# Lumped and distributed activation energy modelling of biomass pyrolysis

Zoé Mercier, Marion Carrier, Frédéric Marias

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Zoé Mercier, Marion Carrier, Frédéric Marias. Lumped and distributed activation energy modelling of biomass pyrolysis. Chemical Engineering Journal Advances, 2026, 25, pp.100998. 10.1016/j.ceja.2025.100998. hal-05450008v2

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Submitted on 12 Jan 2026

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intensified over the last two decades. Only an approach that integrates different spatial scales can offer an accurate representation of biomass thermochemical processes [8]. In this sense, predictive simulation tools for the thermochemical conversion of biomass require the use of pertinent models of all these underlying phenomena, integrated into a unified framework. Several reviews of biomass conversion modeling [3] report the same strategy, structuring modeling in three stages and thus in three main sub-models: the molecular model, the thermochemical model (at particle scale), and the thermodynamic model (at reactor scale). The molecular scale corresponds to the chemical decomposition of biomass. A reaction pathway and an associated kinetic scheme are needed to track the evolution of the species yields. The particle scale model describes the transformation of the biomass particles such as shrinking and shape transformation resulting from the reaction extent. Internal heat and mass transfer should be described at this scale. Particle properties such as thermal conductivity of particle volume can impact the phenomena occurring at the molecular scale. Finally, the reactor model representing the macroscopic scale describes the behavior of a population of particles, the flow of gases in the reactor, and the complex interactions between particles. The challenge in modeling fast pyrolysis is therefore to correctly model each of these scales and to couple them together. Numerous authors have highlighted the importance of intrinsic reaction kinetics, operating at the molecular level, to effectively model lignocellulosic biomass pyrolysis at the reactor scale [3,8,9]. Without adapted kinetic model selection and reliable kinetic parameters determination, reactor design relies mainly on assumptions, making it difficult to correctly predict the system's dynamics. Thus, the use of advanced models based on a detailed description of the relevant physical and chemical phenomena, at every scale, rather than empirical correlations is essential. Developing robust and generalizable kinetic models for fast pyrolysis is challenging due to the chemical complex nature of lignocellulosic biomass, which is primarily composed of cellulose, hemicelluloses and lignins. Each of these components exhibits distinct pyrolytic behavior: hemicelluloses degrade at relatively low temperatures (200–300°C) and have low activation energies (around 100–150 kJ/mol), cellulose decomposes in a narrow intermediate range (300–370°C), dominated by the formation of levoglucosan, with activation energies between 200 and 240 kJ/mol and lignin pyrolysis spans a broad temperature window (250–600°C), governed by complex and overlapping mechanisms, with activation energies varying widely (150–300 kJ/mol) depending on its origin and structure [10,11]. These values might vary significantly across biomass types, experimental conditions, and analytical methods used. Because of this variability, determining universal kinetic parameters that remain valid across different pyrolysis conditions is extremely difficult.

To account for the intrinsic variability in activation energies caused by the structural and chemical heterogeneity of biomass, the Distributed Activation Energy Model (DAEM) was introduced as a refinement over traditional kinetic approaches. Unlike the Constant Activation Energy Model (CAEM), which assumes a single activation energy for each reaction involved in the overall degradation scheme, the DAE approach considers this overall pyrolysis scheme as a superposition of parallel first-order reactions, each of them being characterized by its own activation energy, statistically distributed—most often as a Gaussian distribution. Originally proposed by Van Krevelen and Chermin [12], the DAEM has since become a widely accepted method for modeling the thermal degradation of lignocellulosic biomass and other complex solids. It has been successfully applied to various materials, including: coal pyrolysis [13,14], biomass torrefaction and pyrolysis [11, 15], agricultural residues such as rice husk, corncob, and straw [16], and other organic materials like tobacco [17]. In many of these studies, DAEM has been shown to accurately predict mass loss curves obtained from thermogravimetric analysis (TGA), including overlapping reaction zones, which are poorly captured by single-activation-energy models. For example, Sonobe and Worasuwannarak [16] demonstrated that

DAEM could predict the experimental conversion behavior of several biomass types under TGA conditions. Similarly, Xiong et al. [18] showed that integrating DAEM kinetics in CFD simulations of biomass pyrolysis in a bubbling fluidized bed significantly altered the predicted tar flux distribution, emphasizing the importance of determining the activation energy distribution parameters (i.e., average activation energy and fixed standard deviation) better. Because DAEM accurately describes the pyrolysis of individual components such as cellulose, several researchers have extended it to multi-component models, where a separate DAEM is applied to each biomass pseudo-component, typically cellulose, hemicelluloses, and lignins. These components are treated as thermally independent, with no chemical interaction among them [11, 19]. However, for components like hemicelluloses and lignins, which have more complex and heterogeneous structures, a single Gaussian distribution is often insufficient. For instance, Zhang et al. [20] found that while a single Gaussian DAEM could accurately model cellulose pyrolysis (with $E_0 = 240.23$ kJ/mol), it failed to describe the more complex decomposition patterns of hemicellulose and lignins. They proposed a double Gaussian DAEM (2G-DAEM) model, representing two distinct reaction pathways: the volatile formation reactions, involving lower activation energies and condensation/char-forming reactions, requiring higher activation energies. The rationale is that pyrolysis involves both devolatilization and secondary polymerization reactions, particularly in hemicellulose and lignins, where multiple bond types lead to a broader distribution of reactive sites. For example, their model identified two peaks in the activation energy distribution for hemicellulose ($E_1 = 180$ kJ/mol, $E_2 = 145.45$ kJ/mol) and for lignins ($E_1 = 159.59$ kJ/mol, $E_2 = 174.19$ kJ/mol), in line with experimental data. This double-distribution approach has also been applied successfully to coal pyrolysis, where two main decomposition stages are commonly observed [14]. Although Gaussian distributions are commonly used, alternative distribution shapes have also been explored to better capture asymmetric reactivity profiles. As noted by Burnham and Braun [21] and reviewed in [22], Weibull and Gamma distributions can better reflect the skewed nature of experimental conversion profiles. Biomass pyrolysis often produces asymmetric E-distributions, due to the variety of functional groups and bond energies in lignocellulosic materials. Hence, assuming a symmetric Gaussian distribution may limit the model's fidelity. Weibull distributions, in particular, have been used to address this issue [21]. More accurate distributions can also be derived experimentally, using integral or differential isoconversional methods (e.g., Miura-Maki), which relate the evolution of the degree of conversion, $\alpha$, to the effective activation energy, $E$. These methods enable the reconstruction of the probability density function $f(E)$ by differentiating the $\alpha(E)$ relation, providing a more physically grounded model input, especially useful when extending to semi-detailed or mechanistic models. Despite its success, DAEM remains mostly confined to TGA conditions, which involve heating rates (0.01–1.66°C/s) far below those of fast pyrolysis (>100°C/s) [23]. Moreover, most studies applied DAEM to a global model where the biomass is assumed to be degraded into volatiles and char [24,25]. These models are simple and computationally efficient but lack chemical specificity.

From a chemical kinetics standpoint, this study proposes a novel methodology: the coupling of DAEM with a stoichiometric, semi-detailed degradation scheme. We will demonstrate how this approach allows (i) a better representation of reaction-specific activation energy distributions, rather than a single global distribution; (ii) tracking of individual species' evolution over time, through explicit stoichiometric relationships, and (iii) an improved understanding of reaction pathways. In this study, our goal is not to propose new reaction pathways, but to examine how a more physically flexible representation of activation-energy variability can improve the fidelity of existing mechanisms when applied to complex, multi-step devolatilization phenomena. To do this, we investigate whether the DAEM can effectively capture the thermal decomposition behavior of cellulosic, hemicellulosic, and lignin materials. By focusing on these biopolymers, we
aim to assess the applicability and limitations of the DAEM approach when applied to both well-defined (cellulose) and structurally diverse (lignins and hemicelluloses) biomass constituents. The proposed approach opens the door to reaction-resolved modeling, improving interpretability and accuracy in detailed models, including an accurate representation of the degradation scheme.

## 2. Model description

### 2.1. Reaction scheme

Biomass thermal degradation involves simultaneously a wide range of reactions such as dehydration, depolymerization, and fragmentation, transforming biomass into bio-oil, gas, and solid residues. Various modeling approaches exist to capture the dynamics of these reactions [26,27]. Nonetheless, it is assumed that the true reaction system is too complex to be characterized in any fundamental way. Traditional approaches have often relied on global kinetic schemes, where biomass is treated as a single reactant converting into a lumped product (e.g., volatiles and biochar) [28]. While computationally efficient, these models lack the resolution needed to capture the diversity of reaction pathways and intermediates [29]. To address this issue, multi-component and multiple-step models incorporate successive chemical reactions, considering the sequential and parallel pathways of intermediate species [30]. For example, biomass decomposes into biochar, bio-oil, and gases in the primary stage, while secondary reactions further crack intermediates into non-condensable gases and secondary biochar. Products are also more detailed and lumped according to their functional group. These models offer improved accuracy but still struggle to reflect the overall chemical complexity of biomass. Lumped models go further by treating biomass as individual groups of compounds in order to describe the lignocellulosic nature of plants. Biomass pyrolysis products are derived as a linear combination of the pyrolysis products from reference species, which are usually cellulose, hemicelluloses, and lignins. Regarding biomass pyrolysis, the CRECK model developed by Ranzi et al. [31] is one of the most adopted models by the scientific community. This model groups molecules into functional groups, which simplifies the reaction network while maintaining accuracy. It describes the primary decomposition reactions, which involve the initial breakdown of high molecular weight compounds and secondary reactions such as gas-phase cracking, recombination, and cyclization. More recently, adaptation of this scheme has been proposed by Anca-Couce et al. [32] and Gouws et al. [33]. Both of them account for secondary reactions, Anca-Couce et al. [32] introduce an adjustable parameter to vary the amount of initial fragmentation products. For this part, Gouws et al. [33] also considers the formation of high molecular weight components and the effect of pressure on the reactions. Semidetailed schemes are particularly attractive for reactor-scale modeling, where the integration into CFD platforms requires a balance between chemical details and computational feasibility.

The kinetic scheme of cellulose pyrolysis is extracted from Debiagi et al. [34] and displayed in Table 1. Cellulose pyrolysis is described through a multi-step kinetic scheme, starting with a primary depolymerization of cellulose (CELL) into an active intermediate (CELLa). Active cellulose then follows two parallel decomposition pathways: one toward anhydrosugars, and another toward light oxygenates, gases, and biochar. To improve chemical resolution, delayed release of some species is added through two temperature-dependent reactions. Lignins and hemicelluloses pyrolysis models are both taken from the multiple kinetic mechanism of biomass pyrolysis from Debiagi et al. [35]. Because of its variability in elemental composition, lignins are characterized in terms of three reference components (LIG-C, LIG-H and LIG-O). These units correspond to the three primary structural units found in lignin: LIG-C represents the guaiacyl unit (G), LIG-H denotes the p-hydroxyphenyl unit (H), and LIG-O corresponds to the syringyl unit (S). These monolignol-derived units differ in their degree of

methoxylation and aromatic substitution, which significantly influences the thermal reactivity and degradation behavior of lignin [36]. Hemicelluloses are described by using two reference components: HCE1 and HCE2. Three different combinations of these two reference components allow for distinguishing hemicelluloses in softwoods (GMSW, mainly representative of glucomannan: HCE1/HCE2 = 70/30), hardwoods (XYHW, representative of arabinoxylan: HCE1/HCE2 = 35/65), and grasses (XYGR: HCE1/HCE2 = 12/88) feedstocks [35]. The degradation scheme considers the delayed release at high temperatures of metaplastic species represented as $G\{\}$. The reaction kinetic model employed in this study was based on Carbon, Hydrogen and Oxygen mass balances to ensure the atomic mass conservation.

