Particle breakage processes

Mishra, B. K. 2000 Monte Carlo simulation of particle breakage process during grinding. 

From this it can be concluded that the wide distribution of fragment sizes from milhng is inherent in the breakage process itself and that attempts to improve grinding efficiency by weakening the particles will result in coarser fragments which may reqiiire a further break to reach the desired size. 

All of the chemical species, except one, will be assumed to be completely soluble. The one partially insoluble species will nucleate and grow a solid phase. A typical example is A + B ->P where P is a sparingly soluble compound. The rates of nucleation J and molecular surface growth G can be functions of the local concentration vector c, the particle size l, and the local turbulence properties. Neglecting aggregation and breakage processes, a microscopic PBE for this system can be written as follows ... 

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. 

Computer simulation, based on population-balance models [Bass, Z. Angew. Math. Phys., 5(4), 283 (1954)], traces the breakage of each size of particle as a function of grinding time. Furthermore, the simulation models separate the breakage process into two aspects a... 

The limiting steps in the model development are the formulation of closure relations or closure laws determining turbulence effects, interfacial transfer fluxes and the bubble coalescence and breakage processes. When sufliciently dilute dispersions are considered, only particle - fluid interactions are significant and the two-fluid closures can be employed. In these particular cases, only the interaction between each of the dispersed gas phases (d) and the continuous liquid phase (c) is considered parameterizing the last term on the RHS of (8.12) ... 

The fundamental derivation of the population balance equation is considered general and not limited to describe gas-liquid dispersions. However, to employ the general population balance framework to model other particulate systems like solid particles and droplets appropriate kernels are required for the particle growth, agglomeration/aggregation/coalescence and breakage processes. Many droplet and solid particle closures are presented elsewhere (e.g., [96, 122, 25, 117, 75, 76, 46]). 

A fairly general framework has been formulated for the source terms considering particle breakage, fluid particle coalescence, solid particle agglomeration/aggregation and similar processes (e.g., [109, 80, 81, 37, 114, 43, 25, 94]). Detailed discussions of the particle breakage and coalescence modeling and the mathematical properties of the constitutive equations can be found in the papers by Barrow [4], Laurencot and Mischler [64, 65]. 

Experiments have shown that in beds consisting of B type particles the bubbles increase steadily, whereas in beds consisting of type A particles the bubbles grow rapidly until they reach a stable size determined by a state of equilibrium between the coalescence and breakage processes [82]. 

As it is possible to see, the drift term has disappeared since the continuous growth of particle size does not change the total number concentration (if Gl > 0). However, N is influenced by the rate of formation of particles (e.g. nucleation), and the rates of aggregation and breakage, which cause appearance and disappearance of particles. These processes are all contained in the source term /tL.o The third-order moment mL,3 is related to the fraction of volume occupied by particles with respect to the suspending fluid and can be easily found fromEq. (2.18) ... 

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