Breakage Processes

The reader is invited to revisit the examples in Section 2.11 to develop a proper appreciation for the birth and death rates in population balance equations that appear through the boundary conditions. In this regard, the example in Section 2.11.5 presents the boundary condition (2.11.23), which is a particularly interesting example of a birth process occurring at a boundary. 

We shall now turn our attention to the birth and death functions associated with breakage and aggregation processes. 

Consider the problem in the general setting of the vector particle state space of Section 2.1 in an environment with a continuous phase vector as 

If breakup of particles occurs independently of each other, we let fo(x, r, Y, t) be the specific breakage rate of particles of state (x,r) at time t in an environment described by Y. It represents the fraction of particles of state (x, r) breaking per unit time. Then we have 

The average number of particles formed from the breakup of a single particle of state (x, r ) in an environment of state Y at time t. 

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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. 

Even in the loading regime in which inherent flaw effects dominate the fracture process, further clarification of the fracture activation and growth process is needed. For example, dynamic crack branching leading to multiple fracturing is expected to constitute an important part of the breakage process. Such a cooperative and collective fracture process does not fit well within a... 

A further complication in the breakage process is that depending on the breakage mechanism a breakage event can lead to... 

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 ... 

Marchisio, D. L., R. D. Vigil, and R. O. Fox (2003). Quadrature method of moments for aggregation-breakage processes. Journal of Colloid and Interface Science 258, 322-334. 

Valentas and Amundson (V3) studied the performance of continuous flow dispersed phase reactors as affected by droplet breakage processes and size distribution of the droplets. Various reaction cases with and without mass transfer were studied for both completely mixed or completely segregated dispersed phase. Droplet size distribution is shown to have a considerable effect on the efliciency of a segregated reaction system. They indicated that polydispersed drop populations require a larger reactor volume to obtain the same conversion as a monodispersed system for zero-order (or mass-transfer-controlled) reactions in higher conversion regions. As the dispersed phase becomes completely mixed, the distribution of droplet sizes becomes less important. These interactions are un-... 

It is observed, even in the partial solution to the problem, that realistic models of the droplet coalescence and breakage processes as discussed in Section V,D,2 have yet to be employed. A parallel development has occurred. The work is currently at the point where the realistic model of the droplet dynamics can be applied to the pertinent problems of extent of reaction and solute depletion in dispersions. The success of this effort would permit the researcher and designer to predict dispersed-phase reactor performance from fundamental properties of the dispersion, operation conditions of the vessel, and knowledge of the intrinsic kinetics. 

An illustrative comparison of abstraction, H elimination and C-C breakage processes for the reactions of O( P) with methane, ethane and propane is depicted in Figure 1. The graph shows energies calculated at the B3LYP/6-31G level. For H abstraction processes (Figure 1(a)) the reaction with methane seems to be the more... 

C breakage process. The oxyradical receives more energy than the alkyl radical, the latter being barely above its zero point in all cases. In addition, the oxyradical energy increa.ses with collision energj-, a trend akin to that reported for H chrnination. 

Buwa, V. V. and Ranade, V. V. (2000), Modeling of bubble coalescence and breakage processes in gas-liquid flows, NCL Internal report, August 2000. 

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) ... 

Several extensions of the two-fluid model have been developed and reported in the literature. Generally, the two-fluid model solve the continuity and momentum equations for the continuous liquid phase and one single dispersed gas phase. In order to describe the local size distribution of the bubbles, the population balance equations for the different size groups are solved. The coalescence and breakage processes are frequently modeled in accordance with the work of Luo and Svendsen [74] and Prince and Blanch [92]. 

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]). 

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