Invariant growth

For systems following invariant growth the crystal population density in each size range decays exponentially with the inverse of the product of growth rate and residence time. For a continuous distribution, the population densities of the classified fines and the product crystals must be the same at size Accordingly, the population density for a crystallizer operating with classified-fines removal is given by... 

AA and BB monomers and also AB monomers invariably react to form predominantly linear structures in all but the rather special case where the ring structure in reaction (5.CC) has a value of 1 = 5 or 6. This explains why so many of the monomers in step-growth polymerizations are tetra-, hexa-, and decamethylene compounds. 

The dominant crystal size, is most often used as a representation of the product size, because it represents the size about which most of the mass in the distribution is clustered. If the mass density function defined in equation 33 is plotted for a set of hypothetical data as shown in Figure 10, it would typically be observed to have a maximum at the dominant crystal size. In other words, the dominant crystal size is that characteristic crystal dimension at which drajdL = 0. Also shown in Figure 10 is the theoretical result obtained when the mass density is determined for a perfectiy mixed, continuous crystallizer within which invariant crystal growth occurs. That is, mass density is found for such systems to foUow a relationship of the form m = aL exp —bL where a and b are system-dependent parameters. 

If the crystallizer is now assumed to operate with a cleat feed (n = 0), at steady state (dn jdt = 0), and if the crystal growth rate G is invariant and a mean residence time T is defined as then the population balance can be written as... 

The effects of each selective removal function on CSD can be described in terms of the population density function n. It is convenient to define flow rates in terms of clear Hquor, which requires the population s density function to be defined on a clear-Hquor basis. In the present discussion, only systems exhibiting invariant crystal growth are considered. 

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. 

For exampie, if the crystai growth rate is invariant with size, i.e. G G(L), then in the absence of particie breakage and aggiomeration... 

If crystal agglomeration and breakage can be neglected and crystal growth is invariant then at steady state the familiar analytic form is... 

If the water is found fit for consumption, with respect to both its mineral and biological content, the problem of sanitization can still arise. Public supply invariably has a very small residual chlorine level. This suppresses biological growth and maintains water quality even when the line is stagnant. As with other forms of treatment, the scale of private supply is usually too small to allow good control of chlorinating equipment. 

As indicated earlier, protective oxide scales typically have a PBR greater than unity and are, therefore, less dense than the metal from which they have formed. As a result, the formation of protective oxides invariably results in a local volume increase, or a stress-free oxidation strain" . If lateral growth occurs, then compressive stresses can build up, and these are intensified at convex and reduced at concave interfaces by the radial displacement of the scale due to outward cation diffusion (Fig. 7.7) . 

It is shown that an increase in the heat flux is accompanied by an increase in the liquid and vapor velocities, the meniscus displacement towards the outlet cross-section, as well as growth of vapor to liquid forces ratio and heat losses. When is large enough, the difference between the intensity of heat transfer and heat losses are limited by some final value, which determines the maximum rate of vaporization. Accordingly, when is large all characteristic parameters are practically invariable. 

Kapur and Fuerstenau (K6) have presented a discrete size model for the growth of the agglomerates by the random coalescence mechanism, which invariably predominates in the nuclei and transition growth regions. The basic postulates of their model are that the granules are well mixed and the collision frequency and the probability of coalescence are independent of size. The concentration of the pellets is more or less fixed by the packing... 

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