Wai Man Chan
M.S. Ch.E., University of Texas at Austin, 1988
Crystallization in the chemical industry is a unit operation of enormous economic importance, with applications to production of fertilizer chemicals, sucrose, pharmaceuticals, pesticides, catalysts and proteins. There are three general types of crystallization frequently used in industries: evaporative, cooling, and precipitation. Evaporative crystallizations are usually carried out under vacuum and depend on the removal of solvent by evaporation, causing the deposition of solute. Systems exhibiting flat solubility curves are generally handled in this way. Cooling crystallizers are those, which are fed a concentrated preheated solution. The crystallizer is then cooled by a refrigerant, resulting in solute deposition due to the reduced solubility at lower crystallizer temperature. Systems exhibiting steep curves can be handled in this way. Precipitation crystallizers are those in which supersaturation is generated by adding a third component in which the solute is insoluble or by performing a chemical reaction, the product of which is insoluble. Among all the characteristics in the crystallization process, the crystal size distribution (CSD) is of overriding importance in determining the ease and efficiency of subsequent solid/liquid separation steps and suitability of the crystals for further processing or sales. In spite of its importance, the fundamental theory of CSD control is largely undeveloped, and industrial control practice is often unsuccessful in stabilizing cyclic CSD behavior. Sustained oscillations in continuous crystallizers are an important industrial problem and can lead to off-specification product, overload of dewatering equipment, and increased equipment fouling.
Population balance formulations have been used extensively to analyze crystallizers' open loop dynamic behavior, and is typically represented in term of partial differential equations. Probably the most popular technique presented in the literature for the solution of population balance equations is the method of moments. The moment method simply takes integrals of the distributed model, and reduce it to a set of ordinary differential equations in the CSD moments. The difficulty in using the moment technique is in reconstructing the complete crystal size distribution based only on the values of the moments. Furthermore, when the model of the crystallizer is complex enough, like size dependent growth rate, fines destruction and classified product removal, the moment method fails to apply. Since most of the industrial crystallizers are usually not simple, therefore, the method of moments has limited application.
The goal of this work is to develop a general model and perform a stability analysis for cooling crystallizer with fines destruction and classified product removal. Chapter 2 shows the development of the formulation. The model consists of one partial differential equation (population balance), two ordinary differential equations (energy balance and mass balance), and several algebraic equations, which include growth and nucleation functions. In order to keep the generality, both functions take into account the Arrhenius type temperature dependency. The growth rate expression also includes the size dependent factor, and the nucleation function accounts for the effect of secondary nucleation as well.
The local asymptotic stability of the nonlinear integro-differential equation model is analyzed by the operator approach, and is shown in chapter 3. First, the nonlinear operator is linearized, and then the stability of the system is determined by the linearized operator's spectrum. This method is more generally applicable than the moment technique.
In chapter 4, a numerical method for the nonlinear model's solution is formulated. The general method of weighted residuals (MWR) is used to approximate the solution of the integro-differential equation model, and the population density function is approximated as a linear combination of basis functions. This method leads to a reduction of the partial differential equation into balances, then form a complete lumped model, which can be solved numerically. Also in this chapter, the general characteristic equation for the eigenvalues of the operator developed in chapter 3 is reduced to an algebraic equation. One can therefore study the stability of the system by analyzing the location of the eigenvalues. Furthermore, a comparison between the loci of eigenvalues obtained from both the linearized operator and Jacobian matrix is shown for the case when moment closure problem is present. The chapter ends with a numerical stability analysis for a simple isothermal MSMPR system, where regions of instability are studied in some chosen parameter spaces.
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University of Wisconsin
Department of Chemical Engineering
Madison WI 53706