Quadratic mixture model

Of the 14 formulations listed in Table 1, six experimental runs were required to fit the quadratic mixture model, four additional distinct runs were used to check for the lack of fit, and finally four runs were replicated to provide an estimate of pure error. Design-Expert used the vertices, the edge centers, the overall centroid, and one point located halfway between the overall centroid and one of the edge centers as candidate points. Additionally, four vertices of the design region were used as check points [106],... 

Determine the b, and 6 2 values and their standard errors for the quadratic mixture model. Assume that the variance is the same for any mixture, so the variances of the three compositions can be pooled. 

The second solution is to reduce the size of the experimental domain. Usually the smaller the domain, the less the total curvature, and therefore there exists a simpler satisfactory model within the reduced domain. Third- or higher-order designs are rarely used (except for mixture models) and for the quadratic model to be sufficient the extent of the experimental region must normally be restricted. 

We are sure you will not be very surprised when we say that fitting mixture models is nothing more than a special case of least-squares fitting. What we did in the last section, actually, was the same as solving Eq. (5.12) for a set of 17 observations the 15 appearing in Table 7.1, plus the two observations made at the center point of the triangle. We used two models for this the quadratic one, with six parameters, and the special cubic one, with seven. The statistical significance of these models can be evaluated by an analysis of variance. 

Figure 5.15 Mixture properties modeled by quadratic interaction parameters k...

The model Mr+1 contains the r-th degree term in the mixture components only along with the product of this term with the first degree terms in the Zj s. For example, a planar or first-degree model in the mixture components, and a main effects only model in the process variables, is y=M1+i+e. A planar model in the Xj s, combined with a main effect plus first-order interaction effects model in the Zj s, would be y=Mi+i+Mi+2+ . The model containing up to quadratic blending terms by main effects in the Zfs is defined as y=Mi+i+M2+l+H. This continues, up to the complete 2q+n-2n term model that is defined as ... 

Naturally, the number of initial experiments required to start the optimization procedure will increase if either the number of parameters considered or the complexity of the model equations increases. As far as the number of parameters is concerned, we have seen this to be true with any optimization procedure, and hence the number of parameters should be carefully selected. In order to avoid a large number of initial experiments, the complexity of the model equations may be increased once more data become available during the course of the procedure. For example, retention in RPLC may be assumed to vary linearly with the mixing ratio of two iso-eluotropic binary mixtures at first. When more experimental data points become available, the model may be expanded to include quadratic terms. However, complex mathematical equations, which bear no relation to chromatographic theory (e.g. higher order polynomials [537,579]) are dangerous, because they may describe a retention surface that is much more complicated than it actually is in practice. In other words, the complexity of the model may be dictated by experimental... 

As an example, the design [q = 3, v = 2 for three mixture variables and a quadratic (v = 2) model consists of all possible combinations of the values ... 

Any apparently more general relationship involving an intercept term and pure quadratic terms can by use of (5-15) be shown to be equivalent to (5-19) in the mixture context.) Relationships of the type of (5-19) are often called Scheffe models, after the first author to treat them in the statistical literature. Other more complicated equation forms are also useful in some applications, but we will not present them in this chapter. The interested reader is again referred to Cornell25,26 for more information on forms that have been found to be tractable and effective. 

Of course, some general aspects of our treatment could be easily extended to a general form of f b ireJ as in the semi-infinite case [226],but for explicit numerical work a specific form of fs(b ire) ((()) is needed. Equation (10) can be justified for Ising-type lattice models near the critical point [216,220], i.e. when ( ) is near ( >crit=l/2, as well as in the limits f]>—>0 or <()—>1 [11]. The linear term —pj( ) is expected due to the preferential attraction of component B to the walls, and to missing neighbors for the pairwise interactions near the walls while the quadratic term can be attributed to changes in the pairwise interactions near the walls [144,216,227]. We consider Eq. (10) only as a convenient model assumption to illustrate the general theoretical procedures - there is clear evidence that Eq. (10) is not accurate for real polymer mixtures [74,81,82,85]. 

The Kumar equation of state (Kumar, 1986 Kumar et al., 1987) is a modification of the Panayiotou-Vera model that was developed to simplify the calculations for multicomponent mixtures. Since the Panayiotou-Vera equation is based on the lattice model with the quasichemical approach for the nonrandomness of the molecules in the mixture, the quasichemical expressions must be solved. For a binary system the quasichemical expressions reduce to one quadratic expression with one unknown, but the number of coupled... 

Another possibility is to obtain physico-chemical models. Consider for instance the optimization of pH and solvent strength. The subject was studied among others by Marques and Schoenmakers [66], Schoenmakers et al. [67,68] and Bourguignon et al. [62] for the optimization of the separation of a mixture of chlorophenols. Marques and Schoenmakers proposed the following model based on a quadratic relationship between log k and the solvent composition () and on dissociation equilibria ... 

Figure 4.12 Competitive isotherms of the (-)- and (+)- enantiomers of Troger base on microcrystalline cellulose triacetate, with ethanol as mobile phase, (a) Single-component adsorption isotherm of (+)-TB (squares) and (-)-TB (triangles) at 40° C. Experimental data and best fit to a Langmuir, (+)-TB, and a quadratic, (-)-TB, isotherm model, (b) Competitive isotherms of (-)-TB in enantiomeric mixtures for increasing concentrations (0, 0.5,1,1.5,2, 2.5, 3 g/L) of (+)-TB, calculated with IAS theory, (c) Competitive isotherms of (+)-TB in enantiomeric mixtures for increasing concentrations (0, 0.5, 1, 1.5, 2, 2.5, 3 g/L) of (-)-TB calculated with IAS theory. Reproduced from A. Seidel-Morgenstem and G. Guiockon, Chem. Eng. Sci., 48 (1993) 2787 (Figs. 4, 6, and7).

Figure 12.29 Comparison of theoretical and experimental displacement separations of resorcinol and catechol by phenol. Calculations using the equilibrium-dispersive model, the LeVan- Vermeulen isotherm model, and single-component adsorption data. Experimental results on a 4.6x250 CIS Nucleosil 5 fim column, F = 0.4 carrier, water, Fj, = 0.2 mL/min, T = 20°C 1 1 mixture, = 0.5 mL displacer, 80 g/L phenol in water = 30%, Lf = 16.5%. (a) Calculation with LeVan-Vermeulen isotherm, (b) Calculation with quadratic isotherm, three floating parameters, (c) Calculation with competitive Langmuir isotherm, single-component isotherm parameters, (d) Calculation with Langmuir isotherm, best adjusted parameters. Reproduced with permission from. C. Bellot and J.S. Condoret, J. Chromatogr., 657 (1994) (Figs. 3c, 4c, 6c, 8c) 305.

The above argument enables us to select experimental designs for the binary mixture. For a linear model, values for the pure substances are required. For the quadratic model, 3 equally spaced points are needed. For the cubic model, 4 points are required. From the point of view of sequentiality, one can go from a first-order... 

Each of the coefficients of these models may be a function of composition of the mixture, the dependence being linear, quadratic, or other. Suppose there is one process variable, as in model 9.14, for which the coefficients have a quadratic dependence on the composition, so that ... 

Fig. 7.3. The dashed line represents a linear model for a mixture of the two components, y = h x + 62 2- The h and coefficients are response values for the pure components 1 and 2. The quadratic model is represented by the solid curve. In addition to the terms of the linear model, it contains a term describing the interaction of the two components, 612 1X2.

If the linear model does not prove satisfactory, we can try to fit a quadratic model, as we have already seen. For a mixture of three components, the general expression for the quadratic model contains 10 terms ... 

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