Box - Behnken design

Table 1. Three-variable Box-Behnken design for PS foams. Fixed parameters (1) batch size 25 g (2) target density 0.050 g/cm ...

A Box-Behnken design was employed to investigate statistically the main and interactive effects of four process variables (reaction time, enzyme to substrate ratio, surfactant addition, and substrate pretreatment) on enzymatic conversion of waste office paper to sugars. A response surface model relating sugar yield to the four variables was developed on the basis of the experimental results. The model could be successfully used to identify the most efficient combination of the four variables for maximizing the extent of sugar production. 

Cellulase and all chemicals used in this work were obtained from Sigma. Hydrolysis experiments were conducted by adding a fixed amount of 2 x 2 mm oflSce paper to flasks containing cellulase in 0.05 M acetate buffer (pH = 4.8). The flasks were placed in an incubator-shaker maintained at 50 °C and 100 rpm. A Box-Behnken design was used to assess the influence of four factors on the extent of sugar production. The four factors examined were (i) reaction time (h), (ii) enzyme to paper mass ratio (%), (iii) amount of surfactant added (Tween 80, g/L), and (iv) paper pretreatment condition (phosphoric add concentration, g/L), as shown in Table 1. Each factor is coded according to the equation... 

The resulting data of the Box-Behnken design were used to formulate a statistically significant empirical model capable of relating the extent of sugar 3deld to the four factors. A commonly used empirical model for response surface analysis is a quadratic polynomial of the type... 

This model is capable of estimating both linear and non-linear effects observed experimentally. Hence, it can also be used for optimization of the desired response with respect to the variables of the system. Two popular response surface designs are central composite designs and Box-Behnken designs. Box-Behnken designs were not employed in the experimental research described here and will therefore not be discussed further, but more information on Box-Behnken designs can be obtained from reference [15]. 

Near the optimum both the step width and the model of the hyperplane are changed, the latter mostly from a first order model to a second order model. The vicinity of the optimum can be recognized by the coefficients fli,a2,... of Eq. (5.14) which approximate to zero or change their sign, respectively. For the second order model mostly a Box-Behnken design is used. 

FIGURE 8 Separation dashboard describing the influence of the factors on each of the studied responses based on the results of the Box Behnken design. The closeness to a target response that is aimed for each response in this study is represented by a desirability value (d). Reproduced with permission from reference 16. 

Three- or more-level response surface designs, such as three-level full factorial, central composite (CCD), and Box-Behnken designs, are applied in some case studies. 

Another alternative to the 3 full factorial is the Box-Behnken design (Box and Behnken [19]). These designs are a class of incomplete three-level factorial designs that either meet, or approximately meet, the criterion of rotatability. A Box-Behnken design for p=3 variables is shown in Table 2.7. This design will estimate the ten coefficients of the second-order... 

Table 2.8 gives the runs for a Box-Behnken design in four variables. In this table, the runs are grouped in sets of four, each set of four being... 

As was mentioned above for central composite designs, the experimenter can modify these designs if they believe that certain two-factor interactions can be assumed negligible. Box and Jones [20,21] show how this can be done to yield what they call a modified Box-Behnken design that requires fewer runs than the standard Box-Behnken design. 

To construct the central composite design to estimate the coefficients of the second-order model (equation (14)), usually a fractional factorial design of at least resolution V is used. In this case, if the model is valid, then all of the estimates of the main effect coefficients, p., and the interaction coefficients, p. are imbiased. An alternative to the central composite designs for estimating the coefficients of the second-order model are the Box-Behnken designs or the designs referenced in Section 2.2.5. 

BOX-BEHNKEN DESIGN FOR THREE DESIGN VARIABLES AND FOUR ENVIRONMENTAL VARIABLES... 

Note that even if the environmental variables are at two levels so that a 2 fractional factorial design can be run for the Taguchi environmental array, the complete crossed design has 9x8 = 72 runs, still more than the Box-Behnken design of Table 2.12, while providing estimates of fewer coefficients of the second-order model. 

For this reason one prefers to apply an experimental design. In the literature a number of different designs are described, such as saturated fractional factorial designs and Plackett-Burman designs, full and fractional factorial designs, central composite designs and Box-Behnken designs [5]. 

There are a number of other designs that have advantages in particular circumstances. Many are just variants on the theme of maximally spanning the factor space. The Box-Behnken design is a two-level, spherical, rotatable design (Box and Behnken 1960). For three factors it has experiments at the center and middle of each edge of a cube (figure 3.14). 

Figure 3.14. Box-Behnken design for three factors. The design is spherical and rotatable.

The wall of a knotted reactor was covered with a chelating agent which retains Cd. A factorial design was employed to ascertain the main factors influencing the system and then the final experimental conditions were determined using a Box-Behnken design... 

A. S. Souza, W. N dos Santos and S. L. C. Ferreira, Applieation of Box Behnken design in the optimisation of an on-line pre-eoneentration system using knotted reactor for cadmium determination by flame atomic absorption spectrometry, Spectrochim. Acta, Part B, 60(5), 2005, 737-742. 

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