we have shown that in the stationary state and for a given value of the thermal conductivity ratio (d = 29), the maximum separation is almost equal for walls of the same thickness than the one of porous cavity or for the case of porous cell delimited by the infinitely thin walls.

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