A modelling approach for derivation of the breakage functions

Abstract

Theoretical analysis of the particle fragmentation process requires a proper choice of the breakage functions. In this work we construct the breakage functions on a basis of numerical modelling. Four types of the breakage functions are obtained. We demonstrate that if the average number of fragments at a single breakage event is larger than two, then the breakage kernel is asymmetric. Also a possibility of double-peaked kernels is shown.

Introduction

Particle fragmentation occurs in many natural phenomena and technological processes. Typical examples of such systems include a breakage of liquid droplets in aerosols or emulsions subject to agitation, and milling of solid particles in powder production. The dynamics of the size distribution of particles is described by the population balance equation (Ramkrishna, 2000). This is an integro-differential equation for the distribution function. The validity of the equation has been proven in a variety of experiments and applications (Ramkrishna, 2000, Redner, 1990).

Let the population of particles be described by the number density distribution function $f(x,t)$, which denotes the number of particles of size from x to $x+\mathit{dx}$ per unit volume. Then, particle fragmentation is modelled by the following population balance equation (PBE) (Ramkrishna, 2000, Kostoglou, 2007):

$$\frac{\partial f(x,t)}{\partial t}={\int }_x^{\infty }\eta (y)P(x,y)b(y)f(y,t)\mathit{dy}-b(x)f(x,t),$$

where x is a volume (mass) of a particle, b(x) is the breakage rate, $P(x,y)$ (the breakage kernel) is the size distribution of fragments appearing after a breakage of a particle of size y, $\eta (y)$ is the mean number of fragments from a particle of size y. The integral of $f(x,t)$ on the whole range of the particle sizes, $N(t)={\int }_0^{\infty }f(x,t)\mathit{dx}$, gives the total number of particles per unit volume at a given time. The aim of our work is to present a numerical approach for obtaining the breakage functions $P(x,y)$ and $\eta (x)$.

Functions $P(x,y)$ and $\eta (x)$ satisfy some special conditions. The most important of them are (Ramkrishna, 2000)

$$P(x,y)=0\mathrm{for}x>y,$$

$${\int }_0^{y}P(x,y)\mathit{dx}=1,$$

$$\eta (y){\int }_0^y\mathit{xP}(x,y)\mathit{dx}=y.$$

The last equation is the mass conservation condition so that the total mass of fragments from a breaking particle is equal to the mass of that particle.

A choice of specific forms for the breakage functions is determined by a type of a process. Different examples of the breakage functions are derived from the analysis of underlying physical mechanism (see e.g.Coulaloglou and Tavlarides, 1977, Lehr et al., 2002, Wang et al., 2003). These functions are usually complicated, and they are valid for a particular process and in narrow region of the parameters. Another motivation for a choice of the breakage functions is a possibility to solve exactly the PBE (Reid, 1965, Ziff and McGrady, 1985, Peterson, 1986, Ziff, 1991). Though such functions can be different from the actual ones, they can be useful for testing of numerical techniques for the PBE. In many cases, the form of the breakage functions is selected on a basis of different assumptions and heuristic arguments (Hesketh et al., 1991, Hill and Ng, 1996, Diemer and Olson, 2002a, Diemer and Olson, 2002b, Kostoglou et al., 1997). We use the latter approach, applying numerical modelling.