### 2.2. Mathematical formulation of reactions

In this study, biomass particles are assumed to be homogeneous and thermally thin. Moreover, the shrinkage of particles is neglected. Particle volume is constant. The reactor is considered to be swept by an inert gas, which is homogeneously distributed. Furthermore, it is assumed that there is no injection of solid or gas. Pseudo-unimolecular reactions are assumed to be overall first order. Under this assumption, the evolution of the number of moles of species, $j$, inside the reactor is described by the following equation:

$$ \frac{dn_j}{dt} = \sum_i v_{ij} k_i n_{reactant,i} - l \frac{n_j}{V} Q \quad (1) $$

where $v_{ij}$ refers to the stoichiometric coefficient of the species $j$ in the $i$th reaction, $k_i$ in the constant rate, $Q$ represents the flow rate of the inert gas, $V$ is the reactor volume and $l$ refers to the phase of the species: $l = 0$ for solid and $l = 1$ for gas or liquid. Indeed, in this context, it is assumed that the gas phase may exit the system, whereas the solid phase remains within the system.

#### 2.2.1. Distributed activation energy model

The Arrhenius law was considered to define both degradation and production dynamics.

$$ k_i = A \exp \left( - \frac{E_i}{RT} \right) \quad (2) $$

where $A$ is the pre-exponential factor, $E_i$ is the activation energy, $R$ is the gas constant, and $T$ the particle sample temperature.

Here, a constant activation energy is assumed. In the following, this model is referred as constant activation model (CAEM). When a distributed activation energy is assumed (DAEM), the reaction rate is rewritten (See the mathematical development in Appendix B) as follows:

$$ k_i = \int_0^\infty A \exp \left( -\frac{E}{RT} \right) f_i(E) dE, \quad (3) $$

where $f$ is the probability density function describing the distribution of activation energies and $\int_0^\infty f(E) dE = 1$. The pre-exponential factor, $A$, is assumed constant for all reactions. In this study, a Gaussian distribution, has been assumed since its successful application to describe the pyrolysis kinetics of cellulose, hemicelluloses, and lignins [25]. The Gaussian distribution is expressed as:

$$ f(E) = \frac{1}{\sigma \sqrt{2\pi}} \exp \left( -\frac{(E - E_0)^2}{2\sigma^2} \right), \quad (4) $$

where $E_0$ and $\sigma$ are respectively the average energy and the standard deviation, both parameters of the distribution, which values are unknown.
<table><caption>**Table 1**
Kinetic scheme of biomass pyrolysis [34,35]- Acronym definitions in Appendix A.</caption><thead><tr><th>N°</th><th>Réaction</th><th>A [1/s]</th><th>E<sub>CAEM</sub> [cal/mol]</th></tr></thead><tbody><tr><td>1</td><td>CELL → CELLa</td><td>7.00 × 10<sup>13</sup></td><td>45 000</td></tr><tr><td>2</td><td>CELLa → 0.6 LVG + 0.2 C<sub>6</sub>H<sub>10</sub>O<sub>5</sub> + 0.2 G{LVG}</td><td>6.92 × 10<sup>4</sup></td><td>15 000</td></tr><tr><td>3</td><td>CELLa → 0.026 C<sub>3</sub>H<sub>6</sub>O<sub>2</sub> + 0.3705C<sub>3</sub>H<sub>6</sub>O<sub>2</sub> + 0.036C<sub>2</sub>H<sub>2</sub>O<sub>2</sub> + 0.049C<sub>2</sub>H<sub>4</sub>O + 0.088C<sub>6</sub>H<sub>6</sub>O<sub>3</sub> + 0.1657CH<sub>4</sub>O + 0.1058CH<sub>2</sub>O + 0.1595CO + 0.28CO<sub>2</sub> + 0.204H<sub>2</sub> + 1.5007H<sub>2</sub>O + 0.162CH<sub>2</sub>O<sub>2</sub> + 0.0371CH<sub>4</sub> + 0.901CHAR + 0.18FURAN + 0.1323C<sub>3</sub>H<sub>6</sub>O + 0.287FURFURAL + 0.05G{COH<sub>2Loose</sub>} + 0.07G{CO} + 0.024G{H<sub>2</sub>}</td><td>8.50 × 10<sup>5</sup></td><td>19 100</td></tr><tr><td>4</td><td>CELL → 0.125H<sub>2</sub> + 4.45H<sub>2</sub>O + 5.45CHAR + 0.12G{COH<sub>2Stiff</sub>} + 0.18G{COH<sub>2Loose</sub>} + 0.25G{CO} + 0.125G{H<sub>2</sub>}</td><td>8.00 × 10<sup>7</sup></td><td>31 000</td></tr><tr><td>5</td><td>LIGH → LIGOH + 2C<sub>2</sub>H<sub>4</sub>O<sub>2</sub> + 0.5C<sub>3</sub>H<sub>6</sub>O + 0.1CO + 0.4C<sub>2</sub>H<sub>4</sub> + 0.1C<sub>2</sub>H<sub>6</sub></td><td>6.70 × 10<sup>12</sup></td><td>37 500</td></tr><tr><td>6</td><td>LIGO → LIGOH + CO<sub>2</sub></td><td>3.30 × 10<sup>8</sup></td><td>25 500</td></tr><tr><td>7</td><td>LIGC → 0.35 LIGCC + 0.21CO + 0.27C<sub>2</sub>H<sub>4</sub> + 0.1CO<sub>2</sub> + 0.1C<sub>7</sub>H<sub>8</sub>O + 0.22CH<sub>2</sub>O + H<sub>2</sub>O + 0.1VANILLIN + 5.85CHAR + 0.4G{COH<sub>2Stiff</sub>} + 0.17G{COH<sub>2Loose</sub>} + 0.1G{H<sub>2</sub>} + 0.36G{CH<sub>4</sub>} + 0.2G{C<sub>2</sub>H<sub>6</sub>}</td><td>1.00 × 10<sup>11</sup></td><td>37 200</td></tr><tr><td>8</td><td>LIGCC → 0.35C<sub>2</sub>H<sub>4</sub>O<sub>2</sub> + 1.15CO + 0.3C<sub>2</sub>H<sub>4</sub> + 0.4C<sub>2</sub>H<sub>6</sub> + 0.15C<sub>7</sub>H<sub>8</sub>O + 0.7H<sub>2</sub>O + 0.25VANILLIN + 6.8CHAR + 0.7H<sub>2</sub> + 0.45CH<sub>4</sub> + 0.15CRESOL + 0.40.36G{CO}</td><td>1.00 × 10<sup>4</sup></td><td>24 800</td></tr><tr><td>9</td><td>LIGOH → 0.9LIG + 0.65CO + 0.05CO<sub>2</sub> + H<sub>2</sub>O + 4.25CHAR + 0.1CH<sub>4</sub> + 0.025C<sub>24</sub>H<sub>28</sub>O<sub>4</sub> + 0.1C<sub>3</sub>H<sub>4</sub>O + 0.6CH<sub>4</sub>O + 0.05CH<sub>2</sub>O<sub>2</sub> + 0.4G{COH<sub>2Stiff</sub>} + 0.45G{COH<sub>2Loose</sub>} + 0.6G{CO} + 0.25G{CH<sub>4</sub>} + 0.1G{C<sub>2</sub>H<sub>4</sub>} + 0.15G{C<sub>2</sub>H<sub>6</sub>} + 0.3G{CH<sub>3</sub>OH}</td><td>1.50 × 10<sup>8</sup></td><td>30 000</td></tr><tr><td>10</td><td>LIG → 0.6CO + 0.5C<sub>2</sub>H<sub>4</sub> + 0.1C<sub>7</sub>H<sub>8</sub>O + 0.1CHAR + 0.3C<sub>2</sub>H<sub>4</sub>O</td><td>4.00 × T(t)</td><td>12 000</td></tr><tr><td>11</td><td>LIG → 0.3CO + 0.1CO<sub>2</sub> + 0.4CH<sub>2</sub>O + 0.6H<sub>2</sub>O + 6.1CHAR + 0.2CH<sub>4</sub> + 0.65G{COH<sub>2Stiff</sub>} + 1.25G{COH<sub>2Loose</sub>} + 0.2G{CO} + 0.1G{H<sub>2</sub>} + 0.4G{CH<sub>4</sub>} + 0.5G{C<sub>2</sub>H<sub>4</sub>} + 0.4G{CH<sub>3</sub>OH}</td><td>8.30 × 10<sup>-2</sup> × T</td><td>8000</td></tr><tr><td>12</td><td>LIG → 2.6CO + 0.75C<sub>2</sub>H<sub>4</sub> + 0.5C<sub>2</sub>H<sub>6</sub> + 0.4CH<sub>2</sub>O + 0.6CO + 4.5CHAR + 0.6CH<sub>4</sub> + 0.4CH<sub>4</sub>O</td><td>1.50 × 10<sup>9</sup></td><td>31 500</td></tr><tr><td>13</td><td>GMSW → 0.7 HCE1 + 0.3 HCE2</td><td>1.00 × 10<sup>10</sup></td><td>129 704</td></tr><tr><td>14</td><td>HYHW → 0.35 HCE1 + 0.65 HCE2</td><td>1.25 × 10<sup>11</sup></td><td>131 377</td></tr><tr><td>15</td><td>HYGR → 0.12 HCE1 + 0.88 HCE2</td><td>1.25 × 10<sup>11</sup></td><td>125 520</td></tr><tr><td>16</td><td>HCE1 → 0.06C<sub>2</sub>H<sub>6</sub>O<sub>2</sub> + 0.16FURFURAL + 0.1C<sub>2</sub>H<sub>2</sub>O<sub>2</sub> + 0.13C<sub>6</sub>H<sub>6</sub>O<sub>3</sub> + 0.09CO<sub>2</sub> + 0.02H<sub>2</sub> + 0.54H<sub>2</sub>O + 0.25C<sub>6</sub>H<sub>10</sub>O<sub>5</sub> + 0.1CH<sub>4</sub> + 0.25C<sub>5</sub>H<sub>8</sub>O<sub>4</sub> + 0.1CHAR</td><td>16 × T</td><td>46 024</td></tr><tr><td>17</td><td>HCE1 → 0.39CO<sub>2</sub> + 0.1H<sub>2</sub> + 0.4H<sub>2</sub>O + 0.3CH<sub>4</sub> + 0.975CHAR + 0.4CH<sub>3</sub>O + 0.49CO + 0.05CH<sub>2</sub>O<sub>2</sub> + 0.1C<sub>2</sub>H<sub>4</sub> + 0.37G{COH<sub>2Stiff</sub>} + 0.43G{COH<sub>2Loose</sub>} + 0.01G{CO} + 0.05G{H<sub>2</sub>} + 0.51G{CO<sub>2</sub>} + 0.325G{CH<sub>4</sub>} + 0.075G{C<sub>2</sub>H<sub>4</sub>} + 0.2G{C<sub>2</sub>H<sub>6</sub>}</td><td>3 × 10<sup>-3</sup></td><td>15 062</td></tr><tr><td>18</td><td>HCE2 → 0.275CO<sub>2</sub> + 0.2H<sub>2</sub>O + CHAR + 0.5CH<sub>2</sub>O + 0.175CO + 0.025CH<sub>2</sub>O<sub>2</sub> + 0.1EtOH + 0.2HAA + 0.725G{COH<sub>2Stiff</sub>} + 0.2G{COH<sub>2Loose</sub>} + 0.4G{CO<sub>2</sub>} + 0.25G{CH<sub>4</sub>} + 0.275G{C<sub>2</sub>H<sub>4</sub>} + 0.3G{CH<sub>3</sub>OH}</td><td>7 × 10<sup>9</sup></td><td>127 612</td></tr><tr><td>19</td><td>G{CO<sub>2</sub>} → CO<sub>2</sub></td><td>1.00 × 10<sup>6</sup></td><td>24 500</td></tr><tr><td>20</td><td>G{CO} → CO</td><td>5.00 × 10<sup>12</sup></td><td>52 500</td></tr><tr><td>21</td><td>G{CH<sub>4</sub>O} → CH<sub>4</sub>O</td><td>2.00 × 10<sup>12</sup></td><td>50 000</td></tr><tr><td>22</td><td>G{COH<sub>2Loose</sub>} → 0.2CO + 0.8H<sub>2</sub>O + 0.8CHAR + 0.2H<sub>2</sub></td><td>6.00 × 10<sup>10</sup></td><td>50 000</td></tr><tr><td>23</td><td>G{C<sub>2</sub>H<sub>6</sub>} → C<sub>2</sub>H<sub>6</sub></td><td>1.00 × 10<sup>11</sup></td><td>52 000</td></tr><tr><td>24</td><td>G{CH<sub>4</sub>} → CH<sub>4</sub></td><td>1.00 × 10<sup>11</sup></td><td>53 000</td></tr><tr><td>25</td><td>G{C<sub>2</sub>H<sub>4</sub>} → C<sub>2</sub>H<sub>4</sub></td><td>1.50 × 10<sup>12</sup></td><td>55 000</td></tr><tr><td>26</td><td>G{COH<sub>2Stiff</sub>} → 0.8 CO + 0.2H<sub>2</sub>O + 0.2CHAR + 0.8H<sub>2</sub></td><td>1.00 × 10<sup>9</sup></td><td>59 000</td></tr><tr><td>27</td><td>G{H<sub>2</sub>} → H<sub>2</sub></td><td>1.00 × 10<sup>8</sup></td><td>70 000</td></tr><tr><td>28</td><td>G{LVG} → LVG</td><td>9.50 × 10<sup>-2</sup></td><td>10 000</td></tr><tr><td>29</td><td>G{LVG} → LVG</td><td>6.00 × 10<sup>11</sup></td><td>45 000</td></tr></tbody></table>

