# A generic framework of submodels for soot evolution during fuel pyrolysis

Leonardo Pachano, Damien Aubagnac

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## To cite this version:

Leonardo Pachano, Damien Aubagnac. A generic framework of submodels for soot evolution during fuel pyrolysis. Fuel, 2026, 407 (Part B), pp.137203. 10.1016/j.fuel.2025.137203. hal-05454312

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Submitted on 12 Jan 2026

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Full length article

# A generic framework of submodels for soot evolution during fuel pyrolysis

Leonardo Pachano<fnref n="1" />, , Damien Aubagnac<fnref n="2" />,*,

IFP Energies nouvelles, 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France

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## HIGHLIGHTS

* Soot model validated for pyrolysis of ethylene and acetylene in PFRs.

* Extended validation includes oxygenated fuels and high-pressure shock tubes.

* Importance of surface growth and particle maturity in soot modeling during pyrolysis.

* Surface radical conservation is crucial for accurate acetylene pyrolysis modeling.

* PAH-soot condensation reversibility improves PAH mole fraction predictions.

## ARTICLE INFO

### Keywords:

Pyrolysis

0D modeling

Soot

## ABSTRACT

The study of soot production remains relevant not only because of the interest in decreasing its emissions within the transportation sector but also because of the growing interest in processes involving pyrolytic or quasi-pyrolytic conditions. Even in the context of soot emissions from gasoline direct injection (GDI) engines, soot is produced under high-temperature, low- to no-oxygen conditions. However, soot models are rarely validated in such conditions. Therefore, in this work, the predictability of a soot model based on a sectional approach representative of the current standard is assessed under pyrolysis conditions. A detailed study is carried out for ethylene and acetylene pyrolysis (fuels for which a comprehensive experimental dataset is available for validation) in atmospheric plug flow reactors (PFR) with temperatures ranging from 1073 to 1622 K. This initial validation highlights the importance of modeling surface growth, in particular the consumption of surface radicals, as well as the influence of the particle maturity description. The PAH concentration results also show the importance of the reversible PAH-soot interaction. An extended validation is then presented, on more varied but less documented cases, to assess the model's ability to capture trends in the soot propensity of different fuels including various concentrations of oxygenated fuels (ethanol and dimethyl ether), their binary mixtures with ethylene and acetylene, and binary mixtures of ethylene with propane and methane. This extended validation comprises the pyrolysis of fuel in atmospheric PFR cases with temperatures in the range 975–1475 K and behind reflected shock waves in shock tubes with temperatures in the range 1961–2600 K and pressures in the range 3–5 bar. With a good quantitative agreement with experimental data, results evidence the validity of the proposed formulation of surface growth and PAH-soot interactions to serve as a baseline for further modeling research on pyrolysis-formed carbon, including computational fluid dynamics (CFD) work since the model remains simple enough for such applications.

## 1. Introduction

The combined harmful effects of soot and the polycyclic aromatic hydrocarbons (PAHs) adsorbed onto its surface, impacting human health and the environment, are major reasons for studying soot production. On

the opposite side of the spectrum, the study of soot production is also relevant for numerous processes relying directly on soot or facing issues of clogging because of soot. Such processes can be as varied as processes to synthesize nanoparticles in spray pyrolysis [1,2] or in plug flow reactors (PFR) [3], production of hydrogen and black carbon from methane

<footnote>* Corresponding author.
Email address: damien.aubagnac@ifpen.fr; Damien.AUBAGNAC@3ds.com (D. Aubagnac).</footnote>

<footnote n="1">¹ Current address: Barcelona Supercomputing Center (BSC), Plaça d'Eusebi Güell 1-3, 08034 Barcelona, Spain.</footnote>

<footnote n="2">² Current address: Dassault Systèmes, 10 rue Marcel Dassault, 78140 Vélizy-Villacoublay, France.</footnote>
pyrolysis [4] or chemical looping combustion (CLC) processes facing coke clogging [5]. Depending on the application, the chosen technology and the operating conditions, the formation of soot can be observed in a wide range of temperatures from a few hundred to a few thousand Kelvin.

In Europe, with strict legislation on the limit of the amount and number of particles emitted by gasoline engines [6], and increasing attention on emissions from the aviation sector [7], the study of soot production is of the utmost importance to meet particle emission limits. Of particular relevance for this work, soot production in gasoline direct injection (GDI) engines has the particularity of taking place under quasi-pyrolysis conditions. The combination of fuel impingement (leading to local fuel-rich regions) and high-temperature, low-oxygen conditions induced by the combustion process drives the formation of soot [8,9].

The production of soot resulting from hydrocarbon pyrolysis has been the subject of study in many previous publications [10]. Among different fuels, ethylene and acetylene are commonly chosen due to their relative simplicity which facilitates the study of the already highly complex soot production process [11]. As with the study of many other such processes, the study of soot production relies on both experimental and numerical works. Numerically, Mei et al. [12] and Abian et al. [13] relied on modeling results to complement the analysis of their experimental study on the effect of carbon dioxide and sulfur dioxide addition on the production of soot in PFR ethylene pyrolysis. Similarly, Eremin et al. [14] and Drakon et al. [15] used numerical results to support some of the experimental trends observed for soot production in acetylene pyrolysis with the addition of methane, hydrogen, and oxygen in a shock tube. Also relying on numerical results, Ruiz et al. [16] studied soot production in PFR ethylene and acetylene pyrolysis. Nevertheless, none of the aforementioned studies accounted for soot production modeling.

Several approaches to account for soot production in numerical studies have been reported. For instance, Krestinin [17] proposed a soot model based on the condensation of supersaturated polyyne vapor (C₂nH₂ for n > 1) which was validated for soot induction time, particle number density, and volume fraction in the pyrolysis of several fuels. On their part, Agafonov et al. [18–20] proposed a model based on the description of heterogeneous polymerization kinetics accounting for soot nucleation, growth, and coagulation. They validated it focusing on soot yield, induction time and temperature in the pyrolysis of acetylene mixtures with acetone and propane [18] and pyrolysis (as well as oxidation) of single aliphatic (including ethylene and acetylene) and aromatic molecules [19,20]. Previously, Vlasov and Warnatz [21], also studying soot production in ethylene and acetylene pyrolysis (among other fuels), had validated soot yield and induction time results with the modeling approach later used by Agafonov et al. [18–20,22]. More in line with the soot modeling approach presented in this work, Utsav et al. [23] based the modeling of soot on the Method of Moments (MoM) to study its production in ethylene and benzene pyrolysis through validation of acetylene mole fraction, soot volume fraction, mean particle diameter, particle density, and soot induction time results.

These works provide valuable insights into the phenomena driving soot formation in a pyrolytic context. However, they mostly focus on details of soot particle formation while the phenomena driving their global evolution remain very uncertain mostly because of the uncertainties regarding the accuracy of gas-phase models accuracy. In a different approach, Naseri and Thomson [3] used a particle population balance equation model to study methane oxidation and ethylene pyrolysis from the perspective of carbon black synthesis. In this regard, the model was validated for major species and PAH mole fractions, soot volume fraction, particle size distribution and soot aggregate shape. More recently, Naseri et al. [24] and Juan et al. [25] published results on carbon black synthesis from ethylene pyrolysis in a PFR focusing on the relevance of inception and condensation and the possible effect that radial diffusion might have on the gas phase and soot predictions. Similar to Naseri et al. [24], this study primarily investigates the particle phase, drawing gas-phase data from the literature. Crucially, both works model the overall

distribution, a scale distinct from the molecular dynamics approaches used in other studies.

Within this context, the main objective of this work is to investigate and highlight the specific requirements, related to phenomena modeling, for accurately predicting soot production in pyrolysis cases. The model actually assessed is referred to as Sectional Soot Model (SSM) in this paper. It is based on a sectional approach and was initially developed and validated for automotive emissions predictions [26] before being updated with new developments concerning PAH-soot interactions validated on academic one-dimensional flames [27]. It describes soot formation and evolution as the sum of five main phenomena commonly used for this purpose: three collision-based phenomena which are particle inception due to PAH collisions, PAH adsorption growth represented by PAH collisions on soot particles, soot particle coagulation; and two surface chemistry phenomena which are surface growth and oxidation reactions which in the case of fuel pyrolysis are only relevant for oxygenated fuels. Except for the reversible PAH-soot interaction which is only considered in this model and the work of Eaves and co-workers [28] pursued by Kholghy and co-workers [24,29], the rest of the modeled phenomena are very similar to soot models commonly used in literature for simulations ranging from 0D/1D cases coupled with chemistry solvers for 3D computational fluid dynamics (CFD) simulations. Thus, the proposed model is representative of common approaches within the soot modeling community, including sectional methods, the MoM, and the more recent three-equation models from Franzelli et al. [30] or Kholghy and Kelesidis [31]. The conclusions drawn are therefore expected to remain relevant for any modeling teams using such models. Moreover, since the objective of this study is to provide insights relevant to various types of models, including those that do not predict size distributions, we report only soot yields and gas-phase interactions in the main text. Nevertheless, as the simulations were carried out with a sectional method, size distributions were also obtained. The distributions for the validation cases are presented in the Supplementary Material, together with the distribution from an additional literature case where experimental measurements are available.

