Optimization nonlinear isotherm

To achieve this goal a new optimization strategy has to be applied. The most difficult step is to adjust the flow-rate ratios in sections II and III, especially for strongly nonlinear isotherms with competitive interaction. Therefore, the mn-mln plane is divided into segments by parallel lines to the diagonal. Each of these lines represents combinations of mn-mm values of constant feed flow - the so-called iso-feed lines (Fig. 7.25). 

Figure 7.26 Optimized axial concentration profile (system with nonlinear isotherms).

Optimization of the Operating Conditions for a Nonlinear Isotherm Using the... 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

The main difference between the chromatographic process carried out in the linear and the nonlinear range of the adsorption isotherm is the fact that in the latter case, due to the skewed shapes of the concentration profiles of the analytes involved, separation performance of a chromatographic system considerably drops, i.e., the number of theoretical plates (N) of a chromatographic system indisputably lowers. In these circumstances, all quantitative models, along with semiquantitative and nonquantitative rules, successfully applied to optimization of the linear adsorption TLC show a considerably worse applicability. 

This chapter presents an introduction to the key issues of reactor-based and reactor-separator-recycle systems from the mixed-integer nonlinear optimization perspective. Section 10.1 introduces the reader to the synthesis problems of reactor-based systems and provides an outline of the research work for isothermal and nonisothermal operation. Further reading on this subject can be found in the suggested references and the recent review by Hildebrandt and Biegler (1994). 

Section 10.2 describes the MINLP approach of Kokossis and Floudas (1990) for the synthesis of isothermal reactor networks that may exhibit complex reaction mechanisms. Section 10.3 discusses the synthesis of reactor-separator-recycle systems through a mixed-integer nonlinear optimization approach proposed by Kokossis and Floudas (1991). The problem representations are presented and shown to include a very rich set of alternatives, and the mathematical models are presented for two illustrative examples. Further reading material in these topics can be found in the suggested references, while the work of Kokossis and Floudas (1994) presents a mixed-integer optimization approach for nonisothermal reactor networks. 

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... 

Nonlinearity of the Langmuir adsorption isotherms is observed even in noncompetitive chromatographic processes. Individual adsorption isotherms can be found experimentally using frontal analysis at overload conditions however, the adsorption isotherms in the separation of mixtures are different because of the interference of other compounds in the mixture. In PHPLC method development, it is necessary to optimize separation conditions and column loading experimentally. 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

Before starting a preparative chiral separation it is essential to identify a chiral stationary phase (CSP) exhibiting good chiral recognition ability. This is usually done with an analytical column because it is less substance- and time-consuming. A stationary phase mostly composed of silica gel with only a few chiral elements will be rapidly overloaded. In this event, even if the phase exhibits useful properties for analytical purposes, it will not be appropriate for preparative applications this is the case for protein-based phases [102, 103], Most chiral stationary phases have relatively low saturation capacity, so the enantiomer separations are usually done under strongly nonlinear conditions [103], Accordingly, the accurate determination of the adsorption isotherms of the two enantiomers on a CSP is of fundamental importance to allow computer-assisted optimization to scale up the process. 

This chapter introduces fundamental aspects and basic equations for the characterization of chromatographic separations. Starting from the simple description of an analytical separation of different compounds the influences of fluid dynamics, mass transfer and thermodynamics are explained in detail. The important separation characteristics for preparative and process chromatography, e.g. the optimization of resolution and productivity as well as the differences compared with chromatography for analytical purposes, are described. Especially, the importance of understanding the behavior of substances in the nonlinear range of the adsorption isotherm is highlighted. 

Most determination methods finally lead to discrete loading versus concentration data that have to be fitted to a continuous isotherm equation. For this purpose it is advised to use a least-squares method to obtain the parameters of the isotherm. Nonlinear optimization algorithms for such problems are implemented in standard spreadsheet programs. To select an isotherm equation and obtain a meaningful fit,... 

A problem often encountered in nonlinear optimization is the necessity to provide suitable initial guesses. Therefore, it should be tested if different initial guesses lead to drastically different sets of isotherm parameters. This is connected to the sensitivity problem, which is pronounced in the case of multi-component systems where several parameters need to be fitted at once. Substantial initial guesses are often difficult to find and the sensitivity is often low, which demands much experimental data or leads to the selection of other isotherm equations. 

For preparative chromatography, where we almost always have to deal with high feed concentrations and nonlinear adsorption isotherms, the following approach to the appropriate choice of feed concentration during model-based optimization is recommended ... 

---