Regression Analysis of TPR data

The non-linear regression analysis in this study was performed using MATLAB. The Eq. (38) and (47) were solved using a built-in ordinary differential equation solver (ODE45). The kinetic parameters and of these two equations were estimated using least square curve fitting tool (LSQCURVEFIT), which minimizes the error between the experimental observations and numerical solutions of ODE45. The data points for the evaluation of these kinetic parameters were obtained using the conversion values at every ten seconds interval for individual heating rates as shown in Figure 28. Thus, for each heating rate, the data set contains more than 700 points, which was adequate to adjust the model parameters. The conversion values were collected within the range of .

The rationale for the selected range of solid conversion has already been discussed in Section 6.5.1 considering the linearity of the activation energy obtained assuming that the rate of reaction at a constant conversion is a function of the temperature only. Apart from these, the reaction rate was found to be very slow for (Figure 28 (b)). This could be explained by the characteristics of induction period in NNGM. However, USCM does not include the mechanistic induction period in their formulation. On the other hand, for

the rate of conversion is decreased. This can be described by USCM assuming that the ash layer becomes very thick and is resisting the diffusion of gas phase reactants. NNGM accounts for this in their mechanistic approach as a deceleratory or decay period. Both models regard this range as diffusion control regime. Thus, only the range chosen for regression analysis follows the intrinsic reaction control regime, for which the Eq. (38) and (47) are valid, and, applicable for model discriminations with a meaningful conclusion.

Model discrimination among different A-E models was based on correlation coefficients

, lowest SSQ (sum of square) of residuals and smallest individual parameter confidence intervals. Given the values and parameters spans obtained, it was concluded that the best fitting was with Avrami exponent (n) value of one. Thus, a random nucleation model (“n” value of 1) was selected. This is also consistent with the constant crystallite size observed in between cyclic TPR and TPO experiments using H2

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pulse chemisorption (Table 11). The results of regression analysis with NNGM (random nucleation) and USCM are summarized in Table 18 and Table 19 respectively. The estimated values using both models are plotted in Figure 31 in parallel to the experimental data for comparing goodness of fit. To avoid Figure 31 of being overcrowded with data, only a few experimental data points are reported.

Figure 31: Conversion plot for non isothermal reduction of HMF oxygen carrier at different heating rates and under 10% H2 balance Ar flow. Symbols are for experimental data points and lines represent the estimated values using NNGM and

USCM.

It is noticeable from Figure 31 that both models are capable of predicting the experimental values. The , SSQ and values as reported in Table 18 and Table 19 are slightly more favorable for NNGM. However, model discrimination based on these parameters is not statistically significant. The band of 95% confidence interval, which is one of the most used criteria for model discrimination in kinetic studies, is found large enough for USCM. The activation energy values at different conditions are observed to

Temperature (oC) 260 280 300 320 340 360 380 400 420 440 460  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2.5 oC/min 5.0 oC/min 10.0 oC/min 15.0 oC/min NNGM USCM

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vary from 56 to 40 , wheras in the case of NNGM, the activation energies remain close to constant levels.

The model free approach, iso-conversion kinetics, has wide acceptance in the reaction engineering field for its superiority and accuracy in estimating the activation energy from non-isothermal data. Therefore, the results extracted by this method have been used extensively to discriminate models that are mechanistically different (Hossain 2007; Kanervo et al. 2001; Wimmers et al. 1986). Section 6.5.1 reports the average activation energy of at different values of using iso-conversion technique. These values are very similar to the ones calculated using NNGM . On the other hand; the average activation energy

estimated from USCM varies almost 8% from the iso-conversion kinetics. The offset at different heating rates is observed much higher, almost 27-30%.

Another interesting observation is that the obtained activation energies using different techniques varies between and , which is within the range reported by many researchers (Hossain et al. 2010a; Richardson et al. 2003). According to these studies, the activation energy falls into four groups depending on the reduction rate-controlling regime. A first group exhibiting low activation energy

points toward the external or very strong pore diffusion resistance limiting the overall reaction.

A second group with activation energies of indicates that the pore diffusion is controlling the overall reduction reaction. A third group, with activation energy in the range points towards a chemically reaction controlled reduction. The fourth group covers the same temperature range as the third but exhibits activation energies from . This group includes results in which water was added, a possible explanation for the higher activation energy is retention of H2O molecules on the surface.

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Table 18: Kinetics and statistical parameters obtained from TPR data for different heating rates using NNGM

Estimated 95 % confidence interval Estimated 95 % confidence interval

Lower Upper Band Lower Upper Band

2.5 66,747 65,546 67,948 2,402 0.0918 0.0913 0.0924 0.0011 306.78 0.0052 0.9982 5.0 63,762 62,673 64,851 2,177 0.1737 0.1727 0.1747 0.0019 327.99 0.1032 0.9982 10.0 62,176 61,271 63,081 1,810 0.3302 0.3286 0.3317 0.0030 358.91 0.0501 0.9988 15.0 62,407 61,459 63,355 1,896 0.4987 0.4964 0.5010 0.0046 368.08 0.0559 0.9988 All β 62,779 62,204 63,354 1,150 0.2331 0.2316 0.2346 0.0030 346.38 0.0577 0.9897

Table 19: Kinetics and statistical parameters obtained from TPR data for different heating rates using USCM

Estimated 95 % confidence interval Estimated 95 % confidence interval

Lower Upper Band Lower Upper Band

2.5 44,123 41,044 47,202 6,158 0.0308 0.0304 0.0312 0.0008 306.78 0.0027 0.9964

5.0 42,299 39,488 45,110 5,622 0.0567 0.0559 0.0574 0.0015 327.99 0.1095 0.9963

10.0 40,731 38,261 43,200 4,940 0.1058 0.1046 0.1070 0.0024 358.91 0.0530 0.9972

15.0 40,582 38,007 43,156 5,149 0.1586 0.1567 0.1604 0.0037 368.08 0.0599 0.9974

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One should notice that the activation energy estimated using NNGM falls within the chemically reaction controlled region values, which is consistent with the hypothesized absence of internal or external mass transport limitations. In contrast, the activation energy from USCM represents the diffusion controlling regime.

Hossain et al. (Hossain et al. 2010a) reported the activation energy in the range of

for Ni/La-γAl2O3. It is noticeable that the activation energy observed in this study is considerably lower. As discussed earlier (Section 6.3.2.1), this decreased activation energy is observed because of the addition of Co as promoter, which directly contributes to the enhancement of the reactivity of the HMF oxygen carrier. Enhanced reducibility for the addition of Co has also been reported in case of Ni over α-Al2O3 (Hossain et al. 2007, 2010).

These results confirm both the adequacy of the NNGM and of the determined parameters over the USCM. The inadequacy of the USCM can be related to the idealization included in the definition of a characteristic length of the grain or pores. On the other hand, NNGM with random nucleation fits better in defining the mechanistic of reduction process and is also consistent with the physicochemical properties of the HMF oxygen carrier as shown in the SEM pictures (Figure 25).