Growth constant, crystallization

Crystal growth rate may be constant, which could happen if temperature is decreasing or if there is convection. Smith et al. (1956) treated the problem of diffusion for constant crystal growth rate. In the interface-fixed reference frame, the diffusion equation in the melt is... 

Diffusive crystal growth at a fixed temperature would not result in a constant crystal growth rate (see below). However, under some specific conditions, such as continuous slow cooling, or in the presence of convection with diffusion across the boundary layer, time-independent growth rate may be achieved. Similarly, time-independent dissolution rate may also be achieved. 

Figure 4-9 Calculated (a) concentration profiles in the melt and (b) crystal surface concentration for the case of constant crystal growth rate.

Figure 4-9 Concentration profile with constant crystal growth rate 359...

In conventional batch-cooling-crystallization a saturated solution is cooled from an initial temperature at which the solute has a high solubility to a final lower operating temperature (with lower solubility) along an optimal cooling curve. This cooling is used to maintain a reasonably constant level of supersaturation, and a constant crystal growth-rate. 

Process Stability and Control. Operationally, automatic control of the crystal radius by varying either the input power to the heater or the crystal pull rate has been necessary for the reproducible growth of crystals with constant radius. Techniques for automatic diameter control have been used since the establishment of Czochralski growth. Optical imaging of the crystal or direct measurement of the crystal weight has been used to determine the instantaneous radius. Hurle (156) reviewed the techniques currently used for sensing the radius. Bardsley et al. (157,158) described control based on the measurement of the crystal weight. 

Blaurock and Carothers (1990) and Blaurock and Wan (1990) described a simple way, valid for butteroil, of analyzing isothermal DSC data to characterize the kinetics of early crystallization in a supercooled oil. This approach yielded a single crystallization-temperature dependent combined nucleation/crystal growth constant (which they called NG). The temperature dependence of NG could be modeled with the Arrhenius equation. 

N = number of crystals K = growth constant AC = supersaturation = C — CB TTP = weight of product crystals TPs = weight of seed crystals... 

For an isothermal run, the growth constant K may be evaluated by determining the initial and final weights of the seeds, the number of crystals, and the variation of supersaturation with time. Values of the growth constant obtained at several different temperatures may be used with Eq. (26) to predict nonisothermal operation. Palermo s work is in agreement with McCabe s (Ml) earlier work, for it is in essence an analysis limited to a single crystal size, rather than a distribution of sizes. 

The growth of potassium chloride from an aqueous solution of sodium and potassium chlorides was studied by Davion (D3). Using uniform-size seeds, growing at a constant level of supersaturation, the crystallization growth constant K, was determined by the expression below. 

Jang s work (Jl) with a rotating drum crystallizer pointed out the limitations of an expression such as Eq. (26) for evaluating a growth constant when two widely different seed sizes were used in the same run. The growth constant for the larger crystals fell below the expected value, while the constant for the small crystals was higher than expected. 

Crystallization temperature (K) Filler concentration (% vol) Speed growth constant z c xl0 Degree index n of equation (17)... 

These observations have been summarized in a statistical-mathematical equation called the Constant Crystal Growth (CCG) Model [. The CCG Model provides a generalized engineering approach to account for distribu-... 

Once initial nucleation has been achieved successfully, the control of secondary nucleation becomes important. Since crystal growth is a surface phenomenon, each nuclei formed is available to absorb the supersaturation generated by the cycle. This means that only one nuclei is to be formed for each single crystal removed if a constant crystal size is to be maintained. 

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