*2.3. Distributed Activation Energy Models (DAEMs)*

Two types of DAEMs (the Gaussian and the Miura–Maki [33]) were developed in software codes to model the decomposition of hydrochars, grape seeds, and grape seed oil.

In particular, when a first-order reaction is assumed to model biomass degradation, the variation of α vs. time can be expressed as the product between the reaction rate k and the conversion function f:

$$\frac{d\alpha}{dt} = \mathbf{k}(\mathbf{T})\mathbf{f}(\alpha)\tag{2}$$

By adopting the Arrhenius theory, k(T) can be expressed by Equation (3):

$$\mathbf{k}(\mathbf{T}) = \mathbf{k}\_0 \exp\left(-\frac{\mathbf{E}}{\mathbf{RT}}\right) \tag{3}$$

where k0 represents the pre-exponential factor, E is the activation energy, and R is the universal gas constant.

To deduce the kinetic parameters, f(α) is usually assumed in advance (in the so called model-fitting methods) and then forced to fit experimental data [23]. In particular, nonisothermal models usually adopt a constant heating rate β equal to dT/dt, allowing the re-arrangement of Equation (3) into (4), which can be solved using thermogravimetric data [23].

$$\frac{d\,\mathrm{d}\,\alpha}{\mathrm{d}\,\mathrm{T}} \stackrel{\sim}{\mathrm{d}\,\mathrm{d}} \frac{d\,\mathrm{d}\,\mathrm{d}\,\mathrm{t}}{\mathrm{d}\,\mathrm{T}} = \frac{\mathrm{d}\,\alpha}{\mathrm{d}\,\mathrm{t}} \frac{1}{\beta} = \frac{\mathrm{k}(\mathrm{T})}{\beta} \mathrm{f}(\alpha) = \frac{\mathrm{k}\_0}{\beta} \exp\left(-\frac{\mathrm{E}}{\mathrm{RT}}\right) \mathrm{f}(\alpha) \tag{4}$$

Among model-fitting methods, DAEM assumes that pyrolysis can be described through an infinite number of irreversible first-order parallel reactions, with different rate parameters, which occur simultaneously [23,34]. In particular, the DAEM for a reaction order equal to 1 can be obtained by re-arranging Equations (2)–(4) as:

$$1 - \alpha = \int\_0^\infty \exp\left(-\int\_{\mathbf{T}\_0}^\mathbf{E} \frac{\mathbf{k}\_0}{\mathfrak{J}} \exp\left(-\frac{\mathbf{E}}{\mathbf{RT}}\right) \mathbf{dT}\right) \mathbf{f}(\mathbf{E}) d\mathbf{E} \tag{5}$$

Equation (5) does not have an analytical solution and there are two families of methods to solve it. One is referred to as distribution-fitting method and the other as isoconversional methods. The distribution-fitting method assumes f(E) (like a Gaussian, Weibull, or Boltzmann distribution) and forces to fit the TGA data by applying a certain numerical method [23]. Meanwhile, isoconversional methods do not assume f(α), but adopt a series of thermogravimetric data at different heating rates to directly compute the activation energy. The reaction rate at a certain α is a function of temperature. Among these distribution-free methods, the Miura–Maki integral method is a common one [35].

#### 2.3.1. Gaussian Model

In the Gaussian DAEM, the distribution function f(E) is assumed to be a Gaussian distribution with mean activation energy E0 and standard deviation σ, which is:

$$\mathbf{f}(\mathbf{E}) = \frac{1}{\sigma\sqrt{2\pi}} \exp\frac{\left(\mathbf{E} - \mathbf{E}\_0\right)^2}{2\sigma^2} \tag{6}$$

Applying the Coats–Redfern [36] and Fisher et al. [37] approximations, Equation (7) is obtained.

$$1 - \mathfrak{a} = \int\_0^\infty \left( \exp\left( -\frac{\mathrm{k}\_0 \mathrm{RT}^2}{\beta \mathrm{E}} \exp\left( -\frac{\mathrm{E}}{\mathrm{RT}} \right) \right) \right) \frac{1}{\sigma \sqrt{2\pi}} \exp\frac{\left( \mathrm{E} - \mathrm{E}\_0 \right)^2}{2\sigma^2} \mathrm{dE} \tag{7}$$

To solve Equation (7), the authors applied the simplification reported by Anthony and Howard [38], which consists in keeping k0 constant and equal to 1.67 × 1013 <sup>s</sup><sup>−</sup>1, and the minimization procedure proposed by Güne¸s and Güne¸s [39] that solves the integral using the Simpson's 1/3 rule. To avoid oscillations in the results, dE was kept constant at 50 kJ/mol, while the extremes of integration were iteratively adjusted according to E0. Then, the best solution was calculated by iterating the solution of the integral for several values of E0 and *σ* and minimizing h, i.e., the root mean quadratic error between the model output and the experimental data:

$$\mathbf{h} = \sqrt{\frac{\sum\_{\mathbf{j}=1}^{n} \left(\alpha\_{\text{TGA},\mathbf{j}} - \alpha\_{\text{DAEM},\mathbf{j}}\right)^{2}}{\mathbf{n}}} \tag{8}$$

where αTGA,j and αDAEM,j are the experimental and calculated extent of conversion, respectively, and *n* is the number of available data points.

