# A Generalized Virtual Chemistry Formalism for Direct Integration into Reactive Flow Solvers: Application to Hydrogen-Air and Dodecane-Air Laminar Flame Computations

Matthieu Préteseille, Tan-Phong Luu, Étienne Espada, Nasser Darabiha, Benoît Fiorina

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Matthieu Préteseille, Tan-Phong Luu, Étienne Espada, Nasser Darabiha, Benoît Fiorina. A Generalized Virtual Chemistry Formalism for Direct Integration into Reactive Flow Solvers: Application to Hydrogen-Air and Dodecane-Air Laminar Flame Computations. 2025. hal-05301985

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# A Generalized Virtual Chemistry Formalism for Direct Integration into Reactive Flow Solvers: Application to Hydrogen-Air and Dodecane-Air Laminar Flame Computations

Matthieu Préteseilleª, Tan-Phong Luuª, Étienne Espadaª, Nasser Darabihaª and Benoît Fiorinaª

ªEM2C Laboratory, CNRS, CentraleSupélec, Université Paris-Saclay, 8-10 rue Joliot-Curie, 91190 Gif-sur-Yvette, France

## ARTICLE HISTORY

Compiled July 27, 2025

## ABSTRACT

Accurately and efficiently modeling combustion remains a central challenge in the transition to low-carbon energy systems. Virtual chemistry has emerged as a powerful chemistry reduction strategy, relying on optimized species and reactions to reproduce reference flames behaviors at a fraction of the computational cost. However, previous formulations depend on tabulated parameters, limiting their extrapolation capabilities and integration into standard reactive flow solvers.

To address these limitations, a new standardized virtual chemistry formalism has been introduced, bridging virtual and detailed chemistry through chemical equilibrium between real and virtual species. The present work extends the framework to accurately capture laminar flame structures by jointly optimizing the thermodynamic, kinetic, and transport properties. Special attention is given to validating and generalizing virtual mechanism structures, with the development of a novel architecture dedicated to pyrolyzing fuels. The resulting highly-reduced optimized mechanisms adopt architectures compatible with CHEMKIN and CANTERA solvers.

The methodology is applied to representative decarbonized fuels, Hydrogen and Dodecane, a key component of Sustainable Aviation Fuels (SAF). Particular emphasis is placed on reproducing laminar flame structures across a wide range of combustion regimes, including premixed and non-premixed, adiabatic and non-adiabatic conditions. The results demonstrate the ability of the proposed virtual mechanisms to replicate detailed combustion behavior with drastically reduced complexity and chemical stiffness, paving the way for straightforward and efficient high-fidelity reactive flow simulations.

## KEYWORDS

Virtual chemistry; Flames; Kinetic mechanism; Transport processes; Optimization

## 1. Introduction

The contemporary need to reduce carbon emissions has driven major progress in combustion technology. A key focus is the development of net-zero combustion chambers that use alternative fuels like Hydrogen and Sustainable Aviation Fuels (SAF). However, accurately modeling these new combustion processes while keeping simulations
efficient remains a challenge. In this context, reduced chemistry techniques remain essential to simplify the complex kinetic mechanisms involved in combustion [1].

The field has seen the emergence of various methodologies aimed at simplifying combustion chemistry, including skeletal reduction [2], tabulated chemistry [3, 4], optimized chemistry [5, 6] and virtual chemistry [7–9]. The latter has gained prominence for its effectiveness in capturing critical flame characteristics, such as auto-ignition [10] or pollutants formation [8, 9] with minimal computational resources [7]. The underlying principle of virtual chemistry involves the creation of virtual species and reactions, whose properties are optimized using optimization algorithms trained on a comprehensive database of reference flame properties [11].

Despite its advantages, the traditional formulation of virtual chemistry requires the tabulation of burnt gases proportions and kinetic constants, based on local parameters such as mixing fraction and enthalpy. This requirement, while ensuring accuracy and efficiency, complicates the integration of virtual chemistry into existing CFD codes like CHEMKIN [12] and CANTERA [13]. Therefore, a new formalism has been recently introduced to optimize virtual species and reactions in a manner similar to detailed chemistry [14]. By optimizing the entropy and elemental composition of virtual species, the framework now ensures compliance with the second law of thermodynamics. Compatible with existing reactive flow solvers, the resulting optimized schemes do not require any tables, but instead relies on chemical equilibrium between real and virtual species. This new methodology has been successfully applied to model both Hydrogen and SAF combustion in 0-D reactors.

