Crystal growth rate, correlation

It is possible to determine overall crystal growth rates by adding a known mass of sized seeds to a supersaturated solution in an agitated vessel, following a similar procedure to that outlined above for the fluidized bed method. To correlate the data, however, it is necessary to estimate the particle-fluid slip velocity as a function of impeller speed in the agitated vessel using relationships of the type described in section 9.4.1. 

The CSD of the product from a continuous crystallizer is determined by a direct relationship between nucleation and crystal growth rates and magma residence time distributions. Since a priori prediction of crystallization kinetics is not yet possible, however, experimentally determined and statistically correlated nucleation and growth rates are needed for the design and analysis of industrial crystallizers. Analysis of the CSD from continuous MSMPR (mixed-suspension, mixed-product-removal) crystallizers has proved to be a popular way of inferring such crystallization kinetics. This approach has been widely... 

The difficulty with each of the theoretical approaches to date, however, is that they cannot yet predict crystal growth rate coefficients and exponents for a particular substance a priori. Thus as with nucleation kinetics, crystal growth rate data from industrial crystallizers are usually correlated empirically with environmental conditions, such as concentration and temperature using a power law model of the form... 

Size-dependent Crystal Growth. A number of empirical expressions correlate the apparent effect of crystal size on growth rate (30). The most commonly used correlation uses three empirical parameters to correlate growth rate with crystal size ... 

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... 

In the primary nucleation stage of crystallization at small supercoolings and high pressures, the growth rate G and net transition rate J can be correlated by the following relation ... 

J. Prywer, Correlation between growth of high-index faces, relative growth rates and crystallographic structure of crystal, Eur. Phys. J. B 25, 61-68 (2002). 

Linear regression results show that b is approximately 0.5 for each experimental condition and that a increases with the rate of crystal growth due to environmental conditions. The refrigerant subcooling, aT, was seen to directly effect the value of a, whereas, since the bulk subcooling changed with time, there was no clear correlation between a and aT),. Further work will be necessary to delineate these relationships for both batch and semi-batch conditions. 

The linear correlation indicates that Equation 3 is satisfied and crystal growth obeys the AL law. From the slope of CSD, the growth rate G is obtained. The dominant particle size Im and the nucleation rate are related as follows ... 

Results from a quasi steady-state model (QSSM) valid for long crystals and a constant melt level (if some form of automatic replenishment of melt to the crucible exists) verified the correlation (equation 39) for the dependence of the radius on the growth rate (144) and predicted changes in the radius, the shape of the melt-crystal interface (which is a measure of radial temperature gradients in the crystal), and the axial temperature field with important control parameters like the heater temperature and the level of melt in the crucible. Processing strategies for holding the radius and solid-... 

All seven steps require time, resulting in a rate of incorporating clusters into the growing crystal surface, which is called crystal growth kinetics. The following two sections consider translation of such a rate into a macroscopic equation for correlation and prediction. It is difficult to say which of the steps control the process, or even if the conceptual picture is valid. However, the first step—species transport to the solid surface—is well established and a brief description is given in Section 3.2.1.2. 

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