A multiple-scales framework for branched channel filters

TL;DR

This work addresses the filtration challenge of microfibres by proposing ricochet separation through a branched-channel device to divert water while retaining fibres. It develops a high-$Re$ two-dimensional flow model and uses multi-scale analysis to derive an effective leakage boundary condition, yielding an explicit outer flow solution and a composite flow that accounts for the discrete sinks. The authors validate the asymptotic predictions against full numerical simulations and couple the flow to a simple wall-bounce particle model to quantify filtration efficiency as a function of design and particle inertia, revealing trade-offs between throughput and fibre capture. The approach provides a fast, analytical framework for designing branched-channel filters and can be extended to more complex geometries and particle dynamics for practical washing-machine applications.

Abstract

Fibres shed from our clothes during a washing machine cycle constitute around 35% of the primary microplastics in our oceans. Current conventional dead-end washing machine filters clog relatively quickly and require frequent cleaning. We consider a new concept, ricochet separation, inspired by the feeding process of manta rays, to reduce the cleaning frequency. In such a device, some fluid is diverted through branched channels whilst particles ricochet off the wall structure, forcing them back into the main flow and then into the dead-end filter. In this paper, we consider a simple branched-channel filter structure beneath a high-Reynolds-number laminar flow, in the case where the branch separation is much larger than the thickness of the viscous boundary layer. We use multiple-scales techniques to derive an effective leakage boundary condition, which smooths out localised effects in the flow velocity and pressure that arise due to the discrete branched channels, and then use this boundary condition to explicitly determine the flow away from the boundary. We find that our explicit solution compares well with an analogous numerical solution containing a discrete set of branched channels. We further consider the behaviour of individual spherical particles in the device, with their trajectories determined via a simple force balance model with a wall-bounce condition. We explore the dependence of the fraction of particles that flow into the branched channels on the particle's Stokes number. The resulting combined model is able to predict the relationship between the efficiency of a ricochet filter device and the design and operating parameters, avoiding the need to conduct extensive numerically challenging simulations.

Figures (29)

• Figure 1: Layout schematic of a branched channel filter preceding a dead-end filter. Microfibre particles, trajectories and foulant are indicated in red and water flow is indicated in blue. The operating directions are indicated by black arrows.
• Figure 2: $2$-dimensional repeatable T-junction domain, $\hat{\Omega}$, given by a main channel compartment with $N$ perpendicular branched channels on the bottom wall. Inlet and outlets are indicated by dashed black lines, the T-junction spacing is indicated by dashed red lines and boundary walls are denoted by $\partial \hat{\Omega}_w$, in solid black lines. The domain design parameters are indicated as $h_1$, $h_2$, $L$, $L_1$, $L_2$ and $N$.
• Figure 3: Reduced dimensionless geometry, with point-sinks replacing each branched channel. Each point-sink has coordinates $(x_i, 0)$, where $x_i = (i-1/2) \epsilon$ for $i = 1, 2, \cdots, N$, and has strength $2 Q_i^{\text{branch}}$, where $Q_i^{\text{branch}}$ is the flux through a single channel, as in equation (\ref{['eqn:DimlessFluxStrength']}), batchelor2000introduction. The outer problem views the point-sinks as an effective boundary condition, capturing the overall average behaviour. Both boundary layers are indicated in the regime $\epsilon \gg 1/\sqrt{\Rey}$.
• Figure 4: Outer flow domain with boundary conditions, including the effective boundary condition, $v^{(o)} (x, 0) = - v^* (x)$.
• Figure 5: Conformal map of the semi-infinite half strip inner region to the positive imaginary half plane via the conformal map $\zeta = \sin{(\pi Z)}$.