The validation splits into two phases. The first phase focuses on cases of ethylene and acetylene pyrolysis where the gas-phase accuracy can be thoroughly assessed. This allows the modeling choice justification and sensitivity analysis to be performed on cases where the error induced by gas-phase modeling is minimal and can be quantified. In this context, the impact of surface kinetics and PAH-condensation on soot predictions will be quantified. This investigation is motivated by the fact that the evaluated model initially exhibited deficiencies in its surface kinetics, and its PAH predictions consistently lacked acceptable accuracy across various modeling scenarios. This phase relies on the experimental works of Sánchez and co-workers on acetylene [32,33] and ethylene [33] pyrolysis. These cases provide a very useful resource to assess the predictivity of soot models in conditions where the gas-phase modeling error can be quantified since gas species measurements up to large PAH are available as well as soot yield measurements. These works also include modeling results for which a sectional description of the soot phase is used. However, these studies were focused on the chemical pathways driving gas species formation [32], the soot modeling results being only shown in the work of Sánchez [33]. Similarly to this study's initial model, their model failed to reproduce the trends of soot yields and it closed the modeling effort on these cases. Yet, it sets a reference to compare with on these cases. A second validation phase further evaluates the proposed model's ability to reproduce soot production trends across a wider range of conditions. These cases include ethanol PFR measurements by Viteri et al. [34] at various residence times, PFR measurements by Zhao and Liu [35] for ethylene, ethanol, dimethyl ether (DME), and their blends, and acetylene/ethanol blend PFR measurements at various blending ratios from Esarte et al. [36]. Collectively, these new cases introduce two additional fuels (ethanol and DME), along with their blends, and incorporate variations in residence time, expanding beyond the fuels previously studied in this work. Furthermore, ethylene shock
tube measurements from Agafonov et al. [20] and Utsav et al. [23] are utilized to assess the model's performance at higher temperatures compared to the PFR cases. Notably, the data from Utsav et al. [23] also provide time-resolved information, enabling evaluation of the model's transient behavior.

In the following, Section 2 is divided into two parts: Section 2.1 provides detailed descriptions of the validation cases and discusses their relevance for validating the influence of the assessed soot modeling points on key features of soot production in various applied pyrolysis scenarios; Section 2.2 briefly describes the model reactors used to simulate these cases. The soot model and the assumptions it relies on are described in Section 3 with a focus on the phenomena whose modeling will be assessed in this paper. Then, the gas-phase kinetic mechanism choice is discussed in Section 4. Finally, results are presented and discussed in Section 5.

## 2. Methodology

This section describes the target conditions against which the initial model will be tested and refined, along with the rationale for their selection. Additionally, the modeling methodology employed for these cases is briefly presented.

### 2.1. Target conditions

#### 2.1.1. Detailed validation: soot and gas species

The choice of relevant validation cases is of utmost importance in the methodology. All conclusions drawn further in the paper concerning the soot model rely on the fact that the accuracy of gas-phase predictions can be assessed. Sánchez and co-workers [33,37] provided data allowing such assessment for ethylene and acetylene pyrolysis in PFR. The first step of validation will focus on these two fuels, which enables the use of a single gas-phase chemical mechanism whose choice is justified in Section 4. Moreover, the considerable amount of other experimental data available on ethylene and acetylene pyrolysis [38,39] allows for extending this detailed validation phase with more parametric variations than the ones performed by Sánchez and co-workers while keeping conditions similar to the ones where soot and gas-phase will be validated together. Thus, the data set from Sánchez et al. [33,37] for which fuel, acetylene and PAH mole fractions are reported along with soot results from the pyrolysis of 3 % ethylene and various concentrations of acetylene diluted in nitrogen, will be considered as the main validation cases. Within this set of experiments, fuel was pyrolyzed in a PFR at low-temperature, atmospheric pressure conditions and results were retrieved at long residence times (in comparison to shock tube data sets) in the order of $\mathcal{O}(10^3)$ ms according to the expression $t_r = 1706/T$ for T in K and $t_r$ in s with the constant numerator in Ks. To extend the variation on fuel concentration, cases from Tesner and Shurupov [38] and Krestinin et al. [39] will also be simulated. They report soot yield results for PFR pyrolysis of 20 % ethylene and 14.29 % acetylene diluted in helium at atmospheric pressure, low-temperature conditions and residence times in the order of $\mathcal{O}(10^2)$ ms. This set of cases will provide insights into the soot model's ability to reproduce sooting trends depending on fuel concentration. The target conditions from these two data sets are summarized in Table 1.

<table><caption>**Table 1**
Initial conditions and residence time for the atmospheric PFR cases selected for the first validation step.</caption><thead><tr><th>Fuel</th><th>T [K]</th><th>t<sub>r</sub> [ms]</th><th>Reference</th></tr></thead><tbody><tr><td>3 % C<sub>2</sub>H<sub>4</sub></td><td>1223–1423</td><td>Θ(10<sup>3</sup>)</td><td>[33,37]</td></tr><tr><td>20 % C<sub>2</sub>H<sub>4</sub></td><td>1423–1622</td><td>Θ(10<sup>2</sup>)</td><td>[38,39]</td></tr><tr><td>1 % C<sub>2</sub>H<sub>2</sub></td><td>1073–1323</td><td>Θ(10<sup>3</sup>)</td><td>[33,37]</td></tr><tr><td>2 % C<sub>2</sub>H<sub>2</sub></td><td>1073–1323</td><td>Θ(10<sup>3</sup>)</td><td>[33,37]</td></tr><tr><td>3 % C<sub>2</sub>H<sub>2</sub></td><td>1073–1323</td><td>Θ(10<sup>3</sup>)</td><td>[33,37]</td></tr><tr><td>14.29 % C<sub>2</sub>H<sub>2</sub></td><td>1423–1622</td><td>Θ(10<sup>2</sup>)</td><td>[38,39]</td></tr></tbody></table>

<table><caption>**Table 2**
Initial conditions and residence times for the PFR and shock tube cases selected for the second validation step.</caption><thead><tr><th>Fuel</th><th>T [K]</th><th>t<sub>r</sub> [ms]</th><th>Reference</th></tr></thead><tbody><tr><td>5 % C<sub>2</sub>H<sub>5</sub>OH</td><td>975–1475</td><td>Θ(10<sup>3</sup>)</td><td>[34]</td></tr><tr><td>2 % C<sub>2</sub>H<sub>4</sub></td><td>1273–1473</td><td>Θ(10<sup>3</sup>)</td><td>[35]</td></tr><tr><td>2 % C<sub>2</sub>H<sub>5</sub>OH</td><td>1273–1473</td><td>Θ(10<sup>3</sup>)</td><td>[35]</td></tr><tr><td>2 % DME</td><td>1273–1473</td><td>Θ(10<sup>3</sup>)</td><td>[35]</td></tr><tr><td>1 % C<sub>2</sub>H<sub>4</sub>–1 % C<sub>2</sub>H<sub>5</sub>OH</td><td>1273–1473</td><td>Θ(10<sup>3</sup>)</td><td>[35]</td></tr><tr><td>1 % C<sub>2</sub>H<sub>4</sub>–1 % DME</td><td>1273–1473</td><td>Θ(10<sup>3</sup>)</td><td>[35]</td></tr><tr><td>4.5 % C<sub>2</sub>H<sub>4</sub>–0.5 % C<sub>2</sub>H<sub>5</sub>OH</td><td>975–1475</td><td>Θ(10<sup>3</sup>)</td><td>[36]</td></tr><tr><td>4 % C<sub>2</sub>H<sub>2</sub>–1 % C<sub>2</sub>H<sub>5</sub>OH</td><td>975–1475</td><td>Θ(10<sup>3</sup>)</td><td>[36]</td></tr><tr><td>3 % C<sub>2</sub>H<sub>2</sub>–2 % C<sub>2</sub>H<sub>5</sub>OH</td><td>975–1475</td><td>Θ(10<sup>3</sup>)</td><td>[36]</td></tr><tr><td>5 % C<sub>2</sub>H<sub>4</sub></td><td>2000–2600</td><td>Θ(10<sup>3</sup>)</td><td>[20]</td></tr><tr><td>3 % C<sub>2</sub>H<sub>4</sub></td><td>2000–2600</td><td>Θ(10<sup>3</sup>)</td><td>[20]</td></tr><tr><td>1.5 % C<sub>2</sub>H<sub>4</sub></td><td>2000–2600</td><td>Θ(10<sup>3</sup>)</td><td>[20]</td></tr><tr><td>2.35 % C<sub>2</sub>H<sub>4</sub></td><td>2179</td><td>Θ(10<sup>1</sup>)</td><td>[23]</td></tr><tr><td>2.23 % C<sub>2</sub>H<sub>4</sub>–0.12 % C<sub>3</sub>H<sub>8</sub></td><td>2132</td><td>Θ(10<sup>1</sup>)</td><td>[23]</td></tr><tr><td>2.23 % C<sub>2</sub>H<sub>4</sub>–0.12 % CH<sub>4</sub></td><td>1961–2136</td><td>Θ(10<sup>1</sup>)</td><td>[23]</td></tr></tbody></table>

#### 2.1.2. Extended validation: soot trends on a wider range

As mentioned in the introduction, the proposed model has also been tested, as is, on an extended set of cases including two additional fuels, namely ethanol and DME. In general terms, cases selected for the extended validation can be classified into two groups. First, low-temperature, atmospheric pressure PFR cases. These comprise ethanol pyrolysis at two residence times according to $t_r = 4168/T$ and $t_r = 1563/T$ [34], pyrolysis of ethylene, ethanol, DME and their mixtures at residence time $t_r = 2605/T$ [35], and lastly pyrolysis of acetylene and ethanol mixtures at residence time $t_r = 4550/T$ [36]. In the expressions for residence time, $t_r$ and $T$ are in units of K and s, respectively. Second, high-temperature, high-pressure shock tube cases from the work of Agafonov et al. [20] and Utsav et al. [23]. The first group of authors reported soot yields from the pyrolysis of various concentrations of ethylene at 1.5 ms in a pressure range of 3–5 bar. The second group of authors reported time-resolved acetylene and soot volume fraction profiles from the pyrolysis of ethylene at 3.8 bar, ethylene blended with propane at 3.37 bar, and ethylene blended with methane at 3.70–3.89 bar. Time-resolved measurements from this data set facilitate the evaluation of the predictability of the modeling approach under transient conditions. The target conditions for the extended validation are reported in Table 2.

