REE11-1: Mathematical modelling of membrane filtration of water

Background

To meet the increasing demand for clean water, it is imperative to improve techniques and develop new strategies for the desalination of water and removal of heavy metals.

Membrane filtration uses a pressure difference applied across a semi-permeable membrane to filter particles via a combination of size-exclusion and charge-exclusion from the passing water. The use of magnetic separation with membrane filtration is a popular new technology for the removal of heavy metals from water, promising a more selective and efficient method of particle removal.

In magnetic separation, a wire mesh is used to generate high magnetic field gradients to capture particles. However, the smaller the particle size, the weaker the magnetic force they experience and thus the harder it is to control and trap them. If the particles are too small, the magnetic attractive forces experienced in a field gradient will not be large enough to overcome the random, Brownian, motion of the particles, and so, in principle, separation should not occur.

Magnetic separation experiments have yielded results that are better than expected and enable the collection of nano-sized particles. It is believed that particle aggregation plays a decisive role in the unexpected success of magnetic separation for these small particles.

To better understand this unexpected behaviour, researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) have developed mathematical models that describe the behaviour of magnetisable particle systems to provide insight and quantitative predictions about the system behaviour, which can crucially lead to the optimisation of filtration devices that use magnetic separation.

Techniques and Challenges

Aggregation of paramagnetic particles: Upon the application of a magnetic field, magnetisable particles will attract one another to form aggregates. The most favourable configuration of particles is a long chain, since in this configuration their magnetic moments, or dipoles, may be aligned with the external magnetic field (see Figure 1). This new models focus on this configuration.

Simple models for the behaviour of isolated magnetic particles break down as the density of particles becomes large enough for the particles to interact. The new model captures both the effect of an external magnetic field on a particle’s behaviour and the interaction of each particle with its neighbouring particles. The model also incorporates the key effects that contribute to a particle’s motion, including hydrodynamic interactions and viscous effects of the fluid.

Whilst these new models are complicated in their full form, they may be made more tractable by making various simplifying assumptions, for example, by assuming that only a particle’s nearest neighbours affect its behaviour.

The behaviour of magnetic chains and fibres: As the number of particles that constitute an aggregate chain increases, the system behaviour mimics that of a continuous fibre. The researchers exploited this feature to develop reduced models that describe simply the curvature of a continuous chain of particles rather than the individual particle behaviour.

The researchers further exploited the analogy between a chain of magnetic particles and a magnetic fibre by examining the behaviour of magnetite-seeded polyvinylsiloxane fibres placed in water when subjected to an external magnetic field (see Figure 2). As a magnet is moved closer to a clamped fibre, the fibre buckles at a critical distance. The researchers derived a new model that predicts this buckling threshold and provides insight into the key parameters that influence buckling.

Results

Aggregation of paramagnetic particles: The resulting aggregate model can help to provide information about how, for example, a spatially-varying applied magnetic field will improve the particle separation and capture in filtration.

The behaviour of magnetic chains and fibres: This model provides insight into how the paramagnetic particles can offer a resistance to bending when placed in a magnetic field. An experimental test of these predictions is expected shortly.

The Future

The mathematical models developed in this project provide a solid fundamental basis on which subsequent modelling and analysis of the behaviour of magnetic particle aggregates may be developed.

The results of such theories provide crucial insight in guiding the design and operation of new magnetic separation technologies, leading to faster, more efficient methods for the production of clean water.