FLOCCULATION: A GENERAL AGGREGATION-FRAGMENTATION FRAMEWORK Alex Sheremet1, Cihan Sahin2, Andrew J. Manning3;1 We present a brief review of the key assumptions at the basis of the kinetic description of aggregation-fragmentation processes. The goal is to provide a consistent foundation for using the aggregation-fragmentation equation as a frame- work for modeling flocculation. The derivation presented here is illustrated using a simple form of the equation that describes the dynamics of a 3-class distribution subject to elementary collisions of order 2, with arbitrary interaction coefficients. Keywords: flocculation; kinetic equations; aggregation-fragmentation INTRODUCTION The primary mineral component of cohesive muds are clay particles with a plate-like structure, and a diameter of less than 2 µm. Cohesion arises through a combination of electrostatic charging of pri- mary particles and various biogenic polymer molecules which adhere to individual particles, and causes the aggregation of primary particles into larger agglomerates called flocs which behave as an identifiable sedimentary unit (e.g., Winterwerp and Van Kesteren, 2004; Mehta, 2013). Under typical estuarine condi- tions, most suspended particulate matter occurs in the form of flocs (Kranck and Milligan, 1992). In given environmental conditions, observations suggest that flocs have a wide distribution of sizes that can span over four orders of magnitude, from individual clay particles to stringer-type floc structures several cm in length. An individual floc might comprise up to of order 106 primary particles. As a floc grows in size, its effective density (i.e., bulk density minus water density) generally decreases (e.g., Tambo and Watanabe, 1979; Klimpel and Hogg, 1986; Droppo et al., 2000). Because of the size dependency of the Stokes Law (Gregory, 1978; Mehta and Lott, 1987; Stolzenbach and Elimelich, 1994; Dyer and Manning , 1999; Win- terwerp and Van Kesteren, 2004) the settling velocity of flocculated sediments is significantly greater than the constituent particles and can range over four orders of magnitude (0.01 mm/s up to cm/s; Lick, 1994). Flocculation, i.e., the formation of flocs, has long been recognized as a dynamic process (e.g., Win- terwerp and Van Kesteren, 2004) driven by environmental conditions, primarily through flow-sediment interaction (e.g., Tsai et al., 1987; Burban, 1987; Burban et al., 1990; Manning, 2004). The physical and biochemical factors affecting flocculation include turbulence intensity, differential settling, sediment availability, primary-particle composition, salinity, and content of organic matter and polymers, and others (Burban et al., 1989; Lick et al., 1993; Manning and Dyer, 1999; Mietta et al., 2009; Manning et al., 2013; Parsons et al., 2016; Yang et al., 2016). Flocculation has a significant effect on deposition, erosion and consolidation rates, which makes it a primary mechanisms controlling sediment transport in fine-grained sedimentary environments. Because the characteristic floc size and settling velocity are considered key parameters for cohesive sediment transport models (Mehta and Lott, 1987), the interest in modeling the flocculation process has been growing. All flocculation models are based on some version of the aggregation-fragmentation equation (AFE e.g., Drake, 1972; Ball and Carr, 1990), which expands the coagulation equation originally proposed by Smoluchovsky (1917). While the basic flocculation principles are generally agreed upon, the flocculation debate and contributions (e.g., McAnally and Mehta, 2000; Maggi et al., 2007; Lee et al., 2011; Verney et al., 2011) seem to be focused on the role and form of particular terms, such as turbulence-induced collision and breakup processes (Maggi et al., 2007), or shear aggregation (Verney et al., 2011). To reduce computational costs, some models focus on simplified distributions (e.g., bimodal, Lee et al., 2011). With a few exceptions (e.g., Lee et al., 2011), most models assume homogeneous and isotropic suspensions (so-called 0-D models), neglect settling effects and boundary sediment sources, and neglect all size-class interactions that involve more than 3 classes. Remarkably, most authors caution about the effort to integrate the AFE in its basic form, and stress the need for numerically efficient simplifications (e.g., bimodal distribution of Lee et al., 2011). Simplifications, however, should preserve the essential behavior of the full AFE. Even the simple notion of “characteristic 1Engineering School for Sustainable Infrastructure and Environment, 365 Weil Hall, University of Florida, Gainesville FL 32611, USA 2Department of Civil Engineering, Yildiz Technical University, Esenler, Istanbul 34210, Turkey 3HR Wallingford Ltd, Coasts & Estuaries Group, Wallingford, Oxfordshire, OX10 8BA, UK 2 COASTALENGINEERING 2016 floc size” implies that “essential features” of the dynamics of the floc-size distribution can be represented by a low-dimensional model (a few size classes). Is this true? Under what conditions? What are these “essential