Mathematical Modeling of the Separation Process of Chromatography and Estimation of Parameters
University of Kansas
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Chromatography is widely used as a technology for separating mixtures of compounds by partitioning into the mobile and stationary phases. A mathematical model is essential not only for predicting the retention time and the peak shape of the chromatography analyte concentration distribution, but also for understanding the separation mechanism of chromatography and detecting whether the conditions were correct (e.g., whether there was an overload of the sample). A variety of statistical distribution functions such as exponential, Gaussian (normal), exponential modified Gaussian, Weibull, log-normal have been used to approximate the chromatography analyte concentration distributions, and were further applied to the deconvolution of stacked peaks. The dissertation consists of five chapters. The first chapter presents an overall introduction of the current prevailing mathematical models of chromatography analyte concentration distributions, the generalized chromatography theorem derived from chromatography table and its proof, the relation between on-chromatography analyte concentration distributions and out flow analyte concentration distributions, the asymptotic distribution of on-chromatography analyte concentration distributions and out flow analyte concentration distributions and their applications. The second chapter presents the mathematical model for the separation process of chromatography. In this chapter the first generalized theorem for modeling almost all types of chromatography was developed, and was found to match the mathematical formulas for well-known discrete distribution functions. These empirical formulas were rigorously proven by mathematical induction based on chromatography principle and chromatography process assumptions. The outflow chromatography analyte concentration distributions are demonstrated by simulation to be better approximated by the mathematical model that matches the negative binomial distribution function versus using a Gaussian distribution function, which currently is widely used for approximation. The third chapter establishes the mathematic bridge between on-chromatography and outflow analyte concentration distributions. In following with the previous chapter, which found the on chromatography and outflow analyte concentrations distributions to mathematically match the binomial and negative binomial distributions, respectively, this mathematical bridge can apply to relate these statistical distributions given they mathematical formulas are the same. This theorem is rigorously proved by mathematical induction. This relation is also demonstrated by 3D-plot of on-chromatography and outflow analyte concentration distributions for the first several stages. The fourth chapter proposed the transformation of data collected by chromatography (i.e., the analyte concentration distributions from chromatography experiments) into data that can be used for estimation to the approximate the underlying parameters that govern a particular chromatography process. Outflow chromatography analyte concentration distribution from original data were used to demonstrate this process and to compare the approach derived in this work using parameter estimated by method of moment (MOM) to the currently approach based on the Gaussian statistical distribution’s formula The fifth chapter is the summary of my dissertation work.
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