Analysis of Dispersed - Phase Fresh Perspective Systems :

single-particle process Consider a single particle of volume V ( t ) at time t which undergoes random, discontinuous “erosion” to smaller sizes in accord with the probabilistic law: Pr( V ( t + d t ) I u I V ( f ) = u’ 1 = B ( u I u’ )dt We let F( u, t ) be the cumulative distribution function for the size of the single particle undergoing random erosion. Suppose that the cumulative probability distribution of the particle volume at time t + dt is desired from consideration of its size at time t . If the particle has to have at time t+dt volume less than or equal to u, it can arise from either of two mutually exclusive possibilities. One is that the particle had volume I u at time f, and the other is that it had volume 2 u at time t and underwent random erosion during the time interval t to t + dt to a size I v. Mathematically, this becomes F ( u , t + d t ) = F ( u , t ) + B(ulu’)aF(~’, t)dt s : which, on transposing and letting dt tend to 0, clearly leads to Eq. 6. Simulation of a breakup process The above single-particle process can be simulated in a very simple manner. In view of the equivalence of the single-particle erosion process to a population of particles undergoing breakage alone, the simulation procedure is applicable to the latter with a minimum of the bookkeeping that is characteristic of simulation strategies. Suppose that the size of the particle is known at any instant t . Following Shah et al. (1977), we may define a quiescence interval Tas the period in which the particle does not undergo random erosion (but that it certainly does during the instant following T). Using the methods of the cited o p , . . m ixio-’ 1x104 1 x i o 5 1 x l 0“