Numerical and Analytical Studies of Soret-Driven Convection Flow Inside an Annular Horizontal Porous Cavity

This paper studies the species separation of a binary fluid in a porous cavity between two horizontal concentric cylinders, submitted to a temperature gradient. The thickness of the cavity is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mo>=</mo><msub><mi>R</mi><mi>o</mi></msub><mo>−</mo><msub><mi>R</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>o</mi></msub></mrow></semantics></math></inline-formula> are the internal and external radius, respectively. The numerous previous experiments performed in thermogravitational vertical columns (TGCs) showed that in order to obtain a significant separation, the thickness of the cell must be very small, compared with its height. Therefore, in our configuration, we considered <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>e</mi><mo>≪</mo><msub><mi>R</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>. The solution is assumed to be axisymmetric. Under the assumptions of parallel flow and forgotten effect, an analytical solution is obtained using Maple software, and the results are compared with those found numerically using Comsol Multiphysics. In natural convection, our results are in very good agreement with those evaluated with a regular perturbation method in powers of the dimensionless gap width <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>=</mo><mfrac><mi>e</mi><mrow><msub><mi>R</mi><mi>i</mi></msub></mrow></mfrac><mo> </mo></mrow></semantics></math></inline-formula> of order 15, and with the Galerkin method. The species separation calculated for our configuration is very close to the one obtained in a TGC column of height: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mi>π</mi><msub><mi>R</mi><mi>i</mi></msub><mo>.</mo></mrow></semantics></math></inline-formula> One of the main interests of the analytical solution presented here is that it can be used as a basic solution for a stability study analysis.