### 2.2.2. Thermal model

The precise knowledge of the sample's T-t profile is a prerequisite to conduct kinetic modeling of solid-state reactions. Particle temperature is usually assumed to remain constant and uniformly distributed throughout the sample during the entire reaction. While this simplification makes kinetic modeling easier, it is often not valid in real experimental conditions, especially during fast pyrolysis of biomass. In practice, temperature gradients can develop within the sample due to heat transfer limitations. When the sample is not thermally thin, the core temperature of the particle may lag behind the surface temperature, leading to non-uniform heating. This can result in an inaccurate estimation of kinetic parameters, as the reaction rate is highly temperature-dependent via the Arrhenius law. Moreover, in systems with fast heating (e.g., micropyrolyzers), the assumption that the sample instantly reaches the set temperature is unrealistic. In such cases, transient thermal effects become significant and must be accounted for. For accurate kinetic analysis, it is essential to either use very small sample sizes to ensure the sample behaves as thermally thin or adopt more realistic thermal models that include temperature ramping or heat transfer corrections. Nasfi et al. [37] have investigated this matter and shown that the biomass temperature history in a micropyrolyzer could be fitted to a lumped-capacitance model describing the conducting heat resistance. This model is given by:

$$ \frac{dT}{dt} = \frac{h_{cond} S_p}{C_p \rho V_p} (T_f - T) \quad (5) $$

where $h_{cond}$ is the overall heat transfer coefficient at its surface (device specific), $S_p$ is the surface of the particle, $V_p$ the volume of the particle, $\rho$ its density and $C_p$ its specific heat. Because all the experiments data used are not conducted in the same apparatus, the heat transfer coefficient is not known and a simpler thermal model will be used until the desired temperature ($T_f$) is reached:

$$ \frac{dT}{dt} = \beta \quad \forall T \in [0, T_f] \quad (6) $$

where $\beta$ is a constant heating rate and $T$ the sample temperature.

## 3. Material and methods

### 3.1. Parameters estimation

To accurately model the observed kinetic behavior, a DAEM is solved for each individual reaction. The values of the kinetic constants ($A$, $E_0$, and $\sigma$) of the Arrhenius law for each reaction need to be determined, since these parameters cannot be measured experimentally.

The average activation energy of the DAEM ($E_0$) is determined based on the constant activation energy measured experimentally in the case of the CAEM ($E_{CAEM}$). During kinetic measurements, fast events tend to dominate over slower reactions, which may become negligible on the timescale considered. This implies that the experimentally measured constant activation energy, $E_{CAEM}$, corresponds
<figure><img src="image_2.png" /><figcaption>Fig. 1. Illustration of a typical distributed activation energy (DAE) profile against the constant activation energy (CAE).</figcaption></figure>

to the minimum activation energy among the ensemble of distributed reactions described by the DAEM. Accordingly, it is assumed that the probability of encountering an activation energy lower than this value is only 0.001 (Fig. 1). Based on this assumption, the mean activation energy can be expressed as:

$$E_0 = \frac{E_{CAEM}}{1 + 1.96 \times CV} \qquad (7)$$

with CV = σ/E₀ the coefficient of variation.

To switch between DAE and CAE models, the pre-exponential factor is typically assumed to remain constant despite the existence of the compensation law, a strong, linear and positive correlation between the Arrhenius activation energy, E, and the frequency factor, A, even though these parameters are supposed to be independent. This simplification is justified by the need to reduce the number of adjustable parameters while still capturing the dominant kinetic behavior, but also as a physically grounded constraint that ensures consistency between the single-value and distributed formulations of the same kinetics. Our objective in this study is not to provide the most general multi-parameter estimation framework, but to examine the consequences of introducing a distribution of activation energies in a manner that preserves continuity with the classical model.

Finally, the fixed standard deviation, σ, needs to be determined. This approach leads to N unknown parameters for each of the N reactions. The coefficients of variation are determined by fitting to experimental data. The objective function is defined based on the relative error at each time point (summed over time t) and for each species of group of species (summed over j) for which experimental data are available. The objective function can be written as:

$$\epsilon = \sum_j \sum_t \left( \frac{n_j^{exp}(t) - n_j^{num}(t)}{n_j^{exp}(t)} \right)^2 \qquad (8)$$

In the case of lumped species, n_j corresponds to the sum of the contributions of each individual species within that group. This formulation allows the model to account for all species simultaneously while emphasizing accurate reproduction of both temporal evolution and relative magnitudes. It also ensures that no single species disproportionately influences the optimization due to differences in absolute quantities.

To assess whether the parameters associated with each reaction are coupled or can be fitted independently, we conducted a sensitivity analysis based on the structure of the objective function. Specifically, we examined how variations in the coefficient of variation for one reaction influenced the error landscape with respect to another. This analysis, presented in Fig. 2, clearly demonstrates strong coupling between parameter sets: the optimal value for one reaction depends significantly on the parameter values chosen for the other. Indeed, for CV₂ = 0.06, the error reaches a maximum at CV₁ = 0.04, whereas for CV₂ = 0.01, the maximum error occurs at CV₁ = 0.06. As a consequence, independent fitting of the DAEM parameters for each reaction leads to suboptimal results and inconsistent error behavior. Therefore, it is essential to perform a joint optimization of all N parameters simultaneously to accurately capture the underlying kinetics and ensure robust convergence of the model.

To ensure the robustness and generalizability of the fitted parameters, three different fitting strategies have been employed, each reflecting a different level of experimental integration and constraint. The first method (FM 1) involves the independent optimization of parameters for each individual dataset. This approach allows for flexibility and provides insights into the variability of fitted parameters across different experimental conditions, but it may lead to inconsistencies due to a lack of global coherence. The second method (FM 2) seeks to overcome this limitation by performing a joint optimization over all datasets related to mass loss. This global approach ensures that the obtained parameters are consistent across conditions and better represent the overall degradation behavior observed experimentally. Finally, the third method (FM 3) further extends this strategy by including not only mass loss data, but also the final yields of various chemical species (or groups of species). This adds an additional layer of constraint to the fitting process, enabling a more chemically realistic model and helping to refine the distribution of activation energies by leveraging more complete experimental information. The comparison of these three methods allows us to evaluate the trade-off between fitting flexibility and physical consistency, and to determine the most appropriate level of constraints for robust kinetic parameters identification. The fitted values of E₀ and σ have been computed for each feedstock and added to the supplementary informations in Appendix D.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Objective function over different coefficients of variation of the reaction 1 for various coefficient of variations of the reaction 2.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Final mass yield of lumped species of cellulose pyrolysis for Exp 1. The white bars stands for experimental data, — for CAEM, ___ for DAEM with FM1, ____ for DAEM with FM2 and ____ for DAEM with FM3.</figcaption></figure>

## 3.2. *Experimental data*

A wide range of experimental data has been selected for this study (Table 2). Each biopolymer is analyzed separately before considering the complete biomass system. For each biopolymer, two sets of data are used: one for parameter estimation via fitting, and a second independent set for model validation. In the fitting process, both residual mass yields over time and final yields of different species are considered. Due to the limited availability of experimental data for some biopolymers, pyrolysis data from slow pyrolysis experiments had to be included to supplement the fast pyrolysis dataset.

For the vast majority of experimental data, the heating ramp is assumed to be linear. However, in the case of fast pyrolysis in a micropyrolyser, such as experiments 1, 8, and 18, it has been shown

by Nasfi et al. [37] that the heating ramp can be modeled by an exponential law (see Eq. (5)). Thus, for these data, this thermal model will be considered. The time constant is being equal to $C_p\rho V_p / h_{cond} S_p = 1.36$ s.

