On the Nondimensionalization Process in Complex Problems: Application to Natural Convection in Anisotropic Porous Media

Abstract

The nondimensionalization of the equations governing a given problem is a direct, relatively easy, and low-cost way to provide interesting information, the dimensionless groups that rule the problem and define its final patterns of solution. In complex problems, however, this technique frequently does not provide the precise and complete set of monomials we are looking for. The use of discrimination in the process of nondimensionalization, the detailed application of which is explained in this paper, always leads to a minimum set of parameters, which, separately, determine the solution of the problems. In addition, the physical meaning and order of magnitude of these discriminated monomials are also provided by the discrimination. The technique is applied to the coupled problem of natural convection between horizontal plates heated from below, containing an anisotropic porous medium.

The dimensional character of permeability: Dimensionless groups that govern Darcy's flow in anisotropic porous media

“…In these two problems, it was also verified that the group $$\frac{{{{\rm{k}}_{\rm{x}}}}}{{{{\rm{k}}_{\rm{y}}}}}\frac{{{\rm{l}}_{\rm{y}}^{{\rm{*}}2}}}{{{\rm{l}}_{\rm{x}}^{{\rm{*}}2}}}$$ behaves as an independent dimensionless monomial. In addition, Alhama et al 29 . explained that the groups deduced in their work cannot be obtained with the classical non‐dimensionalization technique, while Cánovas et al 30 .…”

“…In this work, they suggested better references for benchmarking in order to obtain patterns that cover the whole domain, and not just a small region. Alhama et al 29 . and Cánovas et al.…”

“…In this work, they suggested better references for benchmarking in order to obtain patterns that cover the whole domain, and not just a small region. Alhama et al 29 and Cánovas et al repeated the procedure for considering the length of the characteristic cell in the 2D anisotropic Bènard problem. In these two problems, it was also verified that the group…”

“…x behaves as an independent dimensionless monomial. In addition, Alhama et al 29 explained that the groups deduced in their work cannot be obtained with the classical non-dimensionalization technique, while Cánovas et al 30 approached the problem introducing a subdomain instead of the whole scenario, showing a deep understanding of the phenomenon. Finally, the anisotropic Yusa problem, in which, again,…”

Sizing of reactors by charts of Damköhler's number for solutions of dimensionless design equations

“…The latter discriminates the dimensions of space and the components of a variable, creating a greater number of variables and, by the Pi theorem, then a less quantity of dimensionless numbers is obtained. This approach has been used in free convection [9] and mixed convection [10] processes. An alternative way of obtaining dimensionless numbers is by non-dimensionalization of the differential equations that govern the phenomenon, using reference quantities that allow defining dimensionless variables [7].…”