### 2.2. Modeling strategy

OD simulations are carried out using Cantera [40], an open-source suite of tools including a chemical kinetics solver. For either shock tube or PFR setups, the reactor system is composed of an ideal gas, constant-pressure reactor. The same formulation is valid for both setups assuming that the PFR can be characterized by a Lagrangian particle traveling through the reactor. The reactor is also assumed to have no mixing in the axial direction and to be perfectly mixed in the radial direction. Juan et al. recently published results supporting the plug flow idealization for the gas-phase based on 2D results of a laminar flow reactor [25]. The authors also reported variations in soot predictions induced by radial diffusion and proposed the discretization of the PFR into different streamlines treated as individual idealized PFR. The possible effect of radial diffusion on soot results is neglected in this work as the sensitivity found under similar conditions [24,25] is small in comparison to sensitivity to surface growth and PAH-soot mass exchanges further discussed in the next sections. Lastly, it is also worth mentioning that, for the PFR configuration, the energy conservation equation is not solved as the target temperature is reported to be constant in the experiments. Also for this setup, at conditions between 1273 and 1423 K the experimentally measured temperature profile is available from the work of Sánchez et al. [33,37]. Thus, it is imposed in the simulations to initialize the reactor until reaching the target temperature. Lastly, shock tube cases account for heat losses to reproduce the temperature decrease observed by Agafonov
et al. [20] during ethylene pyrolysis under conditions similar to those of this study.

### 3. Soot model

Within the SSM, soot particles are considered a distinct dispersed phase composed of solid particles interacting with the gaseous phase through a two-way coupling. Solid particle populations are assessed using a sectional method, where soot particles are discretized by mass into sections, with the $i^{th}$ section spanning the range [$m_{min,i}$; $m_{max,i}$]. As in the work of Aubagnac-Karkar et al. [27], soot size discretization is done in terms of mass and not volume as in most other section-based soot models. In this case, it does not influence results since the density of soot ($\rho_{soot}$) is assumed to be constant at a value of 1.86 g.cm$^{-3}$. As for the number of sections, a high-resolution approach, accounting for 200 sections, is used in this work to minimize numerical diffusion in the space of mass and prevent any doubt concerning section convergence. Each of these sections is described by its mass fraction $Y_{soot}$ whose evolution is computed using Eq. (1).

$$ \frac{\partial}{\partial t} (Y_{soot,i}) = \dot{\omega}_{soot,i} = \dot{\omega}_{nucl,i} + \dot{\omega}_{cond,i} + \dot{\omega}_{sg,i} + \dot{\omega}_{coag,i} \quad (1) $$

In Eq. (1), $\dot{\omega}_{soot,i}$ is the section i soot mass fraction source term. It includes source terms for collisional (nucleation, condensation and coagulation, $\dot{\omega}_{nucl,i}$, $\dot{\omega}_{cond,i}$ and $\dot{\omega}_{coag,i}$, respectively) and surface chemistry (surface growth, $\dot{\omega}_{sg,i}$) phenomena according to the formulation described by Aubagnac-Karkar et al. [27]. The model also accounts for oxidation, which in this study is only relevant for the pyrolysis of oxygenated fuels like ethanol and DME.

#### 3.1. PAH-soot interaction

Under the current formulation, condensation is assumed to be a reversible process, while particle inception is damped by a factor relying on a thermodynamic equilibrium constant similar to the one used to compute condensation reversibility. The description of PAH-desorption from soot, modeled by considering condensation reversible through the computation of an equilibrium constant, was first proposed by Eaves et al. [28]. In the present model, equilibrium constants are computed as proposed by Totton et al. [41]. The application of a damping factor instead of considering particle inception completely reversible was introduced in Aubagnac-Karkar et al. [27]. Practically, this factor is equivalent to a collision efficiency as can be found in many soot models in various forms [42]. The difference only relies on the way this multiplicative factor is computed. Lastly, the choice of the PAH considered as soot precursors depends on the gas-phase mechanism the SSM is coupled. All PAH with strictly more than three rings described by the gas-phase mechanism are considered as precursors. The effect of PAH-soot interaction is assessed on the 3 % ethylene PFR pyrolysis case [37] and results are provided in Section 5.2.2.

#### 3.2. Surface reactions modeling

Surface reaction modeling requires two key elements described in the following order: first, a method to convert resolved data, such as particle number or mass for a given size, into an equivalent concentration; second, a surface reaction mechanism that represents the processes occurring at the soot particle surface.

##### 3.2.1. Evaluation of soot reacting surface

Under the SSM formulation, the concentration of soot particles within the size range represented by section i is deduced from $Y_{soot,i}$ using Eq. (2):

$$ [C_{s,i}] + [C_{s,i}^*] = \int_{m_{min,i}}^{m_{max,i}} \alpha N_A^{-1} \left( \frac{m}{m_{c2}} \right)^{\frac{\theta(m)}{3}} n_i(m) dm \quad (2) $$

In Eq. (2), $[C_{s,i}]$ and $[C_{s,i}^*]$ represent the equivalent concentrations of stable and radical sites, respectively. The ratio between $[C_{s,i}]$ and $[C_{s,i}^*]$ is estimated by assuming that $[C_{s,i}^*]$ is at quasi-steady state. $(m/m_{c2})^{\frac{\theta(m)}{3}}$ arises from the product of the site surface density $\lambda_{site}$ and the particle surface area, under the assumption of a reactive site surface density $\lambda_{site} = 1/S_{c2}$, where $S_{c2}$ is the surface area of a sphere whose volume corresponds to that of a pair of carbon atoms as proposed by Netzell [43].

A surface-to-volume correction factor in the form of a fractal function $\theta(m)$, dependent on particle size, is introduced, as commonly done in similar models. However, the exact definition of this fractal function remains unknown, as is the case for the fractal dimension of soot particles, as noted by Rodrigues [44]. Rodrigues et al. [45] proposed a correlation to apply a surface-to-volume scaling that matches results from MoM simulations [46,47] themselves validated against Monte-Carlo simulations. While the accuracy of this correlation cannot yet be fully assessed due to current limitations in soot modeling knowledge, it remains the most validated correlation available to date. Therefore, this correlation is adopted for all simulations presented in this study.

The final parameter of Eq. (2) to be described is the proportion of reactive sites $\alpha$. Due to its high uncertainty, it is considered to potentially compensate for errors in other parameters [48]. Functions for $\alpha$ dependent on temperature, particle size or thermal age have been proposed recently in the literature [49–51]. In numerous other studies, $\alpha$ is treated as a constant, with values ranging from approximately 0.08 [27,48] to 1 [48,52]. However, as discussed further in Section 5.1, models that use constant values for $\alpha$ cannot reproduce experimental results with sufficient accuracy. To address this, a new formulation is proposed, which depends on particle size and is similar to the one suggested by Appel et al. [53]. The proposed formulation also draws inspiration from Veshkini et al. [50], who introduced a correction based on particles’ thermal age, reflecting the accepted idea that soot particle reactivity decreases with maturity [50,54]. Additionally, Charoensin-O-Larn et al. [55] proposed a model to account for soot thermodeactivation based on char thermodeactivation [56] though with a different formulation. Their model also heavily depends on the definition of particle thermal age. The primary reason for adopting the proposed formulation, as opposed to the others mentioned, is the goal of developing a model that computes all necessary data on-the-fly, making it applicable to various scenarios without requiring prior knowledge of the maximum temperature or a post-processing step to compute thermal age. Consequently, the assumption made in this work is that particle maturity increases with particle size. While this assumption is rough, it is acceptable for cases where soot particles grow consistently over time such as in the case of fuel pyrolysis. A more detailed description of soot maturity would require structural information such as the C/H ratio, the surface composition, or even data typically accessible only in molecular dynamics simulations. While the former could, in principle, be tracked within a model like the one proposed here, this would dramatically increase computational cost by multiplying the number of variables to solve. This would also render the proposed approach unsuitable for use in simpler two- or three-equation models. Multi-variable approaches have been reported in the literature to track primary particles or concentration of PAH molecules on the surface of the soot particles [28,57], including applications up to 2D simulations of Santoro burners, but adopting such a formalism would strongly limit the model’s applicability to larger-scale or potentially industrial CFD cases. Hence, in the proposed model $\alpha$ is expressed as:

$$ \alpha(d_p) = \alpha_0 \left( 0.5 - 0.5 \cdot \tanh \left( \log_{10} \left( \frac{d_p}{d_{cut}} \right) \right) \right) \quad (3) $$

Where $\alpha_0 = 0.2$ is the maximum value of $\alpha$, $d_{cut} = 25$ nm is a transition diameter for which $\alpha = \alpha_0/2$ and $d_p$ is the section mean diameter. $\alpha_0$ and $d_{cut}$ are the only two parameters introduced and their values have been fitted against the detailed validation database as described in
<caption>**Table 3**</caption>

Soot surface growth rate constants in the Arrhenius form, $k = AT^b e^{-E_a/RT}$, given by the units of mol, m, s, K, kcal.

