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Collisions and Crystal-Crysta IThe rate of secondary nucleation of ice, assumed to be proportional to the product of collision frequency and impact energy, has been quantitatively modeled using idealized representations of collisions between crystals and either other crystals or surfaces in the crystallizer. The crystal-crystallizer collisions were assumed to be driven by either steady or turbulent fluid motion and the crystal-crystal collisions were assumed to be driven by either gravitational forces or turbulent eddies. The models predict to a good approximation the experimentally determined dependence of the secondary nucleation of ice on crystal size, ice concentration, and agitation power. T. W. EVANS SCOPEIn many practical crystallizers the nucleation rate is dominated by interaction of crystals with their environment, that is, by secondary nucleation processes. Secondary nucleation is poorly understood and heavy reliance has been placed on empirical correlations of nucleation kinetics, Without a fundamental understanding of the processes governing secondary nucleation, the form of the function that can best correlate experimental results cannot be well established. Hence there is no sound basis for extrapolating results from one set of experimental conditions to others. As examples of the wide choice of functions available to correlate secondary nucleation, simplistic models based on the assumptions that secondary nucleation is proportional to the number of crystals, crystal perimeter, area, or mass would suggest that the nucleation rate should be proportional to the zeroth, first, second, or third moment of the crystal size distribution, that is,where N is the nucleation rate and p, the nth moment of the crystal size distribution. Other models of crystallization involve collision of crystals with each other, in which case the nucleation rates would be expected to be proportional to the square of the moments of the particle size distribution. Little can be said about which moments should be used without a statement of mechanism. Additional motivation for developing mechanistic models of secondary nucleation is provided by the need to determine the dependence of nucleation rate on design parameters such as agitation rate, scale of equipment, etc.In a previous paper (Evans et al., 1974), it has been shown that the factors influencing crystal surface morphology can be separated from those influencing collisions. In this paper, following leads provided by Clontz and McCabe (1971) and Ottenj and co-workers (1972, 1973), this concept is extended. Expressions are derived for the kinetics of secondary nucleation under the assumption that the rate of nucleation is proportional to collision frequency and collision energy for each of four idealized collision mechanisms. These include (1) collision of crystals with an impeller as the crystals are swept by the impeller in steady flow, (2) collision between crystals in a turbulent flow field and various surfaces in the crystallization, (3) collision between crystals ...

Experiments were designed to identify the mechanism of the secondary nucleation of ice in a vigorously agitated crystallizer. It has been shown that the nucleation rate is proportional to the product of two factors, one characterizing crystal morphology and the other the rate of removal of potential nuclei from the surfaces of the existing crystals. Consequently, the nucleation rate attributable to different mechanisms is additive and the rate is proportional to the number of collisions per crystal. The contribution to the secondary nucleation of ice, by collisions of crystals with the impeller, baffles, and other crystals in an agitated crystallizer have been identified by measurements in a batch crystallizer in which each of the different collision mechanisms could be suppressed.

used in this study, but it is obvious that other procedures could be employed.The convergence criterion used was that With this criterion, convergence occurred, at best, with 50 iterations. At worst more than 1000 iterations may be required.Computations were performed on a CDC 6400. On this machine the computing time was roughly 0.003 sediteration per component. As indicated, computing time increases linearly with the number of components.The advantages of this procedure lie in its ease of formulation and programming, its tie to thermodynamic principles, and its reliability. The formulation presented here can easily be extended to moR than three phases which makes it a candidate for use in computations where solid phases need be considered, as in cryogenic applications.Since the procedure minimizes free energy, it is possible to avoid spurious solutions to the equilibrium problem such as those pointed out in the discussion of Figure 5e. In the liquidliquid-vapor computations the starting point was that almost all the water was in a water rich liquid with a few percent of the total water being present in the two hydrocarbon rich phases. The hydrocarbons were distributed so that the vapor phase was rich in the light components and the liquid phase was rich in the heavy components. This assured convergence to the correct equilibrium conditions. Determination of the Kinetics of Secondary Nucleation in Batch CrystallizersThe kinetics of secondary nucleation have been determined from measurement of the supersaturation as a function of time following the addition of seed nuclei to a supercooled solution in a well-stirred batch crystallizer. Population balance mathematics have been used to show that the secondary nucleation kinetics may be inferred from the supersaturationtime curve. The method has been applied to the determination of the kinetics of the secondary nucleation of ice and found to give results in excellent agreement with those obtained from tedious particle counts. In addition, it has been shown that the moment of the particle size distribution that best correlates the nucleation rate data can be inferred from the initial transient of the supersaturation-time history. SCOPEThe use of batch crystallizers for the determination of the kinetics of secondary nucleation is complicated by the variation during an experiment in both the number of crystals and the supersaturation. Motivation for the use of batch crystallizers is provided, however, by the relative ease with which a large range of operational variables may be studied in a short time. Particle counts, however, may be tedious or difficult, particularly for systems such as ice where crystals are relatively unstable. One major objective in this paper is therefore the development of methods for inferring nucleation kinetics from the supersaturation measured at times greater than the induction time.It should be noted that the inference of nucleation kinetics is complicated by the fact that the rate of secondary nucleation may be proportional to the nth mome...