8
0
A.J. Bonon et al. / Journal of Catalysis 319 (2014) 71–86
where n is the number of variables in study. If each variable occurs
at only two levels, we deal with a so-called factorial design, mea-
suring a response at two levels (for example the product yield
or/and selectivity). In a catalytic reaction, parameters such as con-
centrations of substrates, oxidants and catalysts, along with the
temperature can be chosen. In the notation for the design matrix,
the symbol +1 (or +) is used for the upper level of each factor
and the symbol ꢁ1 (or ꢁ) stands for the corresponding lower level.
A third level called central point, denoted as 0, may be introduced
to increase resolution of the design and also to measure the repro-
ducibility of the experiments, as the central point is usually per-
formed in triplicate. It is possible to extend the experimental
the response is the Pareto Chart. In this graph, the p-value of the
linear and quadratic variables and its interactions are displayed
as bars in crescent order; a line, representing the standard p-value
(the SL), is plotted to indicate the desired limit of significance. The
bars with absolute value greater than the SL cross this line, thus
causing a strong influence on the studied response. Another very
important tool, once the model has proved to be representative
by the F-test, is a response surface. This graphical representation
demonstrates the system behavior for a response as a function of
the studied variables, through which the optimum point of the sys-
tem can be determined.
In the present work, we studied the optimization of the limo-
3
design adding axial points, denoted as
a. In these assays, one vari-
nene epoxidation using a 2 experimental design with 17 assays
able is extended beyond the limits of the design (ꢁ1 and +1),
(six axial points and a center point in triplicate). The initial
amounts of limonene (substrate), hydrogen peroxide (oxidant)
and alumina (catalyst) were taken as variables. Initial rates,
denoted W0, of the accumulation of internal and external epoxides
as well as diepoxides were taken as responses. The values used for
design evaluation were obtained from the samples after it reacted
for 10 h. The established values for the DoE are shown in Table 3.
The studied interval was chosen by considering the initial variation
results. The coded values and the responses obtained experimen-
tally are summarized in Table 4. The responses obtained for the
central point show insufficient variation, indicating a good repro-
ducibility of the process.
whereas the others variables are fixed in the central point. The
pffiffiffiffiffiffiffiffiffi
4
n
coded value of ±
a
is calculated by the formula
a
¼
ð2 Þ, where
n is the number of studied variables in the design. Thus, for a
qffiffiffiffiffiffiffiffiffi
4
3
design with three variables ðꢅ
a
¼
ð2 ÞÞ, the values ꢁa and +a
are equal to ꢁ1.68 and +1.68, respectively. In addition to the infor-
mation about the effect of each variable on the response, the two-
level factorial design gives information about the so-called interac-
tion (that is synergic or antagonistic) effects between two or more
variables, which cannot be obtained using the classical ‘‘one-fac-
tor-at-a-time’’ method. It is clear, however, that if many variables
are involved, it is necessary to carry out a huge number of experi-
ments, and the method becomes cumbersome without any practi-
cal usefulness. For example, if we consider seven variables, we
3
.5.1. Evaluation of initial accumulation rate for internal and external
epoxides
This section discusses the results obtained for the responses of
initial accumulation rate of internal (Table S1) and external
7
should perform 2 = 128 experiments. However, there are statisti-
cally smart ways to select experiments, by creating suitable frac-
tions of the complete set. Application of fractional design at two
levels allows us to reduce the number of experiments. For example,
(Table S2) epoxides. Analyzing the p-values and Pareto chart
(Figs. S14 and S15), it is possible to notice that, in the studied inter-
4ꢁ1
by statistically choosing only 2
= 8 experiments from the com-
2
val, in both cases, only the variable x (alumina) was statistically
4
plete set consisting of 2 = 16 runs, it is possible to obtain almost
the same amount of information and use the obtained results in
a further study of more depth. A statistical hypothesis test is a
method of making decisions using data. The obtained data are eval-
uated by analysis of variance (ANOVA), which statistically assesses
the representability of a mathematical model by the application of
a F-test at a certain confidence interval (CI) or confidence limit.
Typically, the confidence interval used for experimental data is
significant for the accumulation of monoepoxides. This differs from
the interval established based on the initial results.
2
The regression of the models gave the R = 0.85 (Table S3) and
R = 0.67 (Table S4) for internal and external epoxides, respec-
2
Table 3
The established values for the design of experiments.
