Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer: the effect of conducting bounding plates

The aim of this study was to investigate the effect of conducting boundaries on the onset of convection in a binary fluid-saturated porous layer. The isotropic saturated porous layer is bounded by two impermeable but thermally conducting plates, subjected to a constant heat flux. These plates have identical conductivity. Moreover, the conductivity of the plates is generally different from the porous layer conductivity. The overall layer is of large extent in both horizontal directions. The problem is governed by seven dimensionless parameters, namely the normalized porosity of the medium ε, the ratio of plates over the porous layer thickness δ and their relative thermal conductivities ratio d, the separation ratio δ, the Lewis number Le and thermal Rayleigh number Ra. In this work, an analytical and numerical stability analysis is performed. The equilibrium solution is found to lose its stability via a stationary bifurcation or a Hopf bifurcation depending on the values of the dimensionless parameters. For the long-wavelength mode, the critical Rayleigh number is obtained as Racs=12(1+2dδ )/[1+ψ (2dδLe+Le+1)] and kcs=0 for ψ> ψ uni> 0. This work extends an earlier paper by Mojtabi and Rees (2011 Int. J. Heat Mass Transfer 54 293-301) who considered a configuration where the porous layer is saturated by a pure fluid

Influence of acoustic streaming on thermo-diffusion in a binary mixture under microgravity

An analytical and numerical study of the influence of acoustic streaming on species separation of a binary mixture under microgravity is presented. A rectangular cell filled with binary fluid is submitted to an ultrasonic propagating wave along a portion of one of its small walls while the opposite wall is perfectly absorbing. A temperature gradient is applied between the two other walls. The unicellular flow induced by the Eckart streaming may lead to significant species separation. In a first part, the hypothesis of parallel flow is used to determine the analytical solution which describes the unicellular flow and the separation is calculated analytically based on the acoustic streaming parameter, A, the acoustic beam width, e, and the Schmidt number, Sc. Theses analytical results are corroborated by direct numerical simulations. In a second part, a linear stability analysis of the unicellular flow is performed. The eigenvalue problem resulting from the temporal stability analysis is solved by the Galerkin method, a spectral Tau–Chebychev method and by a finite element method. The thresholds for the stationary and oscillatory instability depend on the normalized acoustic beam width

Influence of vertical vibrations on the separation of a binary mixture in a horizontal porous layer heated from below

We study the influence of vertical high-frequency and small-amplitude vibrations on the separation of a binary mixture saturating a shallow horizontal porous layer heated from below. The monocellular flow obtained for a separation ratio w > wmono > 0 leads to a migration of the species towards the two vertical boundaries of the cell. The 2D direct numerical simulations and the linear stability of the averaged governing equations analysis show that the vertical vibrations delay the transition from monocellular flow to multicellular flow. The vibrations also decrease the value of wmono, which allows species separation for a wide variety of binary mixtures

Separation of a binary fluid mixture in a porous horizontal cavity

Thermogravitational separation has, until now, been used in differentially heated vertical cells, called thermogravitational columns (TGCs). The cell can be either a slit or an annular cavity whose two isothermal faces are maintained at different temperatures T1 and T2. In this study, we show contrary to what has been done until now, that it is possible to carry out the separation of the species of a binary mixture in the classical configuration of Rayleigh-Bénard (horizontal cell heated from below with the separation ratio ψ>ψmono>0 or from above with ψψmono at the onset of convection. The species separation, in a horizontal cell, is obtained without fear of the remixing observed in vertical thermogravitational cells for fluids with negative Soret coefficients. In this situation, the heaviest component concentrates in the upper part of the cell creating an unstable physical situation. We improve the efficiency of separation for the development of an industrial process of separation. We study the thermogravitational separation of the components of a binary fluid mixture saturating a horizontal porous layer, in the presence of the Soret effect. The horizontal walls of the cavity are impermeable and maintained at constant and different temperatures. The system of equations governing the problem has an equilibrium solution with a vertical stratification of temperature and concentration. This solution loses its stability via a stationary bifurcation for values of the separation ratio ψ>ψ0 while, for ψψmono>0 and leads to a separation of the species between the two ends of the cell. We propose to determine whether the monocellular flow remains stable until the optimum of separation and if we can obtain high separation in the case of a horizontal cavity. We verify that the critical Rayleigh number Rac2, associated with the transition between monocellular flow and multicellular flow, is higher than the optimum Rayleigh number leading to maximum separation of species. Thus this study reveals that it is possible to obtain optimal separation before the monocellular flow loses its stability. We show also that the separation inside the horizontal cell in a Rayleigh-Bénard configuration permits us to produce the same degree of separation but with a greater quantity of each species compared to a thermogravitational vertical colum

Numerical simulation of two- and threedimensional free convection flows in a horizontal porous annulus using a pressure and temperature formulation.

