Particle-based modeling of aggregation and fragmentation processes: Fractal-like aggregates
- Theoretical Physics/Complex Systems, ICBM, University of Oldenburg, 26129 Oldenburg, Germany
- Received 18 December 2008
- Revised 5 January 2011
- Accepted 6 January 2011
- Available online 13 January 2011
- Communicated by A. Pikovsky
Abstract
The incorporation of particle inertia into the usual mean field theory for particle aggregation and fragmentation in fluid flows is still an unsolved problem. We therefore suggest an alternative approach that is based on the dynamics of individual inertial particles and apply this to study steady state particle size distributions in a 3D synthetic turbulent flow. We show how a fractal-like structure, typical of aggregates in natural systems, can be incorporated in an approximate way into the aggregation and fragmentation model by introducing effective densities and radii. We apply this model to the special case of marine aggregates in coastal areas and investigate numerically the impact of three different modes of fragmentation: large-scale splitting, where fragments have similar sizes, erosion, where one of the fragments is much smaller than the other and uniform fragmentation, where all sizes of fragments occur with the same probability. We find that the steady state particle size distribution depends strongly on the mode of fragmentation. The resulting size distribution for large-scale fragmentation is exponential. As some aggregate distributions found in published measurements share this latter characteristic, this may indicate that large-scale fragmentation is the primary mode of fragmentation in these cases.
Research highlights
► We introduce a new, inertial particle-based model for aggregation and fragmentation. ► Using this model, we investigate numerically different modes of fragmentation. ► We find a strong dependence of the steady state size distribution on fragmentation.
Keywords
- Inertial particles;
- Aggregation;
- Fragmentation;
- Fractal aggregates
1. Introduction
In recent years there has been a great effort to investigate the advection of inertial particles in fluid flows [1], [2], [3], [4], [5] and [6]. Understanding the behavior of inertial particles like aggregates, dust or bubbles moving in incompressible flows plays an important role in such diverse fields as cloud physics [7], engineering [8], marine snow and sediment dynamics [9] and [10] as well as wastewater treatment [11]. The dynamics of individual inertial particles is dissipative. This leads to a behavior that is very different from passive tracers, for example to a preferential accumulation in certain regions in space, i.e. on attractors [12], [13] and [14]. Previous studies concentrated mainly on non-interacting particles, in spite of the fact that accumulation leads unavoidably to mutual interactions of different kinds.
It is well known that as a result of collisions between particles, aggregates can be formed that consist of a large number of primary particles. In many areas of science the formation of such aggregates and their break-up due to shear forces in the flow plays a very important role, e.g. in sedimentation of particles in oceans and lakes [15], chemical engineering systems such as solid–liquid separation [16] and [17], aggregation of marine aggregates [18] and flocculation of cells [19].
Most approaches of aggregation and fragmentation models are based on the pioneering work of Smoluchowski [20] and use usually a mean field approach with kinetic rate equations to model these processes (see e.g. Jackson [9]). However, for particles with inertia a field theory for the particle velocity has not yet been formulated. The existence of caustics, meaning that the dynamics of inertial particles would lead at some points to a multi-valued particle velocity field [21] and [22], has prevented such an approach so far. While attempts have been made to incorporate particle inertia in approximate ways in a mean field approach [23] and [24], no completely satisfying solution has been found yet.
Here we therefore choose a different, individual particle-based approach, where the dynamics of finite size particles are taken directly into account. The approach has recently been introduced in Ref. [25], and discussed in more detail with respect to different flows in [26]. In the present study we adopt this approach to study the long-term behavior of particle size distributions that develop from a balance between aggregation and fragmentation. In particular, we examine the influence of fragmentation and aggregate structure on these size distributions.
In most previous works the particles were considered to be spheres with a specific density. In many realistic cases, for example for marine aggregates, this is a crude approximation. The complex structure of particles can have a great influence on particle dynamics as well as aggregation and fragmentation processes. Both the actual motion of aggregates and the probabilities for aggregation and fragmentation are influenced by the structure of the particles. In the context of a mean field approach, a complex particle structure has been incorporated in the past in terms of a density modification for the particles, e.g. by Kranenburg [27] or Maggi et al. [28]. However, so far there have been very few attempts to treat this problem for inertial particles in a flow. Wilkinson et al. [29] used a model for fractal particles in an aggregation model for dust particles during planet formation. Our present work expands the consideration of spherical particles to model more realistic aggregates. We focus specifically on the problem of aggregation and fragmentation in systems where the aggregates can be described as having a fractal-like structure, as is for example the case for marine aggregates [27]. By this we mean that on average there exists a power-law relationship between the characteristic length and the mass of such aggregates. The exponent of the power law is called the fractal dimension. Such a characterization in terms of a fractal dimension leads to a modification of the radii and effective densities of the aggregates compared to a solid sphere of the same mass. Nevertheless, we still treat them as effectively spherical for the particle motion, allowing us to apply the Maxey–Riley equations of motion [30] with modified parameters.
In this work we choose a parameterization of our model for the case of a suspension of marine aggregates in the ocean. In this way we can study our modeling approach for a specific case, but we emphasize that our model is a general one that can in principle be used for a wide range of applications where aggregation and fragmentation of solid particles play a role. The concept of a fractal-like structure has been found to be a reasonable first approximation in many different applications, ranging from colloidal systems to the flocculation of cells [31]. A different application would require a different parameterization of the model, but the general approach introduced here would remain the same.
Since the fractal dimension of marine aggregates can vary greatly in natural systems [28], we examine its effect on the steady state particle size distributions in our model. We find that while the shape of the size distributions does not depend strongly on the fractal dimension, the average particle size and relaxation time towards the steady state depend strongly on this parameter.
Even though to a certain extent methods from dynamical systems theory can usefully be applied, we mention that the entire problem is much more complex than that of any usual dynamical system. While particles of a single size move on specific attractors, aggregation and fragmentation lead to repeated transitions from one attractor to another one, depending on the aggregate size. The skeleton of the new dynamics is therefore a superposition of the different attractors, with transient motion in between. The structure of the individual attractors and their superposition in turn influence the aggregation probabilities due to different local concentrations of particles. Fragmentation is also affected by the particle dynamics, because shear forces can be locally different in the flow. Therefore, break-up may depend on whether an attractor for a certain particle size lies in a region with high shear or not.
