Mixture space

As the number of components increases, so does the dimensionality of the mixture space. Physically meaningful mixtures can be represented as points in this space  

There are a number of common designs which can be envisaged as ways of determining a sensible number and arrangement of points within the simplex. 

These designs are probably tire most widespread. For k factors they involve performing 2k — 1 experiments, i.e. for four factors, 15 experiments are performed. It involves all 

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Figure 12.29 Conversion of an orthogonal mixture space to an equilateral triangular mixture space. See text for details.

Figure 12.30 Alternative visualization of the equilateral triangular mixture space.

The results of three-component mixture designs are often presented as response surfaces over the triangular mixture space as shown in Figure 12.34. The Scheffe model parameters are seen to be equivalent to the responses at the vertexes. 

The construction of an experimental design for this separation problem is complicated because both mixture and process variables are present. The former variables, which describe the composition of a mixture in terms of fractions, usually result in design spaces which are a subspace of a simplex (e.g. of a triangle or a tetrahedron). Process variables, on the other hand, are really independent. The design space is often a square or a cube. In this paper there are four mixture variables and two process variables. The design space is therefore a part of a tetrahedron in the mixture space, and a square in the process variables space. 

A simulation experiment is performed to validate the method, which can be formulated in algorithms. The applicability of the algorithms to practice is tested by means of the performance of extraction experiments for a group of sulphonamides. The response (partition coefficient) is modelled versus the composition of the extraction liquid. The models are used to predict the criteria within the entire mixture space. 

The most economical procedure for a liquid-liquid extraction would be a single step extraction, since extraction procedures including several steps with the same or with different solvents are laborious and economically disadvantageous. Optimisation of extraction of more than one solute, which give different selective interactions (different response surfaces in the same mixture space), may require several extraction steps with different optimal extraction solvents or separate analysis of each analyte. However, procedures can be used, which select a composition of the extraction liquid that provides satisfactory partition coefficients or extraction yields for all solutes to be extracted. 

In an optimisation procedure involving the minimal partition coefficient, the minimal partition coefficient is calculated using all compositions of the extraction liquid (all possible combinations of x, X2 and within the mixture space). The highest value calculated for this minimal partition coefficient (the maximal minimal partition coefficient) is the optimal value and hence the composition where the partition coefficient of the worst extractable substance is highest. 

Any design can be checked for orthogonality, simply by determining the correlation coefficients between the concentrations of the various compounds. If the correlations are 0, then the design is a good one, and will result in a training set that spans the possible mixture space fairly evenly, whereas if there... 

For a mixture with q variables (i.e., q components), the mixture factor space is a subspace of the respective -variables in Euclidean space. In Figure 8.5, Figure 8.6, and Figure 8.7, we see the relationship between the mixture coordinate system and the respective Euclidean space. Figure 8.5 illustrates the case of a binary mixture. The constraint described by Equation 8.11 holds for points A, B, and C however, only point B and all points on the heavy line in Figure 8.5 are points from the mixture space satisfying the conditions described by the constraints in Equation 8.10 as well. 

One can also combine the process-variable factor space with a constrained mixture space. Figure 8.14 shows an example of the combined space constructed from three mixture variables with lower and upper bounds and one process variable. 

Most chemists represent their experimental conditions in mixture space, which corresponds to all possible allowed proportions of components that add up to 100%. A three component mixture can be represented by a triangle (Figure 2.31), which is a two-dimensional cross-section of a three-dimensional space, represented by a cube, showing the allowed region in which the proportions of the three components add up to 100 %. Points within this triangle or mixture space represent possible mixtures or blends ... 

These experiments are represented graphically in mixture space of Figure 2.33 and tabulated in Table 2.34. 

Another class of designs called simplex lattice designs have been developed and are often preferable to the reduced simplex centroid design when it is required to reduce the number of interactions. They span the mixture space more evenly. 

The experiments fall in exactly die same pattern as the original mixture space. Some authors call the vertices of die mixture space pseudo-components , so the first pseudo-component consists of 70% of pure component 1, 10% of pure component 2 and 20% of pure component 3. Any standard design can now be employed. It is also possible to perform all the modelling on the pseudo-components and convert back to the true proportions at die end. 

An upper bound is placed on each factor in advance. The constrained mixture space often becomes somewhat more complex dependent on the nature of die upper bounds. The nick is to find the extreme corners of a polygon in mixture space,... 

If this condition is not met, the constrained mixture space will resemble an irregular polygon as in Figure 2.35(b). An example is illustrated in Table 2.40(b). 

Note that in some circumstances, a three factor constrained mixture space may be described by a hexagon, resulting in 12 experiments on the comers and edges. Provided that there are no more than four factors, the constrained mixture space is often best visualised graphically, and an even distribution of experimental points can be determined by geometric means. 

Double the number of experiments, by taking the average between each successive vertex (and also die average between the first and last), to provide 2v experiments. These correspond to experiments on die edges of die mixture space. 

Represent the constrained mixture space, diagrammatically, in the original mixture space. Explain why the constraints are possible and why the new reduced mixture space remains a triangle. 

We introduce an abstract n-dimensional vector space U and we call it the mixture space. In it we select bases q, and l for now it is sufficient to assume that these bases are othonormal, i.e. Iq, = l , cf. Appendix A.4. In the space U, we define the vectors of molar masses M and reaction rates J by... 

Figure 2 Representation of three-factor mixture space.

Catalyst Catalys t weight (mg) Gas flow rate (cm /min) Inlet gas mixture Space velocity, X, (cm /g catmin) Ref. 

P. Bravais, Y. Batonneau, D. Amariei, C. Kappenstein and M. Theron, Experimental investigation of catalytic ignition of cold 02/H2 mixtures, Space Propulsion 2008, Heraclion, Greece, May 2008, 3AF Publisher, Paper S51. 

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