Filters. The Number of Channels That Can Clog in a Network

Abstract

Abstract.We model filters as two-dimensional networks of channels. As a suspension (fluid with particles) flows through the filter, particles clog channels. We assume that there is no flow through clogged channels. In this paper, we compute a sharp upper bound on the number of channels that can clog before fluid can no longer flow through the filter. Fluid suspensions (or suspensions, for short) are fluids with small solid particles in them. According to their size and properties, these particles are called fines or colloids. As a suspension flows through a permeable porous material, some fines are trapped within the material. In fact, the function of the filters we consider in this paper is to clean suspensions by capturing most particles bigger than a certain size.The removal of particles from fluid suspensions is of importance in a wide range of industrial and technological applications such as waste water treatment [18] and other filtration processes [4,31]. Our studies are motivated by the filters used in the process known as deep bed filtration. As a suspension flows through a filter composed of granular or fibrous materials, fines or colloidal particles penetrate the filter and deposit at various depths [32]. As a result, the fluid suspension is cleaner when it exits the filter (i.e., it exits the filter with many fewer solid particles than it originally had when it entered the filter).Theoretical models to study transport in porous media can be classified as either macro-scale [5,15,22,23,24,26,32] or pore-scale [9,19,28] models. Within the latter group, the class of network models, in which the pore space is modeled as a network of channels, is very popular. Network models provide flexibility in modeling different geometries of pore space while keeping the computational cost low. Our work belongs to this class of models.Network models to study transport in porous media were introduced by Fatt in 1956 [10,11,12]. Donaldson, Baker, and Carrol in 1977 [8] were the first to use networks to study particle transport within porous media. The clogging of particles has been studied in networks with different geometries including bundles of parallel tubes [8], square networks [14,16,21], triangular networks [3,25], cubic networks

Clogging of Multigraphs as Toy Models of Filters

“…We are able to consider both two-dimensional and three-dimensional networks. When the networks are two-dimensional, the results in [18,19] are recovered. Our results for three-dimensional networks are new.…”

“…This two-dimensional result was previously obtained in [18,19] In an attempt to shed light on the answers to these important questions, we introduced a network model and obtained a bound on the number of channels that clog (Theorem 5.2). This bound is given in terms of the properties of the network and is sharp (in the sense explained in section 7).…”

“…In section 9, we consider a three-dimensional example. In section 10, we obtain an alternative description of our bound valid for two-dimensional networks and we make the connection with the work in [18,19]. A final discussion is given in section 11.…”

“…The problem considered in this paper was previously studied in [18,19] for twodimensional networks. In those works, the authors use graph theory techniques that make sense but are valid only for planar graphs.…”