<table><thead><tr><th></th><th>Reaction</th><th>A</th><th>b</th><th>E<sub>a</sub></th><th>Ref</th><th>Cases</th></tr></thead><tbody><tr><td>R1<sub>f</sub></td><td>C<sub>s</sub> + H &harr; C<sub>s</sub><sup>*</sup> + H<sub>2</sub></td><td>4.17 &middot; 10<sup>7</sup></td><td>0.0</td><td>13.0</td><td>[53]</td><td>A,B,C</td></tr><tr><td>R1<sub>b</sub></td><td></td><td>3.9 &middot; 10<sup>6</sup></td><td>0.0</td><td>11.0</td><td>[53]</td><td>A,B,C</td></tr><tr><td>R2</td><td>C<sub>s</sub><sup>*</sup> + H &harr; Cs</td><td>2.0 &middot; 10<sup>7</sup></td><td>0.0</td><td>0.0</td><td>[53]</td><td>A,B,C</td></tr><tr><td>R3</td><td>C<sub>s</sub><sup>*</sup> + C<sub>3</sub>H<sub>3</sub> &harr; Cs + H</td><td>8.0 &middot; 10<sup>1</sup></td><td>1.56</td><td>3.8</td><td>[53]</td><td>A,B,C</td></tr><tr><td>R3*</td><td>C<sub>s</sub><sup>*</sup> + C<sub>2</sub>H<sub>2</sub> &harr; C<sub>s</sub><sup>*</sup> + H</td><td>same as R3</td><td></td><td></td><td>[62]</td><td>all</td></tr><tr><td>R4</td><td>C<sub>s</sub> + C<sub>2</sub>H<sub>2</sub> &harr; C<sub>s</sub><sup>*</sup> + C<sub>2</sub>H<sub>2</sub></td><td>2.0 &middot; 10<sup>7</sup></td><td>0.0</td><td>0.0</td><td>[60]</td><td>B</td></tr><tr><td rowspan="2">R5<sub>f</sub></td><td rowspan="2">5f.1<br>5f.2</td><td>C<sub>s</sub> + CH<sub>3</sub> &harr; C<sub>s</sub><sup>*</sup> + CH<sub>4</sub></td><td>2.0 &middot; 10<sup>7</sup></td><td>0.0</td><td>0.0</td><td>[60]</td><td>B</td></tr><tr><td>3.99 &middot; 10<sup>-7</sup></td><td>3.933</td><td>11.771</td><td>[23]</td><td>C</td></tr><tr><td>R5<sub>b</sub></td><td></td><td>4.48 &middot; 10<sup>-8</sup></td><td>4.248</td><td>4.277</td><td>[23]</td><td>C</td></tr><tr><td rowspan="2">R6<sub>f</sub></td><td rowspan="2">6f.1<br>6f.2</td><td>C<sub>s</sub> + C<sub>3</sub>H<sub>3</sub> &harr; C<sub>s</sub><sup>*</sup> + AC<sub>3</sub>H<sub>4</sub></td><td>5.0 &middot; 10<sup>6</sup></td><td>0.0</td><td>0.0</td><td>[60]</td><td>B</td></tr><tr><td>9.538 &middot; 10<sup>-6</sup></td><td>3.529</td><td>24.4492</td><td>[23]</td><td>C</td></tr><tr><td>R6<sub>b</sub></td><td></td><td>1.36 &middot; 10<sup>-6</sup></td><td>3.761</td><td>1.0889</td><td>[23]</td><td>C</td></tr><tr><td>R7<sub>f</sub></td><td>C<sub>s</sub> + OH &harr; C<sub>s</sub><sup>*</sup> + H<sub>2</sub>O</td><td>1.0 &middot; 10<sup>4</sup></td><td>0.734</td><td>1.43</td><td>[53]</td><td>all</td></tr><tr><td>R7<sub>b</sub></td><td></td><td>3.68 &middot; 10<sup>2</sup></td><td>1.139</td><td>17.1</td><td>[53]</td><td>all</td></tr><tr><td>R8</td><td>C<sub>s</sub><sup>*</sup> + O<sub>2</sub> &harr; Cs + 2 CO</td><td>2.2 &middot; 10<sup>6</sup></td><td>0.0</td><td>7.5</td><td>[53]</td><td>all</td></tr><tr><td>R9</td><td>C<sub>s</sub> + OH &harr; C<sub>s</sub><sup>*</sup> + HCO</td><td>1.62 &middot; 10<sup>5</sup></td><td>0.5</td><td>0.0</td><td>[63]</td><td>all</td></tr></tbody></table>

Section 5.1. The absence of temperature dependence in Eq. (3) is discussed in Section 5.1 and 5.3. A sensitivity analysis of the results with respect to $\alpha_0$ and $d_{cut}$ is provided in the Supplementary Material.

### 3.2.2. Soot surface reaction kinetics

The soot surface reaction mechanism describing soot interaction with the gas phase is also very uncertain to date. Although several PAH growth processes are considered in the literature as listed by Shukla and Koshi [58], the HACA mechanism [54,59] and slight variations of it are mostly used for soot growth. Thus, the approach retained in this study is to evaluate the accuracy of the main mechanisms proposed in the literature in the context of pyrolysis, since most of them are designed for combustion. This evaluation is presented in Section 5.1 as well as the process that led to the choice of a preferred mechanism. The most relevant reactions assessed in the present work for soot surface growth are summarized in Table 3. In addition to reactions in the reference HACA mechanism [53] (Case A), hydrocarbon radicals' surface-growth-enhancing reactions proposed by Hwang and Chung [60] are also considered (Case B). So are the reactions proposed in the works of Utsav et al. [23] (Case C), Saggese et al. [61] (Case D), and Agafonov, Bilera and Vlasov [22] (Case E). The chosen set of reactions used in this work is indicated as "p.w." and it is a subset of the mechanism proposed by Hwang and Chung [60]. Table 3 also includes reactions accounting for oxidation which, as stated before, only play a role in the pyrolysis of ethanol and DME.

The consumption of surface radicals through reaction R3 is another highly uncertain aspect of surface reaction mechanisms. Although all the mechanisms referenced include R3, Wang et al. [62] suggested that the radical-based pathway R3* could dominate in the fuel-rich regions of counterflow non-premixed flames. In fuel pyrolysis, where temperatures typically remain around or below 1500 K, the conditions are expected to resemble those of fuel-rich soot-forming regions in such flames. This similarity motivates the inclusion of R3* in the present study. The impact of this modeling choice is evaluated in Section 5.1 through a comparison of simulation results using different surface reaction mechanisms, incorporating either R3* or R3.

## 4. Gas-phase kinetic mechanisms

Six different gas-phase mechanisms, summarized in Table 4, have been selected from the literature because they describe ethylene and acetylene pyrolysis and the subsequent gas-phase species evolution up to PAH which can be considered soot precursors. Two of these gas-phase mechanisms were developed and validated for fuel pyrolysis (Mech-1 from Norinaga and Deutschmann [64] and Mech-4 from Tao et al. [65]). The other four mechanisms (Mech-2 from Wang et al. [66], Mech-3 from Chernov et al. [67], Mech-5 from Wang et al. [68], and Mech-6 from

CRECK Modeling Group [69–71]) were developed and used to describe fuel oxidation over a wide range of operating conditions and flame types.

Some particularities are worth mentioning. First, even if the development of Mech-4 [65] only targeted acetylene pyrolysis results, it is still relevant to assess its performance for ethylene pyrolysis as Mech-1 [64] is one of its main building blocks. Second, Mech-5 [68] is a recently introduced version of Mech-3 [67] optimizing $C_{<3}$ chemistry as a first step toward an improved PAH sub-model. Third, Mech-6 [69–71] is a detailed mechanism with an embedded soot model (labeled "$C_1-C_{16}$ HT + Soot mechanism (Version 2003, March 2020)" by the authors) comprising 24,041 reactions and 452 species. In order to carry out a fair comparison, the mechanism has been modified and all reactions coming from the soot model have been removed. Results for the different gas-phase mechanisms are reported in the following.

