where the moment forms of B and D due to agglomeration are defined as:

A solution is only obtained for a size-independent kernel ft0. For a batch-agglomerating system (without growth), only solutions for m0 and mt (which is equal to the m3 in size coordinates) are obtained:

where S(v) is a selection function that describes the rate at which particles fall apart, and b is a breakage function that describes how many particles of size v are formed on disruption of a particle of size u.

Often, the disruption terms are loaded into the agglomeration kernel ft, which then represents the effective agglomeration kernel. In that case terms expressed by the eqns [43] and [44] can be left out and only eqns [41] and [42] are needed for substitution in the PBE. The main disadvantage of this lumped description of the whole agglomeration process is that prediction of the agglomeration behaviour for different hydrodynamic conditions (and thus for different scales and geometries) becomes virtually impossible because ft and a have a different dependency on s.

The PBE with agglomeration and disruption terms can rarely be solved analytically. A moment transformation is again required with the moments defined as:

where v represents crystal size volume v. The zero moment is again the total number of particles, but the first moment of the volume-based CSD already equals the third moment of the length-based CSD.

Substitution of the above expressions into the PBE (eqn [36]) for a system with agglomeration and growth and no nucleation, with a constant crystal-lizer volume, only one crystal-free inlet stream and a nonclassified product stream, results after a moment transformation in:

The numerical solution of the population balance for cases where nucleation, growth and agglomeration is present often imposes a problem, because no analytical solution exists and the generation of numerical solutions is not easy. It is essential to divide the size axis into proper size intervals to obtain reliable results and to transform the population balance, a partial differential equation, into a set of ordinary differential equations. Hounslow et al. presented a numerical scheme based on a geometrical discretization of the size axis in which for each size interval the ratio between the upper and lower boundary r is exactly equal to ^2. Although accurate results are obtained in the absence of crystal growth and nuclea-tion, the method is in general not accurate enough in the presence of these crystallization phenomena and gives rise to an overestimation of the higher moments. An improved finite-element technique has been introduced to solve the steady-state solution of the population balance for nucleation, growth and agglomeration.

Solar Panel Basics

Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

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