# 4. Results and discussions

## 4.1. *Cellulose*

### 4.1.1. *Kinetic parameters identifications*

The coefficients of variation are determined by fitting experimental data from different processes. In the case of cellulose, data from drop-tube furnace (Exp 1), wire mesh reactors (Exp 4), laboratory-scale batch reactors (Exp 2 and 3), and TGA are used (Exp 5). The
<table><caption>**Table 2**
Experimental data considered for the study with TGA as thermogravimetric analysis and RMY as residual mass yield.</caption><thead><tr><th>Experiment</th><th>Reference</th><th>Biomass</th><th>Equipment</th><th>T init [K]</th><th>T final [K]</th><th>Heating rate [K/s]</th><th>Data type</th></tr></thead><tbody><tr><td>1</td><td>Patwardhan et al. [38]</td><td>Micro-crystalline cellulose</td><td>Drop-tube furnace (Frontier)</td><td>723</td><td>723</td><td>Non linear heating rate</td><td>Final yields</td></tr><tr><td>2</td><td>Hajaligol et al. [39]</td><td>Dried cellulose</td><td>Laboratory scale?</td><td>300</td><td>1173</td><td>350</td><td>RMY over temperature</td></tr><tr><td>3</td><td>Hajaligol et al. [39]</td><td>Dried cellulose</td><td>Laboratory scale?</td><td>300</td><td>1173</td><td>100</td><td>RMY over temperature</td></tr><tr><td>4</td><td>Milosavljevic and Suuberg [40]</td><td>Acid-washed glucose</td><td>Wire-mesh reactor</td><td>300</td><td>900</td><td>16.7</td><td>RMY over temperature</td></tr><tr><td>5</td><td>Qiao et al. [41]</td><td>Microcrystalline cellulose</td><td>TGA</td><td>300</td><td>1173</td><td>8.33</td><td>RMY over temperature</td></tr><tr><td>6<sup>a</sup></td><td>Qiao et al. [41]</td><td>Microcrystalline cellulose</td><td>TGA</td><td>300</td><td>1173</td><td>5</td><td>RMY over temperature</td></tr><tr><td>7<sup>a</sup></td><td>Qiao et al. [41]</td><td>Microcrystalline cellulose</td><td>TGA</td><td>300</td><td>1173</td><td>1.67</td><td>RMY over temperature</td></tr><tr><td>8</td><td>Patwardhan et al. [42]</td><td>Lignin isolated from corn stover</td><td>Drop-tube furnace (Frontier)</td><td>723</td><td>723</td><td>Non linear heating rate</td><td>Final yields</td></tr><tr><td>9</td><td>Ojha et al. [43]</td><td>Alkali lignin</td><td>Filament-pyrolyzer (CDS Pyroprobe)</td><td>300</td><td>673</td><td>131</td><td>RMY over time</td></tr><tr><td>10</td><td>Ojha et al. [43]</td><td>Alkali lignin</td><td>Filament-pyrolyzer</td><td>300</td><td>723</td><td>131</td><td>RMY over time</td></tr><tr><td>11</td><td>Ojha et al. [43]</td><td>Alkali lignin</td><td>Filament-pyrolyzer</td><td>300</td><td>773</td><td>131</td><td>RMY over time</td></tr><tr><td>12</td><td>Ojha et al. [43]</td><td>Alkali lignin</td><td>Filament-pyrolyzer</td><td>300</td><td>823</td><td>131</td><td>RMY over time</td></tr><tr><td>13<sup>a</sup></td><td>Ojha et al. [43]</td><td>Alkali lignin</td><td>Filament-pyrolyzer</td><td>300</td><td>873</td><td>131</td><td>RMY over time</td></tr><tr><td>14<sup>a</sup></td><td>Ojha et al. [43]</td><td>Alkali lignin</td><td>Filament-pyrolyzer</td><td>300</td><td>923</td><td>131</td><td>RMY over time</td></tr><tr><td>15</td><td>Qiao et al. [41]</td><td>Dealkaline lignin</td><td>TGA</td><td>300</td><td>1173</td><td>8.33</td><td>RMY over time</td></tr><tr><td>16<sup>a</sup></td><td>Qiao et al. [41]</td><td>Dealkaline lignin</td><td>TGA</td><td>300</td><td>1173</td><td>5</td><td>RMY over temperature</td></tr><tr><td>17<sup>a</sup></td><td>Qiao et al. [41]</td><td>Dealkaline lignin</td><td>TGA</td><td>300</td><td>1173</td><td>1.67</td><td>RMY over temperature</td></tr><tr><td>18</td><td>Patwardhan et al. [44]</td><td>Hemicelluloses extracted from switchgrass</td><td>Drop-tube furnace (Frontier)</td><td>723</td><td>723</td><td>Non linear heating rate</td><td>Final yields</td></tr><tr><td>19</td><td>Wang et al. [45]</td><td>Hemicelluloses from rice straw</td><td>TGA</td><td>300</td><td>1073</td><td>20</td><td>RMY over temperature</td></tr><tr><td>20<sup>a</sup></td><td>Yeo et al. [46]</td><td>Xylan from beechwood</td><td>TGA</td><td>323</td><td>1173</td><td>1.67</td><td>RMY over temperature</td></tr><tr><td>21<sup>a</sup></td><td>Wang et al. [45]</td><td>Hemicelluloses from rice straw</td><td>PyGC/MS A Frontier a CDS?</td><td>300</td><td>1073</td><td>20</td><td>Final yields</td></tr><tr><td>22</td><td>Qiao et al. [41]</td><td>Xylan</td><td>TGA</td><td>300</td><td>1173</td><td>8.33</td><td>RMY over temperature</td></tr><tr><td>23<sup>a</sup></td><td>Qiao et al. [41]</td><td>Xylan</td><td>TGA</td><td>300</td><td>1173</td><td>5</td><td>RMY over temperature</td></tr><tr><td>24<sup>a</sup></td><td>Qiao et al. [41]</td><td>Xylan</td><td>TGA</td><td>300</td><td>1173</td><td>1.67</td><td>RMY over temperature</td></tr><tr><td>25<sup>a</sup></td><td>Carrier et al. [47]</td><td>Corn stover</td><td>Fluidized bed reactor</td><td>300</td><td>773</td><td>100</td><td>Final yields</td></tr><tr><td>26<sup>a</sup></td><td>Carrier et al. [47]</td><td>Corn stover</td><td>TGA</td><td>313</td><td>823</td><td>2.50</td><td>RMY over time</td></tr><tr><td>27<sup>a</sup></td><td>Moore et al. [48]</td><td>Pinus radiata sawdust</td><td>TGA</td><td>300</td><td>1173</td><td>1.33</td><td>RMY over temperature</td></tr><tr><td>28<sup>a</sup></td><td>Moore et al. [48]</td><td>Pinus radiata sawdust</td><td>Fluidized bed reactor</td><td>300</td><td>973</td><td>100</td><td>Final yields</td></tr><tr><td>29<sup>a</sup></td><td>Luo et al. [49]</td><td>Cellulose</td><td>Heat radiation reactor</td><td>300</td><td>370-1370</td><td>168</td><td>Final yields</td></tr></tbody></table>

heating rate is going from 8.33 K/s to 350 K/s. CAEM and DAEM predictions of residual mass yield for the dried cellulose and micro-crystalline cellulose are presented in Fig. 4. The three fitting methods are compared. When the fitting is performed individually for each experiment (FM 1), the prediction outperforms CAEM across all cases, with a significantly lower error (see Table 3). However, this approach requires a specific coefficient of variation dataset for each experiment and operating conditions, which implies a very large set of data and parameters. Since the ultimate goal is to couple the kinetic model with other physics and apply it to various samples, this method is not suitable. Therefore, a single dataset must be established to represent all experiments collectively. Although this fitting method has not been retained, it nevertheless allowed us to study the influence of the heating ramp on the coefficient of variation. Based on the optimal coefficients of variation obtained for the different heating rates, no clear trend is observed between coefficients of variation and the heating ramp (See Supplementary Table C.14).

A first approach was to determine these kinetic parameters from a dataset based solely on residual mass for different equipment and therefore different heating ramps (from 8.33 K/s to 350 K/s corresponding to Exp 2 to 5) (see Fig. 3). This approach corresponds to the fitting method 2 (FM2). When considering the experiments used for fitting, the DAEM correctly captures the temporal evolution of mass loss (Fig.

<table><caption>**Table 3**
Relative errors obtained for Exp 1 to 5 describing cellulose pyrolysis.</caption><thead><tr><th></th><th>Exp1</th><th>Exp2</th><th>Exp3</th><th>Exp4</th><th>Exp5</th></tr></thead><tbody><tr><td>CAEM</td><td>5.6895</td><td>0.0414</td><td>0.0107</td><td>0.0165</td><td>0.0111</td></tr><tr><td>DAEM FM 1</td><td>1.412</td><td>0.006</td><td>5.12e-4</td><td>8.00e-4</td><td>0.0016</td></tr><tr><td>DAEM FM 2</td><td>15.2831</td><td>0.0082</td><td>0.0076</td><td>0.0018</td><td>0.0021</td></tr><tr><td>DAEM FM 3</td><td>1.4121</td><td>0.0091</td><td>0.0081</td><td>0.0018</td><td>0.0028</td></tr></tbody></table>

4), with lower error compared to CAEM (Table 3). However, final yield predictions are not consistent with experimental results. For instance, in the case of micro-crystalline cellulose pyrolysis in a drop-tube furnace (experiment 1), the prediction of $C_6$ is highly underestimated, whereas the production of water and biochar is overestimated. As a result, a third approach was considered. To ensure optimal simulations, the model has to be corrected with both the final residual mass yields of different species and the yield over time. This method is the fitting method 3 (FM3). In this case, the results appear consistent regardless of the dataset considered and better than CAEM predictions (Table 3).

Regardless of the fitting method used, the DAEM proves to be more effective than the CAEM overall (Table 3). As expected, DAEM slows down the mass loss rate, which allows for a more accurate prediction of the mass loss evolution over time. Indeed, by statistically

<footnote>a Datasets used exclusively for validation.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Residual mass yields of cellulose pyrolysis for Exp 5 (top left) and Exp 4 (top right), Exp 3 (bottom left), Exp 2 (bottom right). The * stands for experimental data, — for CAEM, — for DAEM with FM1, — for DAEM with FM2 and — for DAEM with FM3.</figcaption></figure>

picking activation energies systematically equal to or higher than that of CAEM, the conversion rate is decreased. It is important to note that in the case of fast pyrolysis (Exp 2 and 3), the error is greater than for pyrolysis with a lower heating ramp (Exp 4 and 5). This is most likely due to the use of common degradation schemes for both pyrolysis modes. Since the predominance and type of reactions differ between slow and fast pyrolysis, it can be assumed that the distribution functions should be different. This error may also come from the thermal model. Indeed, a thermal delay can occur within large particles for high heating rates. The thermal model adopted does not correct the inevitable thermal lag due to the thermal inertia of the sample (low conductivity) and the endothermic character of pyrolysis counteracted by the applied heat supply. The experimental datasets, most specifically the sample temperature used, have not been systematically corrected [50]. Let us now focus on experiment 3, i.e., the rapid pyrolysis (100 K/s) of dried cellulose. Figs. 5 and 6 respectively present the reaction rates, v, and the reaction rate constants k for the primary reaction of cellulose pyrolysis. For the sake of clarity, the rate constants have been adimensionalized by dividing each curve by the final value obtained with the CAEM for the corresponding reaction (Fig. 6). The thermal decomposition of cellulose (CELL) begins with the formation of an active intermediate (CELLa) (Reaction 1). This intermediate can follow different competitive pathways. In one route, CELLa mainly yields levoglucosan (LVG), oligomeric fragments, and light volatiles (Reaction 2). In a more complex pathway, CELLa decomposes into a wide spectrum of low-molecular-weight oxygenates (e.g., hydroxyacetone, glycolaldehyde, acetaldehyde, furfural), permanent gases (CO, CO₂, H₂, CH₂), water, and significant amounts of char (Reaction 3). In parallel, cellulose may also undergo direct degradation to form predominantly char, along with water, hydrogen, and minor gaseous fragments (Reaction 4). The kinetic parameters indicate that the char-forming route (Reactions 3 and 4) is associated with higher activation energies, whereas volatilization pathways (Reaction 2) proceed through lower-energy barriers. Regardless of the reaction, the DAEM systematically reduces the value of the rate constant compared to the CAEM; delaying the LVG production (see Fig. 7). This reduction becomes more pronounced when the temperature increases. A similar

<caption>Table 4</caption>

Relative errors obtained for Exp. 6 and 7 for microcrystalline cellulose pyrolysis.

<table><thead><tr><th></th><th>CAEM</th><th>DAEM</th></tr></thead><tbody><tr><td>Exp6</td><td>1.6122</td><td>1.2862</td></tr><tr><td>Exp7</td><td>1.4427</td><td>1.3579</td></tr></tbody></table>

trend is observed when comparing the reaction rates (Fig. 5): the peak values reached with the DAEM are consistently lower, and the maxima are shifted towards higher temperatures for each reaction. These shifts indicate that the DAEM slows down the overall reaction progress and spreads it over a broader temperature range.

These findings support the notion that the DAEM provides a more physically realistic description of thermal decomposition by tempering excessively sharp kinetic transitions and accounting for the distributed nature of activation energies more representative of the chemical heterogeneity of lignocelluloses. Despite a slightly more algorithmically intricate approach, the DAEM calculation time is similar to that of the CAEM. Under similar calculation conditions, the ratio of calculation times is 1.33. All simulations were executed on a standard laptop (Dell Precision 3571) equipped with an Intel Core i5-12500H processor and 16 GB of RAM. Average CPU time for CAEM simulations is 0.60 s and 0.80 s for DAEM simulations.

### 4.1.2. Validation

Compared to CAEM, DAEM has an advantage: one of its parameters is directly fitted to experimental results. To assess the model's predictive capability, we have evaluated its performance on experimental data that were not included in the fitting process. These experiments are slow pyrolysis in TGA (experiments 6 and 7). The coefficients of variation determined using Fitting Method 3 have been considered for this analysis.

In these cases as well, the DAEM proves to be more predictive than the CAEM (Table 4). While the final yields are less accurately predicted, the DAEM provides a better estimation of the reaction rate over time,
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Reaction rates of reaction 1 to 5 for Exp 3. --- stands for CAEM, — for DAEM. Black stand for reaction 1, red for reaction 2, blue for reaction 3 and orange for reaction 4.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Scaled reaction rate constants of reactions 1 to 4 and 20 for Exp 3. --- stands for CAEM, — for DAEM. Black stands for reaction 1, red for reaction 2, blue for reaction 3, orange for reaction 4 and green for reaction 20.</figcaption></figure>

and consequently offers a more accurate prediction of the mass loss evolution (Fig. 8).

It is interesting to observe that the first loss stage, starting at 100K and certainly corresponding to the loss of moisture content, has not been captured by our model.

In the case of fast pyrolysis of cellulose, the use of DAEM enables better prediction of mass loss and final yields. From this study, it appears that to ensure robust fitting parameters, time data on a species, as well as detailed data on several products, must be considered. It is therefore this fitting method (FM3) that will be used in the subsequent sections.

## 4.2. Lignins

### 4.2.1. Kinetic parameters identifications

A similar modeling approach is now applied to lignins. As previously mentioned, the lignins containing different monolignol units are assumed to decompose into three representative pseudo-components: LIG-C, LIG-H, and LIG-O. Table 5 summarizes the chemical composition of the lignin samples considered in this study. The values presented were derived from the ultimate analysis data as reported in each of the respective source articles.

