Discrete models, continuum models and scale transitions for drying porous media

Drying is of great practical importance but difficult to understand and describe, owed to transport phenomena in three different, complexly intermingled phases (solid skeleton, liquid-filled pores, emptying and emptied pores). Quickly solvable continuum models are desirable, but also models with high spatial resolution, which necessarily are discrete. Continuum models may be algebraic or based on differential equations, and they are indeed not unsuccessful. For example, the simple and empirical characteristic drying curve model often achieves to capture the global drying kinetics of porous bodies, in some cases even when solid matrix diffusion is the real mechanism of moisture transport [1]. The so-called reaction engineering approach [2] decreases the liquid (typically water) vapor pressure at the free surface of the convectively drying material in kind of a sorption isotherm, even if the material is non-hygroscopic. Most sophisticated in conventional theory, the homogenized one-equation continuum model [3] still expresses the water transport in the drying medium in terms of diffusion. However, and contrary to a simple diffusion model, the effective diffusivity is interconnected with and derivable from more fundamental properties, such as absolute and relative permeability, effective gas phase diffusivity, and capillary pressure curve [4]. 

Especially when thoroughly derived by homogenization, versions of the sophisticated one-equation continuum model raise expectations of accuracy and reliability. In fact, with many model parameters and some adjustment, coarse data from common drying experiments can usually be reproduced. The real challenge for the model arises when it is confronted with comprehensive synthetic data for fictitious drying porous media, which stem from discrete modeling and are resolved down to every single pore. Recent results from this kind of test are shown in Figure 1 [5]. The shown non-equilibrium functions, connect local relative humidity (i.e., water vapor pressure) with the local saturation in the material and on its surface. Albeit for completely non-hygroscopic material, they look, again, like sorption isotherms. They don’t change too much if volumeless pores are assumed (throat-node model, TNM), but depend, at least inside the material, on the level of global (here, network) saturation. And, they are necessary to achieve accordance between pore network and continuum model – kind of complex and hardly derivable closures for the underlying scale transition.  Correspondingly, effective liquid and gas phase diffusivities are also strong and non-unique functions of local saturation. 

So, negatively seen, pore scale resolution may destroy our belief in models. Positively seen, it breaks path to new and superior continuum models: Models that have simpler, unique and easier to identify parameters, and don’t need non-equilibrium functions to work, or at least not too strong functions of this kind. There are indications that two-equation (heterogeneous) continuum models, which distinguish between liquid and gas (wet air) phase in the material may go in this direction [6]. In those, an interrelation between volumetric liquid-gas interface area and local saturation seems to play an important role, in analogy to modern descriptions of capillarity during drainage or imbibition [7]. And, distinction between mobile (spatially connected) and immobile (disconnected) liquid phase in three-equation continuum models might further enforce the positive effects. 

In general, we have started realizing how brute conventional homogenization in drying theory has been. And, we have also started aiming at less destructive scale transitions that preserve and transfer more microscale information to new and better continuum models. With this as a big goal, it should be clear that coarsening always has a price. In drying, there are effects which will certainly be difficult, perhaps even impossible to reflect at the macroscale. For example, and to only name few: The influence of secondary capillary structures (films in the corners of capillaries with edged cross section, rings around the contact points of particles) [8]; local recondensation in presence of strong temperature fields in the material or in case of superheated steam drying – a process that gains great significance in terms of energy savings and CO2 footprint reduction [9]; local capillary instabilities that result in sudden phase redistribution with the formation of ganglia or bubbles [10]; multiphase thin media which offend by far conventional scale separation rules, but are crucial for modern electrochemical devices (batteries, electrolysers) [11]. However, we can try to keep the price of scale transition as low as possible. And, we can hope for a big reward, namely better structure-property relations in the sense of predictability of the influence that structural features of porous materials are expected to have on their drying behavior. Fortunately, we need not start from scratch to this respect, because even simple models reveal the property which is most influential in keeping the surface of drying porous media wet, and their drying rate high. Usually, this is not average porosity, and also not average pore diameter; but, the coefficient of variance of the pore size distribution [12]. 

Figure 1. Right bottom: Regular three-dimensional pore network model used to generate highly resolved synthetic drying data. Starting from full saturation, the network dries to air from the top, the bottom is sealed, the sides are periodic. Right top: Detail of used throat-pore model (TPM) with cylindrical throats and spherical pores, both normally distributed in size. Left: Non-equilibrium functions of relative humidity in dependence of local saturation within the dring material (top) as well as on its free surface to the air (bottom).