Fragmentation energy

Abstract

Motivated by a problem arising in the mining industry, we estimate the energy ε(η) that is needed to reduce a unit mass to fragments of size at most η in a fragmentation process, when η→0. We assume that the energy used in the instantaneous dislocation of a block of size s into a set of fragments (s1,s2,…) is sβφ(s1/s,s2/s,…), where φ is some cost function and β a positive parameter. Roughly, our main result shows that if α>0 is the Malthusian parameter of an underlying Crump-Mode-Jagers branching process (with α = 1 when the fragmentation is mass-conservative), then there exists a c∈(0,∞) such that ε(η)∼cηβ-α when β<α. We also obtain a limit theorem for the empirical distribution of fragments of size less than η that result from the process. In the discrete setting, the approach relies on results of Nerman for general branching processes; the continuous approach follows by considering discrete skeletons. In the continuous setting, we also provide a direct approach that circumvents restrictions induced by the discretization.

Energy efficiency of consecutive fragmentation processes

“…Proof. The proof is an extension of the arguments given in [6] corresponding to the case 'η = 1 + ' or η = η 0 and which we repeat here for convenience. By the compensation formula for the Poisson point process ( (u), k(u)) associated with the first fragmentation process X, we obtain, for η 0 ∈…”

“…We observe that homogeneous fragmentation processes are self-similar fragmentation processes with zero index of self-similarity (see [5,Chapter 2]). Since self-similar fragmentation processes with different indices are related by a family of random time changes (depending on the fragments), there is no loss of generality in working here in the homogeneous case as the quantities we study are only size dependent (see also [6]).…”

“…= ∞ n=1 s β n − 1, which corresponds to the law of Charles, Walker, and Bond [10]. Within this mathematical framework, the asymptotic behavior of the energy required by a single fragmentation process to reduce all fragments to sizes less than η was studied in [6]. It was shown that the mean energy behaves as 1/η α−β when η → 0, where α denotes the Malthusian exponent of the fragmentation process and α > β in physically reasonable cases.…”

“…We will formulate problem (A) in mathematical terms, adopting the same mean energy point of view as in [6]. First, we will explicitly compute the objective function, which we express in terms of the Lévy and renewal measures associated with the 'tagged fragment' of each of the two fragmentation processes (see [4]).…”

On the Information Content of Coarse Data with Respect to the Particle Size Distribution of Complex Granular Media: Rationale Approach and Testing

“…Fragmentation of particles together with other coupled processes, suggest that the use of energy and its storage takes place in the form of "information" or disorder in the particle sizes. There are two constrains: the available energy for fragmentation is limited and also the energy needed to fragment a particle has a power law dependence of the size of the particle [4]. The maximum entropy principle [5] states that, under certain rules of optimality and randomness, the system thus would reach the maximum level of disorder conditional to the constrains imposed on the process.…”