3.4.2. Miura–Maki Model

Figure 6 shows the values of E, while Tables S1 and S2 (Supplementary Materials) report the values of both E and k0 with the correlation coefficient R2. Figure 7 shows the plot of ln(β/T2) vs. −1/RT at different values of <sup>α</sup>. Each line corresponds to a certain <sup>α</sup>, while the three points on each line correspond to different heating rates. Each α corresponds to a certain temperature and the sequence Tβ=1 ◦C/min< T<sup>β</sup>=3 ◦C/min Tβ=10 ◦C/min has to be respected. The maximum value of conversion for which the correct sequence of temperature was respected is defined with αmax < 1. This was found in the range of α 0.1–0.8 for pyrolysis and 0.1–0.9 for oxidation. Then, for the valid range of α, E and k0 were determined graphically using the schemes in Figure 7; E is given by the slope of each line for each α, while k0 is obtained from the intercept and by applying Equation (13).

**Figure 6.** Distribution of E at different α of hydrochars, computed through the Miura–Maki model during (**a**) pyrolysis and (**b**) oxidation; per mol of dry feedstock.

**Figure 7.** Miura–Maki diagrams of pyrolysis of: (**a**) hydrochar 180 ◦C; (**b**) hydrochar 220 ◦C; (**c**) hydrochar 250 ◦C.

Under pyrolysis, the fitting is satisfactory, with a R<sup>2</sup> always higher than 0.95, except for one outlier at α = 0.7 for the 220 ◦C hydrochar. Results show that α hugely affects the kinetic parameters. All the hydrochars present an E that increases with the degree of conversion: it rapidly passes from 100–147 kJ/mol (at α = 0.1) to 371–386 kJ/mol in a range of α 0.6–0.7. This phenomenon can be explained by the progressive increase of the degree of carbonization with temperature. As pyrolysis proceeds, the biomass undergoes volatilization and is converted into a high-carbon content matrix. This char is more difficult to degrade than the volatile matter and therefore causes a higher E. Generally, values are considerably higher than those commonly found in literature for biomass substrates (for example, for lignin it is 237–267 kJ/mol, and for pine wood it is 186–271 kJ/mol [27]), which can be explained by the nature of the substrate. Meanwhile, k0 reaches its maximum at an α of 0.5–0.6, with maximum values of 1024–27 for the 180 and 250 ◦C hydrochars and 10<sup>38</sup> for the 220 ◦C one.

Figure 6b shows how E of different hydrochars vary during oxidation. For the 180 and 220 ◦C hydrochars, E increases up to α = 0.3–0.5 to values of 357–379 kJ/mol. Meanwhile, the 250 ◦C hydrochar shows a very similar trend to the 220 ◦C one up to an α of 0.35, and then stabilizes at much lower values of 221–225 kJ/mol. The 180 ◦C hydrochar shows higher values of E at the beginning of the conversion. Therefore, the HTC severity decreases the kinetic parameters of both volatile matter and char produced during the process. Similar trends were observed by Bach et al. [20] on hydrochars produced from wood residues. Meanwhile, pre-exponential factors k0 show a very similar trend to E (they reach their maxima in correspondence of the maximum of α, arriving up to 1024–27 for the 180 and 220 ◦C hydrochars, and 10<sup>14</sup> for the 250 ◦C one).

3.4.3. Comparison and Suggestions for Future Work

Table 6 reports a comparison table with the average values of E, σ, and k0, computed through the Gaussian and Miura–Maki models.


**Table 6.** Average values of E, σ, and k0 computed through the Gaussian and Miura–Maki models (k0 constant and equal to 1.67 <sup>×</sup> 1013 <sup>s</sup>−<sup>1</sup> for the Gaussian model).

Due to its starting hypothesis, i.e., approximating the profile with a single Gaussian curve, the Gaussian model does not highlight the effect of the severity of the HTC process on E, which ranges between 193 and 206 kJ/mol for pyrolysis. This does not occur for the Miura–Maki model, which assumes a distribution of E that varies on the entire range leading to more diverse average values. Indeed, the Gaussian approach models a decomposition involving several stages through a single decomposition stage characterized by a Gaussian profile of the activation energy. Therefore, this assumption "averages" the profiles, canceling the differences among the samples obtained at different severities. Meanwhile, the Miura–Maki model avoids this averaging since it does not impose a similar strong assumption on the distribution.

Since the literature lacks the modeling of the kinetics of pyrolysis of hydrochar derived from grape seeds, the comparison of the results obtained was possible considering typical values of activation energy for decomposition of the main constituents of lignocellulosic materials. In particular, cellulose decomposition activation energy ranges between 175 and 279 kJ/mol [21,27,50] hemicellulose is in the range 132–186 kJ/mol [27,51,52], whereas lignin ranges from 62 to 271 kJ/mol [27,50–52], with the broadest range. These values seem to prove the consistency of the results obtained in the present work.

In general, the Gaussian model has the advantage of being valid over the entire range of conversion, while the Miura–Maki model requires always that the temperature sequence must be satisfied, an aspect that is not obvious (in this study this occurs only for α in the range of 0.1–0.8). Overall, both the models suffer from a strong starting hypothesis: k0 fixed for the Gaussian model and a strong integral simplification in the Miura–Maki model that can lead to an overestimation of the average values of the kinetics parameters [53].

To improve the prediction, the authors suggest extending the single Gaussian model to a Multi Gaussian, in which the entire decomposition profile is divided into single decomposition peaks, each one approximated by a Gaussian curve [21]. Indeed, the single approach suits well homogeneous substrates (like the oil) characterized by a single decomposition peak, but seems too simple for complex substrates like biomasses. As a result, future works should include a comparison with other kinetics models.

Then, applying other DAEMs (like Kissinger–Akahira–Sunose method and the Coats– Redfern method) on the same feedstock could help to validate the results. Apart from contributing to understanding the mechanisms behind the decomposition, kinetic parameters can help in technological design and optimization. Moving to a larger scale, integration with other modeling techniques is necessary. Among these, it is worth mentioning models that consider the effects of particle distribution and geometry on heat transfer phenomena (like fractal models [54,55]), statistical models [56], and comprehensive computational models [57].
