Separation conditions simplex optimization

The function of RPS can aid setting up the chromatographic separation conditions. The optimization of a separation can be done experimentally based a chromatographer s experience and intuition, or with computer control based on the "Chemometrics", i.e., "Simplex" procedure (4,5). Although such approaches are reliable and promising enough to use for practical purposes, this is a time-consuming task because a number of experiments required with the increase factors for optimization. On the other hand, it is possible to optimize a separation in RPS with almost the same accuracy as obtained in the conventional Chemometrics approaches within a few minutes. 

Figure 3. Four parameter, simplex-optimized SFC separation of a 12-component mixture. Chromatographic conditions as in Vertex 13 of Table II. Sample components isoquinoline, n-octadecane (n-CigH3g), naphthalene, quinoline, acetophenone, undecylbenzene, benzophenone, 2 -acetonaphthone, diphenylamine, o-dioctylphthalate, unidentified impurity, N-phenyl-1-naphthylamine, phenanthrene quinone. Other conditions as described in the experimental section.

Later, Felinger and Guiochon [4] discussed the four-dimensional simplex optimization of a separation by overloaded elution. This permits the simultaneous optimization of the column design parameters (column length and average particle size), and the operating conditions (mobile phase flow velocity and sample size). Systematic calculations were made to study the influence of the retention factor, which is not usually considered as an optimizable factor but has a profound... 

The alternative is to employ a multivariate optimization procedure such as Simplex. Simplex is an algorithm that seeks the vector of parameters that corresponds to the separation optimum within an n-dimensional experimental space. For example, a two-parameter CE separation optimized by Simplex would begin with three observations of the separation response at three different electrolyte conditions. These conditions are chosen by the analyst, often his or her best guess. From the evaluation of the response of each observation, the algorithm chooses the next experimental condition for investigation (4). As with the univariate method, the experiments continue until an optimal separation condition is determined. The disadvantage of such an approach is that it is unknown how many experiments are required to achieve an optimum, or if the optimum is local or global as the entire response surface is not known. 

Window Diagram" or "Simplex" methods (1—5). This approach is very useful to get optimal separation conditions however, it has one major limitation. It necessitates all materials of interest be available. 

Termination of the Simplex. The decision of when to stop the simplex is somewhat subjective. Upon reaching the vicinity of the optimum conditions, it is advisable to discontinue the simplex, as many experiments could be wasted as described by Schoenmakers (33). On the other hand, it is important to conduct a sufficient number of experiments to establish that the simplex is in this vicinity. Although statistical comparisons of the best results (e.g., best 20% as in Table II) may prove to be helpful, it should also be recognized that it is not always necessary to fully optimize a given separation. In our view, whenever a set of experimental conditions that provides the desired separation within the maximum specified analysis time has... 

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

The main disadvantage of the simplex method consists in the laige number of experiments required to find optimal working conditions. Further, the optimisation criterion characterises the separation of the sample mixture by a single number, so that the detailed information on the separation of the individual sample components is lost and because of the high probability that the search method will slide into a region with a local maximum of the optimisation criterion, the simplex optimisation method can be expected to be fully successful only with the separations of relatively simple samples. 

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