Box-Wilson

Fig. 5.6. Schematic representation of the Box-Wilson optimization with step width h... 

The well-known Box-Wilson optimization method (Box and Wilson [1951] Box [1954, 1957] Box and Draper [1969]) is based on a linear model (Fig. 5.6). For a selected start hyperplane, in the given case an area A0(xi,x2), described by a polynomial of first order, with the starting point yb, the gradient grad[y0] is estimated. Then one moves to the next area in direction of the steepest ascent (the gradient) by a step width of h, in general... 

The optimum found by sequential proceeding, both by Box-Wilson and simplex technique, is that local optimum situated nearest the starting point. It must not inevitably be identical with the global optimum. Therefore, it may be useful to repeat the optimization procedure one or several times. 

Initial screens can be distinguished between methods that are used to determine what factors are most important, and follow-up screens that allow optimization and improvement of crystal quality (Table 14.1). In experimental design, this is known as the Box-Wilson strategy (Box et al., 1978). The first group of screens is generally based on a so-called factorial plan which determines the polynomial coefficients of a function with k variables (factors) fitted to the response surface. It can be shown that the number of necessary experiments n increases with 2 if all interactions are taken into account. Instead of running an unrealistic, large number of initial experiments, the full factorial matrix can... 

Table 2 Box-Wilson Type Fractional Factorial Design...

Second-order Rotatable Design (Box-Wilson Design)... 

Statistical optimization methods other than the Simplex algorithm have only occasionally been used in chromatography. Rafel [513] compared the Simplex method with an extended Hooke-Jeeves direct search method [514] and the Box-Wilson steepest ascent path [515] after an initial 23 full factorial design for the parameters methanol-water composition, temperature and flowrate in RPLC. Although they concluded that the Hooke-Jeeves method was superior for this particular case, the comparison is neither representative, nor conclusive. 

In this study, ethanol extraction using SC CO2 was achieved from 15 to 75 %(v/v) synthetic ethanol solutions and also from fermentation broth. Effects of temperature, pressure, extraction time and initial ethanol concentration on extraction yield were investigated in the ranges of 313 to 333 K and 80 to 160 atmospheres. Optimum conditions for the batch extraction of 15%(v/v) ethanol solution were found using the Box-Wilson optimization method in a linear form. In addition, the effect of consecutive solvent feeding on extraction yield were investigated both with 15 %(v/v) ethanol solution and with fermentation broth. 

In literature, inconsistent results were given with respect to the effect of temperature and pressure on extraction yield at low ethanol concentrations [4,7,12]. For this reason, an experimental design was performed utilizing the Box-Wilson method to observe this effect more clearly and for the optimization of the experimental conditions. 

Experimental design by Box-Wilson method and the experimental results obtained... 

The Box-Wilson experimental designs are a general series of experiments that have been developed to efficiently serve as a basis for deriving the mathematical model of a physical process. Their usefulness is enhanced in the study of industrial applications because most physical situations can usually be approximated by a quadratic function over a reasonable range of the factors. For a two-factor system, the generally used form of this model is... 

Boiling liquids, heat-transfer coefficients for, 597 Bonds, interest rate on, 248-249 Book value definition of 277 in replacement analysis, 332 Box-Wilson design, 766, 769 Break-even chart, 155-156... 

Modeling of n-decane dehydrogenation using Box-Wilson experimental design... 

Linear alkyl benzene (LAB) is manufactured by catalytic dehydrogenation of C10-C13 n-parafifins, followed by alkylation with benzene. High product selectivity, and reasonable catalyst life, in the dehydrogenation reaction, are obtained at the expense of conversion, by adjusting reaction parameters. Proper choice of reaction parameters is thus of paramount importance in this reaction. The present study, was carried out with n-decane, as model feed, and a promoted Pt/ALOs catalyst. A composite Box-Wilson experimental design was adopted to develop an empirical model for predicting monoene yield as a function of reaction conditions. Further, the model was used for determination of optimum reaction parameters. 

Key words n-decane dehydrogenation, optimization of reaction parameters, model for n-decene yield, Box-Wilson design... 

Four important reaction parameters, viz. reactor inlet temperature, reactor inlet pressure, hydrocarbon feed LHSV and H2 HC mole ratio were chosen for the study. A full factorial experimental design is desirable, but four variables at 5 levels requires 625 experimental points even a 1/5 fractional factorial constitutes 125 experiments. Hence a Box-Wilson... 

In conclusion, the Box-Wilson composite design is a convenient method for modeling of product yield as a function of reaction parameters (independent variables) especially when their number exceeds 2. In the dehydrogenation of n-decane, effect of reaction parameters on monoene selectivity / yield and n-decane conversion are represented satisfactorily by full II degree polynomial equations. The canonical form of the equations in the present study is indicative of an approximately stationary ridged system, with the reaction parameters close to center of design being optimum for monoene yield at conversion levels of 12 - 13 %. The polynomial equations were found to be consistent, with mechanistic considerations. 

---