Growth rate supersaturation

Eq. (2.52) is a rather complex expression relating the crystal growth rate, supersaturation, and the two constants, kd and ki. Normally this equation is approximated by the simple relation... 

Over 50 acidic, basic, and neutral aluminum sulfate hydrates have been reported. Only a few of these are well characterized because the exact compositions depend on conditions of precipitation from solution. Variables such as supersaturation, nucleation and crystal growth rates, occlusion, nonequilihrium conditions, and hydrolysis can each play a role ia the final composition. Commercial dry alum is likely not a single crystalline hydrate, but rather it contains significant amounts of amorphous material. 

The screw dislocation theory (27), often referred to as the BCE theory (after its formulators), shows that the dependence of growth rate on supersaturation can vary from a paraboHc relationship at low supersaturation to a linear relationship at high supersaturation. In the BCE theory, growth rate is given by... 

Both supersaturation and temperature can have different effects on the growth rates of different faces of the same crystal. Such occurrences have implications with respect to crystal habit, and these are dealt with in a later section. 

In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to... 

While Eq. (18-26) has been popular among those attempting correlations between nucleation rate and supersaturation, recently it has become commoner to use a derived relationship between nucleation rate and growth rate by assuming that... 

Here it can be seen that the nucleation rate is a decreasing function of growth rate (and supersaturation). The physical explanation is believed to be the mechanical influence of the crystallizer on the growing suspension and/or the effect of Bujacian behavior. 

It is imphcit that increasing the value of Ly will raise the supersaturation and growth rate to levels at which mass homogeneous nucleation can occur, thereby leading to periodic upsets of the system or cycling [Randolph, Beer, and Keener, Am. In.st. Chem. Eng. J., 19, 1140 (1973)]. That this could actually happen was demonstrated experimentally by Randolph, Beckman, and Kraljevich [Am. In.st. Chem. Eng. J., 23, 500 (1977)], and that it could be controlled dynamically by regulating the fines-destruction system was shown by Beckman and Randolph [ibid., (1977)]. Dynamic control of a ciystaUizer with a fines-destruction baffle and fine-particle-detection equipment... 

MeCabe s (1929a,b) AL law states that erystals of the same substanee growing under the same eonditions should grow at the same rate. Experimental evidenee has shown that this law is frequently violated. The growth rate of a erystal faee, for example, and the instantaneous veloeity of steps spreading aeross the surfaee of a erystal have been shown to fluetuate with time, even though external eonditions, e.g. temperature, supersaturation and hydrodynam-ies, remain eonstant. 

Growth rate fluetuations appear to inerease with an inerease in temperature and supersaturation leading to erystals of the same substanee, in the same solution at identieal supersaturation, exhibiting different growth rates this is thought to be a manifestation of the phenomenon of either size-dependent crystal growth or alternatively, growth rate dispersion. 

Figure 6.20 Growth rate of calcium oxalate versus supersaturation at 21 °C (Zauner and Jones, 2000a)...

The seeond-order dependenee of the growth rate on the supersaturation ean be explained by a number of growth theories. The most eonvineing, however, is that of Burton etal. (1951). In their BCF theory about the serew disloeation eentred surfaee spiral step, it is assumed that growth units enter at kinks with a rate proportional to cr and that the kink density is also proportional to cr whieh gives the faetor cr in the rate expression. 

The disruption experiments were earried out at cr = 0 ( S = 1) and therefore did not aeeount for any effeets of the supersaturation on the disruption proeess. Flartel and Randolph (1986b) and Wojeik and Jones (1997) reported a deerease in the disruption rate of ealeium oxalate and of ealeium earbonate, respeetively, with inereasing growth rate. Based on these findings, a linear deerease of the disruption kernel with the growth rate was assumed giving... 

Agglomeration rates also depend on the level of supersaturation in the reaetor and on the power input. Wojeik and Jones (1997) found a linear inerease of the agglomeration kernel with the growth rate. Therefore, the level of supersaturation was aeeounted for by Zauner and Jones (2000a) using the relation... 

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 level of supersaturation (S) also ehanges. If it is assumed, for simplieity, that the erystal growth rate exhibits linear kineties i.e. 

Thus, because less surface area is present in this case, the supersaturation increases and hence the growth rate also increases. 

The dependence of the growth rate on supersaturation is modelled using the power law expression... 

Growth rates and ultimate sizes of crystals are controlled by limiting the extent of supersaturation at any time. 

Typical surfaces observed in Ising model simulations are illustrated in Fig. 2. The size and extent of adatom and vacancy clusters increases with the temperature. Above a transition temperature (T. 62 for the surface illustrated), the clusters percolate. That is, some of the clusters link up to produce a connected network over the entire surface. Above Tj, crystal growth can proceed without two-dimensional nucleation, since large clusters are an inherent part of the interface structure. Finite growth rates are expected at arbitrarily small values of the supersaturation. 

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