Analytical solutions of drying in porous media for gravity-stabilized fronts

Abstract

We develop a mathematical model for the drying of porous media in the presence of gravity. The model incorporates effects of corner flow through macroscopic liquid films that form in the cavities of pore walls, mass transfer by diffusion in the dry regions of the medium, external mass transfer over the surface, and the effect of gravity. We consider two different cases: when gravity opposes liquid flow in the corner films and leads to a stable percolation drying front, and when it acts in the opposite direction. In this part, we develop analytical results when the problem can be cast as an equivalent continuum and described as a one-dimensional (1D) problem. This is always the case when gravity acts against drying by opposing corner flow, or when it enhances drying by increasing corner film flow but it is sufficiently small. We obtain results for all relevant variables, including drying rates, extent of the macroscopic film region, and the demarkation of the two different regimes of constant rate period and falling rate period, respectively. The effects of dimensionless variables, such as the bond number, the capillary number, and the Sherwood number for external mass transfer are investigated. When gravity acts to enhance drying, a 1D solution is still possible if an appropriately defined Rayleigh number is above a critical threshold. We derive a linear stability analysis of a model problem under this condition that verifies front stability. Further analysis of this problem, when the Rayleigh number is below critical, requires a pore-network simulator which will be the focus of future work.

Pore Network Modeling of Drying Processes in Macroporous Materials: Effects of Gravity, Mass Boundary Layer and Pore Microstructure

“…Throat radius in the mass boundary layer, r t,S+ r t,S+ = √ λt t Effects of gravity, capillarity and mass boundary layer are expressed in terms of the three dimensionless numbers: A film-based capillary number Ca f (again not to be confused with Ca f defined in Yiotis et al (2012a) for a square capillary with sharp corners), that expresses the ratio of viscous forces to capillary forces in the films;…”

“…At later times, when the film tips have detached from the product surface, Eq. (15) remains valid, but now the Φ profile becomes continuous at S (Yiotis et al 2012a). …”

“…9-(right)], the capillary pressure gradients become larger in order to overcome the opposing effect of gravity and the film profiles are more complex. As described in the 1-D analysis (Yiotis et al 2012a), this effect eventually leads to shorter film regions and shorter CRP's, especially in the case of fast drying conditions (i.e., high Sh numbers). The latter case will be discussed in more detail below.…”

“…The case where gravity enhances drying can be further subdivided into two categories (Yiotis et al 2012a), depending on the relative strengths of gravity, viscous forces and mass transfer;…”

Structure of drying fronts in three-dimensional porous media

“…A large body of literature exists that has focused on various aspects of drying of porous media (see, for example, Refs. [1][2][3][4][5][6][7][8][9][10][11][12]). The majority of such studies provided evidence that the early stages of evaporation from porous media include a relatively high evaporation rate that is limited by the atmospheric conditionsthe so-called stage-1 evaporation -which is supplied by the capillarity-induced liquid flow that hydraulically connects the wet region to the evaporating surface.…”