Central composite design response surface

Appropriate designs might be based on factorial designs (full or fractional) or a central composite design. Response surface methods frequently rely on visualization of the data for interpretation. 

The latest DoE was focussing on the axial and swirl stream temperature (Tswiri) the rotation speed (n) in a face-centred central composite (CCF) response surface design with those three factors (/=3) on three levels. Levels were set linearly as mentioned in section Improved Experimental Setup so that N = 2 +2/+1 = 15 experiments were required for this model. The centre point was repeated five times to ensure reproducibility and reasonable model validity. Particle size, span, particle shape, surface roughness, flowabDity and BET surface area were chosen as responses to evaluate the significant effects of the factors on these particle properties [34, 35]. 

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]. 

Replication is often included in central composite designs. If the response surface is thought to be reasonably homoscedastic, only one of the factor combinations (commonly the center point) need be replicated, usually three or four times to provide sufficient degrees of freedom for s. If the response surface is thought to be heteroscedastic, the replicates can be spread over the response surface to obtain an average purely experimental uncertainty. 

Based on the obtained response surface, a second roimd of optimization follows, using the steepest ascent method where the direction of the steepest slope indicates the position of the optimum. Alternatively, a quadratic model can be fitted around a region known to contain the optimum somewhere in the middle. This so-called central composite design contains an imbedded factorial design with centre... 

If the F-test is significant then there is evidence of a quadratic effect due to at least one of the variables. With the present design, however, the investigator will not be able to determine which of the variables has a quadratic effect on the response. Additional experimentation, perhaps by augmenting the current design with some star points to construct a central composite design (see section on central composite designs below), will need to be conducted to fully explore the nature of the quadratic response surface. 

A class of augmented designs, first proposed by Box and Wilson [6] and frequently applied in response surface work, is the central composite design. Composite designs consist of ... 

Now if each of the design points in the central composite design is replicated five times, so that the complete design has 75 runs, then at each design point we can calculate the average response and the standard deviation of the response. The analysis techniques associated with response surface methodology can then be applied to fit separate models to... 

Figure 3.8. Possible configurations of nine sets of factor values (experiments) that could be used to discover information about the response of a system. Dotted ellipses indicate the (unknown) response surface, (a) A two-factor, three-level factorial design (b) a two-factor central composite design.

Five factors for hydride generation were studied to develop a method to determine As in gasolines. A central composite design was used to develop the response surface and, thus, optimise the extraction procedure... 

A screening design detected significant instrumental and chemical variables to volatilise and measure Sb. They were optimised using response surfaces derived from central composite designs. Findings were confirmed using artificial neural networks... 

FAAS, ETAAS The variables implied on an ultrasound-assisted acid leaching procedure were evaluated by experimental designs. The relevant variables were subsequently optimised by a central composite design and response surface... 

The conditions to determine the three metals were assessed by experimental design and, Anally, optimised by response surfaces developed after applying central composite designs... 

The full factorial central composite design includes factorial points, star points, and center points. The corresponding model is the complete quadratic surface between the response and the factors, as given by Eq. 1 ... 

Many designs for use in chemistry for modelling are based on die central composite design (sometimes called a response surface design), die main principles of which will be illustrated via a three factor example, in Figure 2.29 and Table 2.31. The first step,... 

Cocaine has been extracted from coca leaves and the optimization procedure was investigated by means of a central composite design [17]. Pressure, temperature, nature, and percentage of polar modifier were studied. A rate of 2 mL/min CO2 modified by the addition of 29 % water in methanol at 20 M Pa for 10 min allowed the quantitative extraction of cocaine. The robustness of the method was evaluated by drawing response surfaces. The same compound has also been extracted by SEE from hair samples [18-20]. 

The central composite design was often selected because of the limited number of experiments needed to sample the response surfaces. In the separation of As and Se species in tap water, the analysis of isoresponse curves allowed the determination of optimum chromatographic conditions and the robustness of the method [77]. The same design was also used to study the influence of an organic modifier and IPR concentration on retention of biogenic amines in wines. To obtain a compromise between resolution and chromatographic time, optimization through a multi-criteria approach was followed [78]. 

Probably, the very long time used in most cases could be significantly reduced by using a multivariate optimization approach focusing on interrelated variables. Figure 5.4 shows a surface response from a central composite design for [HCI]-ultrasound exposure for the determination of tin in coal acidified slurries [6]. 

Figure 5.4. Response surface estimated from the central composite design obtained for the pair [HCIJ-US exposure time In the determination of tin in coal acidified slurries. (Reproduced with permission of Elsevier, Ref [6].)...

They may allow extension of the experiment at a subsequent stage to a central composite design for modeling of response surfaces (shown in the following sections). 

Example Optimization of a synthetic procedure by response surface modelling from a central composite design. Enamine synthesis by a modified TiCl -method... 

Central Composite Designs. CCDs are the most often used response surface designs (1,7,17).These designs are constructed by combining a two-level full factorial design (2 experiments), a star design (2/experiments), and a center point, which is often replicated a number of times. Thus, to examine / factors, at least = 2 + 2/ + 1 experiments are required. For more... 

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