Soot predictions are very dependent on the gas phase description. However, the scarcity of data concerning soot precursors makes any choice of gas-phase mechanism questionable. This section aims to explain how this choice is made amongst the mechanisms listed in Table 4 in the present work as well as highlighting the resulting uncertainties.

The validation of PAH and major species mole fraction results is the main criterion for the gas-phase mechanism choice. The suitability of any given mechanism is evaluated for PFR ethylene pyrolysis. Experimental data available allow evaluating mole fraction modeling results for several PAH, fuel, and acetylene which is a species of paramount importance for soot surface growth [72]. In a first approach, in the task of choosing the most suitable gas-phase mechanism, modeling results for four rings or larger PAH (validation of smaller PAH can be found in the Supplementary Material) are compared against experimental data in Fig. 1. Since the production of soot is expected to impact PAH predictions, modeling results are presented with soot production not being taken into account to isolate the inherent differences between the gas-phase mechanisms.

PAH mole fractions reported in Fig. 1 hardly allow discrimination between mechanisms. Although they behave differently, especially concerning pyrene dependency on temperature, none actually outperforms the others. In more concrete terms, Mech-1 [64] and Mech-4 [65] PAH predictions seem quite similar. This observation might be expected as the development of the latter is largely based on Mech-1 [64]. With these two gas-phase mechanisms, most predictions seem reasonable at temperatures higher than 1300 K. Fluoranthene, chrysene (only for Mech-1), benzo[k]fluoranthene, and benzo[a]pyrene are overpredicted but this is expected since soot is not taken into account at this point and significant soot yields are produced at these temperatures. Other PAHs predictions are in line with the measurements which means they will probably be underpredicted in simulations with soot modeling enabled. On the contrary, at lower temperatures, results are underpredicted
<table><caption>**Table 4**
Assessed gas-phase mechanisms.</caption><thead><tr><th>Mechanism</th><th>N° reactions/species</th><th>Validation</th><th>Ref</th></tr></thead><tbody><tr><td>Mech-1</td><td>827/248</td><td>C<sub>2</sub>H<sub>4</sub>, C<sub>2</sub>H<sub>2</sub> and C<sub>1</sub>H<sub>6</sub> pyrolysis</td><td>[64]</td></tr><tr><td>Mech-2</td><td>1351/202</td><td>C<sub>2</sub>H<sub>4</sub> oxidation</td><td>[66]</td></tr><tr><td>Mech-3</td><td>831/102</td><td>C<sub>2</sub>H<sub>4</sub>, CH<sub>4</sub> and C<sub>2</sub>H<sub>6</sub> oxidation</td><td>[67]</td></tr><tr><td>Mech-4</td><td>1266/290</td><td>C<sub>2</sub>H<sub>2</sub> pyrolysis</td><td>[65]</td></tr><tr><td>Mech-5</td><td>971/115</td><td>C<sub>2</sub>H<sub>4</sub> oxidation</td><td>[68]</td></tr><tr><td>Mech-6</td><td>14,233/367</td><td>C<sub>1</sub>–C<sub>16</sub> oxidation</td><td>[69–71]</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Soot precursors PAH mole fractions from the pyrolysis of 3 % C<sub>2</sub>H<sub>4</sub> in a PFR as a function of initial temperature. Modeling results with all gas-phase mechanisms listed in Table 4. Experimental data (black symbols) from Sánchez et al. [37].</figcaption></figure>

across all measured PAH except for pyrene for which Mech-4 [65] matches the experimental observations. Mech-2 [66] also shows reasonable results for pyrene and benzo[g,h,i]perylene. Moreover, it predicts exactly the expected trends for benzo[a]pyrene with a good agreement at low temperatures, where consumption due to soot formation is expected to be low and overprediction at temperatures where taking soot into account is expected to lower the concentration. However, it underpredicts the rest of measured PAH (species for which Mech-2 [66] results are not shown are not present in the gas-phase mechanism). Mech-3 [67] provides predictions for pyrene only, while Mech-5 [68] includes both pyrene and benzo[a]pyrene. Given the simplified particle inception model used in this study, and commonly used in similar soot models, lumping soot precursors into a limited number of PAHs is acceptable, even if this can lead to their overprediction, as these species effectively gather the mass of unrepresented aromatics. Mech-3 predictions for pyrene generally agree with experimental data, except at the lowest temperature, where they unexpectedly underestimate pyrene concentrations. This underprediction is not favorable for future use in soot modeling, since an overprediction would typically be expected for two reasons: first, pyrene acts as a surrogate for multiple larger aromatic species, and should therefore be overestimated to account for missing pathways; and second, the absence of a soot model means no consumption of pyrene through soot formation occurs, which should otherwise lower its predicted concentration. In contrast, Mech-5 significantly overpredicts both pyrene and benzo[a]pyrene across most conditions. At low temperatures, however, Mech-5 predictions align with measurements. In this regime, no soot formation is expected, so aromatic consumption through soot-related pathways is negligible—rendering the influence of a soot model irrelevant. While these trends

qualitatively match expectations, the quantitative accuracy of the mechanisms cannot be fully assessed from this comparison alone. Lastly, Mech-6 [69–71] results are reported in the Supplementary Material as this mechanism was originally developed for use with an integrated soot sub-model.

Because the selection of the most appropriate gas-phase mechanism cannot be based solely on PAH results, the analysis is supplemented with predictions of fuel and acetylene mole fractions (under the same initial conditions as Fig. 1), as shown in Fig. 2. In the bottom panel of Fig. 2, a value is introduced to enable a direct comparison between gas-phase measurements and the overall carbon balance (grey markers labeled as “Measurement + equivalent”). This corrected acetylene mole fraction accounts for carbon sequestered in soot, based on the assumption that acetylene addition is the primary source of soot growth. To calculate this value, the measured soot yield was first converted into an equivalent number of moles of acetylene by assigning two carbon atoms per acetylene molecule. These acetylene equivalent moles were then added to the measured gas-phase acetylene mole fraction. The resulting reconstructed profile represents a virtual concentration that incorporates the contribution of soot formation, allowing for a more complete picture of the carbon budget.

Fuel and acetylene mole fraction results demonstrate that differences among gas-phase mechanisms extend beyond PAH predictions. Unlike PAH predictions, which did not allow for a clear distinction between mechanisms, these results offer a basis to identify the most suitable mechanism for this study. Fuel mole fraction predictions (top panel in Fig. 2, dashed lines for numerical results and black square markers for experimental measurements) highlight Mech-1 [64], Mech-5 [68], and Mech-6 [69–71] as the most appropriate candidates, showing good
<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Ethylene (top) and acetylene (bottom) mole fractions from the pyrolysis of 3 % C₂H₄ in a PFR as a function of initial temperature. Modeling results with all gas-phase mechanisms listed in Table 4. Experimental data (black symbols) from Sánchez [33]. The equivalent acetylene mole fraction is derived as the contribution from carbon bound in soot based on the soot yield measurement.</figcaption></figure>

agreement with experimental data unlike Mech-3 [67] and Mech-4 [65], which overpredict fuel consumption, and Mech-2 [66], which underpredicts it. Among the three mechanisms with acceptable fuel predictions, Mech-1 appears to be the most compatible with soot modeling based on the acetylene predictions (bottom panel in Fig. 2, dashed lines for numerical results and black square markers for experimental measurements). Mech-1 overpredicts acetylene concentrations compared to measurements, with the discrepancy increasing with temperature—mirroring the trend of expected soot yields. This suggests that Mech-1 provides an adequate acetylene reservoir for soot formation when coupled with a soot model. In contrast, while Mech-5 and Mech-6 offer closer agreement with experimental acetylene levels in the absence of soot modeling, they are likely to significantly underpredict acetylene once the soot model is activated. Furthermore, Mech-1 predicted acetylene concentration is consistently lower than the corrected acetylene mole fraction (“Measurement + equivalent”). Therefore, the comparison between the predicted and the corrected acetylene mole fraction further supports the interpretation that the apparent overprediction of acetylene originates from neglecting soot formation. At the same time, the fact that the corrected acetylene mole fraction remains higher than the predicted values suggests that soot mass is not derived exclusively from acetylene. First, soot mass can also grow through PAH adsorption, and some authors have proposed additional surface reactions that contribute carbon beyond acetylene addition. Second, soot formation perturbs the concentrations of several major species, thereby modifying the rates at which they interact.

Consequently, Mech-1 [64] is chosen as the reference gas-phase mechanism for the results presented in Section 5 based on its reasonable performance in predicting PAH, fuel (ethylene) and acetylene mole fractions. As previously mentioned and discussed further below, the increasing overprediction of acetylene and PAH concentrations with temperature aligns with the absence of soot production in the simulations. However, the significant underprediction of PAH at low temperatures raises concerns about the ability of a soot model coupled with this mechanism to accurately capture soot yields, particularly regarding the transition between non-sooting and sooting conditions. This issue is further examined in Section 5. Finally, Mech-6 is retained for the

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Soot yield from ethylene (top left, center) and acetylene (remaining panels) pyrolysis in a PFR as a function of initial temperature, for cases listed in Table 1. Panels (top left to bottom right): 3 % C₂H₄, 20 % C₂H₄, 14.29 % C₂H₂, 3 % C₂H₂, 2 % C₂H₂, and 1 % C₂H₂. Modeling results with Mech-1 [64] and all surface reaction mechanisms listed in Table 3. Experimental results (black markers) from sources listed in Table 1.</figcaption></figure>

study of ethanol and DME pyrolysis in Section 5.3, following a similar comparative analysis among mechanisms that include the chemistry of these oxygenated fuels.

