Estimation of batch precipitation kinetics by a simplified differential method

Abstract

Crystallization from solution is a unit operation of great economic importance in the chemical industry. The uses of this separation and purification technique range from the small-scale manufacture of fine chemicals to bulk production, such as the precipitation of aluminum trihydroxide in the Bayer process. In addition to the purity and shape of crystals, Ž . the crystal-size distribution CSD is usually a key quality consideration for a crystallization product. The product CSD is determined by the relative rates of crystal growth, agglomeration and nucleation, which are controlled by the process conditions and the chemistry of the system. A population-bal-Ž . ance PB -based model, the theory for which can be found in Ž . Randolph and Larson 1988 , provides a mathematical tool to describe the relationship between the CSD and process conditions.This research note describes a method based on a discretized PB to estimate the rates of growth, agglomeration, and nucleation from experimental batch CSD data. This socalled ''inverse'' problem is posed as a nonlinear optimization, where an objective function based on the difference between the experimental and model-predicted values is minimized with respect to the kinetic parameters. It is shown that in the case when the objective function is formulated in terms of the time derivatives of the measured variables, the original problem can be reduced into three smaller subproblems that can be rapidly solved.The method is illustrated using data from a batch aluminum trihydroxide precipitation experiment. The validity of the estimates is evaluated by comparing the experimental CSD with that obtained from the simulation using the estimated kinetics. Estimates from our method are also compared to the estimates obtained using the differential tech-Ž . nique of Bramley et al. 1996 .

Influence of the Estimation Procedure on the Accuracy and Precision of Aluminum Trihydroxide Crystallization Kinetics from Dynamic Data

“…More recently, Bramley et al . and Livk et al have presented techniques which take advantage of a discretized population balance (DPB) approach to determine crystallization kinetics.…”

“…They occur simultaneously, making it difficult to identify the individual contributions to the overall CSD. This is a common problem in the study of industrial crystallization systems, which is illustrated by the, all too common, occurrence of different values for crystallization kinetic parameters being reported by different authors studying the same system. − Taking the aluminum trihydroxide system, for example, White and Bateman and Veesler and Boistelle tabulate the activation energies for the growth rate reported by different laboratories and show that they vary from 53 to 120 kJ mol -1 . In a further example of this problem, Li et al found that predictions of the aluminum trihydroxide CSD obtained using available literature correlations for the crystallization kinetics in this system − were in poor agreement with their experimental CSD data.…”

Optimal parameter selection for nonlinear multistage systems with time-delays

“…An optimal parameter selection problem is a type of optimal control problem in which a finite set of parameters need to be chosen optimally subject to a given dynamic system. Many common dynamic optimization problems can be formulated as optimal parameter selection problems, such as parameter identification problems [6][7][8] and controller parameter design problems [9,10]. There are many useful theories and algorithms for solving optimal parameter selection problems for nonlinear systems with a single stage and no time-delays; see, for example, [11][12][13].…”

Incremental and simultaneous identification of reaction kinetics: methods and comparison

“…This shows the preference for methods that are easy to apply such as the di erential method. New procedures are therefore being developed (Gonzalez-Tello et al, 1996;Kamenski and Dimitrov, 1993;Yeow et al, 2003) and the di erential method ÿnds also use in other areas of chemical engineering such as crystallization (Livk et al, 1999).…”