9
5% or 90%. The chosen CI value corresponds to a level of signifi-
Variable
Code
ꢁ1.68 (ꢁ
0.32
0.02
2.64
a
)
ꢁ1
1.00
0.05
4.00
0
1
1.68 (+a)
cance, i.e. a 90% confidence interval reflects a significance level
Limonene (mmol)
Alumina (g)
x
1
2.00
0.10
6.00
3.00
0.15
8.00
3.68
0.18
9.36
(
SL) of 0.1. The significance level is compared with the p-value
x2
x
3
(
probability value) obtained in the regression coefficients table. If
2
H O
2
(mmol)
the p-value is lower than the SL, it means that the factor has a sta-
tistically significant impact in the evaluated response. Once the
analysis of all factors in the ANOVA table is constructed, F-test
evaluations assure the representability of the model. The F-test
Table 4
The design matrix.
Assay Variables
Responses, W0 (mmol min 1
ꢁ
)
(
or Fisher test) is a statistical hypothesis test used to evaluate
the equality of two variances by taking the ratio of two variances
in this case the data obtained in the lab and the data from the pro-
x
1
x
2
x
3
Int. epoxide Ext. epoxide Diepoxides
(
ꢁ0.90 ꢁ0.98 ꢁ0.96 3.08 ꢀ10ꢁ
3
6.00 ꢀ10
6.81 ꢀ10
7.76 ꢀ10
1.38 ꢀ10
5.51 ꢀ10
7.32 ꢀ10
9.02 ꢀ10
1.79 ꢀ10
1.47 ꢀ10
6.81 ꢀ10
4.20 ꢀ10
1.26 ꢀ10
6.65 ꢀ10
1.00 ꢀ10
8.17 ꢀ10
8.96 ꢀ10
9.37 ꢀ10
ꢁ4
ꢁ4
ꢁ4
ꢁ3
ꢁ4
ꢁ4
ꢁ4
ꢁ3
ꢁ3
ꢁ4
ꢁ4
ꢁ3
ꢁ4
ꢁ3
ꢁ4
ꢁ4
ꢁ4
1.97 ꢀ10
9.79 ꢀ10
8.06 ꢀ10
0
ꢁ4
ꢁ5
ꢁ4
1
2
3
4
5
6
7
8
9
posed mathematic model) and verifying if this ratio does not
exceed a certain theoretical value found in the Fisher table (or F-
table). If the calculated F (Fcal) value of the regression residues [cal-
culated from the Degrees of Freedom (DF) with the sum of squares
ꢁ3
ꢁ3
0.79 ꢁ0.98 ꢁ0.96 2.96 ꢀ10
ꢁ0.90
0.79
0.97 ꢁ0.96 5.73 ꢀ10
0.97 ꢁ0.96 8.83 ꢀ10
ꢁ
ꢁ
ꢁ
ꢁ
3
3
3
3
ꢁ4
ꢁ4
ꢁ3
ꢁ4
ꢁ3
ꢁ5
ꢁ5
ꢁ4
ꢁ4
ꢁ4
ꢁ4
ꢁ4
ꢁ4
ꢁ0.90 ꢁ0.98
0.91 3.41 ꢀ10
0.91 4.14 ꢀ10
0.91 4.30 ꢀ10
0.91 1.01 ꢀ10
2.89 ꢀ10
1.92 ꢀ10
1.96 ꢀ10
6.60 ꢀ10
1.31 ꢀ10
9.79 ꢀ10
8.21 ꢀ10
6.07 ꢀ10
1.35 ꢀ10
4.87 ꢀ10
3.47 ꢀ10
3.73 ꢀ10
3.55 ꢀ10
0.79
ꢁ0.90
0.79
0.97
0.97
0.97
(
(
SS) resulting in the mean square (MS)] is larger than the table F
tab) value, it means that the mathematic model appropriately
F
ꢁ2
ꢁ
ꢁ
ꢁ
ꢁ
3
3
3
3
describes the experimental data. There is also a relationship
ꢁ1.64
0.04 ꢁ0.07 4.54 ꢀ10
between the F-test and a parameter t (taken from Student’s t-test),
10
11
12
1.93
0.04 ꢁ0.07 2.96 ꢀ10
2
0.16 ꢁ1.67 ꢁ0.07 1.93 ꢀ10
where F = t . When regression coefficients are generated, it is pos-
2
0.16
0.16
0.16
0.16
0.16
0.16
1.73 ꢁ0.07 8.15 ꢀ10
0.04 ꢁ1.63 3.91 ꢀ10
sible to obtain R to give us an idea of the data quality, as is regu-
ꢁ3
ꢁ3
1
1
3
4
2
larly done with R in a standard calibration curve in analytical
0.04
1.59 5.59 ꢀ10
ꢁ
ꢁ
3
3
3
chemistry. The DoE generates large amounts of data and offers sev-
eral ways to analyze them and display the results. One visual
option to evaluate how each variable and its interactions impact
15
0.04 ꢁ0.07 5.38 ꢀ10
0.04 ꢁ0.07 6.12 ꢀ10
1
1
6
7
0.04 ꢁ0.07 6.00 ꢀ10ꢁ