This paper considers aggregation during precipitation from sobeek (DLVO) theory, as presented, for example, by Hielution in a batch system. Unlike the aggregation of colloidal partimenz (1). However, aggregation during precipitation from cles in ionic solutions which has been extensively studied, aggregasupersaturated solutions in which other size enlargement tion during precipitation from supersaturated solutions in which mechanisms such as crystal growth occur is less well underother size enlargement mechanisms occur is less well understood.
stood.
Of particular interest is the development of a method to determine
Aggregation is an important size enlargement mechanism the rates of two active size enlargement mechanisms, namely in various precipitating systems ranging from industrial progrowth and aggregation, from experimental data. We describe a cesses such as the precipitation of aluminium trihydroxide novel technique which is capable of performing two main tasks: extracting the rates of growth and aggregation from experimental from caustic aluminate solutions as part of the Bayer process data and simulating changes in a particle size distribution during (2) to calcium oxalate crystallization which leads to human an experiment. A differential technique is developed which uses kidney stone formation (3).
a discretized population balance to determine the rates from exper-
In most precipitating systems, when both growth and imental data. The same discretized population balance is used to aggregation occur simultaneously, it is very difficult to simulate changes in a particle size distribution for given growth determine the rates of both these size enlargement mechaand aggregation rates. Both these operations, unlike other more nisms. The reason for this is that the mathematical descripcomplicated techniques, can be performed on a personal computer tion of such systems is intractable analytically for all but and give results in a few minutes. The use of this method is illusthe most idealized situations. The lack of analytical solutrated by a study of the batch precipitation of calcium oxalate tions requires the use of numerical methods, in which the monohydrate. It is shown that growth rates can be determined particle size domain is discretized into a large number of without specific knowledge of the aggregation kernel and that aggregation rate constants can be determined for different aggre-intervals and any of a variety of integration methods are gation kernels. Comparison of simulated and experimental particle applied, a procedure which often entails substantial comsize distributions and their moments show it is possible to distinputational effort ( 4 ) .
guish between the kernels and determine which is appropriate for
The objective of this paper is to introduce a general modeling the aggregation of calcium oxalate monohydrate; it is method that uses the numerical technique of Hounslow et found that a size-independent kernel is most appropriate. α§ 1996 al. ( 4 ) to solve a population balance model and determine Academic Press, Inc.
the rates of growth and aggregation in a batch system.