Clogging time of a filter.

We study the time until a filter becomes clogged due to the trapping of suspended particles as they pass through a porous medium. This trapping progressively impedes and eventually stops the flow of the carrier fluid. We develop a simple description for the pore geometry and the motion of the suspended particles which, together with extreme-value statistics, predicts that the distribution of times until a filter clogs has a power-law long-time tail, with an infinite mean clogging time. These results and its consequences are in accord with simulations on a square lattice porous network. PACS numbers: 47.55.Kf, 47.55.Mh, 64.60.Ht, 83.70.Hq In this Letter, we investigate the time for a filter to clog. In a typical filtration process, a dirty fluid is “cleaned” by passing it through a porous medium to remove the suspended particles. The medium enhances filtering efficiency by increasing both the available filter surface area for trapping suspended particles as well as the exposure time of the suspension to the active surfaces. Such a mechanism is the basis of water purification, air filtration, and many other separation processes [1,2]. As suspended particles become trapped, the fluid permeability of the medium gradually decreases, and eventually the filter clogs. Determining the time dependence of this clogging is basic to predicting when a filter is no longer useful, either because of reduced throughput or reduced filtration efficiency, and should be discarded. While clogging has been extensively investigated by engineers, much of the previous literature on this phenomenon is either empirical or incorporates so many microscopic details that it is difficult to draw general conclusions about clogging kinetics [3]. In contrast, we develop a minimalist model that provides an intuitive understanding for clogging which should apply to simple and easily prepared porous media, such as unconsolidated bead packs [4]. Our model is based on approximating the clogging of a porous medium by the clogging of a single parallel array of pores which are blocked in decreasing size order (Fig. 1). The use of a single-layer system is based on previous studies of filtration which showed that clogging preferentially occurs in the upstream end of the network when particles and pores are of comparable size [5,6]. The single-layer system represents the ultimate limit of this gradient-controlled process. The approximation of size-ordered blocking is based on the commonly used picture that a particle has a probability proportional to the relative flux to enter a given open pore from among the outgoing pores at a junction [2,5,7]. For Poiseuille flow, this flux is proportional to r 4 , where r is the pore radius. Size-ordered blocking arises by replacing the exponent 4 in this flow-induced entrance probability by `. This approximation is relatively accurate for a broad distribution of pore radii. The size ordering also provides a direct correspondence between the radius of the currently blocked pore and the time until it is blocked. By using extreme value statistics [8] to compute the radius distribution of the last few open pores, together with the connection between the radius of the currently blocked pore and the network permeability, we determine the distribution of clogging times. The predictions of this idealized modeling agree well with detailed simulations of the flow field evolution and the attendant clogging kinetics on a square lattice porous network. In our network simulations, each unit length bond corresponds to a pore and sites represent pore junctions. We assume Poiseuille flow, in which the fluid flux through a bond of radius ri is proportional to 2r 4 i =p, where =p is