# 5. Results and discussions

## 5.1. *Modeling strategy: systematic elimination via targeted validation*

As noted in Section 3, several modeling parameters have significant uncertainties. The consumption of surface radicals due to acetylene addition (reactions R3 or R3') and the proportion of reactive sites $\alpha$ are particularly critical. Nonetheless, these two parameters can initially be assigned specific values to determine the most appropriate soot surface reaction mechanism for modeling soot pyrolysis within the proposed framework. This preliminary step should be taken before delving deeper into the investigation of these parameters.

This approach is illustrated in Fig. 3, which shows the soot yields predicted by the SSM for the detailed validation cases using all soot surface reaction mechanisms listed in Section 3.2, with $\alpha = 1$ (the upper bound for this parameter) and R3'. Black markers depict experimentally measured soot yields from Sánchez and co-workers [33,37]. Regarding modeling results, only the mechanism proposed by Hwang and Chung [60] (Case B) produces suitable results for further use within the proposed modeling framework, as simulations with $\alpha = 1$ and R3' represent a high bound of potential reactivity. Therefore, any mechanisms that result in underestimated soot yields are unsuitable for describing soot surface reactions in a pyrolysis context, at least within the present model.

The results presented in Fig. 3 show that using the full Hwang and Chung [60] mechanism leads to a significant overprediction of soot yields. In the model proposed in this work, only one of the three reactions originally added to the standard HACA mechanism by Hwang and Chung [60] is retained. The systematic elimination process that supports this choice is illustrated by the results in Fig. 4. Soot yield predictions are compared for simulations run on the detailed validation database with $\alpha = 0.02$ (a deliberately low value) and R3 (accounting for surface radical depletion), corresponding to a lower-bound configuration. Three surface reaction schemes are considered:

* Case A, which includes R4, R5<sub>f,1</sub>, and R6<sub>f,1</sub> (representing the full Hwang and Chung [60] extension to HACA),

* HACA with only R5<sub>f,1</sub> added,

* HACA with only R6<sub>f,1</sub> added.
<figure><img src="image_8.png" /><figcaption>Fig. 4. Soot yield from ethylene (top left, center) and acetylene (remaining panels) pyrolysis in a PFR as a function of initial temperature, for cases listed in Table 1. Panels (top left to bottom right): 3 % C₂H₄, 20 % C₂H₄, 14.29 % C₂H₂, 3 % C₂H₂, 2 % C₂H₂, and 1 % C₂H₂. Modeling results with Mech-1 [64] and surface reactions modeled using HACA extended with reactions proposed by Hwang and Chung [60] as listed in Table 3. Experimental results (black markers) from sources listed in Table 1.</figcaption></figure>

The reaction R4 is not included in Fig. 4, as its addition was found to have no impact on the results compared to the baseline HACA mechanism. In contrast, any configuration including R5f.1 consistently leads to a significant overprediction of soot yields in ethylene cases at high temperatures. Consequently, R5f.1 was excluded from the surface reaction mechanism. A second set of rate constants for R5 and R6, proposed by Utsav et al. [23], includes backward rates, but their influence on soot yield is negligible, as shown in Fig. 3. By process of elimination, the resulting surface reaction mechanism, summarized in Table 3, is based on the baseline HACA mechanism proposed by Appel et al. [53], extended to include reaction R6f.1 from Hwang and Chung [60], and adopting R3* in place of R3 as recommended by Wang et al. [62].

The following step concerns the existence of conserved surface radicals or persistent free radicals discussed by Wang [73] who cites experimental evidence, including some results from pyrolysis cases [74]. The effect of accounting for this conservation in models describing soot at a similar scale as in the present study has been assessed by Mehta et al. [75] or Roy and Haworth [76] on atmospheric ethylene premixed and diffusion flames. In this study, the effect of surface radical conservation was assessed through simulations using either reaction R3 (with surface radical depletion) or R3* (with surface radical conservation), while keeping $\alpha = 1$. The results, presented in Fig. 5, show that using R3 leads to a dramatic underestimation of soot yields in the acetylene cases, even with $\alpha = 1$, whereas soot yields in the ethylene cases at temperatures above 1350 K are overestimated. Since surface radical depletion is expected to be primarily temperature-dependent, it should be similar for different fuels under the same temperature conditions. This observation supports the use of R3*, which conserves surface radicals and provides predictions that are more consistent with experimental data across both fuel cases. Although conserving surface radicals results in a notable overprediction of soot yields, this is considered acceptable in these cases, given that the simulations were conducted with $\alpha = 1$, representing an upper bound for this highly influential parameter. This choice is consistent with the previous studies of Mehta et al. [75] or Roy and Haworth [76] who used models describing soot at a similar scale as in the present study. However, under the pyrolysis conditions in this work, the influence of surface radical depletion is much more noticeable on acetylene cases than on ethylene ones, an observation that can be readily explained. In principle, it is logical that consuming [C*<sub>s,t</sub>] lowers the

<figure><img src="image_9.png" /><figcaption>Fig. 5. Soot yield from ethylene (top left, center) and acetylene (remaining panels) pyrolysis in a PFR as a function of initial temperature, for cases listed in Table 1. Panels (top left to bottom right): 3 % C₂H₄, 20 % C₂H₄, 14.29 % C₂H₂, 3 % C₂H₂, 2 % C₂H₂, and 1 % C₂H₂. Modeling results with Mech-1 [64] and surface reactions modeled using HACA extended with reaction R6 proposed by Hwang and Chung [60] accounting full depletion (blue) or conservation (red) of surface radicals. Experimental results (black markers) from sources listed in Table 1.</figcaption></figure>

rate of acetylene addition reaction since [C*<sub>s,t</sub>] is a reactant of this reaction. In the model this appears with the quasi-steady state assumption giving the ratio between [C<sub>s,t</sub>] and [C*<sub>s,t</sub>] writing

$$[C_{s,t}^*] = \left( \frac{k_{1f}[H] + k_{2b} + k_{6f}[C_3H_3]}{k_{1b}[H_2] + k_{2f}[H] + k_{3f}[C_2H_2]} \right) [C_{s,t}], \quad (4)$$

with $k_f$ the rate constant of the i'th reaction, following the reaction list given in Table 3. It appears clearly from Eq. (4) that removing R3 will favor [C*<sub>s,t</sub>] and thus lead to more soot. The counterintuitive fact that a term related to acetylene concentration might lower soot production in some cases also appears explicitly in Eq. (4). This point is the reason why two fuels, one of them being acetylene, were needed in the detailed validation database. This allows for analysing model results both in a case where acetylene is present in large quantities before soot production (showing a sort of asymptotic behavior of this part of the model) and in a more common case where C₂H₂ is produced (simultaneously with soot) in much lower quantities. Having two fuels in the database also permitted discrimination of which surface reaction had to be considered. As explained by Wang et al. [62], R3 or R3* are the two limiting cases and should be reconciled with a model leaning toward R3 at high temperature. However, the fact that soot yields were overpredicted using R3 for ethylene pyrolysis discouraged further investigations on that point because it indicates that other aspects of the model needed to be improved before being able to assess the relevance of any modeling attempt concerning surface radical depletion.

## 5.2. Detailed validation database results: insights on gas-soot interactions

A similar validation could be expected concerning the computation of the parameter $\alpha$. However, as explained in Section 3, usual formulations for $\alpha$ are unfit for the purpose of this paper leading to the proposition of Eq. (3). Therefore, $\alpha_0$ and $d_{cut}$ were directly fitted to match the soot yields on the detailed validation database. The obtained results are reported in Fig. 6 and 7. In Fig. 6 results labeled as “Sánchez” [33] and “Krestinin” [17] correspond to the only other modeling results published so far for the cases from Table 1 to the best of the authors’ knowledge. Soot yield predictions reported in this figure only bear the information
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Soot yield from ethylene (top left, center) and acetylene (remaining panels) pyrolysis in a PFR as a function of initial temperature, for cases listed in Table 1. Panels (top left to bottom right): 3 % C₂H₄, 20 % C₂H₄, 14.29 % C₂H₂, 3 % C₂H₂, 2 % C₂H₂, and 1 % C₂H₂. Modeling results with Mech-1 [64] and surface reactions proposed in this work (p.w. in Table 3). Experimental results (black markers) from sources listed in Table 1. Reference modeling results from Krestinin [17] (green) and Sánchez [33] (blue).</figcaption></figure>

that the formulation proposed in Eq. (3) can be fitted to match experimental results with a good agreement. Nevertheless, two points are to be noted concerning this. First, the results' agreement with measurements is similar to or slightly better than previous modeling attempts on these data, even though these other studies usually focused on sub-samples of the selected database while the proposed model predicts satisfactory results on a wider range of fuel concentrations (1 % to 20 %) and temperatures (1000–1700 K). Second, these results show that neglecting both the temperature dependence of $\alpha$ and the depletion of surface radicals is acceptable as a first approximation for pyrolysis cases in the

temperature range of 1000–1700 K. This approximation is challenged for higher temperature cases in Section 5.3.