The objective of the present work is to extend the recently-introduced virtual chemistry formalism [14] to enable accurate predictions of flame structures and dynamics. Achieving this requires to define and optimize virtual species transport properties, alongside their thermodynamic and kinetic characteristics. Furthermore, the architectures of the reduced mechanisms are refined to capture shifts in dominant chemical pathways driven by variations in equivalence ratio, which govern both the flame structures and consumption rates.

Section (2) briefly reviews the virtual products optimization procedure for recovering reference mixtures at chemical equilibrium under both adiabatic and non-adiabatic conditions. Section (3) introduces the generalized virtual reaction model architecture, with particular emphasis on a novel pyrolysis-dedicated design to accurately represent the complex chemistry of large hydrocarbons. Validation of the methodology is provided in Section (4) for both premixed and non-premixed laminar flames, even under non-adiabatic conditions. A comparison of chemical stiffness and computational costs between the virtual schemes and their detailed counterparts is also presented. This study applies the chemistry reduction methodology to two fuels of particular relevance for decarbonized combustion: H₂ and *n*C₁₂H₂₆, a key constituent of Sustainable Aviation Fuels.

## 2. Thermodynamics optimization

The thermodynamic optimization approach introduced by Préteseille & al. [14] is applied here to model the equilibrium properties of virtual H₂-Air and *n*C₁₂H₂₆-Air mixtures.
## 2.1. Procedure

A reduced ensemble of real and virtual species is considered to model both fresh and burnt gases of the reactive mixture. The reduced set of real species includes the fuel F, the oxidizer Ox, the dilutant D and $N_P^r$ real combustion products of the user's interest, such as CO₂ and H₂O. Combustion products are completed by $N_P^v$ optimized virtual products so that the equilibrium thermochemical states of the reduced mixture reproduce those of the detailed mechanism across a given range of target conditions $\psi_c \in [1..N_c]$. These conditions cover variations in equivalence ratio, enthalpy, and pressure. As explained in [14], the total number of products, $N_P$, is assumed equal to the number of chemical elements $N_e$ involved in the chemical reactions:

$$N_P = N_P^r + N_P^v = N_e. \qquad (1)$$

A reference thermodynamical state is defined by computing the equilibrium properties at constant pressure and enthalpy for each target condition $\psi_c$. Virtual product properties are then optimized to retrieve the burnt gases mean mixture molecular weight $\bar{M}$, heat capacity $C_p$, enthalpy $\bar{H}$, entropy $\bar{S}$ and Gibbs free energy $\bar{G}$. Accurately retrieving these thermodynamic properties naturally ensures the correct prediction of equilibrium temperatures $T_{eq}(\psi_c)$.

## 2.2. Application to H₂-Air and nC₁₂H₂₆-Air mixtures

Throughout this study, nitrogen $N_2$ is always considered as an inert dilutant D in the developed virtual schemes. Under this assumption, the chemical elements involved in the combustion reactions of H₂-Air are H and O. According to Eq. (1), two combustion products are needed. By including H₂O as a real product, only one optimized virtual product, P₁, is required. Similarly, for nC₁₂H₂₆, the three products needed are split in two real species CO₂, H₂O and a single optimized virtual product P'₁. The compositions of the reduced mixtures for each case are summarized in Table (1). Reference detailed mechanisms are those of Zhu & al. [15] for H₂ and CRECK C0-C16 [16] for nC₁₂H₂₆ chemistry.

Non-adiabatic effects are incorporated via a heat loss ratio $\Delta h$ expressed in %:

$$\Delta h = 100 \left( 1 - \frac{T_{eq}(\psi_c) - T_0}{T_{ad}(\psi_c) - T_0} \right) \qquad (2)$$

where $T_{eq}(\psi_c)$ and $T_{ad}(\psi_c)$ respectively represent the equilibrium and adiabatic temperatures for the condition $\psi_c$. Optimizations of virtual products are carried out following the procedure proposed in [14] for target conditions presented in Table (1).

<caption>**Table 1.** Comparison of reduced and detailed mixture compositions for H₂-Air & nC₁₂H₂₆-Air combustion under their respective reduction target conditions.</caption>

$\phi$: Equivalence ratio, $\Delta h$: Heat losses ratio (in %), $T_0$: Fresh gas temperature, $p$: Pressure.

