Breakage kernel particle

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 (—)... 

Solsvik J, Tangen S, Jakobsen HA (2013) On the evaluation of breakage kernels for fluid particles. Accepted for publication in Rev Chem Eng... 

However, in other cases the model predictions deviate much more from each other and were in poor agreement the experimental data considering the measurable quantities like phase velocities, gas volume fractions and bubble size distributions. An obvious reason for this discrepancy is that the breakage and coalescence kernels rely on ad-hoc empiricism determining the particle-particle and particle-turbulence interaction phenomena. The existing param-eterizations developed for turbulent flows are high order functions of the local... 

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]). 

As an example, consider a monodisperse population of particles characterized by mass as internal coordinate and moments m = mjt(O) = 1 with k = 0,..., 2N 1. This population of particles is continuously fed to a system wherein particles undergo aggregation and symmetric binary breakage with constant kernels. The equations describing the evolution of the moments are... 

In a first modeling approach, a macroscopic population balance is formulated directly on the averaging scales in terms of number density functions [80, 102], A corresponding set of macroscopic source term closures are presented as well. Reviews of numerous fluid particle breakage and coalescence kernels on macroseopie scales can be found elsewhere [60,73,74, 122], This modeling framework resembles the mixture model concept. 

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