The code was developed in MATLAB and a direct search technique was adopted [39]. The code is reported in Supplementary Materials. Firstly, a larger grid procedure was used for both E0 and σ, with a step size equal to 5. The first three values minimizing h were obtained. Then, for these last three values, an iteration process using E0 ± 5 and σ ± 5 with a step equal to 1 was set. Finally, final values were used to solve the integral of Equation (7) and to calculate αDAEM.

#### 2.3.2. Miura–Maki Model

The Miura–Maki model adopts some assumptions to determine f(E) and k0 without any a priori assumption on the energy distribution [33]. In particular, Miura and Maki approximated [33,35] the so called Φ function (Equation (9)) to a step-function E = Es for a selected temperature T.

$$\Phi(\mathbf{E}, \mathbf{T}) = \exp\left(-\mathbf{k}\_0 \int\_0^\mathbf{t} \exp\left(-\frac{\mathbf{E}}{\mathbf{R}\mathbf{T}}\right) \mathbf{dt}\right) \cong \exp\left(-\frac{\mathbf{k}\_0}{\beta} \int\_0^\mathbf{T} \exp\left(-\frac{\mathbf{E}}{\mathbf{R}\mathbf{T}}\right) \mathbf{dT}\right) \tag{9}$$

Thus, Equation (5) can be written as:

$$\alpha \cong 1 - \int\_{\mathcal{E}\_\ast}^{\infty} \mathbf{f}(\mathcal{E}) d\mathcal{E} = \int\_0^{\mathcal{E}\_\ast} \mathbf{f}(\mathcal{E}) d\mathcal{E} \tag{10}$$

with Es so that Φ(Es,T) ∼= 0.58.

By the Miura and Maki approach, only a reaction having an activation energy Es occurs at a given T and β—the model simplifies an actual reaction system by a set of N reactions, characterized by their own activation energy and pre-exponential factor.

Approximating the function Φ(E,T) as:

$$\Phi(\mathbf{E}, \mathbf{T}) \cong \exp\left(-\frac{\mathbf{k}\_0 \mathbf{R} \mathbf{T}^2}{\beta \mathbf{E}} \exp\left(-\frac{\mathbf{E}}{\mathbf{R} \mathbf{T}}\right)\right) \tag{11}$$

the extent of conversion for the j-th reaction is expressed by Equation (12):

$$1 - \frac{\Delta \mathbf{v}}{\Delta \mathbf{v}\_{\infty}} \cong \exp\left(-\frac{\mathbf{k}\_{0} \mathbf{R} \mathbf{T}^{2}}{\beta \mathbf{E}} \exp\left(-\frac{\mathbf{E}}{\mathbf{R} \mathbf{T}}\right)\right) \tag{12}$$

where Δv and Δv<sup>∞</sup> represent the amount of volatiles evolved and the effective volatile content for the j-th reaction.

After some mathematical steps [35], and after imposing 1 − Δv/(Δv∞) = Φ(E, T) ∼= 0.58, the representative equation of the Miura–Maki model is given by Equation (13).

$$\ln\left(\frac{\beta}{T^2}\right) = \ln\left(\frac{\mathbf{k}\_0 \mathbf{R}}{\mathbf{E}}\right) + 0.6075 - \frac{\mathbf{E}}{\mathbf{RT}}\tag{13}$$

Compared to the Gaussian DAEM, the approach of Miura [35] and Miura and Maki [33] relies on the dependence of k0 on E. In particular, they proposed a simple procedure to estimate f(E) and k0, consisting of the following steps:

(1) measurement of α vs. T relationship for at least three heating rates;

(2) calculation of β/T2 at selected α values from the α vs. T relationship obtained in (1) for each heating rate;

(3) plotting of ln (β/T2) vs. −1/(RT) at the selected <sup>α</sup> values, and determination of E and k0 from the Arrhenius plot at different α values using the relationship represented by Equation (13): E can be estimated by the slope, and k0 by the intercept of each line with the y axis; and

(4) plotting the α value against the activation energy E, as calculated in (3), and differentiating the α vs. E relationship to obtain f(E).