features”? For example, one would expect low-dimensional models to work for equilibrium distributions. Do equilibrium distributions exist? What kind of equilibrium can be achieved (local or global)? While the effort to derive and validate coefficients for various AFE terms is important, working with truncations of the equation (e.g., 0-D models) has no solid justification beyond personal intuition, unless the general behavior of the system is understood. Comparisons with observations do not provide a compelling validation because: 1) field observations contain inherent forcing variability on multiple scales, and relaxation times associated with the floc-size distributions are not known; 2) it is almost impossible to control the forcing (most of the existing studies are based on laboratory experiments under controlled forcing conditions); and 3) current observations capabilities to resolve floc sizes are quite limited. The AFE has been studied quite extensively in general (e.g., Drake, 1972; Ball and Carr, 1990; Con- naughton and Krapivsky, 2010, and many others), but so far little is known about its behavior as a floccula- tion model. The goal of this paper is to derive the full AFE framework from the perspective of flocculation, and prove some basic properties, such as mass conservation and the existence of local-balance equilibrium for very simple cases. We hope that this will provide a general framework for the study of flocculation. While exploring the dynamics of the full AFE is a significant effort, we contend that it is an important step in understanding flocculation, defining the “essential features” of its dynamics, and providing a compre- hensive validation framework for future numerically-efficient formulations. THE AGGREGATION-FRAGMENTATION FRAMEWORK Here we formalize the concepts of floc and flocculation, and set the foundation for the AFE framework. The model proposed below refers to the dynamics of aggregation/disaggregation of flocs and largely ignores primary particles. Instead of flocs, we will use below the term “particle”. Particles A particle is a sedimentary unit that has long life time T compared to the characteristic time τ of an aggregation/fragmentation process, i.e., T τ. The hypothesis of separation of time scales is fundamental to the kinetic description of the aggregation-fragmentation process. It implies that 1) particles have to be somewhat resilient to forces in the fluid, and 2) that primary particles should not be included in this definition. Indeed, assume that the concentration of primary particles is high enough that particles float in a “soup” of primary particles, and exchange continuously primary particles with the soup. If primary particles are allowed as components of the input/output state, the identity of a particle changes through absorbing/shedding of a single primary particle. Because the such exchange can be regarded as continuous, the lifetime of a particle shrinks to the same order of magnitude as the duration of collisions, violating the time-scale separation. If the characteristic mass of a primary particle ism ¯ , the mass m of a particle satisfies the relation m m¯ . In other words, particles contain a large number, say O(105), of primary particles (e.g., microflocs, D < 160 µm are resistant to break-up by turbulent shear and are generally considered to be the building blocks of larger flocs; Eisma, 1986; Manning, 2001). For numerical modeling purposes particles are divided into a discrete number of classes based on their mass. Denote by ∆m m¯ the characteristic mass of smallest particles. The set of particles with mass m¯ jm − k∆mj ≤ ∆m (1) is called a k-class. We will approximate the mass of a k-particle as mk. We will refer to particles from class k as k-particles. If p and q are two classes such that mp < mq, q and p will be called upper, and lower class, respectively. In some volume V of fluid, let Mk be the total mass of k-particles, and Nk = Mk=mk the number of P k-particles in V. The total number of particles in V is N = Nk, and the total mass of sediment in V is P P k M = k Mk = k Nkmk. The density of k-particles in V can be defined in the standard way as Mk(t) ρk(r; t) = lim = nk(r; t)mk; (2) Vr!0 Vr where V is a volume containing the point r = (x; y; z), and nk(r; t) is the number of particles per unit volume. In Eq. 2 the limit is taken in a macroscopic sense (i.e., Vr contains a very large number of particles). In the COASTALENGINEERING 2016 3 absence of sources, the total mass M of the sediment, and the total number of elementary particles in the system M=m0 is are conserved. The total number of particles N is not conserved. Collisions A collision is a type of aggregation/fragmentation event that occurs on a time scale τ and redistributes an input set of N pi-particles, i = 1; 2; ··· ; N, into an output set of M q j-particles j = 1; ··· ; M, represented by the transformation PN = fp1; p2; ··· ; pN g = fpigN ! QM = fq1; q2; ··· ; qMg = fqigM: (3) This definition identifies types of collisions, and not individual, physical aggregation or fragmentation events.