Breakage model

The simplest aggregation and breakage models can be formulated in terms of the NDF (o), which uses volume as the independent variable.6 The microscopic transport equation for the NDF has the form (Wang et al., 2005a,b)... 

Hagesffither et al [28, 29, 30] extended the model by Luo and Svendsen [74], but the resulting breakage model was still not completely conservative. To ensure number and mass conservation they thus adopted a numerical procedure redistributing the bubbles on pivot points in accordance with the discrete solution method [94]. 

In an interesting attempt to overcome the limitations found in the turbulent breakage models described above Martmez-Bazan et al [78] (see also Lasheras et al [58]) proposed an alternative model in the kinetic theory (microscopic) framework based on purely kinematic ideas to avoid the use of the incomplete turbulent eddy concept and the macroscopic model formulation. 

In what is to follow, we first discuss Simha s scheme and its exact solution Then a brief discussion of the different breakage models will be given together with a description of the fit procedure Finally, this detailed evaluation of data will be com-... 

In the Gaussian breakage model the standard deviation is set proportional to the degree of polymerization i ... 

To discuss the thermodynamically induced shear degradation in further detail, we first turn to the chemistry of mechanical chain scission. The next step is the comparison of the experimental results with different breakage models according to the method developed in Sect. 3. Then, the dependence of chain scission on various parameters as the applied shear rate, concentration, and temperature will be discussed in detail. 

Theoretical calculations of the stabilities of various hexahaloantimonates have shown that the rupture of an Sb-Cl bond requires more energy than that of an Sb-Br bond, and the relative positions of chlorine and bromine atoms within a mixed halogen counterion can affect the ease of bond breakage. Model compound studies of the reaction between p-methyl benzyl chloride, triethyl aluminium, and 2,4,4-trimethylpent-l-ene (dimeric isobutylene) indicated that the principal chain termination reactions involve hydridation and ethylation resulting from counterion decomposition. ... 

Calculated DSD from described breakage model, and breakage frequencies from DSD data. 

Different variations of the sectional methods for the solution of the governing PBE (12.308) can be obtained from different choices of numerical approximations of Bbj, Db,u Bc,i and Dc,i in terms of n (r, t). It is noted that the breakage problem is generally easier to solve than the coalescence problem because the breakage model is linear whereas the coalescence problem is non-linear. 

---