Abstract--A numerical investigation of two-dimensional and three-dimensional free convection flows in a saturated porous horizontal annulus heated from the inner surface is carried out, using a Fourier-Galerkin approximation for the periodic azimuthal and axial directions and a coUocation-Chebyshev approximation in the confined radial direction. The numerical algorithm integrates the Darcy-Boussinesq's equations formulated in terms of pressure and temperature. This method gives an accurate description of the 2-D multicellular structures for a large range of Rayleigh number and radii ratio. Some considerations about the existence of the various 2-D solutions previously described in the literature are reported. The 3-13 spiral flows are described in the vicinity of the transition from the 2-D unicellular flows. Bifurcation points between 2-D unicellular flows and either 2-D multicellular or 3-D flows are also determined numerically

Etude analytique et numérique de la diffusion thermo-gravitationnelle en cavité poreuse horizontale saturée par un fluide binaire

Lorsqu'un fluide binaire ou multi-constituant, initialement homogène, est soumis à un gradient thermique, un gradient de faction massique apparaît au sein du mélange. Ce phénomène est connu sous le nom de thermodiffusion ou effet Soret. Ce phénomène peut conduire à une séparation des espèces plus importante qu'en thermodiffusion. Dans ce travail nous nous intéressons à l'étude de la diffusion thermogravitationnelle, en présence d'effet Soret, dans le cas d'une cavité poreuse horizontale saturée par un fluide binaire et soumise à un flux de chaleur uniforme sur la paroi inférieure et à une température constante et uniforme sur la paroi supérieure. On présente une étude analytique faite dans l'hypothèse d'un écoulement de type parallèle pour des cellules de grand rapport d'aspect. On s'intéresse en particulier à la séparation des espèces et à l'optimum de séparation entre les deux parois verticales de la cellule en fonction du nombre de Rayleigh. Les résultats analytiques obtenus sont en bon accord avec ceux obtenus par simulations numériques directes 2D et 3D pour des cellules de rapport d'aspect Ax=10et Ay=2 et 3

Nouveau procédé de séparation des espèces d'un fluide binaire par convection mixte

Lorsqu’on soumet une solution initialement homogène, constituée d’au moins deux espèces chimiques, à un gradient thermique, celui-ci induit un transfert des constituants au sein du mélange : ce phénomène est appelé thermodiffusion ou effet Soret. Le couplage entre la convection et la thermodiffusion, appelé diffusion thermogravitationnelle conduit, sous certaines conditions, à une séparation des espèces plus importante qu'en thermodiffusion. Dans les colonnes thermogravitationnelles verticales le gradient thermique imposé induit non seulement la thermodiffusion mais également le mouvement convectif. Nous proposons dans ce travail une nouvelle technique permettant d'améliorer le procédé de séparation en découplant le gradient thermique de la vitesse convective. On considère pour cela, une cavité rectangulaire, horizontale, remplie d’un fluide binaire et soumise à un flux de chaleur vertical. La paroi supérieure est animée d’une vitesse uniforme, ce qui permet de disposer de deux paramètres de contrôle indépendants. Les résultats analytiques et numériques obtenus sont en parfait accord

A new process for species separation in a binary mixture using mixed convection

In this paper, a numerical and analytical analysis is performed in order to improve the species separation process in a binary fluid mixture by decoupling the thermal gradient from the convective velocity. The configuration considered is a horizontal rectangular cavity, of large aspect ratio, filled with a binary fluid. A constant tangential velocity is applied to the upper horizontal wall. The two horizontal impermeable walls are maintained at different and uniform temperatures T1 and T2 with ΔT = T1 − T2 Species separation is governed by two control parameters, the temperature difference ΔT and the velocity of the upper plate Uex. The intensity of the thermodiffusion is controlled by the temperature, while the velocity Uex controls the convective flow. This problem depends on six dimensionless parameters, namely, the separation ratio, ψ, the Lewis number, Le, the Prandtl number Pr, the aspect ratio of the cell, A and two control parameters: the thermal Rayleigh number, Ra and the Péclet number Pe. In this study, the formulation of the separation (mass fraction difference between the two ends of the cell) as a function of the Péclet number and the Rayleigh number is obtained analytically. For a cell heated from below, the optimal separation m = √42/15 is obtained for Pe = √42/Le and Ra = 540/(Leψ). 2D numerical results, obtained by solving the full governing equations, are in good agreement with the analytical results based on a parallel flow approach

Separation in an inclined porous thermogravitational cell

This paper reports a theoretical and numerical study of species separation in a binary liquid mixture saturating a shallow porous layer heated from below or from above and inclined with respect to the vertical axis. It is shown that the separation can be increased using this configuration and the stability of the unicellular flow obtained in this case is investigated. The critical Rayleigh number obtained is much higher than the one leading to the maximum separation. Experiments performed with a solution of CuSO4 give results which are almost in good agreement with the analytical and the numerical results