Optimization Based on Theoretical Considerations

It would be very attractive to derive analytical expressions for the optimum experimental conditions from the solution of a realistic model of chromatography, i.e., the equiUbriiun-dispersive model, or one of the lumped kinetic models. Approaches using analytical solutions have the major advantage of providing general conclusions. Accordingly, the use of such solutions requires a minimum number of experimental investigations, first to validate them, then to acquire the data needed for their application to the solution of practical problems. Unfortunately, as we have shown in the previous chapters, these models have no analytical solutions. The systematic use of these numerical solutions in the optimization of preparative separations will be discussed in the next section. 

A combination of the solution of the ideal model with a simple model of band broadening, following the approach initially suggested by Knox and Pyper [17], permits one to accoimt for the influence of the column efficiency on the production rate of the second component. Some useful conclusions have been reached, on which we report here. 

1 Simultaneous Optimization of the Production Rate and Recovery Yield using the Ideal Model 

FeUnger and Guiochon [18] maximized the product of the production rate and the recovery peld using the ideal model. The recovery yield for the more retained component is [2]  

For the definition of the cycle time, they considered the time required for regeneration and re-equilibration of the column after each nm. Assuming that j column volumes of solvent are needed to regenerate the column, the cycle time will be defined as the analytical retention time of the more retained component plus j times the void time. With this definition, the production rate can be expressed as  

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Optimization Based on Theoretical Considerations where X is given by Eq. 18.27b and... 

Kouvelis et al. (2004) present a relatively simple multi-period MILP plant location model for global production network design with investment decisions only allowed in the first period. The production system consists of component-dedicated manufacturing sites and final assembly sites. It is limited to two production levels and one final product. The objective function maximizes the NPV of the production network. The main purpose of the model is to analyze the effects financing subsidies, tax regimes, tariff structures and local content requirements have on optimal network design. The analysis is based on theoretical considerations and a numerical example. More complex aspects of international trade such as duty drawbacks are not considered. 

Selection of components and their proportions are, in most cases, based on theoretical models and consecutive trials. There is a place for considerable improvement in the material design by the application of the optimization approach, from which objective indications may be obtained (cf. Section 12.7). The optimization of concrete mixtures for multiple design objectives is a challenging and important area of research. Spatial gradation of the material composition and structure adds a new set of design parameters that must be accommodated within such an optimization framework. 

Based on these considerations, it is generally true that the smallest thickness (or radius) that gives a conveniently and accurately measurable AT is preferred. However, even with optimal thickness-to-area ratios for a disk-shaped sample (or radius-to-length for a cylindrical sample), unacceptably high heat losses usually occur. These losses can be corrected theoretically or mathematically only for very special cases. Such corrections are not feasible for high polymers, especially near or above room temperature. 

In the past, several theoretical models were proposed for the description of the reversed-phase retention process. Some theories based on the detailed consideration of the analyte retention mechanism give a realistic physicochemical description of the chromatographic system, but are practically inapplicable for routine computer-assisted optimization or prediction due to then-complexity [9,10]. Others allow retention optimization and prediction within a narrow range of conditions and require extensive experimental data for the retention of model compounds at specified conditions [11]. 

The theoretical description of the reaction kinetics is based on results of ab initio investigations.29,56,57 The most reliable estimates of the heats for both reaction channels were reached using the Gaussian-2 (G2) method.58"61 The results of ab initio calculations by Jodkowski et al.29 show that the reaction mechanism is considerably more complex than was previously expected. This apparently elementary gas-phase reaction proceeds with the formation of intermediate complexes of reactants or products. The optimized structures of the intermediates important for the reaction kinetics are featured in Fig. 2. Three transition state structures were found for the separate reaction paths. 

Nxuneroxis publications have discussed the issue of optimization of the experimental conditions in preparative chromatography. Most papers in this area are based on empirical observations. They report on the conclusions derived by people who have acquired long-term experience and familiarity with the method. Each author has dealt with a variety of problems, but the scope and range of these problems vary considerably from author to author. Without a solid theoretical background to sift through this experience and place it into perspective, the validity of the conclusions and, more importantly, the range in which they are valid are still much in doubt. 

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