Second Order Orthogonal Plan

When we select the good value of the dimensionless a, then the corresponding sequential plan remains orthogonal like its CFE basic plan. At the same time, if we do not have any special request concerning a sequential plan, the number of experiments to determine fundamental factors can be drastically reduced to 

In order to comply with the othogonality property, we have to transform the plan described in Table 5.23. For this purpose, we carry out the quadratic transformations of the data given in Table 5.23 by  

For k = 2, a second order orthogonal matrix plan is the state shown in Table 5.25. Due to the orthogonality of the matrix plan, the regression coefficients will 

The relation (5.104) can be particularized to the general case of the second order orthogonal plan when we obtain the following relation for coefScients variances  

So the regression coefficients have been calculated for an orthogonal composition matrix and as a consequence, for the quadratic effect, we obtain the next expressions  

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Table 5.24 Computed a values for a second order orthogonal plan.

Second Order Orthogonal Plan, Example of the Nitration of an Aromatic Hydrocarbon... 

The presentation of this example has two objectives (i) to solve a problem where we use a second order orthogonal plan in a concrete case (ii) to prove the power of statistical process modelling in the case of the non-continuous nitration of an aromatic hydrocarbon. 

To solve this problem we have to use a second order orthogonal plan based on a 2 CFE plan. According to Table 5.24, we can establish that, for a dimensionless values of factors, we can use the numerical value a = 1.414. Table 5.27 contains all the data that are needed for the statistical calculation procedure of the coefficients, variances, confidence, etc., including the data of the dependent variables of the process (response data). 

Even though the second order orthogonal plan is not a rotatable plan (for instance see Eqs. (5.114) and (5.115)), the errors of the experimental responses (from the response surface) are smaller than those coming from the points computed by regression. It is possible to carry out a second order rotatable plan using the Box and Hunter [5.23, 5.27] observation which stipulates that the conditions to transform a sequential plan into a rotatable plan are concentrated in the dimensionless a value where a = for k<5 and a = for k>5 respectively. Simulta-... 

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