Soret-Driven Convection Inside Concentric Porous Spheres Saturated by Binary Fluid: Comparison with Parallelepipedic Vertical Column

The columns commonly used for the species separation of a binary fluid are very thin vertical parallelepipedic columns, filled with a porous medium (i.e. TGC columns). This species separation is due to the mass flux induces by the thermal gradient applied on walls of the cavity, namely the Soret effect, in the gravity field. In this paper, a new configuration used to study the species separation in a binary fluid was presented. The evolution of species separation in a binary fluid saturating a porous media, in a cavity between two concentric spheres, was studied analytically and numerically. The thickness of the cavity is e=Ro-Ri\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e=R_o-R_i$$\end{document}, where Ri\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_i$$\end{document} and Ro\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_o$$\end{document} are the internal and external radii, respectively. First, an analytical solution was developed using the parallel flow approximation considering e≪Ri\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e\ll R_i$$\end{document}. The analytical results obtained are in a good agreement with the numerical ones performed with a spectral method or Comsol Multiphysics software. Thus, the results in the spherical configuration were compared with those obtained in a porous TGC column of same thickness, e, height, H=πRi,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=\pi R_i,$$\end{document} and depth, L=2Ro\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=2R_o$$\end{document}.The numerical and analytical studies showed that the differences in mass fractions, Cmax-Cmin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\max }-C_{\min }$$\end{document}, and the time needed to obtain the stationary separation starting from a homogeneous binary solution were almost equal in the two configurations. The amount of species separated is greater in the spherical configuration than in the vertical column used.