Overall crystal growth coefficient

Eliminating c and introducing an overall crystal growth coefficient, KG gives the approximate relation ... 

ICg being the overall crystal growth coefficient and n the order of the crystal growth process. For more details reference 2 should be consulted. 

An overall crystal growth coefficient K may he defined using an overall concentration difference (Cm — Cis) ... 

Further, the growth term from the overall crystal growth coefficient JCo in expression (3.4.24) may be used to equate the growth term contribution in (6.4.52) ... 

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... 

To ensure that all the overall crystal-growth rate coefficients are measured under the conditions without nucleation, the metastable region of the solution has to be determined first and therefore the solubility and super solubility need to be measured. 

For comparison, the experiments for measuring the overall crystal-growth rate coefficient are carried out in an impinging stream crystallizer (ISC) and a fluidized bed crystallizer (FBC). 

The values measured in the ISC for the overall crystal-growth rate coefficient of Na2HP04 are listed in Table 12.2. As can be seen, the reproducibility of the data is in a reasonable range and that of most data is very good. 

Overall crystal-growth rate coefficient of Na2HP04 in SCISR... 

Comparison between overall crystal-growth rate coefficients measured in the ISC and the FBC,... 

Except for a few questionable data, the values for the observed active energy measured in the two crystallizers of different types, EiS and EFB, show little difference and can be considered to be more or less identical. On the other hand, the values measured in the impinging stream crystallizer for the overall crystal-growth rate coefficient, KIS, are obviously and systematically larger than those in the fluidized bed crystallizer, A pe. Therefore it can be affirmed without the need for further analysis that, with the observed frequency factors, there must... 

In the ranges of the operating conditions tested, the overall crystal- growth rate coefficient of Na2HP04 measured in ISC, Vs. is higher systematically than that measured in FBC, Vu. by 15 to 20%, while the reaction rate constant of ethyl acetate saponification measured in the SCISR, Vs. is larger systematically than that measured in the STR, Vt> by about 20%. 

A comparative study [10] is made for crystal-growth kinetics of Na2HP04 in SCISR and a fluidized bed crystallizer (FBC). The details of the latter cem be found in [11]. Experiments are carried out at rigorously controlled super-saturations without nucleation. The overall growth rate coefficient, K, are determined from the measured values for the initial mean diameter, t/po, masses of seed crystals before and after growth. The results show that the values for K measured in ISC are systematically greater than those in FBC by 15 to 20%, as can be seen in Table 2. On the other hand, the values for the overall active energy measured in ISC and FBC are essentially the same. 

Crystal growth is a diffusional process, modified by the effect of tlie solid surfaces on which the growth occurs. Solute molecules or ions reach the growing faces of a crystal by diffusion through the liquid phase. The usual mass-transfer coefficient kjf applies to this step. On reaching the surface, the molecules or ions must be accepted by the crystal and organized into the space lattice. The reaction occurs at the surface at a finite rate, and the overall process consists of two steps in series. Neither the diffusional nor the interfacial step will proceed unless the solution is supersaturated. 

INDIVIDUAL AND OVERALL GROWTH COEFFICIENTS. In mass-transfer operations it is generally assumed that equilibrium exists at the interface between phases. If this were true in crystallization, the concentration of the solution at the face of the crystal would be the saturation value y, and the total driving force for mass transfer would be y — y, where y is the concentration at a distance from the crystal face. Because of the surface reaction, however, a driving force is needed for the interfacial step, and the concentration at the interface is therefore y, where y < y < y. Only y — / remains as the driving force for mass transfer. This is illustrated in Fig. 27.8. 

At this point it is appropriate to consider the temperature coefficient of overall crystallization and in particular that of spherulite growth. Given the observation of maxima in either type of rate study, two main factors need to be considered. One is the application of the general concepts of nucleation theory to polymers. The other involves the description of the transport of chain units across the liquid-crystal interface, the transport term. 

The demarcation between Regimes I and II, with the required change in temperature coefficients, has been experimentally observed for several different polymers by studies of either the overall crystallization rate or the spherulite growth Either of these techniques is adequate... 

The analysis of the temperature coefficient of the overall crystallization rate is more complex. In this case both the initiation of the crystallite (or spherulite) and its subsequent growth has to be accounted for. Both of these processes are nucleation controlled. However, they are not necessarily of the same type. There are many possibilities that are consistent with a specific set of experimental results. For example, it could be assumed that the initiating or primary nucleus is three-dimensional and formed either homogeneously or heterogeneously. The further assumption can be made that the secondary or growth nucleus also has three-dimensional characteristics. It can be postulated further that the critical free energy required to form a secondary nucleus is less by a factor a than that for primary nucleation. With these assumptions... 

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