<table><thead><tr><th>Case</th><th>Reduced mixture</th><th>Detailed thermochemistry</th><th>Target conditions</th></tr></thead><tbody><tr><td>H₂-Air</td><td>H₂, O₂, N₂, H₂O, P₁</td><td>Zhu & al. [15] 43 species</td><td>φ ∈ [0.1, 10], Δh ∈ [0, 30%], T₀ = 300K, p = 1 atm</td></tr><tr><td>nC₁₂H₂₆-Air</td><td>nC₁₂H₂₆, O₂, N₂, H₂O, CO₂, P'₁</td><td>CRECK C0-C16 [16] 413 species</td><td>φ ∈ [0.7, 1.4], Δh ∈ [0, 30%], T₀ = 400K, p = 1 atm</td></tr></tbody></table>

As depicted in [14], the thermodynamic properties of the reference burnt gases are
recovered by the reduced mixture of virtual and real products. Fig. (1) shows that, for both fuels, the equilibrium temperature $T_{eq}(\psi_c)$ is accurately modeled by the virtual mixture for the whole range of equivalence ratio $\phi$ and heat loss ratios $\Delta h$ considered as optimization targets.

<figure><img src="image_1.png" /><figcaption>**Figure 1.** Equilibrium temperature $T_{eq}$ for (a) H₂-Air and (b) nC₁₂H₂₆-Air combustion over their respective flammability domains for various heat loss ratios $\Delta h$. Comparison of the results obtained with optimized virtual species with that of detailed mechanisms for the targeted conditions of Table (1).</figcaption></figure>

# 3. Chemical scheme optimization

## 3.1. Virtual reaction model architecture

### 3.1.1. Generalization of the virtual oxidation mechanism

This work extends the reaction model presented in [14] to include $N_I$ intermediate species $I_i, i \in [1..N_I]$ whose thermodynamical properties and elemental composition are optimized to capture combustion dynamics in the flame front. The ensemble of $N_P$ combustion products is noted $\mathcal{P} = \mathcal{P}^r + \mathcal{P}^v$, where $\mathcal{P}^r$ and $\mathcal{P}^v$ respectively represent the set of real and virtual combustion products from the thermodynamics optimization.

#### Initiation reaction

The initiation reaction consumes the fuel F and oxidizer Ox, partially converting them into the first intermediate species $I_1$ along with the combustion products:

$$F + S \text{ Ox} \rightarrow \alpha''_{1,I_1} I_1 + \sum_{P_k \in \mathcal{P}} \alpha''_{1,P_k} P_k \quad (\mathcal{R}_1)$$

where S is the molar stoichiometric coefficient of the global combustion reaction. Among the remaining molar stoichiometric coefficients, only $\alpha''_{1,I_1}$ is optimized, while the $N_P$ coefficients $\alpha''_{1,P_k}$ (equal in number to the chemical elements involved since $N_P = N_e$) are computed by solving a linear system guaranteeing elemental conservation.

#### Chain-propagating reactions

The intermediate species $I_1$ is completely consumed to form new intermediate species and complete the formation of combustion products in the following chain-
propagating reactions:

$$I_1 + \sum_{P_k \in \mathcal{P}} \alpha'_{2,P_k} P_k \longrightarrow \alpha''_{2,I_2} I_2 + \sum_{P_k \in \mathcal{P}} \alpha''_{2,P_k} P_k \quad (\mathcal{R}_2)$$

$$I_2 + \sum_{P_k \in \mathcal{P}} \alpha'_{3,P_k} P_k \longrightarrow \alpha''_{3,I_3} I_3 + \sum_{P_k \in \mathcal{P}} \alpha''_{3,P_k} P_k \quad (\mathcal{R}_3)$$

$\vdots$

$$I_{N_1-1} + \sum_{P_k \in \mathcal{P}} \alpha'_{N_1,P_k} P_k \longrightarrow \alpha''_{N_1,I_{N_1}} I_{N_1} + \sum_{P_k \in \mathcal{P}} \alpha''_{N_1,P_k} P_k \quad (\mathcal{R}_{N_1})$$

where all stoichiometric coefficients $\alpha''_{i,I_i}$ are optimized $\forall i \in [2..N_I]$. This chain-propagating structure has been specifically chosen to decompose the reference combustion kinetics in successive optimized elementary reactions. In line with all subsequent reactions, combustion products stoichiometric coefficients are determined to conserve elements in any reaction $r$. Depending on their sign, the associated species $P_k \in \mathcal{P}$ is considered as a reactant ($\alpha'_{r,P_k} > 0 \& \alpha''_{r,P_k} = 0$) or a product ($\alpha'_{r,P_k} = 0 \& \alpha''_{r,P_k} > 0$) in the reaction.