The technical alkali lignins are well-known to contain relatively low content of aliphatic hydroxyl group and high content of free-phenolic hydroxyl group (Synapyl units and C5 substituted), which levels highly
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Mass yield of levoglucosan (black lines), water (blue lines) and low molecular weight compound (red lines) for cellulose pyrolysis at 100 K/s (Exp 3). — stand for CAEM and — for DAEM.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Residual mass of cellulose pyrolysis at varying heating rates. * stands for experimental data, — for CAEM, — for DAEM. Blue curves stand for $HR = 1.66$ K/s (Exp 6), green for $HR = 5$ K/s (Exp 7)</figcaption></figure>

<table><caption>**Table 5**
Mass fraction distribution of lignin pseudo-components: LIG-H, LIG-O, LIG-C.</caption><thead><tr><th></th><th>LIG-H</th><th>LIG-O</th><th>LIG-C</th></tr></thead><tbody><tr><td>Lignins isolated from corn stover</td><td>0.1173</td><td>0.57</td><td>0.3127</td></tr><tr><td>Alkali lignin</td><td>0.3914</td><td>0.3665</td><td>0.2421</td></tr><tr><td>Dealkaline lignin</td><td>0.3716</td><td>0.3768</td><td>0.2516</td></tr></tbody></table>

depend on the temperature used during alkali treatments; and therefore increases the structural diversity of alkali lignins (Table 5). As a result, the use of a series of four studies (experiments 9 to 12) that correspond

to the fast pyrolysis of different alkaline lignins conducted to mixed results. In these simulations, the final desired temperature is the only parameter that varies between experiments. It goes from 673 K (exp 9) to 823 K (exp 12). The heating rate is constant and equals 131 K/s. Fig. 9 shows the residual mass yield over time obtained from the filament pyrolyzer and predicted by CAEM and DAEM. Experimentally, an increase in target temperature leads to a decrease in the final biochar yield. This trend is well captured by any of the models. However, the evolution is not well captured by either of the models. First of all, during the initial stage of the pyrolysis, the yield evolution is quicker than the predicted one. For a longer time, the DAEM shows better predictability in terms of reaction kinetics. Indeed, once again, using DAEM allows for slowing down the reaction compared to CAEM. With
<figure><img src="image_10.png" /><figcaption>Fig. 9. Relative mass loss of alkaline lignin pyrolysis for Exp 9 to 12. The * stands for experimental data, — — — for CAEM, — — — for DAEM.</figcaption></figure>

<caption>**Table 6**</caption>

Relative errors obtained from different fitting methods on different experiments.

<table><thead><tr><th></th><th>Exp8</th><th>Exp9</th><th>Exp10</th><th>Exp11</th><th>Exp12</th><th>Exp15</th></tr></thead><tbody><tr><td>CAEM</td><td>2.2255</td><td>0.0090</td><td>0.3137</td><td>0.5834</td><td>0.7846</td><td>0.4235</td></tr><tr><td>DAEM F.M. 1</td><td>1.9445</td><td>0.0123</td><td>0.2555</td><td>0.0924</td><td>0.6743</td><td>0.388</td></tr><tr><td>DAEM F.M. 3</td><td>1.8536</td><td>0.0259</td><td>0.0279</td><td>0.0626</td><td>0.3047</td><td>0.3812</td></tr></tbody></table>

the exception of Experiment 9, the error between experimental data and numerical prediction is smaller for DAEM than for CAEM (Table 6); once again demonstrating the benefits of DAEM.

With respect to the fast pyrolysis of alkali lignins conducted in a hot filament type pyrolyzer (experiment 9), at 400°C, the simulation results indicate that the selected mechanistic specificities should be revised. In the case of slow pyrolysis of dealkaline lignins (Fig. 10), more accurate predictions of final product yields were obtained. Regardless of the model employed, DAEM performs better than CAEM.

When considering different reactor designs and various sources of lignins (alkaline or dealkaline), more accurate predictions were obtained for slow conditions (TGA-type reactions) (see Fig. 10) than for fast conditions. Those simulation results indicate that mechanistics details still require some improvements. Regardless of the model employed, DAEM performed better than CAEM. However, the devolatilization physics are not accurately captured (Fig. 11). Numerically, the onset of biochar and gas are overestimated, while C2 and C7 are underestimated. This discrepancy highlights limitations for the current kinetic model to fully capture the process, most probably due to the poor considerations of secondary reactions. Because of its complexity and variability, lignin is a far more complex biopolymer than cellulose, and therefore more complicated to study.

Finally, unlike the case of cellulose pyrolysis (Table 3), the error generated in the case of FM3 reveals that none of the fitting methods outperforms (Table 5), underlying the importance of using a large and robust dataset.

### 4.2.2. Validation

Coefficients of variation determined in the previous section are now used on independent experimental data. Whether for filament-pyrolyzer (experiments 13 and 14) or TGA experiments (experiments

<caption>**Table 7**</caption>

Predictive errors of CAEM and DAEM on external datasets not used during fitting.

<table><thead><tr><th></th><th>CAEM</th><th>DAEM</th></tr></thead><tbody><tr><td>Exp13</td><td>0.5515</td><td>0.3577</td></tr><tr><td>Exp14</td><td>0.7890</td><td>0.6705</td></tr><tr><td>Exp16</td><td>0.2301</td><td>0.1608</td></tr><tr><td>Exp17</td><td>0.4252</td><td>0.2618</td></tr></tbody></table>

16 and 17), DAEM consistently demonstrates greater predictive capability than the CAEM (Table 7), thereby validating its robustness.

However, in the case of alkali lignins converted at both high temperatures and heating ramps 873 and 923 K and 130 K/s, experiments 13 and 14, respectively, the residual mass yields are significantly overestimated (Fig. 12). This discrepancy is not repeated in the case of dealkaline lignins' fast pyrolysis conducted at even higher temperatures (1173 K) and lower heating ramps (5 and 1.7 K/s), respectively, for the experiments 16 and 17 (Fig. 13). This difference may stem from thermal model assumptions made. Indeed in the experiments 13 and 14, Ojha et al. [43] assume the particle to be nearly isothermal, due to the very fast heating rate. Yet, at such high temperatures, the particle may in reality experience a slower internal temperature rise, leading to thermal gradients. This would effectively delay the reaction progress, resulting in the use of a lower reaction rate, and higher predicted mass residual yields.

Once again, the results are less accurate than those obtained for cellulose; however, the DAEM still provides a more satisfactory fit than the CAEM. This highlights the importance of refining kinetic parameters for more complex substrates or at least adopting flexible models when dealing with heterogeneous feedstocks.

### 4.3. Hemicelluloses

In the same way as Debiagi et al. [35], hemicelluloses are classified into three categories based on their monomeric sugar composition and botanical origin: GMSW, XYHW, and XYGR.

This classification enables a more structured comparison of their properties and behaviors. For instance, the sample extracted from
<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Relative mass loss of dealkaline lignin pyrolysis for Exp 15. The \* stands for experimental data, --- for CAEM, — for DAEM.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 11.** Final mass yield of lumped species for lignins (extracted from corn stover) pyrolysis for Exp 8. The white bars stand for experimental data, — for CAEM, and — for DAEM.</figcaption></figure>

switchgrass (Exp 18) is representative of the XYGR group, as are those isolated from rice straw and corn stalk. In contrast, the following samples are categorized under the XYHW group: xylans from beechwood and birchwood. This grouping reflects compositional similarities and facilitates the interpretation of their thermal and chemical behaviors in subsequent analyses.

### 4.3.1. Kinetic parameters identifications

As observed for other biopolymers, the application of the DAEM significantly improves the prediction of fast pyrolysis behavior for hemicelluloses (Table 8). For instance, for Xylan pyrolysis in TGA at 8.33 K/s, the modeling error is reduced by more than a factor of 30 compared to the CAEM. This improvement is mainly due to the

DAEM's ability to better capture the distribution of activation energies, thereby more accurately describing devolatilization kinetics. While the CAEM triggers devolatilization at temperatures significantly higher than those seen experimentally, the DAEM more precisely predicts the onset temperature and the progression of residual yield (Fig. 14). In terms of product yields, DAEM also offers improved predictability (Fig. 15). However, notable discrepancies remain: the production of light gases such as $C_1$ (methane) and $C_2$ (ethane/ethylene) is substantially overestimated, while for Gas, $C_5$, char yield is underestimated. These deviations could be attributed to oversimplified assumptions regarding secondary gas-phase reactions. Additionally, the thermal model may inadequately represent the actual heating rates, which could affect the extent of secondary cracking reactions and char formation.
<figure><img src="image_13.png" /><figcaption>Fig. 12. Residual mass loss of alkali lignin pyrolysis for exp 13 and 14. The * stands for experimental data, --- for CAEM, — for DAEM.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 13. Mass loss curves of dealkaline lignin pyrolysis for exp 16 and 17. The * stands for experimental data, --- for CAEM, — for DAEM.</figcaption></figure>

<table><caption>**Table 8**
Relative errors obtained for different fitting methods on different experiments.</caption><thead><tr><th></th><th>Exp18</th><th>Exp19</th><th>Exp22</th></tr></thead><tbody><tr><td>CAEM</td><td>45.480</td><td>25.07</td><td>22.94</td></tr><tr><td>DAEM F.M. 1</td><td>43.91</td><td>3.41</td><td>0.60</td></tr><tr><td>DAEM F.M. 3</td><td>44.04</td><td>3.49</td><td>0.76</td></tr></tbody></table>

<table><caption>**Table 9**
Prediction errors of CAEM and DAEM on external datasets not used during fitting.</caption><thead><tr><th></th><th>CAEM</th><th>DAEM</th></tr></thead><tbody><tr><td>Exp20</td><td>3.91</td><td>8.48</td></tr><tr><td>Exp21</td><td>2.12</td><td>2.12</td></tr><tr><td>Exp23</td><td>11.70</td><td>1.25</td></tr><tr><td>Exp24</td><td>7.46</td><td>2.72</td></tr></tbody></table>

### 4.3.2. Validation

DAEM predictions are validated over experiments 20, 21, 23, and 24, which correspond respectively to a large range of hemicelluloses-based feedstocks: Xylan extracted from beechwood pyrolysis in TGA, hemicellulose extracted from rice straw pyrolyzed in Py-GC/MS, and Xylan pyrolysis in TGA. For experiments 20, 23, and 24, the DAEM

provides a better fit than the CAEM, particularly in capturing the de-volatilization kinetics (Fig. 16). However, this is not the case for experiment 20, where the CAEM slightly outperforms the DAEM. This result is unexpected, given that experiment 20 is very similar to experiment 24, with only a 23 K difference in the desired final temperature. Despite
<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Mass loss curves of hemicelluloses pyrolysis for exp 19 and 22. The \* stands for experimental data, —— for CAEM, ____ for DAEM.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Final mass yield of lumped species of hemicellulose pyrolysis from exp 18. The white bars stands for experimental data, — for CAEM, and ____ for DAEM.</figcaption></figure>

this minor variation, the mass loss profiles between the two experi-
ments differ significantly, which may suggest potential experimental
inconsistencies or measurement errors. Consequently, the results of
experiment 20 should be interpreted with caution. For the other cases,
especially experiments with the higher heating rates (20 K/s and 5 K/s
corresponding to experiment 21 and 23, respectively), the DAEM more
accurately reproduces the devolatilization rate, likely due to its ability
to represent the variability in activation energies. Nevertheless, when
it comes to predicting final product yields, both DAEM and CAEM
produce nearly identical results Table 9). This indicates that, in these
conditions, the introduction of a distributed activation energy has little
to no influence on the prediction of final yields, and mainly affects

the transient kinetics of devolatilization, which is surprising compared
to the results for fast pyrolysis of cellulose and lignins. The model
clearly underestimates the production of C3 and C6. The production of
these light molecules results from the degradation of heavier molecules
during secondary reactions. This highlights the need to develop a
secondary reaction block in the degradation scheme (see Fig. 17).

