Fitting of the experimental data of response variable (i.e. % v/v H2 content of product gas) to the quadratic regression model using coded values of independent variables listed in Table 2.1B yielded following equation:
1 2 3 1 2 1 3 2 3
2 2 2
1 2 3
40.897 1.097 2.334 3.917 0.374 0.208 1.108
8.264 3.514 2.629
Y X X X X X X X X X
X X X
= + + + − − −
− − − (2.7)
The values of the response variable predicted by the quadratic response model have also
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predicted values of the response variable are in close agreement, which indicates the best fit of the model to experimental data. The statistical analysis of the quadratic response model is given in Table 2A that lists the coefficients of the model (given in eq. 2.1) along with their p– and t–values. Analysis of variance (ANOVA) of the quadratic model is given in Table 2.2B. ANOVA results show that regression model was highly significant (p<
0.01), while the lack of fit was not significant (p> 0.05). Coefficient of determination (R2) was 0.998, which indicated that 99.8% variation in the response variable could be explained using the model. R2 = 0.998 also indicated best fit of the model to the experimental data, which is also corroborated by close agreement of the experimental and model–predicted values of response variable.
The t–test, F–values and p–values of quadratic model coefficients indicate relative significance of corresponding independent variables. A large t–stat value and p–value <
0.05 indicates significance of the coefficient and the corresponding independent variable.
Relative F–values of linear, interaction and quadratic coefficient indicate significance of the individual effects of independent variables and the magnitude of interaction between them. As per ANOVA results given in Table 2B, F–value of overall regression is 537.36, while F–value of linear coefficient is 285.95. F–value of 10.61 for interaction coefficients is much smaller than F–value for linear coefficients, which implies relatively standalone or unrelated effect of independent variables on the % v/v H2 content of product gas. p–
values of all linear and quadratic coefficients are < 0.05 with large absolute t–values, which indicates that all variables have significant effect on % v/v H2 content of product gas. The t–values of interaction coefficients are relatively smaller than linear and square coefficients indicating their relative less significance. Moreover, p–values of interaction coefficients between temperature and pH; and temperature and initial glycerol
is insignificant or in other words, these variables have independent effect on the response variable. The p–value of the interaction coefficient between pH and initial glycerol concentration is < 0.05, which signifies that these variables have an inter–related effect on response variable. Relatively small F–value of 3.03 for The Lack of Fit with p–value of 0.124 implies that Lack of Fit is not significant as compared to the pure error, or in other words the model was significant.
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Table 2.2. Results of central composite design for optimization of fermentation parameters
(A) The coefficients of quadratic regression model and their t– and p– values
Model term Coefficient t–value p–value
Prob >F
Intercept (A0) 40.897 169.802 0.000*
Linear coefficients
Temperature (X1) 1.097 7.477 0.000*
pH (X2) 2.334 14.885 0.000*
Glycerol (X3) 3.917 24.091 0.000*
Square coefficients
Temperature (X12) –8.264 –57.284 0.000*
pH (X22) –3.514 –23.114 0.000*
Glycerol (X32) –2.629 –16.532 0.000*
Interaction coefficients
Temperature and pH (X1 × X2) –0.374 –2.066 0.066 Temperature and Glycerol (X1 × X3) –0.208 –1.071 0.309 pH and Glycerol (X2 × X3) –1.108 –5.140 0.000*
(B) ANOVA for quadratic model
Source DF SS MS F–value p–value
Prob >F
Regression 9 1622.99 180.33 537.36 0.000*
Linear 3 345.67 95.96 285.95 0.000*
Square 3 1266.63 422.21 1258.11 0.000*
Interaction 3 10.68 3.56 10.61 0.002*
Residual (Error) 10 3.36 0.34
Lack of fit 5 2.52 0.50 3.03 0.124
Pure error 5 0.83 0.17
Total 19 1626.34
* Significant p values, p≤ 0.05; R2 = 0.998; Predicted R2 = 0.987; Adjusted R2 = 0.996 (C) Analysis of the contour plots
Fixed parameter (Center point value)
Variable parameter # H2 (% v/v)# Temperature = 36°C pH = 6.69 Glycerol = 7.38 g/L 42.406 Initial pH = 6.5 Temp. = 36.2°C Glycerol = 7.5 g/L 42.259
Glycerol = 5.5 g/L Temp. = 36.2°C pH = 6.79 41.070
Global optimum values of variable parameters: 1. Parameter actual (coded) values:
Temperature = 36.18°C (0.72), pH = 6.7 (0.22), Glycerol conc. = 7.4 g/L (0.70); 2. H2
conc. (% v/v): 42.54; 3. Regression coefficients: R2 = 99.8%; R2 (adj) = 99.6%
# Values of variable parameters corresponding to max. H2 conc. for centre point value of third parameter
Figs. 2.3A, B, C show contour plots that reveal interactions among any two independent variables for the % v/v H2 content of product gas. The contour plots (which essentially are graphical representation of regression eq. 2.7) represent infinitive number of combinations of two test variables, with the third variable maintained at its zero (or center point) level. Analysis of the contour plots is given in Table 2.2C. The shapes of response surface contours, whether elliptical, circular, or saddle point, explain the magnitude of interaction between independent variables (Tanyildizi et al., 2005;
Ravikumar et al., 2005). For strong interactions among the variables, the contour plots have perfectly elliptical shape. The surface confined in the smallest contour represents parameter space for maximum value of response variable. A peculiar facet of the contour plots shown in Fig. 2.3is that each of these plots has a clear highest point, which means that the maximum hydrogen production could be achieved inside the design boundary. The contour plots in Figs. 2.3A (temperature vs. initial pH) and2.3B (temperature vs. initial glycerol conc.) have circular shape, which points to relatively individual effect (or insignificant interactions) between the parameters. This is also corroborated by p–value >
0.05 for the interaction coefficients for these variables, as noted earlier. On the other hand, the contour plot between pH and glycerol shown in Fig. 2.3C has elliptical shape, which reveals significance of interaction between these variables. This result is also corroborated by the ANOVA results, i.e. p–value < 0.05 for the interaction coefficient of glycerol and pH. The parameter space corresponding to maximum % v/v H2 for any two variables (with value of the third variable held at its center point) is given in Table 2.2C. Values of all independent variables corresponding to global optimum of maximum % v/v H2 content of product gas are also listed in Table 2.2C. In the given range of optimization parameters, the highest H2 content of 42.54% v/v in product gas is obtained for crude glycerol
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(A)
(B)
(C)
Figure 2.3. Cotour plots for interactive effects of different parameters on hydrogen content of product gas. (A) temperature and initial pH (B) temperature and crude glycerol (C) initial pH and crude glycerol
It is noteworthy that optimum values of temperature and initial pH for maximum H2
content of product gas with crude glycerol as substrate are very similar to those for other substrates such as palm oil effluent (Chong et al., 2009) and glucose (Ghosh et al., 2010;
Mullai et al., 2013). However, as seen from literature summary in Table 1 from Chapter 1, the optimum crude glycerol concentration in our study (7.4 g/L) is much smaller than the optimum pure glycerol concentration of 15 to 20 g/L reported by previous authors (Kumar et al., 2015; Jitrwung et al., 2011; Sittijunda et al., 2012).