Kinetic Expressions

In order to solve the population balance equation, kinetic expressions are needed to represent the physical processes that take place in the crystallizer, such as nucleation, growth and agglomeration. Their mechanisms and the corresponding equations will be treated here.

Nucleation

Two different nucleation mechanisms can be distinguished: primary and secondary nucleation.

Primary nucleation is new phase formation from a clear liquid or solution. It can be subdivided into homogeneous and heterogeneous nucleation. In the latter case a foreign substrate of tiny invisible particles, e.g. dust or dirt particles, is present in the solution on which nucleation starts. In homogeneous nucleation such a substrate is absent and nuclei are formed by statistical fluctuations of solute entities that cluster together.

Secondary nucleation is the breeding of nuclei from crystals of the crystallizing material that are already present in the solution. These nuclei are in general attrition fragments, and result from collisions of the larger crystals with the hardware of the crystallizer, in particular with the blades of impellers and pumps. At high solid densities in the crystallizer collisions between the larger crystals can create fragments that act as secondary nuclei.

During the start-up phase of evaporative or cooling crystallization of moderately to very soluble compounds, primary nucleation takes place. After their outgrowth to larger crystals, secondary nucleation takes over, and becomes the most important source of nuclei at low supersaturation values. For precipitation of slightly soluble compounds the process generally remains dominated by primary nucleation for two reasons. Supersaturation remains high enough, especially at the inlet points of the feed streams, to produce primary nuclei, and the often agglomerated crystals remain too small to be prone to attrition.

Homogeneous primary nucleation Local fluctuations in concentration induce the formation of numerous clusters that can fall apart again. In under- or just saturated solutions, cluster formation and cluster decay are in equilibrium; it is a reversible process. In supersaturated solutions, however, clusters of a critical size are formed that either fall apart or grow out. In the classical nucleation theory of Volmer, Becker and Do ring, these clusters are formed by the attachment and detachment of single solute entities. Although clusters can also grow by the collision of clusters, their concentration is always so much lower than that of single solute entities that this process of cluster enlargement can be ignored.

The critical size of a cluster that is represented by its critical radius, r*, is given by:

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Solar Panel Basics

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