4.4. *Lignocellulosic biomasses*

Having analyzed each biopolymer individually and validated the
corresponding coefficients of variation, a complete biomass system
can be studied. In this section, two representative types of biomass
<figure><img src="image_17.png" /><figcaption>Fig. 16. Relative mass loss of hemicelluloses pyrolysis for exp 20, 23 and 24. The * stands for experimental data, --- for CAEM, ______ for DAEM.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 17. Final mass yields of lumped species of hemicelluloses pyrolysis for exp 21. The white bars stands for experimental data, the black for CAEM, and the blue for DAEM.</figcaption></figure>

<table><caption>**Table 10**
Lignocellulosic composition of *Pinus radiata* and corn stover.</caption><thead><tr><th>Biomass</th><th>CELL</th><th>HCE</th><th>LIG-H</th><th>LIG-O</th><th>LIG-C</th><th>Ash</th></tr></thead><tbody><tr><td>Pinus radiata</td><td>0.4687</td><td>0.1566 (GMSW)</td><td>0</td><td>0.2931</td><td>0.0738</td><td>0.0078</td></tr><tr><td>Corn stover</td><td>0.4410</td><td>0.3419 (XYGR)</td><td>0.0072</td><td>0.1291</td><td>0.0005</td><td>0.0803</td></tr></tbody></table>

are examined: corn stover and *Pinus radiata* (Softwood). Their respective lignocellulosic composition is detailed in Table 10 and extracted from [47,48], respectively. For the sake of clarity, the proportion of extractives is considered negligible in this analysis.

Both predictions performance of lumped product yields distribution from *Pinus radiata* sawdust fast pyrolysis in a fluidized bed reactor using CAE and DAE models were found equivalent (Fig. 18).

When evaluating residual mass yield predictions based on thermogravimetric (TGA) data for corn stover (exp 26) and *Pinus radiata* sawdust (exp 27), the comparison becomes more nuanced. For instance, in the pyrolysis of *Pinus radiata*, the DAEM demonstrates superior predictability (Fig. 19). It successfully captures the reaction deceleration around 400°C, which aligns with experimental observations. In contrast, the CAEM fails to detect this first drop that corresponds to the moisture loss and only reflects a reduced reaction rate closer to 470°C, suggesting a delay in its kinetic response to temperature changes.

Conversely, during the modeling of fast pyrolysis of corn stover (Fig. 20), the DAEM fails to accurately reproduce the residual mass curve over time. Although it performs well during the initial phase (up to 60 s), capturing the early devolatilization more precisely than the CAEM, it tends to overestimate the reaction rate, leading to premature completion of the pyrolysis. Moreover, a plateau is observed beyond
<figure><img src="image_19.png" /><figcaption>**Fig. 18.** Final mass yield of lumped species of pinus radiata pyrolysis from exp 25. The white bars stands for experimental data, — for CAEM, and — for DAEM.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 19.** Relative mass loss of pinus radiata sawdust pyrolysis for exp 27. The * stands for experimental data, — — for CAEM, — for DAEM.</figcaption></figure>

90 s in the DAEM simulation, which is not present in the experimental data or in the CAEM predictions. This unexpected behavior may be attributed to a misestimated coefficient of variation in one of the modeled reaction pathways, resulting in artificial stabilization of the system.

Despite this temporal discrepancy, the final char yield predicted by the DAEM for corn stover remains closer to experimental values than that of the CAEM (Table 12), reaffirming its strength in capturing global outcomes even if the transient behavior is less accurate. Regarding the prediction of final product yields obtained from slow and fast pyrolysis of corn stover and pine (Exp 25 and 28), the DAEM outperforms the CAEM, offering results that are generally closer to experimental data. Indeed, the DAEM provides improved estimates across all product categories (Table 12). However, in specific cases such as the fast pyrolysis of corn stover (Experiment 25), the difference between the two

models becomes negligible, indicating that under certain conditions, the simpler CAEM may still offer reasonable approximations (Fig. 18).

Finally, simulations of a wide range of molecules as pseudo-intermediate species such as the high molecular weight compounds (HMWC) and individual species, have been completed for the fast pyrolysis of pine conducted in a fluidized bed reactor (Experiment 28) (Table 11). In particular, the prediction of levoglucosan by DAEM is notably very good. We can note that the biochar production is once again underestimated.

### 4.5. Limitations and perspectives of model upgrades

A more detailed study was carried out on the production of levoglucosan (LVG) during the fast pyrolysis of cellulose in a heat radiation
<figure><img src="image_21.png" /><figcaption>Fig. 20. Residual mass loss of corn stover pyrolysis for exp 26. The * stands for experimental data, --- for CAEM, — for DAEM.</figcaption></figure>

<table><caption>**Table 11**
Yields of lumped and individual species for pyrolysis of Pinus radiata (Exp 28).</caption><thead><tr><th></th><th>Exp</th><th>CAEM</th><th>CAEM error</th><th>DAEM</th><th>DAEM error</th></tr></thead><tbody><tr><td><strong>Gases</strong></td><td><strong>13.9</strong></td><td><strong>11.74</strong></td><td></td><td><strong>11.21</strong></td><td></td></tr><tr><td>CO</td><td>6.4</td><td>4.32</td><td>0.10</td><td>4.13</td><td>0.13</td></tr><tr><td>CO<sub>2</sub></td><td>6.2</td><td>5.63</td><td>0.008</td><td>5.34</td><td>0.019</td></tr><tr><td>H<sub>2</sub></td><td>0</td><td>0.06</td><td></td><td>0.04</td><td></td></tr><tr><td>C<sub>2</sub>H<sub>4</sub></td><td>0.8</td><td>1.12</td><td>0.16</td><td>1.06</td><td>0.11</td></tr><tr><td>CH<sub>4</sub></td><td>0.5</td><td>0.61</td><td>0.05</td><td>0.63</td><td>0.07</td></tr><tr><td><strong>Condensables</strong></td><td><strong>65.6</strong></td><td><strong>39.66</strong></td><td></td><td><strong>38.99</strong></td><td></td></tr><tr><td>H<sub>2</sub>O</td><td>11.5</td><td>6.98</td><td>0.15</td><td>5.10</td><td>0.31</td></tr><tr><td>CH<sub>2</sub>O</td><td>2.3</td><td>1.65</td><td>0.08</td><td>1.74</td><td>0.06</td></tr><tr><td>CH<sub>3</sub>OH</td><td>2.9</td><td>1.96</td><td>0.10</td><td>1.72</td><td>0.17</td></tr><tr><td>CH<sub>3</sub>CHO</td><td>0.8</td><td>0.67</td><td>0.025</td><td>0.54</td><td>0.102</td></tr><tr><td>HCOOH</td><td>0.3</td><td>0.94</td><td>4.56</td><td>0.66</td><td>1.45</td></tr><tr><td>Acrolein (C<sub>3</sub>H<sub>4</sub>O)</td><td>0.3</td><td>0.39</td><td>0.09</td><td>0.39</td><td>0.09</td></tr><tr><td>Glyoxal (C<sub>2</sub>H<sub>2</sub>O<sub>2</sub>)</td><td>0.8</td><td>0.51</td><td>0.13</td><td>0.33</td><td>0.34</td></tr><tr><td>Glycoaldehyde (C<sub>2</sub>H<sub>4</sub>O<sub>2</sub>)</td><td>3</td><td>2.66</td><td>0.013</td><td>1.72</td><td>0.18</td></tr><tr><td>Acetol (C<sub>3</sub>H<sub>6</sub>O<sub>2</sub>)</td><td>1.8</td><td>0.42</td><td>0.58</td><td>0.27</td><td>0.72</td></tr><tr><td>Furfural (C<sub>5</sub>H<sub>4</sub>O<sub>2</sub>)</td><td>0.9</td><td>3.37</td><td>7.52</td><td>1.95</td><td>1.37</td></tr><tr><td>Sugars C5</td><td>4.7</td><td>1.84</td><td>0.37</td><td>1.31</td><td>0.51</td></tr><tr><td>Levoglucosan (C<sub>6</sub>H<sub>6</sub>O<sub>3</sub>)</td><td>22</td><td>17.43</td><td>0.04</td><td>22.42</td><td>0.0003</td></tr><tr><td>Anisole (C<sub>7</sub>H<sub>8</sub>O)</td><td>1</td><td>0.16</td><td>0.70</td><td>0.16</td><td>0.70</td></tr><tr><td>HMWC</td><td>0.6</td><td>0.66</td><td>0.009</td><td>0.66</td><td>0.009</td></tr><tr><td><strong>Solid residue</strong></td><td><strong>20.3</strong></td><td><strong>18.91</strong></td><td></td><td><strong>17.86</strong></td><td></td></tr><tr><td>Biochar</td><td>19.6</td><td>18.21</td><td>0.005</td><td>17.16</td><td>0.015</td></tr><tr><td>Ash</td><td>0.7</td><td>0.7</td><td>0</td><td>0.7</td><td>0</td></tr></tbody></table>

<table><caption>**Table 12**
Comparison of CAEM and DAEM modeling errors based on experimental fast pyrolysis data of biomass.</caption><thead><tr><th></th><th>CAEM</th><th>DAEM</th></tr></thead><tbody><tr><td>Exp25</td><td>1.43</td><td>1.30</td></tr><tr><td>Exp26</td><td>1.6248</td><td>1.96</td></tr><tr><td>Exp27</td><td>0.8498</td><td>0.5181</td></tr><tr><td>Exp28</td><td>14.72</td><td>6.37</td></tr></tbody></table>

reactor (Exp 29). The figure shows the final yields of char, gas, and LVG at different target heating temperatures. While the model provides accurate predictions for char, discrepancies arise for LVG and gas (Fig. 21). At high temperatures (above 600 K), the experimental data show a decrease in LVG production, whereas the model predicts a plateau. In a similar way, the production of gas increases with the

heating temperature, which is not predicted by either of the models. These results are due to the degradation scheme limitations. Indeed, in the present scheme, LVG is not subjected to secondary reactions. However, as discussed by Ranzi et al. [31], levoglucosan undergoes successive dehydratation reactions to form 5-hydroxy-methyl-furfural, furfural, furan and radical combinations to form benzene. Accounting for these successive reactions would likely decrease the predicted yield of LVG while increasing the formation of gaseous products. A second limitation of the model is the absence of considerations related to the biomass container or potential catalytic effects, which may influence the degradation pathways. Together, these aspects highlight the need for further research aimed at refining degradation models, with a stronger emphasis on incorporating secondary reactions and catalytic influences.

## 5. Conclusions

In this work, we have demonstrated the superior performance of the Distributed Activation Energy Model (DAEM) over the Constant Activation Energy Model (CAEM) in the context of biomass pyrolysis modeling. By considering a continuous distribution of activation energies, the DAEM better captures the complex, multi-step degradation kinetics of lignocellulosic materials, as opposed to the oversimplified, single-value assumption in the CAEM. A key element in the implementation of the DAEM is the parameter fitting stage, which requires optimization against experimental data. While this step introduces a calibration dependency, the DAEM maintains strong predictive capabilities even on independent datasets, where it consistently outperforms the CAEM. This reinforces its robustness and generalizability across varying thermal conditions and biomass compositions. Our methodology followed a progressive, component-wise strategy, applying the DAEM and CAEM separately to cellulose, hemicelluloses, and lignins, before scaling up to a complete biomass sample. This modular approach allowed for better chemical insight and improved prediction of global pyrolysis behavior through the sum of its parts.

Despite the promising performance of the DAEM, several limitations remain. First, the availability of high-quality experimental data is a significant constraint, as the accuracy of fitting and validation is strongly influenced by the reliability and resolution of measurements, which often carry non-negligible uncertainties. Second, the thermal modeling can be questioned, particularly with regard to temperature input. In this
<figure><img src="image_22.png" /><figcaption>**Fig. 21.** Final yield of levoglucosan, char and gas over the heating source pyrolysis for cellulose pyrolysis. Grey markers stand for experimental data, green for DAEM predictions, and blue for CAEM prediction.</figcaption></figure>

study, the thermal model assumed the use of a constant heating rate, a time-temperature history rather simple. The chemical kinetic model could be directly coupled with a CFD code to get the instantaneous temperature as input and then provide the products yield as output.

