Two-factor design

For example, consider a two-factor design with each factor at two levels. This is also a form of all-possible-combinations experiment. One item we note here is that there is more than one way to describe the form of an experiment, and we include a short digression here to explicate this multiplicity of ways of describing an experiment. In this particular case, we have two factors, each at two levels. We can describe it as a listing of values corresponding to each experiment (Table 10-1). 

Rotatability implies that the confidence in the predictions depends only on the distance from die centre of the design. For a two factor design, this means that all experimental points hi a circle of a given radius will be predicted equally well. This has useful practical consequences, for example, if the two factors correspond to concentrations of acetone and methanol, we know that the further the concentrations are from the central point the lower is the confidence. Methods for visualising diis were described in Section 2.2.5. Rotatability does not depend on the number of replicates in the centre, but only on die value of a, which should equal j/Nj, where Nf is the number of factorial points, equal to 2k if a full factorial is used, for diis property. Note that the position of the axial points will differ if a fractional factorial is used for the cubic part of die design. 

One of die most popular approaches is called simplex optimisation. A simplex is file simplest possible object in A-dimensional space, e.g. a line in one dimension and a triangle in two dimensions, as introduced previously (Figure 2.32). Simplex optimisation implies that a series of experiments are performed on the comers of such a figure. Most simple descriptions are of two factor designs, where the simplex is a triangle, but, of course, there is no restriction on the number of factors. 

For example, let us say that there are four factors of potential interest. We can begin with the two-factor design defined by the first four rows in Table 6.19, taking care to keep the levels of factors 3 and 4 fixed at 0 in all runs. Then, when we wish to study the influence of the two factors that have been kept fixed, we only have to add to the initial design the runs corresponding to rows 5-11 in the table (and their negatives, of course). 

We explain in words how the generating functions seek to make the design. First, multiply the first and the second brackets in Eq. (23.7) to create a two-factor design of A, B, together with all the interactions AB and average (1). Then multiply by the next bracket. This multiplication says... 

A one-factor-at-a-time optimization is consistent with a commonly held belief that to determine the influence of one factor it is necessary to hold constant all other factors. This is an effective, although not necessarily an efficient, experimental design when the factors are independent. Two factors are considered independent when changing the level of one factor does not influence the effect of changing the other factor s level. Table 14.1 provides an example of two independent factors. When factor B is held at level Bi, changing factor A from level Ai to level A2 increases the response from 40 to 80 thus, the change in response, AR, is... 

A 2 factorial design with two factors requires four runs, or sets of experimental conditions, for which the uncoded levels, coded levels, and responses are shown in Table 14.4. The terms Po> Po> Pfc> and Pafc in equation 14.4 account for, respectively, the mean effect (which is the average response), first-order effects due to factors A and B, and the interaction between the two factors. Estimates for these parameters are given by the following equations... 

In Situ Bioremediation. In situ bioremediation can be an aerobic or anaerobic process, or a combination of the two. In designing an in situ bioremediation system, one should consider the types of microorganisms available (naturally in place or added), the stmctural and chemical makeup of the soil matrix, types of contaminants, oxygen and nutrient addition and distribution, and temperature. These factors are discussed prior to introducing the individual techniques for in situ bioremediation. 

Further contrast between metal and composite stiffeners is revealed when we examine the objectives and characteristics of stiffener design. For a metal stiffener of uniform or even nonuniform thickness, we attempt to maximize the moment of inertia of the stiffener in order to maximize the bending stiffness of the stiffener. Those two factors are proportional to one another when we realize that the bending stiffness of metal stiffeners about the middle surface of the plate or shell to which they are attached is... 

As discussed in the preceding section, filter bags must be periodically cleaned to prevent excessive build-up of dust and to maintain an acceptable pressure drop across the filters. Two of the three designs discussed, reverse-flow and reverse-pulse, depend on an adequate supply of clean air or gas to provide this periodic cleaning. Two factors are critical in these systems the clean-gas supply and the proper cleaning frequency. 

Two factors militate against the universal use of water-based fluids. Very severe machining operations call for a lubrication performance that is beyond the capacity of such fluids, and the design of some machine tools means that water cannot be used because of the risk of cross-contamination with machine lubricants. In these instances, neat cutting oil is the only fluid that can provide the required performance. 

The ideal design is one in which ail parts can be operated satisfactorily with water flowing with the least turbulence and aeration, and at a rate of flow within the limits that the materials involved can securely withstand. These limits, with regard to flow-rate limitations, vary with the material, as described in Section 1.2, but turbulence, aeration or presence of suspended particulates can lower these limits considerably, and designs that eliminate these two factors go a long way towards preventing impingement attack, which can be the major cause of failures in sea-water systems. (See also Sections 1.6 and 2.1.)... 

As can be seen, two factors are particularly critical (a) the density of the particle, since heavier particles are more difficult to fluidize, and (b) particle size, since the necessary gas velocity varies as the square of the particle diameter. The design of the reactor is also important since gas velocity at the top must be less than the terminal velocity of the particles, otherwise they would be blown out of the bed.P l... 

---