Pseudocomponents

In practice, mixture optimization problems normally require the presence of aU the components, to obtain a satisfactory product. For example, to produce the membrane for the selective electrode, we need to 

Analyses of variance for the quadratic and special cubic models. The number of distinct mixtures is 10, which permits us to test for lack of fit of the two models 

Source of variation Sum of squares Deg. of freedom Mean square 

Values within parentheses are those of the special cubic model. 

We misled a bit, for didactic purposes. In fact, the components 1,2 and 3 whose proportions appear in Table 7.1 are pseudocomponents, that is, mixtures of the real chemical components. 

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The very basis of the kinetic model is the reaction network, i.e. the stoichiometry of the system. Identification of the reaction network for complex systems may require extensive laboratory investigation. Although complex stoichiometric models, describing elementary steps in detail, are the most appropriate for kinetic modelling, the development of such models is time-consuming and may prove uneconomical. Moreover, in fine chemicals manufacture, very often some components cannot be analysed or not with sufficient accuracy. In most cases, only data for key reactants, major products and some by-products are available. Some components of the reaction mixture must be lumped into pseudocomponents, sometimes with an ill-defined chemical formula. Obviously, methods are needed that allow the development of simple... 

The sulfonate/Genapol pair was assimilated with a pseudocomponent, with the cosurfactant acting only as a solubilizer in the brine used. 

Figure 7. Constrained mixture design, showing relationship between real and pseudocomponents.

The direct optimization of a single response formulation modelled by either a normal or pseudocomponent equation is accomplished by the incorporation of the component constraints in the Complex algorithm. Multiresponse optimization to achieve a "balanced" set of property values is possible by the combination of response desirability factors and the Complex algorithm. Examples from the literature are analyzed to demonstrate the utility of these techniques. 

Computationally, the use of pseudocomponents improves the conditioning of the numerical procedures in fitting the mixture model. Graphically, the expansion of the feasible region and the rescaling of the plot axes allow a better visualization of the response contours. 

The perceptional advantages of response contours in illustrating nonlinear blending behavior and the additional information of the experimental boundary locations were incorporated into a generalized algorithm which determines the feasible region on a tricoordinate plot for a normal or pseudocomponent mixture having any number of constrained components. 

Gorman (7) transformed the mixture variables to pseudocomponents to develop the following equation ... 

Several features make this algorithm particularly attractive for the optimization of a formulation response. The algorithm requires only the input of the lower and upper limits of the individual components and the equation describing the response. Both of these must be expressed in either normal or pseudocomponent form. A randomization procedure generates the initial simplex within the individual component constraints by ... 

Example Optimization of the Four Component Flare Mixture. The formulation of the previously discussed flare example was optimized by the Complex algorithm to yield the maximum intensity. The McLean and Anderson mixture equation (9), fit using normal component values and constraints, produced an optimum formulation at X . 5232, X2 . 2299, Xj . 1669, and X . 0800. The Gorman pseudocomponent equation for the same mixture data (7), constrained using... 

Example Three Component Multiresponse Optimization. An adhesive composition consisting of three components was under developmental evaluation. Twelve formulations were prepared consisting of compositions within the following pseudocomponent ranges ... 

The collective behavior corresponds to the same mixture proportions in bulk phases as at interface, so that the mixture may be treated as a pseudocomponent. This implies the same partitioning for the different species [38]. 

Molal heats of vaporization often differ substantially, as the few data of Table 13.4 suggest. When sensible heat effects are small, however, the condition of constant molal overflow still can be preserved by adjusting the molecular weight of one of the components, thus making it a pseudocomponent with the same... 

The most simple version of our model considers the two-phase nature of the fluidized beds in the reactor and in the regenerator in a simplified way. The kinetic model that we use considers three pseudocomponents in modeling type IV FCC units. This model is a consecutive-parallel model formed of three lumped components and coke as follows ... 

These starting values are used as initial guesses for fitting the model to industrial data and the preexponential factors are changed to obtain the best fit. This is done because the kinetic parameters depend upon the specific characteristics of the catalyst and of the gas oil feedstock. This complexity is caused by the inherent difficulties with accurate modeling of petroleum refining processes in contradistinction to petrochemical processes. These difficulties will be discussed in more details later. They are clearly related to our use of pseudocomponents. But this is the only realistic approach available to-date for such complex mixtures. 

In order that Scheffes simplex lattice designs may be applied to this case, a renormalization is performed and compositions at vertices Aj(j=l, 2,..., q) are taken to be independent pseudocomponents so that for all the range of the local simplex the condition be met ... 

The experimental design is in the pseudocomponent coordinates. All the designs discussed earlier can be built in the new variables Z1 Z2,..., Zq that satisfy the condition of Eq. (3.83). To conduct the experiments it is required to convert the pseudocomponents Z into the initial components with real ratios Xj. For the u-th design point this conversion is defined by the formula ... 

After the design has been realized the coefficients of the regression equation are calculated in pseudocomponent coordinates ... 

The coefficients of the fourth-order regression equation are calculated by Eq. (3.29) using the property of saturated design matrix. The regression equation in pseudocomponent variables has the form ... 

This was the way to select vertices, or their coordinates of the local factor space. It should be noted that those are pseudocomponent coordinates. 

It should be noted that pseudocomponents or coded factors appear in the regression model. A check of lack of fit of the regression model in control points has shown that the regression model is adequate with 95 % confidence. 

The third-order D-optimal design is prepared relative to pseudocomponents Zj, Z2 and Z3 and the content of initial components at the design points is determined by Eq. (3.84). Table 3.40 presents the experimental conditions both in terms of pseudocomponents and on the natural scale (per cent). The sample variance here is S 0.53 and the number of degrees of freedom is f=13. From Eq. (3.105) for viscosity at 0 °C the coefficients have been calculated for the third-order regression equation ... 

A fourth-order D-optimal design is produced with reference to pseudocomponents Zi Z2 and Z3 - Table 3.42. The pseudocomponents satisfy the principal condition for Scheffe s designs. The conversion to initial components at any point within the local simplex studied is carried out from Eq. (3.84). According to this design, an experiment is run with mixtures, each observation being repeated twice. Using Eqs. (3.109)-(3.113) the coefficients of fourth-order regression equation are calculated in pseudocomponents... 

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

Table 1.3 is also an excellent source for critical pressure Pc. If the particular HC compound or mixture is not listed in this table, consider relating it to a similar compound in Table 1.3. If molecular weight and the boiling points are known, you may find a close resemblance in Table 1.3. Also consider the API Technical Data Book, which lists thousands of HC compounds. Grouping as one component per se would also be feasible from Procedure 4A2.1 of the API book. Herein, components grouped together as a type of family could be represented as one component of the mixture. This one representing component may be called a pseudocomponent. Several of these pseudocomponents added together would make up the 100% molar sum of the mixture.

ASTM4 has supplied you with pseudocomponent characterization, molecular weights, acentric factors, critical constants, boiling points, and pseudocomponent gravity. With this database you are prepared to resolve most any crude oil and products database problem, deriving calculated needed results. 

The critical temperature Tc of each pseudocomponent is calculated using the API Technical Data Book, Eq. 4D1.1 [16]. This equation is shown here in Table 1.9, code lines 3260 through 3360. Normally, this equation is good for most all types of hydrocarbons, having an error estimation of 6°F. This equation has been noted to have a maximum... 

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