Crystal growth rate expressions

Table 15.4. Mean over-all crystal growth rates expressed as a linear velocity ...

It has often been observed that the plot of ln(L) versus L results in curvature rendering the method of determining the growth rate from the slope strictly inappropriate, but ways to accommodate such deviations have also been proposed. Thus, if G = G(L) integration of equation 3.15 leads to the following expression for determining crystal growth rates (Sikdar, 1977)... 

Crystal growth rate may be expressed either as a rate of linear inerease of eharaeteristie dimension (i.e. veloeity) or as a mass deposition rate (i.e. mass flux). Expressed as a veloeity, the overall linear erystal growth rate, G (=dL/dt where L is the eharaeteristie dimension that is inereasing). The rate of ehange of... 

The crystal growth rate has been found in many eases to be extremely rapid, more rapid than can be accounted for on the diffusion hypothesis thus Tammann (foe. cit.) found for benzophe-none a maximum crystallisation velocity of 2 4 mm. per minute (Walton and Judd, J. Phys. Ohem. xvni, 722,1914). Much higher values, e.g. 6840 mm. per minute for water and 60,000 mm. per minute for phosphorus (Gernez, O.R, xcv. 1278, 1882) have been recorded. In some cases the rate was found independent of the speed of rotation of the stirrer and occasionally the reaction velocity followed a bimolecular law instead of the simple unimolecular expression which holds true for solution. 

Further, substituting Eq. (12-8) into Eq. (12-3), integrating the resulting equation between t - 0 and t = tf and rearranging yields the expression for the overall crystal-growth rate coefficient as... 

The size of the crystalites produced will depend on the nucleation rate and crystal growth rate at the solidification front. Both are controlled by the local supersaturation. The rate expressions for homogeneous nucleation (equation (6.15)) and heterogeneous nucleation (equation... 

The linear growth rate of a face can be expressed in terms of the step velocity, step height, and step spacing. Techniques used for in situ measurement of crystal growth rates as a function of supersaturation include the following ... 

Incorporating the concepts discussed above, the equation describing the crystal growth rate in a miscible polymer blend can be expressed as ... 

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... 

Crystals form by the deposition of the precipitate constituent ions onto nuclei. Since water and wastewater treatment processes involving precipitation often do not reach equilibrium, the rate of crystal growth is of critical importance. Crystal growth rate can be expressed as... 

The theoretical basis for use of the DSC or DTA for study of crystallization rates was developed independently by Johnson and Mehl and by Avrami. The volume fraction of a sample crystallized, x, as a function of time, t, is expressed in terms of the nucleation rate per unit volume, /, and the crystal growth rate, u, via the equation ... 

According to Mersmann [16], the supersaturation in cooling crystallization is usually small, i.e., relative supersaturation, a, is less than 0.1 and the relative solubility expressed by the ratio of solubility (kg/m ) and crystal density is usually higher than 0.01. As a result, primary nucleation does not take place and the nuclei are formed only as attrition fragments. If a coarse product is desired, the attrition rate should be low and the crystal growth rate at the maximum allowable level with respect to crystal purity, all of which depends mainly on the mean residence time of the slurry. 

One method is to solve the population balance equation (Equation 64.6) and to take into account the empirical expression for the nucleation rate (Equation 64.10), which is modified in such a way that the expression includes the impeller tip speed raised to an experimental power. In addition, the experimental value, pertinent to each ch ical, is required for the power of the crystal growth rate in the nncleation rate. Besides, the effect of snspension density on the nucleation rate needs to be known. Fnrthermore, an indnstrial suspension crystallizer does not operate in the fully mixed state, so a simplified model, such as Equation 64.6, reqnires still another experimental coefficient that modifies the CSD and depends on the mixing conditions and the eqnipment type. If the necessary experimental data are available, the method enables the prediction of CSD and the prodnction rate as dependent on the dimensions of the tank and on the operating conditions. One such method is that developed by Toyokura [23] and discussed and modified by Palosaari et al. [24]. However, this method deals with the CTystaUization tank in average and does not distinguish what happens at various locations in the tank. The more fundamental and potentially far more accurate simulation of the process can be obtained by the application of the computational fluid dynamics (CFD). It will be discussed in the following section. 

The constants and fs may be called volume and surface shape factors, respectively. In the expressions for crystal growth rate in section 6.2.1,/v and /s are given the symbols a and / , respectively. These latter symbols are also used in Chapter 8. 

The crystal growth rate (e.g. mass per unit time) may be expressed as a function of the overall temperature driving force (cf. equation 6.18), by... 

There is no simple or generally accepted method of expressing the rate of growth of a crystal, since it has a complex dependence on temperature, supersaturation, size, habit, system turbulence, and so on. However, for carefully defined conditions crystal growth rates may be expressed as a mass deposition rate Rq (kgm s ), a mean linear velocity v(ms ) or an overall linear growth rate G (ms ). The relationships between these quantities are... 

The overall crystal growth rate, Rq (the mass rate of deposition, dmjdt, per unit crystal surface area. A, see equation 6.61) may thus be expressed as... 

Figure 6.32. Dissolution and growth rates (expressed as a mass increase or decrease, normalized with the initial seed crystal mass) of a single potassium sulphate crystal in the presence o/Fe(III) as trace impurity added as FeNH4(S04)2 2H2O. (After Kubota et al., 1999)...

The crystal growth rate G can be expressed in a similar manner ... 

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