AL law of crystal growth

This generalizaton, developed hy W.L. Me Cahe in 1929, is called the AL law of crystal growth. Correspondingly, over a time period At, the total increase in the crystal dimension, AL or Arp, is given hy... 

A further factor important in the control of phosphate levels by calcium phosphate precipitation is the rate of crystal growth. Ferguson et al. " showed that a rate law of the form... 

A theoretical analysis of an idealized seeded batch crystallization by McCabe (1929a) lead to what is now known as the AL law . The analysis was based on the following assumptions (a) all crystals have the same shape (b) they grown invariantly, i.e. the growth rate is independent of crystal size (c) supersaturation is constant throughout the crystallizer (d) no nucleation occurs (e) no size classification occurs and (f) the relative velocity between crystals and liquor remains constant. 

The linear correlation indicates that Equation 3 is satisfied and crystal growth obeys the AL law. From the slope of CSD, the growth rate G is obtained. The dominant particle size Im and the nucleation rate are related as follows ... 

Furthermore, the solution of a differential population balance requires that the relationship between growth rate and size of the growing crystals be known. When all crystals in the magma grow at a constant and identical rate, the crystal-solvent system is said to follow the McCabe AL law, while systems that do not are said to exhibit anomalous growth. 

Crystal growth is a description of the linear velocity of a growing face, that is, the linear velocity perpendicular to that face. Growth velocity, in the classical view, is considered to be constant (size independent), according to the classical McCabe AL Law (McCabe 1929). 

Defining the liquor and crystal mean residence time r = VjQ and assuming that growth is independent of size (AL law), i.e. dG/dL = 0, then... 

McCabe s (1929a,b) AL law states that crystals of the same substance growing under the same conditions should grow at the same rate. Experimental evidence has shown that this law is frequently violated. The growth rate of a crystal face, for example, and the instantaneous velocity of steps spreading across the surface of a crystal have been shown to fluctuate with time, even though external conditions, e.g. temperature, supersaturation and hydrodynamics, remain constant. 

With the assumption that the growth rate G is not a function of the particle size (McCabe s AL-law), for a constant crystallizer volume and for a crystal-free inlet flow (no seeding) the population balance becomes... 

Let us now focus on an MSMPR crystallizer having a size-independent growth (McCabe s AL law). To determine the effect of t,es, we assume two situations where the suspension density Mp is identical but 4es/ G and n are different. From expression (6.4.18) for Mj and for two situations (identified by subscripts a and h) and from above,... 

One of the first models for crystal growth is McCabe AL law and is given by the following size independent rate. 

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