Kernel breakage

Homogeneous breakage kernels with positive homogeneity index (A>0) produce self-similar size distributions at long times... 

The importance of developing coalescence and breakage kernels based on physical grounds became evident for the goal of predicting liquid-liquid dispersion properties. Subsequent workers devoted efforts in this direction. 

Fig. 9.5. A sketch of the breakage kernel of Luo and Svendsen [74]. An unequal daughter size distribution is predicted by this model. In this case the parent diameter size is, di = 0.006 (m), and the turbulent energy dissipation rate is set at, = 1 Reprinted with permission from [38], Copyright 2005 American Chemical...

To close the population balance problem, models are required for the growth, birth and death kernels. It is required that these kernels are consistent with the inner coordinate used. The coalescence and breakage kernels presented in this chapter are expressed in terms of the particle diameter. 

A particle size ratio in bubble breakage kernel closure, A = d jd (—)... 

Analytical solutions are available for some particular cases for a constant breakage kernel, a( ) = do. For example, in the case of symmetric fragmentation with daughter distribution... 

We have shown how this set of differential equations is closed in some simple cases (i.e. for nucleation at zero size, with constant growth rate, and constant aggregation and breakage kernels). In general, when the system is not closed, closure must be sought through a stable numerical method. 

For this specific example, consider the case of constant aggregation and breakage kernels, namelyPa,y,s,[Pg.326]

Patruno, L. E., Dorao, C. A., Svendsen, H. F. Jakobsen, H. A. 2009 Analysis of breakage kernels for population balance modelling. Chemical Engineering Science 64, 501-508. 

Population-balance analysis has been adapted to both coalescence and dispersion of drops in numerous papers by Calabrese, Ramkrishna, and Tavlarides. The analyses with these tools have led to a considerably better understanding of breakage kernels, breakage rates, coalescence efficiency, and collision rates. However, the description and use of these tools goes beyond the scope of this chapter. For a detailed understanding, see Ramkrishna [66]. 

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