Although assuming a symmetric activation energy distribution has led to good results, this remains a modeling choice that could be challenged. Indeed, the distribution shape could be better characterized from experiments adopting the Muira-Maki's strategy. This approach will avoid assuming a symmetric activation energy distribution and should improve the predictability. Using a single, uniform pre-exponential factor across different biomass types may be invalid. The structural and chemical diversity of feedstocks suggests adapting this factor to specific reactivity, which could improve kinetic parameterization and model fidelity. The catalytic effects of inorganics and related secondary reactions were also not considered. Without such adaptations, the model will fail at predicting accurately behavior across various biomasses.

## CRediT authorship contribution statement

**Zoé Mercier:** Writing – original draft, Software, Methodology, Formal analysis, Data curation, Conceptualization. **Marion Carrier:** Writing – review & editing, Validation, Supervision, Data curation. **Frédéric Marias:** Writing – review & editing, Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors acknowledge both French Institut Carnot ISIFOR and Carnot M.I.N.E.S for the funding.

## Appendix A. Molecular formula

See Table A.13.

<caption>**Table A.13**</caption>

Chemical name and corresponding molecular formula.

<table><thead><tr><th>Chemical name</th><th>Molecules</th><th>Formula</th></tr></thead><tbody><tr><td>Cellulose</td><td>CELL</td><td>C<sub>6</sub>H<sub>10</sub>O<sub>5</sub></td></tr><tr><td>Active cellulose</td><td>CELLa</td><td>C<sub>6</sub>H<sub>10</sub>O<sub>5</sub></td></tr><tr><td>Hydrogen rich lignin</td><td>LIG-H</td><td>C<sub>22</sub>H<sub>28</sub>O<sub>9</sub></td></tr><tr><td>Oxygen rich lignin</td><td>LIG-O</td><td>C<sub>20</sub>H<sub>22</sub>O<sub>10</sub></td></tr><tr><td>Carbon rich lignin</td><td>LIG-C</td><td>C<sub>15</sub>H<sub>14</sub>O<sub>4</sub></td></tr><tr><td>Glucosmannan from Softwood</td><td>GMSW</td><td>C<sub>8</sub>H<sub>8</sub>O<sub>4</sub></td></tr><tr><td>Arabinoxylan from Hardwood</td><td>XYHW</td><td>C<sub>8</sub>H<sub>8</sub>O<sub>4</sub></td></tr><tr><td>Arabinoxylan from Grass</td><td>XYGR</td><td>C<sub>8</sub>H<sub>8</sub>O<sub>4</sub></td></tr><tr><td>Levoglucosan</td><td>LVG</td><td>C<sub>6</sub>H<sub>10</sub>O<sub>5</sub></td></tr><tr><td>Active intermediate</td><td>LIGCC</td><td>C<sub>15</sub>H<sub>14</sub>O<sub>4</sub></td></tr><tr><td>Active intermediate</td><td>LIGOH</td><td>C<sub>19</sub>H<sub>22</sub>O<sub>8</sub></td></tr><tr><td>Lignin</td><td>LIG</td><td>C<sub>11</sub>H<sub>12</sub>O<sub>4</sub></td></tr><tr><td>Active intermediate</td><td>HCE1</td><td>C<sub>8</sub>H<sub>8</sub>O<sub>4</sub></td></tr><tr><td>Active intermediate</td><td>HCE2</td><td>C<sub>8</sub>H<sub>8</sub>O<sub>4</sub></td></tr><tr><td>–</td><td>Char</td><td>C</td></tr><tr><td>–</td><td>FURAN</td><td>C<sub>4</sub>H<sub>4</sub>O</td></tr><tr><td>furan-2-carbaldehyde</td><td>FURFURAL</td><td>C<sub>8</sub>H<sub>4</sub>O<sub>2</sub></td></tr><tr><td>4-hydroxy-3-methoxybenzaldehyde</td><td>VANILLIN</td><td>C<sub>8</sub>H<sub>8</sub>O<sub>3</sub></td></tr><tr><td>methylphenol</td><td>CRESOL</td><td>C<sub>7</sub>H<sub>8</sub>O</td></tr><tr><td>Ethanol</td><td>Etoh</td><td>C<sub>2</sub>H<sub>6</sub>O</td></tr><tr><td>Metaplast specie</td><td>G{CO}</td><td>CO</td></tr></tbody></table>

## Appendix B. Mathematical derivation development

The DAEM considers a set of parallel and independent reactions co-occurring described as a continuous probability distribution in the activation energies, f(E), which satisfies the condition:

$$ \int_{0}^{\infty} f(E)dE = 1, \qquad (B.1) $$

and each volatiles fraction reacts according to this constant rate:

$$ k = Aexp\left(-\frac{E}{RT}\right) \qquad (B.2) $$

the global rate is:

$$ \frac{dV}{dt} = \int k(E)dV^* = \int Aexp\left(-\frac{E}{RT}\right) V^* f(E)dE, \qquad (B.3) $$

Factor out V:

$$ \frac{dV}{dt} = V^* \int_{0}^{\infty} A \exp\left(-\frac{E}{RT}\right) f(E)dE, \qquad (B.4) $$
<table><caption>**Table C.14**
Coefficients of variation for the reactions implied in cellulose pyrolysis obtained from fitting methods 1, 2 and 3.</caption><thead><tr><th></th><th>CV<sub>1</sub></th><th>CV<sub>2</sub></th><th>CV<sub>3</sub></th><th>CV<sub>4</sub></th><th>CV<sub>20</sub></th><th>CV<sub>22</sub></th><th>CV<sub>27</sub></th><th>CV<sub>28</sub></th><th>CV<sub>29</sub></th><th>CV<sub>30</sub></th></tr></thead><tbody><tr><td rowspan="4">F.M. 1</td><td>EXP1</td><td>0.0336</td><td>0.0242</td><td>0.0589</td><td>0.0285</td><td>0.0405</td><td>0.0405</td><td>0.0405</td><td>0.0405</td><td>0.0405</td></tr><tr><td>EXP2</td><td>0.0453</td><td>0.0599</td><td>0</td><td>0.0001</td><td>0.0383</td><td>0.0239</td><td>0.0109</td><td>0.0149</td><td>0.0207</td></tr><tr><td>EXP3</td><td>0.0543</td><td>0.0290</td><td>0.0125</td><td>0.0286</td><td>0.0200</td><td>0.0200</td><td>0.0200</td><td>0.0200</td><td>0.0200</td></tr><tr><td>EXP4</td><td>0.0198</td><td>0.0598</td><td>0.0001</td><td>0.0001</td><td>0.0200</td><td>0.0200</td><td>0.0200</td><td>0.0200</td><td>0.0200</td></tr><tr><td rowspan="2">F.M. 2</td><td>EXP5</td><td>0.0499</td><td>0.0599</td><td>0.0001</td><td>0</td><td>0.0200</td><td>0.0201</td><td>0.0200</td><td>0.0200</td><td>0.0200</td></tr><tr><td></td><td>0.0508</td><td>0.0642</td><td>0.0067</td><td>0.0075</td><td>0.0400</td><td>0.0400</td><td>0.0400</td><td>0.0400</td><td>0.0400</td></tr><tr><td>F.M. 3</td><td></td><td>0.0505</td><td>0.0225</td><td>0.0560</td><td>0.0209</td><td>0.0400</td><td>0.0400</td><td>0.0400</td><td>0.0400</td><td>0.0400</td></tr></tbody></table>

<table><caption>**Table C.15**
Coefficients of variation for the reactions implied in lignin pyrolysis obtained from fitting methods 1 and 3.</caption><thead><tr><th></th><th>CV<sub>5</sub></th><th>CV<sub>6</sub></th><th>CV<sub>7</sub></th><th>CV<sub>8</sub></th><th>CV<sub>9</sub></th><th>CV<sub>10</sub></th><th>CV<sub>11</sub></th><th>CV<sub>12</sub></th><th>CV<sub>19</sub></th><th>CV<sub>20</sub></th><th>CV<sub>21</sub></th><th>CV<sub>22</sub></th><th>CV<sub>23</sub></th><th>CV<sub>24</sub></th><th>CV<sub>26</sub></th><th>CV<sub>27</sub></th><th>CV<sub>28</sub></th></tr></thead><tbody><tr><td rowspan="5">F.M. 1</td><td>EXP8</td><td>0.0597</td><td>0.0471</td><td>0.0257</td><td>0.0163</td><td>0.0484</td><td>0.0216</td><td>0.0370</td><td>0.0255</td><td>0.0262</td><td>0.0469</td><td>0.0271</td><td>0.0334</td><td>0.0170</td><td>0.0292</td><td>0.0227</td><td>0.0369</td><td>0.0315</td></tr><tr><td>EXP9</td><td>0.0452</td><td>0.0600</td><td>0.0000</td><td>0.0337</td><td>0.0028</td><td>0.0000</td><td>0.0596</td><td>0.0523</td><td>0.0300</td><td>0.0300</td><td>0.0300</td><td>0.0297</td><td>0.0299</td><td>0.0300</td><td>0.0300</td><td>0.0300</td><td>0.0300</td></tr><tr><td>EXP10</td><td>0.0308</td><td>0.0497</td><td>0.0313</td><td>0.0341</td><td>0.0483</td><td>0.0132</td><td>0.0337</td><td>0.0435</td><td>0.0300</td><td>0.0301</td><td>0.0300</td><td>0.0316</td><td>0.0263</td><td>0.0306</td><td>0.0299</td><td>0.0300</td><td>0.0296</td></tr><tr><td>EXP11</td><td>0.0494</td><td>0.0573</td><td>0.0516</td><td>0.0538</td><td>0.0600</td><td>0.0001</td><td>0.0597</td><td>0.0593</td><td>0.0300</td><td>0.0300</td><td>0.0301</td><td>0.0260</td><td>0.0324</td><td>0.0300</td><td>0.0299</td><td>0.0315</td><td>0.0299</td></tr><tr><td>EXP12</td><td>0.0257</td><td>0.0511</td><td>0.0442</td><td>0.0041</td><td>0.0498</td><td>0.0337</td><td>0.0003</td><td>0.0599</td><td>0.0080</td><td>0.0521</td><td>0.0115</td><td>0.0136</td><td>0.0552</td><td>0.0099</td><td>0.0532</td><td>0.0180</td><td>0.0110</td></tr><tr><td rowspan="2">F.M. 3</td><td>EXP13</td><td>0.0189</td><td>0.0797</td><td>0.0772</td><td>0.0759</td><td>0.0435</td><td>0.0246</td><td>0.0334</td><td>0.0518</td><td>0.0138</td><td>0.0004</td><td>0.0350</td><td>0.0591</td><td>0.0611</td><td>0.0640</td><td>0.0336</td><td>0.0306</td><td>0.0319</td></tr><tr><td></td><td>0.0281</td><td>0.0794</td><td>0.0703</td><td>0.0182</td><td>0.0645</td><td>0.0352</td><td>0.0003</td><td>0.0519</td><td>0.0493</td><td>0.0466</td><td>0.0388</td><td>0.0475</td><td>0.0516</td><td>0.0433</td><td>0.0171</td><td>0.0285</td><td>0.0240</td></tr></tbody></table>

<table><caption>**Table C.16**
Coefficients of variation for the reactions implied in hemicellulose pyrolysis obtained from fitting methods 1, 2 and 3.</caption><thead><tr><th></th><th>CV<sub>1</sub></th><th>CV<sub>2</sub></th><th>CV<sub>3</sub></th><th>CV<sub>4</sub></th><th>CV<sub>5</sub></th><th>CV<sub>6</sub></th><th>CV<sub>7</sub></th><th>CV<sub>8</sub></th><th>CV<sub>9</sub></th><th>CV<sub>10</sub></th><th>CV<sub>11</sub></th><th>CV<sub>12</sub></th><th>CV<sub>13</sub></th><th>CV<sub>14</sub></th><th>CV<sub>15</sub></th></tr></thead><tbody><tr><td rowspan="4">F.M. 1</td><td>EXP18</td><td>0.0419</td><td>0.0999</td><td>0.0504</td><td>0.0009</td><td>0.0811</td><td>0.0563</td><td>0.0526</td><td>0.0995</td><td>0.0757</td><td>0.1000</td><td>0.0570</td><td>0.0420</td><td>0.0453</td><td>0.1000</td><td>0.0489</td></tr><tr><td>EXP19</td><td>0.0500</td><td>0.0500</td><td>0.0832</td><td>0.0737</td><td>0.0499</td><td>0.0793</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td></tr><tr><td>EXP20</td><td>0.0500</td><td>0.0500</td><td>0.0902</td><td>0.0944</td><td>0.0504</td><td>0.0800</td><td>0.0500</td><td>0.0500</td><td>0.0933</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td><td>0.0500</td></tr><tr><td>EXP23</td><td>0.0508</td><td>0.0529</td><td>0.0751</td><td>0.0909</td><td>0.0247</td><td>0.0788</td><td>0.0522</td><td>0.0490</td><td>0.0503</td><td>0.0496</td><td>0.0503</td><td>0.0498</td><td>0.0485</td><td>0.0499</td><td>0.0500</td></tr><tr><td>F.M. 3</td><td></td><td>0.0504</td><td>0.0852</td><td>0.0877</td><td>0.0998</td><td>0.0537</td><td>0.0794</td><td>0.0524</td><td>0.0490</td><td>0.0884</td><td>0.0999</td><td>0.0499</td><td>0.0507</td><td>0.0536</td><td>0.1000</td><td>0.0504</td></tr></tbody></table>