### Termination reaction

The last intermediate $I_{N_1}$ is finally completely consumed to finish the production of burnt gases throughout the terminating reaction:

$$I_{N_1} + \sum_{P_k \in \mathcal{P}} \alpha'_{N_1+1,P_k} P_k \longrightarrow \sum_{P_k \in \mathcal{P}} \alpha''_{N_1+1,P_k} P_k \quad (\mathcal{R}_{N_1+1})$$

### Equilibrium reactions

As long as D is an inert species and Eq. (1) is satisfied, only 2 independent chemical reactions are needed to link species present at chemical equilibrium [14]. The proposed equilibrium reactions to complete the virtual mechanism are [14]:

$$F + \sum_{P_k \in \mathcal{P}} \alpha'_{eq1,P_k} P_k \rightleftharpoons \sum_{P_k \in \mathcal{P}} \alpha''_{eq1,P_k} P_k \quad (\mathcal{R}_1^{eq})$$

$$Ox + \sum_{P_k \in \mathcal{P}} \alpha'_{eq2,P_k} P_k \rightleftharpoons \sum_{P_k \in \mathcal{P}} \alpha''_{eq2,P_k} P_k \quad (\mathcal{R}_2^{eq})$$

### 3.1.2. Extension to pyrolyzing fuels

Although the previous reaction model is generalized to include any number of intermediate species $I_i$ and chemical reactions in the flame front, it still lacks a mechanism to represent pyrolysis. Yet, it is well established that fuels such as heavy hydrocarbons undergo thermal or oxidative decomposition prior to oxidation [17, 18]. This pyrolysis process, typically endothermic, precedes the oxidation of its fragments and plays a key role in determining the fuel consumption speed, and, as a result, the laminar flame velocity $s_L$. Detailed chemical models like HyChem [18] explicitly separate the chemistry into two stages: a pyrolysis model that produces intermediate fragments, followed by a distinct oxidation model that governs the subsequent reactions of these fragments.
Introducing $N_F$ pyrolysis fragments $F_f, f \in [1..N_F]$, a mechanism dedicated to fuel decomposition is designed:

$$ \begin{array}{lcl} F \longrightarrow \alpha''_{1,pyro} F_1 & (\mathcal{R}_1^{pyro}) & \\ F_1 \longrightarrow \alpha''_{2,pyro} F_2 & (\mathcal{R}_2^{pyro}) & \\ \vdots & & \\ F_{N_F-1} \longrightarrow \alpha''_{N_F,pyro} F_{N_F} & (\mathcal{R}_{N_F}^{pyro}) & \end{array} $$

where stoichiometric coefficients ($\alpha''_{1,pyro}, \dots, \alpha''_{N_F,pyro}$) are optimized. For each fragment $F_f$, its thermodynamic properties are also optimized to reproduce the fuel pyrolysis behavior. Atomic compositions of the fragments are fixed to ensure, together with the optimized stoichiometric coefficients, the conservation of elements in all chemical reactions.

Reactions ($\mathcal{R}_1$) and ($\mathcal{R}_1^{eq}$) are then corrected to involve the averaged pyrolysis fragments $F_{N_F}$ instead of the original fuel $F$. This fragment $F_{N_F}$, which does not have any physical representation, is assumed absent in the burnt gases. Therefore, the reaction ($\mathcal{R}_1^{eq}$) is set irreversible in the forward direction. In scenarios without pyrolysis, the reversibility of ($\mathcal{R}_1^{eq}$) is maintained: A specific notation $\overleftarrow{\overline{\overline{\hspace{2.5cm}}}}$ is introduced for this matter, as seen in (Mech.1).

### 3.1.3. Final mechanism

The virtual scheme integrates the previous chemical reactions into the generalized mechanism (Mech.1). Entitled $V-N_F/N_I-F$, it describes the fuel $F$ combustion with $N_F$ pyrolysis fragments $F_f$ and $N_I$ intermediates $I_i$, as defined below:
Mechanism V-N<sub>F</sub>/N<sub>I</sub>-F:

$$
\begin{align*}
F & \longrightarrow \alpha''_{1,\text{pyro}} F_1 \\
F_1 & \longrightarrow \alpha''_{2,\text{pyro}} F_2 \\
& \vdots \\
F_{N_F-1} & \longrightarrow \alpha''_{N_F,\text{pyro}} F_{N_F}
\end{align*}
$$