### 5.2.1. Major species

Although the agreement between simulations and experiments concerning soot yields reported in Fig. 6 stems from the fit on $\alpha_0$ and $d_{cut}$, the gas-phase species results obtained from the same cases and reported in Fig. 7 offer new insights into the importance of soot modeling in pyrolysis conditions. Fig. 7 shows predictions with and without soot modeling along with experimental measurements and reference modeling results from Sánchez [33]. Unlike combustion cases where taking soot into account usually only impacts PAHs, the coupling of the proposed model to a gas-phase mechanism for pyrolysis cases leads to a significant modification of major species predictions. This observation highlights the importance of the carbon mass balance between soot and the gas phase as also suggested in a previous study by Ruiz et al. [16]. Moreover, the soot model does not influence only acetylene, which is the main source of carbon for soot growth, but also hydrogen and to a lesser extent, methane. This behavior is attributed to the large amount of soot produced in pyrolysis, exceeding 10 % of the total introduced carbon, making soot a major component of the reacting mixture in these cases. Two key conclusions can be derived from the influence of soot on gaseous species concentrations. First, it justifies, a posteriori, the choice of Norinaga and Deutschmann mechanism [64] (Mech-1). Indeed, the consumption of acetylene by the soot model predicting soot yields consistent with the experimental measurements, and consuming a reasonable amount of acetylene, modifies the model predictions to values in good agreement with the experimental measurements. This observation supports the rationale for selecting this gas-phase mechanism, as it accurately predicts acetylene concentrations in low-soot-yield cases and overpredicts them in high-soot-yield cases. Second, the strong influence of soot on gaseous species concentrations highlights the importance of accounting for soot in fuel pyrolysis simulations, even when soot formation is not the primary focus. This is evidenced by the variations in hydrogen and methane concentrations when soot production is considered.

<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Major species: acetylene (left panels), hydrogen (middle), and methane (right) mole fractions from pyrolysis of 3 % C₂H₄ (first row of panels), 3 % C₂H₂ (second row), 2 % C₂H₂ (third row), and 1 % C₂H₂ (fourth row) in a PFR as a function of initial temperature. Modeling results with Mech-1 [64] and surface reactions proposed in this work (p.w. in Table 3). Experimental (black markers) and reference modeling results (blue) from Sánchez [33].</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Soot precursors PAH mole fractions from the pyrolysis of 3 % C₂H₄ in a PFR as a function of initial temperature. Modeling results with Mech-1 [64] and surface reactions proposed in this work (p.w. in Table 3). Numerical results without soot modeling (green) and with reversible (red) and irreversible (blue) condensation. Experimental data (black symbols) from Sánchez et al. [37].</figcaption></figure>

### 5.2.2. PAHs

Lastly, the role that soot plays in the prediction of PAHs is analysed in terms of the effect that condensation has on these species, which in this work is modeled with a PAH-soot balance controlled by an equilibrium constant [27]. A first noticeable effect is that condensation reversibility has only a very low influence on soot yield as observed in Fig. 8. The observed behavior deviates from that documented in prior simulations of low-pressure premixed methane flames [27]. These discrepancies can be primarily attributed to substantial disparities in soot yields between the present simulations and the aforementioned premixed flames. This divergence in soot production leads to significantly different gas-soot interactions. Notably, acetylene profiles in the premixed flames from Aubagnac-Karkar et al. [27] exhibited minimal sensitivity to the soot model. However, the influence of condensation reversibility on PAH soot precursor predictions is significant as shown in Fig. 8 with PAH mole fraction results for PFR ethylene pyrolysis. Modeling results without soot modeling are included for reference, alongside results that either include or omit condensation reversibility. For either of the last two setups, PAH mole fractions are lower compared to the reference modeling case as might be expected due to the two-way coupling between soot and the gas phase and the consequent consumption of PAH soot precursors. However, the cases with condensation reversibility consistently show higher PAH mole fraction values compared to cases where the condensation process is irreversible. For benzo[a]pyrene and indeno[1,2,3-cd]pyrene, the mole fraction is even slightly higher than the reference case at the lowest temperature. Despite large differences compared to experiments, the setup accounting for condensation reversibility yields much more reasonable results for PAH mole fractions, with the results without condensation reversibility always being several orders of magnitude lower than those experimentally observed. Nevertheless, even when condensation reversibility is taken into account, the predicted PAH mole fractions remain more than one order of magnitude lower than the measured ones and they all become of a similar order of magnitude while more differences between PAH are observed. These two considerations indicate that a mechanism balancing PAH and soot concentrations is necessary to accurately predict both. However, they also highlight a limitation of the condensation reversibility model, as its reliance on an equilibrium constant forces all PAH concentrations toward similar values. However, a more detailed analysis

on the subject of condensation reversibility is out of the scope of this work. Although the magnitude of the condensation source term is similar to that of the surface growth source term, current results indicate that the soot yield mostly depends on surface growth modeling while condensation modeling mostly influences PAH profiles.

### 5.3. Extended validation

Throughout this section, validation of the modeling approach is extended to a broader set of initial conditions as summarized in Table 2. Cases selected for this second validation step comprise soot yield measurements from atmospheric PFR fuel pyrolysis between 975–1475 K and residence times on the order of $O(10^3)$ s. PFR cases within this extended validation step have also been selected to facilitate the assessment of the model for soot prediction in the pyrolysis of oxygenated fuels, namely ethanol and DME. The description of the chemistry for the pyrolysis of these fuels relies on the detailed chemical mechanism from CRECK Modeling Group [69–71] (Mech-6 in Table 4) and its choice follows the methodology described in Section 4.

The first set of PFR experiments within the set for extended validation were performed by Viteri et al. [34]. The influence of temperature and residence time on soot yield was studied in the pyrolysis of 5 % ethanol diluted in nitrogen. Results in Fig. 9 depict soot yield measurements in black and the indirect measurement of soot yield derived from the measurement of the gas yield. Measurements are available for a longer residence time (top) and a shorter residence time (bottom) both of which vary as a function of temperature. In general, modeling results (in red) show that the SSM performs fairly well in predicting the effect of temperature, with increasing soot yield as the temperature rises in the studied range, and the effect of residence time with slightly increased soot yield for the cases with longer residence time. The effect of the latter, is more pronounced around the soot onset temperature range 1175–1325 K. With a shorter residence time, the SSM predicts an increase in soot yield up to 0.12, while with a longer residence time soot yield rises up to 0.23 within the same temperature range.

The second set of PFR cases for extended validation comes from the work of Zhao and Liu [35] and results are presented in Fig. 10. The effect of increased oxygen-in-fuel content on soot yield is evaluated at 1273 (top), 1373 (middle), and 1473 K. Measurements (in black) are carried
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Soot yield from the pyrolysis of 5 % C₂H₅OH in a PFR as a function of initial temperature for two residence times: *t*ᵣ = 4168/T (top) and *t*ᵣ = 1563/T (bottom). Modeling results with Mech-6 [69–71] and surface reactions proposed in this work (p.w. in Table 3). Experimental data (black symbols) from Viteri et al. [34].</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Normalized soot yield from the pyrolysis of 2 % fuel in a PFR as a function of the content of oxygen in the fuel for three initial temperatures: 1273 K (top), 1373 K (middle), and 1473 K (bottom). Fuels include C₂H₄, C₂H₅OH, DME, and C₃H₄/C₂H₅OH and C₂H₄/DME blends. Soot yields are normalized with respect to soot yield from 2 % C₂H₄. Modeling results with Mech-6 [69–71] and surface reactions proposed in this work (p.w. in Table 3). Experimental data (black symbols) from Zhao and Liu [35].</figcaption></figure>

out for the pyrolysis of 2 % fuel diluted in argon. The composition of the fuel varies from pure ethylene (reference fuel), a blend of ethylene with an oxygenated fuel (either ethanol or DME) and pure oxygenated fuel

according to the values summarized in Table 2. As in the experiments, numerical results in Fig. 10 are normalized with respect to results for the non-oxygenated fuel (pure ethylene). Across the studied temperature range, the SSM predicts, in line with experimental observations, a drop in soot yield as the oxygen-in-fuel content increases for either ethanol or DME. The drop in soot yield favored by the addition of oxygen to the fuel is more pronounced at the lowest temperature where both measured and predicted soot yields fall below 0.5 when the content of oxygen in the fuel goes from 0 in the reference fuel to 0.88 % in the ethylene/ethanol and ethylene/DME mixtures. Comparing results from ethanol and DME, the latter is predicted to have a better performance in terms of soot mitigation with consistently lower soot yield across all conditions as also observed experimentally.

