Experimental design

22 j_L= ^{g∙ δL2}

3ν (2.7)

𝑄_𝐿, 𝑛, 𝑏, and 𝑔 denote the volumetric flow rate of the liquid phase, the number of channels, the width of the microchannel and the acceleration due to gravity, respectively.

The Schmidt number is used to indicate the ratio between the viscosity (µ_𝐿) and the mass diffusivity (𝐷_𝐺,𝑏) and may be calculated using Equation 2.8.

Sc_L= ^µL

ρL ∙ DG,b (2.8)

FFMRs also afford the opportunity for continuous operation which makes the prospect of industrial scale processes a promising venture. The transition from pilot or laboratory scale systems to an industrial scale process may be achieved by the numbering-up of microstructured units or relevant flow features such as number of channels or channel length (Chen et al., 2008, Ziegenbalg et al., 2010).

23 Box-Wilson central composite design

Box-Wilson central composite design (more commonly referred to as central composite design) involves investigating three classes of experimental conditions, viz. cubic design points, axial design points and central design points. Considering the range of interest for a particular design variable (i.e. temperature, catalyst concentration, flow rate, etc.) and expressing the bounds of this range as the -1 and +1 levels, the centre of this range may be expressed as the 0 level. Cubic design points may then be described as levels -1 and +1 of a particular design variable. Central design points may then be described as the set of experimental conditions at the 0 level of the design variable. The axial design points are experimental conditions which lie outside of the initial parameter range selected and are located at the ±  level.  quantifies the rotatability of the experimental design and is based on the number of factorial points which are investigated, if a full factorial design is used then  may be calculated using Equation 2.9 (Lazić, 2004). 𝐹 is the number of design variables to be varied throughout the experimental study.

α = (F)^{1 4⁄} (2.9)

The concept of rotatability was first introduced by Box and Hunter (1957) and is used to ensure that reasonable predictions of responses are made throughout the investigation region. One way of ensuring this is to ensure that the variances at all points (cubic, central and axial) are equal. When this is achieved the experimental design is then termed a rotatable design (Box and Hunter, 1957, Montgomery, 2008).

From Equation 2.9, the experimental conditions of the axial design points may be calculated by substituting - and + into Equation 2.10. 𝑋_𝑚𝑖𝑛 and 𝑋_𝑚𝑎𝑥 represent the initial range of the design variable chosen (i.e. the -1 and +1 levels).

X_i= ^{α .(Xmax−Xmin)+ Xmax+ Xmin}

2 (2.10)

An important aspect during experimentation is a test of the variability and consistency of the experiment which can be evaluated by repeating a set of experimental conditions and observing the deviations thereof. Central composite design accounts for the variability of experiments by specifying replicate experiments of the central design points. The number of replicates to be performed is at the discretion of the experimenter, but it is recommended that a minimum of 3 replicates be performed (Lazić, 2004).

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There are three types of central composite designs, namely; circumscribed, inscribed and face centred designs. A circumscribed experimental design is the original method of employing central composite designs. It positions the axial design points a distance ±  from the central design point and is based on the number of factors of interest.  for a circumscribed experimental design is greater than 1 thus the axial design points form the new bounds of the experimental region of investigation (Sematech, 2006).

An inscribed experimental design is employed when the limits of investigation are clearly defined and the axial design points are positions at the edges of the design space. An inscribed design is thus a circumscribed experimental design which has been scaled down by  (Sematech, 2006).

Circumscribed and inscribed central composite designs require five levels for each design variable i.e. the -, -1, 0, +1 and + levels of the design variable.

A face centred design has three levels for each process variable where the axial design points are located at the middle of each face thus the value of  is approximately 1. Figure 2.9 provides an illustrative example of the location of these design points for a design which has two factors.

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- 1 + 1

Circumscribed

Inscribed

Face centered

Figure 2.9: Illustrative example of a circumscribed, inscribed and face centred Box-Wilson central composite design for a two factor experiment. Axial design points () and cubic design points ().

It is important to note that the circumscribed central composite design spans the greatest design space as can be seen in Figure 2.9 and is thus the most useful when trying to gauge the effect of experimental conditions over a large design space (Sematech, 2006).

The number of experimental runs required for a full central composite experimental design may be determined by Equation 2.11 where 𝑘 is the number of process variables to be varied while 𝐶_𝑝 refers to the number of central design points chosen (Souza et al., 2005).

N = 2^k+ 2k + C_p (2.11)

Box-Benhken experimental design

A Box-Behnken experimental design is another example of a multivariate technique used in optimization studies and consists of experimental conditions chosen at the midpoint of the chosen

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range for a particular design variable (Figure 2.10). A Box-Behnken experimental design only consists of the central and midpoint design points and thus consists of less experimental runs which utilises resources more efficiently (Sematech, 2006). The number of experimental runs required may be determined using Equation 2.12 (Souza et al., 2005).

N = k²+ k + C_p (2.12)

Figure 2.10: Illustrative example of three factor Box-Behnken experimental design. Central design points () and midpoints design points ().