Central composite designs axial points

Fig. 6. Distribution of experimental points in central composite designs factorial points, O centre point, x axial points...

Three-Factor Central Composite Design with Axial Values a and Four Center Points... 

Table 2.33 Position of the axial points for rotatability and orthogonality for central composite designs with varying number of replicates in the centre.

Fig. 4 Central composite design for three factors. The factorial points are shaded, the axial points unshaded, and the center point(s) filled.

If the experimental region is defined by maximum and minimum values of each factor, then the domain is cubic. The central composite design can be applied to such a situation, the axial points being set then at 1, coded values corresponding to the minimum and maximum allowed values. Other designs for the cubic domain are reviewed in Ref... 

The 2 design with centre point can be extended by adding experiments along each of the axes at values a of the other coded variable (A, = a, Xj = 0 and X, = 0, 2 = a). These are called axial points. The result is the central composite design for 2 factors. If a is set equal to 1, as in figure 5.1b, the design is also a full factorial design at 3 levels (3 ), quite often used for studies on 2 factors. 

The similarity to the central composite designs is clear. These also consist of a factorial and an axial design, but which involve all of the variables and are both centred at the same point. Hybrid designs exist for 3, 4 and 6 variables. They are designed for a second-order model, for the investigation of a fixed spherical zone of interest, most of the experiments being on the surface of a sphere. They are not suitable for a sequential approach. All experiments should be done in one block, in a random order. 

Sometimes known as Box-Wilson designs, these are the same as the central composite designs for the spherical domain, except for the positioning of the axial points. These are face-centred, with a = 1. An example is given for 3 factors in table A3.4. They may be derived easily by applying the rules given in chapter 5. 

The central composite design (CCD) is based on the full quadratic polynomial. Hence it is composed of 2 factorial design, center points and 2k axial portion of design. 

This design locates the star or axial points on the centers of the faces of the cube as shown in Figure A3. This variation of the central composite design requires only three levels of each factor, and in practice it is frequently difficult to change factor levels. 

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]. 

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