The third and last set of PFR cases includes the measurements from the work of Esarte et al. [36] who studied the role of temperature and the content of oxygen in the pyrolysis of 5 % fuel diluted in nitrogen. Fig. 11 depicts results for three fuel mixture compositions where acetylene is the primary component and the concentration of ethanol varies between 0.5 % (top), 1 % (middle), and 2 % (bottom). In Fig. 11, both the “dry” soot yield (black markers) reported by Esarte et al. [36], obtained after removing surface-bound tar from the sampled particles, and the “1-gas yield” (grey markers), calculated by subtracting the measured gas-phase species from the total carbon balance, are shown. Although “1-gas yield” is not a direct soot measurement and may contain undetected hydrocarbons, its inclusion serves a specific purpose. The formulation does not provide an explicit distinction between soot and tar, as it resolves only a single soot mass fraction distribution by particle size, without characterizing the physical state of the dispersed phase. Under these conditions, the predicted soot yield is expected to fall between the dry soot measurement and the “1-gas yield” value. This provides upper and lower

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Soot yield from the pyrolysis of C₂H₂/C₂H₅OH blends in a PFR as a function of initial temperature for three fuel concentrations: 4.5 %C₃H₂-0.5 %C₂H₅OH (top), 4.0 %C₂H₂-1.0 %C₂H₅OH (middle), 3.0 %C₂H₂-2.0 %C₂H₅OH (bottom). Modeling results with Mech-6 [69–71] and surface reactions proposed in this work (p.w. in Table 3). Experimental data (black and grey symbols) from Esarte et al. [36].</figcaption></figure>
bounds for comparison with numerical data. In line with experiments, and the observations made for the pyrolysis of ethylene/ethanol and ethylene/DME mixtures, there's a decreasing trend in soot yield as the amount of ethanol in the fuel mixture increases with predicted peak soot yield dropping from 0.87 with 0.5 % to 0.77 with 2 % ethanol.

In addition to PFR cases, the extended validation of the modeling approach also comprises the evaluation of the predictability of the SSM under high-pressure, high-temperature shock tube conditions. Two shock tube sets are considered. First, the experiments from Agafonov et al. [20] conducted over a wide range of temperatures for various fuel concentrations. Second, experiments from Utsav et al. [23] that provide valuable time-resolved results that enable the validation of transient predictions. Results for the first set of shock tube cases are presented in Fig. 12 where black markers denote the experimental measurements. Ethylene diluted in argon is pyrolyzed in a shock tube for three fuel concentrations namely, 5 % (top), 3 % (middle), and 1.5 % (bottom). Modeling results (in red) correspond to simulations carried out at 4 bar. The vertical lines correspond to numerical results at 3 and 5 bar to highlight the sensitivity of the model to pressure in the same range reported experimentally. Overall, modeling results agree well with the experimental observations, predicting a drop in soot yield with lower fuel concentration accompanied by a shift in peak soot yield toward lower temperatures. Under these lower temperatures, predictions at various pressure levels also reveal a more prominent variation in soot production. Agafonov et al. [20] also conducted experiments on acetylene-fueled cases. However, simulations of these cases revealed that the selected gas-phase mechanism fails to predict PAH formation—necessary to initiate the soot phase—within a timescale consistent with the measured soot induction times. This issue is further discussed in the Supplementary Material.

Lastly, Fig. 13 introduces time-resolved acetylene mole fraction (top) and soot volume fraction (bottom) results from the pyrolysis of ethylene and binary mixtures of ethylene with propane and methane diluted in argon. Experiments (profiles in black) from Utsav et al. [23] also include soot induction time measurements depicted with a blue marker. Reference modeling results [23] are also included for two chemical mechanisms namely, KM2 [77] (in green) and AKP (in light green)

<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Soot yield in the pyrolysis of C₂H₄ in a shock tube at 4 bar for three fuel concentrations: 5 % C₂H₄ (top), 3 % C₂H₄ (middle), and 1.5 % C₂H₄ (bottom). Vertical lines show results at 3 and 5 bar. Residence time 1.5 ms. Modeling results with Mech-1 [64] and surface reactions proposed in this work (p.w. in Table 3). Experimental data (black symbols) from Agafonov et al. [20].</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 13.** Soot onset time (blue markers), C₂H₂ mole fraction (top) and soot volume fraction (bottom) from the pyrolysis 2.35 % fuel in a shock tube. Fuels include 2.35 % C₂H₄ at 2179 K (left), 2.23 % C₂H₄-0.12 % C₃H₈ at 2132 K (middle), and 2.23 % C₂H₄-0.12 % CH₄ at 1961 K (dashed line) and 2136 K (solid line) (right). Modeling results with Mech-1 [64] and surface reactions proposed in this work (p.w. in Table 3). Experimental data (black lines and blue markers) and reference modeling results (green lines) from Utsav et al. [23].</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Predicted C₂H₂ mole fraction (top) and soot precursors mole fractions: A4 (middle) and A7 (bottom). Results from C₂H₄ (red) and its binary mixtures with C₃H₈ (green), and CH₄ (blue) under the same conditions described for Fig. 13. Results with Mech-1 [64] and surface reactions proposed in this work (p.w. in Table 3).</figcaption></figure>

which is a modified version of the latter. Experimentally, the addition of propane or methane induces a reduction in acetylene mole fraction and soot production which is not the case for modeling results with virtually invariant acetylene mole fraction and soot volume fraction profiles. Moreover, for the ethylene/methane mixture, the drop in temperature from 2136 (solid line) to 1961 K (dashed line) is experimentally observed to cause a delay in the appearance of soot. With a longer soot onset time, a higher acetylene mole fraction peak is observed for the lower temperature case while in the simulation results (with virtually no change in soot onset time) the acetylene mole fraction peak is lower for the
1961 K condition due to enhanced soot production. Similar trends are observed from reference simulation results from Utsav et al. [23] (green profiles) who reported marginal changes in acetylene concentration and increased soot production in the pyrolysis of ethylene mixtures.

Discrepancies from simulations can be explained based on acetylene, pyrene, and coronene (as representative PAH soot precursor species) results from Fig. 14. These results do not account for soot production to facilitate the analysis. The first noticeable observation is that peak acetylene mole fraction values barely change with the addition of propane or methane while experimental results in Fig. 13 show lower peak values (all of which take place before the appearance of soot). The second noticeable observation is that without soot production being taken into account, more pyrene and coronene are predicted for ethylene mixtures thus explaining the observed increase in soot production. These two observations highlight, yet again, the importance of a proper description of chemistry. This might not be the case for Mech-1 which was not originally developed nor validated for ethylene mixtures.

## 6. Conclusions

This study introduced and validated a modeling framework for soot production during fuel pyrolysis. Although implemented within Cantera [40] using a sectional based framework [27], the individual sub-models selected or developed for key phenomena—such as reversible PAH adsorption, soot surface growth kinetics, active surface site fraction, and radical site conservation—are formulated in a general way. Therefore, they can readily be adapted to alternative soot modeling approaches commonly employed in 3D CFD simulations, such as two-equation, three-equation, or MoM models.

The validation strategy followed a two-phase approach. First, a detailed validation was performed for ethylene and acetylene pyrolysis under PFR and shock tube conditions, using both soot yield and gas-phase species measurements. This enabled a rigorous assessment of the carbon balance between the dispersed soot phase and the gas phase which, unlike in typical flame environments, was shown to critically affect the accurate prediction of key gas-phase species such as acetylene. In this context, incorporating enhanced soot surface growth reactions involving propagyl radicals, as proposed by Hwang and Chung [60], was essential to simultaneously reproduce soot yields and acetylene concentrations across a broad range of conditions. Moreover, accounting for surface radical conservation, as initially proposed by Wang et al. [62], proved particularly important in acetylene pyrolysis. Neglecting this phenomenon led to major discrepancies and made it impossible to develop a unified model able to predict soot yields for both acetylene and ethylene pyrolysis.

In the second phase, the validation was extended to a wider range of operating conditions—including higher temperatures, pressures, and additional fuels such as ethanol and DME—primarily using soot yield measurements. The model consistently captured the experimentally observed trends, demonstrating its robustness in predicting the sensitivity of soot formation to fuel type, temperature, and fuel-diluent ratios. This highlights the model’s relevance for applications in emerging 3D CFD scenarios of industrial interest, including hydrogen production via small hydrocarbon pyrolysis, mitigation of clogging in CLC processes, and nanoparticle synthesis by spray pyrolysis.

Despite these encouraging results, several limitations were identified, most notably in the modeling of PAH-soot interactions, including nucleation and PAH adsorption kinetics, as well as in the representation of the active surface site fraction through the $\alpha$ parameter. While these aspects are recognized as critical for improving model fidelity, they are inherently linked to molecular-scale processes that would require detailed molecular dynamics or kinetic Monte Carlo simulations for more accurate representation. As such, these refinements fall outside the scope of the present work, which aims to provide a simplified yet physically grounded description of the key sub-models governing soot evolution in pyrolysis conditions, suitable for integration into large-scale

simulations. Nonetheless, incorporating insights from higher-fidelity molecular modeling into 3D CFD-compatible frameworks appears to be a logical next step to further enhance models such as the one proposed in this study.

## CRediT authorship contribution statement

**Leonardo Pachano:** Writing – original draft, Visualization, Formal analysis, Conceptualization. **Damien Aubagnac:** Writing – original draft, Supervision, Software, Formal analysis, Conceptualization.

## 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.

## Appendix A. Supplementary data

Supplementary data for this article can be found online at doi:10.1016/j.fuel.2025.137203.

## Data availability

Data will be made available on request.

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