Particle breakage processes single

Single-Particle Fracture The key issue in all breakage processes is the creation of a stress field inside the particle that is intense enough to cause breakage. The state of stress and the breakage reaction are affected by many parameters that can be grouped into both particle properties and loading conditions, as shown in Fig. 21-58. 

FIGURE 4.6.3 Comparison of simulation of a pure breakage process by the single particle technique of Ramkrishna et al. (1995) with numerical solution of the discretized form of Eq. (4.5.19). (Reproduced with permission of the American Institute of Chemical Engineers. Copyright 1995 AIChE. All rights reserved.)... 

Although the single particle process has been identified, we are faced with the inconvenient circumstance of having to increment the particle size by an amount whose distribution is itself the quantity to be calculated. Thus, the situation here is not as desirable as that in the breakage process considered earlier. However, let us proceed to consider the simulation of this process using the technique of Shah et al (1977). The cumulative distribution function for the quiescence interval T during which time no increment occurs in the particle is given by... 

Show that the single particle simulation of a breakage process can be extended to the case where particle growth occurs in accord with the function X(x). Elucidate the simulation strategy by calculating the quiescence interval distribution. (See Ramkrishna et al 1995 for application to a mass transfer problem in a stirred liquid-liquid contactor). 

There are examples in the literature of fitting parameters to single particle models in both aggregation and breakage processes until an experimentally measured equilibrium particle size distribution is closely matched by the solution to the population balance equation. The rationality of such a procedure is much in question, as it is clearly not sensitive to the time scales of breakage and aggregation. 

The breakage process was simulated by Sathyagal et al (1995) using the single particle technique of Section 4.6.4 to obtain the cumulative volume fraction of drops of various sizes x at different times t. Their test of similarity, made through a plot of In t versus In x for 14 different values of the cumulative fraction is represented in Fig. 6.1.1 below. 

The manner in which a particle fractures depends on (i) the nature of the particle and (ii) the manner in which the fracture force is applied. A number of terms have been used to describe the different mechanisms of single particle fracture. The different terms considered here are abrasion, cleavage, shatter, and chipping. It may be pointed out that in practice these events do not occur in isolation. Real breakage involves a combination of these processes, with the proportions changing, depending on the equipment, and on the manner each particle is stressed within it. 

The breakage theory of spheres is a reasonable approximation of what may occur in the size reduction of particles, as most size-reduction processes involve roughly spherical particles. An equation for the force required to crush a single particle that is spherical near the contact regions is given by the equation of Hertz (Timoschenko and Goodier, Theory of... 

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