<table><caption>**Table D.17**
Fitted values for the reactions implied in cellulose pyrolysis obtained from the fitting method 3.</caption><thead><tr><th></th><th>CV<sub>1</sub></th><th>CV<sub>2</sub></th><th>CV<sub>3</sub></th><th>CV<sub>4</sub></th><th>CV<sub>19</sub></th><th>CV<sub>21</sub></th><th>CV<sub>26</sub></th><th>CV<sub>27</sub></th><th>CV<sub>28</sub></th><th>CV<sub>29</sub></th></tr></thead><tbody><tr><td>F.M. 3 - CV</td><td>0.0505</td><td>0.0225</td><td>0.056</td><td>0.0209</td><td>0.040</td><td>0.040</td><td>0.0405</td><td>0.040</td><td>0.0399</td><td>0.040</td></tr><tr><td>E<sub>CAEM</sub> (cal/mol)</td><td>45 000</td><td>15 000</td><td>19 100</td><td>31 000</td><td>24 500</td><td>52 500</td><td>59 000</td><td>70 000</td><td>10 000</td><td>45 000</td></tr><tr><td>E<sub>0</sub> (cal/mol)</td><td>40 947</td><td>14 366</td><td>17 211</td><td>29 780</td><td>22 719</td><td>48 683</td><td>54 711</td><td>64 911</td><td>92 75</td><td>41 728</td></tr><tr><td>σ (cal/mol)</td><td>2068</td><td>323</td><td>964</td><td>622</td><td>909</td><td>1947</td><td>2188</td><td>2596</td><td>370</td><td>1669</td></tr></tbody></table>

<table><caption>**Table D.18**
Fitted values for the reactions implied in lignins pyrolysis obtained from the fitting method 3.</caption><thead><tr><th></th><th>CV<sub>5</sub></th><th>CV<sub>6</sub></th><th>CV<sub>7</sub></th><th>CV<sub>8</sub></th><th>CV<sub>9</sub></th><th>CV<sub>10</sub></th><th>CV<sub>11</sub></th><th>CV<sub>12</sub></th><th>CV<sub>19</sub></th><th>CV<sub>20</sub></th><th>CV<sub>21</sub></th><th>CV<sub>22</sub></th><th>CV<sub>23</sub></th><th>CV<sub>24</sub></th><th>CV<sub>26</sub></th><th>CV<sub>27</sub></th><th>CV<sub>28</sub></th></tr></thead><tbody><tr><td>F.M. 3 - CV</td><td>0.0281</td><td>0.0794</td><td>0.0703</td><td>0.00182</td><td>0.0645</td><td>0.0352</td><td>0.0003</td><td>0.0519</td><td>0.0493</td><td>0.0466</td><td>0.0388</td><td>0.0475</td><td>0.0516</td><td>0.0433</td><td>0.0171</td><td>0.0285</td><td>0.0240</td></tr><tr><td>E<sub>CAEM</sub> (cal/mol)</td><td>37 500</td><td>25 500</td><td>37 200</td><td>24 800</td><td>30 000</td><td>12 000</td><td>8000</td><td>31 500</td><td>24 500</td><td>52 500</td><td>50 000</td><td>50 000</td><td>52 000</td><td>53 000</td><td>59 000</td><td>70 000</td><td>10 000</td></tr><tr><td>E<sub>0</sub> (cal/mol)</td><td>35 542</td><td>22 067</td><td>32 694</td><td>23 944</td><td>26 632</td><td>11 226</td><td>7995</td><td>28 591</td><td>22 340</td><td>48 102</td><td>46 466</td><td>45 738</td><td>47 221</td><td>48 852</td><td>57 087</td><td>66 292</td><td>9551</td></tr><tr><td>σ (cal/mol)</td><td>999</td><td>1751</td><td>2299</td><td>437</td><td>1718</td><td>395</td><td>3</td><td>1484</td><td>1102</td><td>2244</td><td>1803</td><td>2174</td><td>2438</td><td>2116</td><td>976</td><td>1892</td><td>229</td></tr></tbody></table>

<table><caption>**Table D.19**
Fitted values for the reactions implied in hemicelluloses pyrolysis obtained from the fitting method 3.</caption><thead><tr><th></th><th>CV<sub>1</sub></th><th>CV<sub>2</sub></th><th>CV<sub>3</sub></th><th>CV<sub>4</sub></th><th>CV<sub>5</sub></th><th>CV<sub>6</sub></th><th>CV<sub>7</sub></th><th>CV<sub>8</sub></th><th>CV<sub>9</sub></th><th>CV<sub>10</sub></th><th>CV<sub>11</sub></th><th>CV<sub>12</sub></th><th>CV<sub>13</sub></th><th>CV<sub>14</sub></th><th>CV<sub>15</sub></th></tr></thead><tbody><tr><td>F.M. 3 - CV</td><td>0.0504</td><td>0.0852</td><td>0.0877</td><td>0.0998</td><td>0.0537</td><td>0.0794</td><td>0.0542</td><td>0.0490</td><td>0.0884</td><td>0.0999</td><td>0.0499</td><td>0.0507</td><td>0.0536</td><td>0.1000</td><td>0.0504</td></tr><tr><td>E<sub>CAEM</sub> (cal/mol)</td><td>45 000</td><td>15 000</td><td>19 100</td><td>31 000</td><td>37 500</td><td>25 500</td><td>37 200</td><td>24 800</td><td>30 000</td><td>8 000</td><td>8 000</td><td>31 500</td><td>129 704</td><td>131 377</td><td>125 520</td></tr><tr><td>E<sub>0</sub> (cal/mol)</td><td>40 957</td><td>12 853</td><td>16 298</td><td>25 928</td><td>33 928</td><td>22 067</td><td>33 734</td><td>22 628</td><td>25 569</td><td>10 035</td><td>7288</td><td>28 651</td><td>117 383</td><td>1 098 489</td><td>114 245</td></tr><tr><td>σ (cal/mol)</td><td>2063</td><td>1095</td><td>1430</td><td>2588</td><td>1822</td><td>1752</td><td>1769</td><td>1108</td><td>2260</td><td>1002</td><td>363</td><td>1454</td><td>6286</td><td>10 984</td><td>5752</td></tr></tbody></table>

We obtain by identification:

$$k = \int_{0}^{\infty} A \exp \left( -\frac{E}{RT} \right) f(E) dE, \qquad (B.5)$$

If the solid does not have a distribution of energies but consists of only one type of reactive site with activation energy $E_i$, then the distribution collapses to a Dirac delta:

$$f(E) = \delta(E - E_i) \qquad (B.6)$$

This becomes:

$$k = A \exp \left( -\frac{E_i}{RT} \right) \qquad (B.7)$$

## Appendix C. Coefficients of variation

See Tables C.14–C.16.

## Appendix D. Fitted values of $E_0$ and $\sigma$

See Tables D.17–D.21.
<caption>**Table D.20**</caption>

Fitted values for the reactions implied in biomass pyrolysis obtained from the fitting method 3.

<table><caption>**Table D.21**</caption><thead><tr><th></th><th>CV<sub>1</sub></th><th>CV<sub>2</sub></th><th>CV<sub>3</sub></th><th>CV<sub>4</sub></th><th>CV<sub>5</sub></th><th>CV<sub>6</sub></th><th>CV<sub>7</sub></th><th>CV<sub>8</sub></th><th>CV<sub>9</sub></th><th>CV<sub>10</sub></th><th>CV<sub>11</sub></th><th>CV<sub>12</sub></th><th>CV<sub>13</sub></th><th>CV<sub>14</sub></th><th>CV<sub>15</sub></th></tr></thead><tbody><tr><td>F.M. 3 - CV</td><td>0.0505</td><td>0.0225</td><td>0.056</td><td>0.0209</td><td>0.0624</td><td>0.0798</td><td>0.0723</td><td>0.0722</td><td>0.0680</td><td>0.0359</td><td>0.0005</td><td>0.0597</td><td>0.0504</td><td>0.0852</td><td>0.0877</td></tr><tr><td>E<sub>CAEM</sub> (cal/mol)</td><td>45 000</td><td>15 000</td><td>19 100</td><td>31 000</td><td>37 500</td><td>25 500</td><td>37 200</td><td>24 800</td><td>30 000</td><td>12 000</td><td>8000</td><td>31 500</td><td>129 704</td><td>131 377</td><td>125 520</td></tr><tr><td>E<sub>0</sub> (cal/mol)</td><td>40 957</td><td>14 366</td><td>17 211</td><td>29 780</td><td>33 413</td><td>22 051</td><td>32 581</td><td>21 726</td><td>26 470</td><td>11 211</td><td>7993</td><td>28 201</td><td>118 043</td><td>112 577</td><td>107 109</td></tr><tr><td>σ (cal/mol)</td><td>2063</td><td>323</td><td>964</td><td>622</td><td>2085</td><td>1760</td><td>2357</td><td>1568</td><td>1801</td><td>403</td><td>4</td><td>1683</td><td>5949</td><td>9592</td><td>9393</td></tr></tbody></table>

More fitted values for the reactions implied in biomass pyrolysis obtained from the fitting method 3.

<table><thead><tr><th></th><th>CV<sub>16</sub></th><th>CV<sub>17</sub></th><th>CV<sub>18</sub></th><th>CV<sub>19</sub></th><th>CV<sub>20</sub></th><th>CV<sub>21</sub></th><th>CV<sub>22</sub></th><th>CV<sub>23</sub></th><th>CV<sub>24</sub></th><th>CV<sub>25</sub></th><th>CV<sub>26</sub></th><th>CV<sub>27</sub></th><th>CV<sub>28</sub></th><th>CV<sub>29</sub></th></tr></thead><tbody><tr><td>F.M. 3 - CV</td><td>0.0998</td><td>0.0537</td><td>0.0794</td><td>0.0509</td><td>0.0452</td><td>0.0636</td><td>0.0625</td><td>0.0507</td><td>0.047</td><td>0.0353</td><td>0.0562</td><td>0.0381</td><td>0.04</td><td>0.04</td></tr><tr><td>E<sub>CAEM</sub> (cal/mol)</td><td>46 024</td><td>15 062</td><td>127 612</td><td>24 500</td><td>52 500</td><td>50 000</td><td>50 000</td><td>52 000</td><td>53 000</td><td>55 000</td><td>59 000</td><td>70 000</td><td>10 000</td><td>45 000</td></tr><tr><td>E<sub>0</sub> (cal/mol)</td><td>38 494</td><td>13 628</td><td>110 427</td><td>22 278</td><td>48 227</td><td>44 458</td><td>44 454</td><td>47 300</td><td>48 529</td><td>51 441</td><td>53 146</td><td>65 136</td><td>9273</td><td>41 728</td></tr><tr><td>σ (cal/mol)</td><td>3842</td><td>732</td><td>8768</td><td>1134</td><td>2180</td><td>2828</td><td>2783</td><td>2398</td><td>2281</td><td>1816</td><td>2987</td><td>2482</td><td>371</td><td>1669</td></tr></tbody></table>

## Data availability

Data will be made available on request.

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