A homogenised model for the motion of evaporating fronts in porous media

Abstract

Evaporation within porous media is both a multiscale and interface-driven process, since the phase change at the evaporating interfaces within the pores generates a vapour flow and depends on the transport of vapour through the porous medium. While homogenised models of flow and chemical transport in porous media allow multiscale processes to be modelled efficiently, it is not clear how the multiscale effects impact the interface conditions required for these homogenised models. In this paper, we derive a homogenised model, including effective interface conditions, for the motion of an evaporation front through a porous medium, using a combined homogenisation and boundary layer analysis. This analysis extends previous work for a purely diffusive problem to include both gas flow and the advective–diffusive transport of material. We investigate the effect that different microscale models describing the chemistry of the evaporation have on the homogenised interface conditions. In particular, we identify a new effective parameter, $\mathcal{L}$ , the average microscale interface length, which modifies the effective evaporation rate in the homogenised model. Like the effective diffusivity and permeability of a porous medium, $\mathcal{L}$ may be found by solving a periodic cell problem on the microscale. We also show that the different microscale models of the interface chemistry result in fundamentally different fine-scale behaviour at, and near, the interface.

Mathematical modelling of impurity deposition during evaporation of dirty liquid in a porous material

“…In the pore space occupied by the vapour-gas mixture (behind the evaporating front, i.e. y < h(x, t)), we expect the Reynolds number to be small (Luckins et al 2023) and so we assume that the mixture satisfies the Stokes equations…”

“…(2.5) where D d is the diffusivity of suspended dirt in liquid. As discussed in Luckins et al (2023), the assumption that the liquid does not flow means that capillary effects are neglected from the model.…”

“…Finally, we must also incorporate a condition that describes the chemistry governing the evaporation. In Luckins et al (2023) the effect of different chemistry conditions were considered, and these were shown to affect the form of macroscale boundary conditions derived through a homogenisation analysis. For simplicity, we assume that the liquid and vapour are in chemical equilibrium at the evaporating interface.…”

“…In this paper we systematically derive a homogenised model for the coupled processes of evaporation, transport of liquid vapour, diffusion and deposition of dirt in a drying porous material, starting from a pore-scale model for these processes. This analysis is based on previous work (Luckins et al 2023) for evaporation of a pure liquid in a porous material, extended to incorporate dirt transport and deposition. One benefit of the homogenisation approach is that the pore-scale behaviour is included in the homogenised model through averaged terms.…”

“…This ensures that the model conserves mass of all species, and also results in a different diffusive term in the homogenised equations compared with the model posed by Ji & Sanaei (2023). For simplicity, as in both Luckins et al (2023) and Ji & Sanaei (2023), we assume that capillary flows are negligible and a sharp evaporating interface moves into the porous material. In practice, such systems would be valid when viscous or gravitational forces dominate over surface tension, for instance, if the solid is hydrophobic (Shokri, Lehmann & Or 2008) or the pores are sufficiently large relative to the capillary length scale (Lehmann et al 2008).…”