Fractionated designs

A typical primary distillation product pattern at a coke-oven tar-processing plant is given in Table 1. At some coke-oven distilleries, only one fraction, designated naphthalene oil, is taken between 180 and 240°C. Two fractions, light creosote or middle oil (230—300°C) and heavy creosote or heavy oil (above 300°C), are taken between the naphthalene oil and pitch. 

More shortcut design methods and rules of thumb have been developed for fractionation than probably any other unit operation. For example the paper reprinted in Appendix 5 on development of shortcut equipment design methods contains 18 references for fractionation shortcut methods out of 37 total. Both the process and mechanical aspects of fractionation design have useful rules of thumb. Many of the mechanical design rules of thumb become included in checklists of do s and don ts. 

Van Winkle, M. and W. G. Todd, Optimum Fractionation Design by Simple Graphical Methods, Chem. Eng. Sept 20 (1971), p. 136. 

Plackett and Burman [1946] have developed a special fractional design which is widely applied in analytical optimization. By means of N runs up to m = N — 1 variables (where some of them may be dummy variables which can help to estimate the experimental error) can be studied under the following prerequisites and rules ... 

The conservation of oxygen for the mass fraction designation by Y follows from Equation (3.22). 

A fractional factorial design is often suggested to observe several parameters at the same time. Advantages are, among others, the formal sampling plan, which is easy to evaluate by supervisors and auditors. Moreover, a fractional design only needs a fraction (usually about 50%) of experiments compared to a design that tests the relevant parameters one by one. 

In this chapter we explore factorial-based experimental designs in more detail. We will show how these designs can be used in their full factorial form how factorial designs can be taken apart into blocks to minimize the effect of (or, if desired, to estimate the effect of) an additional factor and how only a portion of the full factorial design (a fractional replicate) can be used to screen many potentially useful factors in a very small number of experiments. Finally, we will illustrate the use of a Latin square design, a special type of fractionalized design. 

As we will see in Section 14.11, fractional factorial designs are often used to look for important factors. If it turns out that one or more factors is unimportant (say factor JCj), then a fractional factorial design can be collapsed into a less fractional design in a lower-dimensional factor space. Figure 14.6 shows an example of collapsing the design. 

R.A. Maclean and V.L. Anderson, Applied Factorial and Fractional Designs, New York, Marcel Dekker, 1984. 

The most commonly used designs are half-fractional designs and saturated fractional designs. 

Half-fractional designs are constructed by assuming that all interaction effects higher than first order can be assumed to be negligible. For a study... 

Half-fractional designs reduce the number of experiments by half for a two level design and can prove very efficient, however the number of experiments can still be prohibitive when a large number of factors require... 

Saturated fractional designs are constructed on the assumption that all interaction effects can be assumed to be insignificant and the number of experiments is now reduced to k +1. 

These limitations can be seen by comparing the experimental procedure for a three factor, three level star design, shown in Table 5.10, with the experimental procedure for a reflected saturated fractional design, which also tests three factors at three levels, shown in Table 5.11. 

The experimental scheme for a three level reflected saturated fractional design for seven factors is shown in Table 5.15 ( note that one factor was retained as a dummy factor to be used as an additional error check). The experimental order of the scheme was sorted on acid type as this required long equilibration times, this ordering loses some of the features of the initial design but is a compromise that can be justified on the fact that... 

For this particular design, there are only as many coefficients as experiments (8), and so there no way to estimate how good the model is (e). As the number of factors increases, the possible experiments that can be done (AT = L ) increases faster than the number of coefficients in the model (see fractional designs below). 

Table 2.4 Matrix design of a 2 fractional design. Generator C=AB.

Sample treatment was studied by the saturated fractional design considering volumes and concentrations of acids, temperatures, ramp time and hold time for the microwave heating. An optimised programme was set after the central composite study... 

The ratios of components both for 10 extreme vertices and for designs of experiments n=4 and 8 are given in Tables 3.7 and 3.8. Levels or ratios of components Xj, X2 and X3 in the design of experiment n=4 are generated from a fractional design 23 1 with generating ratio X3=XjX2, Table 3.9. 

Figure 3.36a Fractional design in each point of a simplex-centroid design...

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