Analysis of the Variances for Two Factors Processes

When we investigate the effect of two factors on a process response, then the collected data will be as shown in Table 5.41. Here the differences between the observed values along one line present the effect of the change of Xj from Uj to whereas the differences between the observed values along one column are the result of the change of Xj from Pj to P . Each value of the table represents an observation that corresponds to a grouping of factors. Here, we can have one or more measurements of the process response, but frequently only one measurement is used. 

In this case, conversely to the residual variance, we can propose two zero hypotheses the first is HjqI the variance of the response values determined by the change of factor Xj has the same value as the residual variance the second one is H20 the variance of the response values (when X2 factor changes) is similar to the residual variance . With these hypotheses we indirectly start the validation of two others assumptions (i) the equality of the mean values of the lines (related to Hjo), (ii) the equality of the mean values of the columns (related to H2o°)-The splitting of the total variance into parts associated to Table 5.41 follows a procedure similar to that for the analysis of the variances of a monofactor process, as previously explained. In this case, we introduce the sums of the squares Sj, S2, 

Sj that are defined using Eqs. (5.155)-(5.159). Then, we compute the variances of the data of the lines (Sj), the variances of the data of the columns(S2) and the residual variance of all data (sj ). Then, the sums for the computation of the analysis of the variances of two factors processes are  

It is not difficult to observe, when we compare this example with the analysis of variances of a monofactor processes, that sum S3 is the only one to be completely new. The other sums, such as Sj and S2, remain unchanged or are named differently (here, S4 is similar to the S3 of the analysis of variances for a monofactor process). The corresponding number of degrees of freedom is attached to S2, S3 

The analysis of the catalytic oxidation of SO2 developed previously in this chapter, can be completed as follows (i) the experiments with catalysts number 2 and number 6 are eliminated (ii) new experiments are introduced in order to consider the temperature as a process factor. All the other factors of the catalytic process keep the values from Table 5.40. In Table 5.43 we present a new set of experimental results in order to obtain more knowledge of the effect of the type of catalyst and the temperature on the degree of oxidation. The correspondence between the different types of catalysts reported in Tables 5.43 and 5.40 are respectively 1 1, 2 3, 3 4, 4 5. As has been explained above, the inlet gas composition, the gas flow rate and the length of the catalytic bed remain unchanged for all experiments, the last limitation is imposed in order to obtain the smallest errors in the measurements for the process response [5.32]. 

---