Crystallization effective distribution coefficient

Aim of the directed crystallization is in most cases to reach a crystal coat of highest possible purity. The most used measure to qualify the purification effect of the crystallization is the effective distribution coefficient k ff. For the case of crystallization processes in which the concentration of the feed remains constant the effective distribution coefficient is defined as the ratio of the impurity concentration in the crystal product C to the impurity concentration in the feed cq ... 

Several investigations (8.9 show, that the effective distribution coefficient can be described as a function of the growth rate. Own experiments show that a model to calculate the purity in dependence of the real growth rate is rather realistic. To considerate irregularities by the calculation of k ff, the effective distribution coefficient is presented as a function of the ideal growth rate of the crystal layer V, 3 and the growth rate deviation dy. ... 

The achieved purity is quite often plotted in diagrams as an effective distribution coefficient. The effective distribution coefficient has been defined by Burton et al. (1953) and Wintermantel (1986) as the ratio of the impurity content in the crystals to the impurity content in the feed melt. Impurities can be incorporated in crystals or crystal layers as a result of a kinetic process. 

Wellinghoff et al. (1995) derived a correlation directly from a pore diffusion model based on Pick s law and obtained an equation which predicts the effective distribution coefficient of diffusion washing as a function of four dimensionless numbers. The apparent differences between the two approaches vanish upon closer examination. Namely, when neglecting the factors that cannot be arbitrarily influenced due to process inherent restrictions the two approaches are essentially alike. They only differ in that Wellinghoff et al. (1995) additionally consider a dimensionless area which corresponds to the ratio of the pore surface to the surface of the crystal layer. This factor can only be determined experimentally yet which is fairly complicated and thus limits the applicability of the approach. 

For realistic mass transfer coefficients (3, Fig. 8.2-13 shows how the effective distribution coefficient depends on the rate of crystal growth. At a certain rate of crystal growth the cleaning effect of the crystallization step fully disappears. During layer crystallization high rates of crystal growth are intended, especially to reach high area specific production rates. 

To account for deviation from thermodynamics, under real conditions an effective distribution coefficient ke F is defined (Equation 7.3). The equation is similar to Equation 7.2, but the effective distribution coefficient results from parameters measured under real crystallization conditions. That is, the parameters Xir,ip s as the impurity content in the solid phase and ximp.i, as the impurity content in the liquid phase are values obtained from the separation process performed. In contrast, the parameters in Equation 7.2 are directly related to the phase diagram. Thus, the effective distribution coefficient also comprises the influence of the crystallization kinetics, in particular the crystal growth rate and mass transfer limitations. 

Figure 7.6 illustrates the influence of yield on product purity on the basis of a system with an effective distribution coefficient of 0.2. Shown is the depletion behavior of a certain impurity for different crystallizations. From the figure, one could simplify that for a yield of <80-90%, the purification is of same level within the limit of the high dispersion in the data and deteriorates drastically above that yield. 

Equation 7.4 describing the effective distribution coefficient contains the crystal growth rate G and the mass transfer coefficient ka characterized by the diffusion coefficient in the liquid D and the thickness of the interfacial diffusion boundary... 

Yao et al. (1996b) investigated 9x9x4mm Zn-doped YBa2(Cui cZnj )30z crystals with X = 0.004-0.03 grown by the SRL-CP method. The effective distribution coefficient of Zn was found to be about 0.35 at growth temperatures 986-1005°C. Tc decreased almost linearly and rapidly with a slope 12 K per 1 at% of Zn (see also sect. 5.1). 

Figure 7 Illustration of the effects of equilibrium (batch) crystallization or melting on trace element abundances, (a) Variation in liquid concentration (Cl) (normalized to unit source concentration Cq= 1) as a function of melt fraction (F) for six elements with different bulk distribution coefficients (D). (b) Change in the ratios of incompatible elements with different Ds as a function of F. Each curve is for a different pair of elements that have the Z)s indicated. Note that when D < 0.1, incompatible element ratios can be changed only at very low extents of melting (or high extents of crystallization) (Langmuir et al., 1992) (reproduced by permission of American Geophysical...

The conditions for stable growth of a crystal are closely associated with segregation of dopants or impurities. Crystal growth theory includes an expression for the effective segregation coefficient K, for a crystal growing from a medium in which the dopant distribution is characterized by a boundary layer of thickness 5 ... 

Figure 3.12 Effect of growth rate and agitation on the distribution coefficient of antimony in germanium crystals. (Reproduced with permission from Burton et al. 1953.)...

Both the intrinsic rate constant and the effective diffusivity (KD) can be extracted from measurements of the reaction rate with different size fractions of the zeohte crystals. This approach has been demonstrated by Haag et al. [116] for cracking of n-hexane on HZSM5 and by Post et al. [117] for isomerization of 2,2-dimethylbutane over HZSM-5. It is worth commenting that in Haag s analysis the equilibrium constant (or distribution coefficient K) was omitted, leading to erroneously large apparent diffusivity values. 

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