Date of Award
Doctor of Philosophy (PhD)
Dr. T.W. Hoffman
The overall objective of this program was to develop mathematical and experimental techniques, and thus obtain a mathematical model, capable of predicting the crystal size distribution as a function of batch time, for a seeded batch crystallizer in which alumina trihydrate was crystallizing in a supersaturated-aqueous solution of sodium aluminate.
In this regard, a 2 bench-scale crystallizer was designed so as to operate in a similar way (with respect to mechanisms and rates) as industrial crystallizers in that gentle mixing allowed significant agglomeration to occur during the crystallization. Thus, nucleation, growth and agglomeration rates had to be predicted in the crystallizer model as a function of the crystallizer operation conditions.
Reproduceable and representative samples were extracted from this crystalliser by a developed sampling system which did not contaminate or affect the sample nor the crystallizer contents. Analytical techniques were developed to obtain sodium aluminate and solids concentration. Crystal size distribution was measured for each sample by a special Coulter counter equipped with a Channelizer and a Log-Transformer apparatus. This equipment measured crystal volume with a resolution of 100 size intervals over a spherical equivalent diameter range of 4 to 80 μm. The lack of measurements below 4 μm was a source of trouble in the modeling program, and was the main source of error in the mathematical analyses using the particle size distribution data.
A special mathematical method based on the population balance was developed to investigate and determine each of the rate equations for the processes growth, agglomeration, and nucleation individually from the obtained batch data. The method is termed here the method of pseudo moments'. It allows for investigation within feasible computer resources and circumvents the confounding of each of these processes with each other.
The growth rate was modeled by a two-dimensional birth and spread mechanism, while the model for the agglomeration rate was based on a mechanism of free-in-space binary collisions with an agglomeration effectiveness kernel. This kernel was modeled by the product of a crystal environment term and a crystal size effect term. The form of the environment term suggests that the agglomeration rate is related to the growth rate lending support to the concept that alumina trihydrate crystals after collision are grown together to form true polycrystalline crystals and are not a 'flock' of crystals held together by physical attractive forces. The size-dependent term is made up of an inertial impaction mechanism term which accounts for the collision frequency, and a term which accounts for the efficiency of the collision. It was shown that crystals in the intermediate size range (ca. 10 to 30 μm) agglomerate most effectively. The expression for the nucleation rate, which was also developed using the pseudo-moment method suggests that the formation of nuclei in this experimental crystallizer at 85°C was not via normal homogeneous or heterogeneous nucleation mechanisms but rather occurred through attrition of very small particles from the seed particles (a sort of 'dusting-off' of nuclei-sized particles). These rate expressions still need to expanded for the effect of temperature, of intensity of agitation, and of impurities.
Although the method of pseudo moments allows for the determination of rate expressions from size distribution data, it does not provide a means of predicting the evolution of size distributions. To this end a model based on the population balance, mass balances, and rate equations was formulated. However, such a model, if it includes agglomeration, is impractical for investigation of rate equations due to long numerical solution times.
Because of the significance of the three rate processes the resulting model was made up of a set of algebraic equations and an integro-first order non-linear hyperbolic partial differential equation, with one of the integrals being a convolution integral. Numerical solution schemes of such an equation are prone to exhibit stability, accuracy, and long solution time problems. The stability and accuracy problems were solved and the long solution time problem minimized by the development of two numerical solution schemes. One was based on a developed weighted central difference approximation for the partial derivate with respect to crystal size and integrating by the Runge-Kutta Merson technique along a rectangular grid with respect to batch time. The other was based on solution along the characteristics of the equation and interpolating between the 'characteristic grid' and a specially developed rectangular grid for the integral terms. This grid allows for the convolution nature of one of the integrals. The resultant quadrature was fast and accurate. Depending on the type of crystallization process and the shape of the evolving size distribution either one or the other is best suited with regard to speed of solution and accuracy.
The method of pseudo moments and numerical solution schemes allow for any growth, agglomeration, and nucleation expressions and any form of size distributions. Thus the numerical methods developed here for the determination of rate equations and prediction of size distributions are applicable to any particulate process, such as polymerization, microbial reactions, etc.
Groeneweg, Pieter G., "Development of an Al(OH)₃ Crystallization Model Based on Population Balance" (1980). Open Access Dissertations and Theses. Paper 603.