$$
\begin{align*}
& \prod_{k=1}^{N_F} \alpha''_{k, \text{pyro}} F_{N_F} + S \text{Ox} \longrightarrow \alpha''_{1, I_1} I_1 + \sum_{P_k \in \mathcal{P}} \alpha''_{1, P_k} P_k \\
& \qquad I_1 + \sum_{P_k \in \mathcal{P}} \alpha'_{2, P_k} P_k \longrightarrow \alpha''_{2, I_2} I_2 + \sum_{P_k \in \mathcal{P}} \alpha''_{2, P_k} P_k \\
& \qquad I_2 + \sum_{P_k \in \mathcal{P}} \alpha'_{3, P_k} P_k \longrightarrow \alpha''_{3, I_3} I_3 + \sum_{P_k \in \mathcal{P}} \alpha''_{3, P_k} P_k \tag{Mech.1} \\
& \vdots
\end{align*}
$$

$$
I_{N_1-1} + \sum_{P_k \in \mathcal{P}} \alpha'_{N_1, P_k} P_k \longrightarrow \alpha''_{N_1, I_{N_1}} I_{N_1} + \sum_{P_k \in \mathcal{P}} \alpha''_{N_1, P_k} P_k
$$

$$
I_{N_1} + \sum_{P_k \in \mathcal{P}} \alpha'_{N_1+1, P_k} P_k \longrightarrow \sum_{P_k \in \mathcal{P}} \alpha''_{N_1+1, P_k} P_k
$$

$$
F_{N_F} + \sum_{P_k \in \mathcal{P}} \alpha'_{eq1, P_k} P_k \underset{\text{if no pyro.}}{\overleftarrow{\overbrace{\phantom{\sum}}^{\sum}}} \sum_{P_k \in \mathcal{P}} \alpha''_{eq1, P_k} P_k
$$

$$
\text{Ox} + \sum_{P_k \in \mathcal{P}} \alpha'_{eq2, P_k} P_k \rightleftharpoons \sum_{P_k \in \mathcal{P}} \alpha''_{eq2, P_k} P_k
$$

While highly-compact, this virtual scheme architecture presents a structure simi-
lar to detailed mechanisms, including initiation, chain-propagating, termination, and
equilibrium reactions. The scheme is composed of N = 3 + N<sub>F</sub> + N<sub>I</sub> + N<sub>P</sub> species and
N<sub>R</sub> = 3 + N<sub>I</sub> + N<sub>F</sub> elementary chemical reactions, where the number of intermediate
species N<sub>I</sub> and pyrolysis fragments N<sub>F</sub> are user's defined. The mechanism structure
is unified by optimizing the vector x<sub>Scheme</sub>, defined accordingly:

$$
\begin{equation}
\begin{split}
\boldsymbol{x}_{\text{Scheme}} = \left( \underbrace{(a_1^i, a_2^i, a_3^i, a_4^i, a_5^i, a_6^i, a_7^i)}_{\substack{\text{Thermodynamic coefficients} \\ \text{of } I_i}} , \underbrace{(N_C^i, N_O^i, N_H^i, \dots)}_{\substack{\text{Atomic composition} \\ \text{of } I_i}} \right) \dots \forall i \in [1..N_I], \\
\underbrace{(a_1^f, a_2^f, a_3^f, a_4^f, a_5^f, a_6^f, a_7^f)}_{\substack{\text{Thermodynamic coefficients} \\ \text{of } F_f}} \dots \forall f \in [1..N_F],
\end{split}
\tag{4}
\end{equation}
\begin{center}
\underbrace{(\alpha''_{1,I_1}, \dots, \alpha''_{N_1,I_{N_1}}, \alpha''_{1,\text{pyro}}, \dots, \alpha''_{N_F,\text{pyro}})}_{\substack{\text{Stoichiometric coefficients}}}
\end{center}
$$
## 3.2. Reaction rates

For a given elementary reaction $r \in [1..N_R]$ of (Mech.1), the rate of progress variable $q^r$ is classically defined as the difference between forward and backward contributions:

$$q^r = k_f^r \prod_{k=1}^{N_{reac}} [X_k]^{n_{f,k}} - k_b^r \prod_{k=1}^{N_{prod}} [X_k]^{n_{b,k}} \quad (5)$$

where $k_f^r$ and $k_b^r$ are the forward and backward Arrhenius rates, $N_{reac}$ and $N_{prod}$ the numbers of reactants and products in the reaction, $[X_k]$ the concentration of species k and $n_{f,k}$, $n_{b,k}$ the corresponding reaction orders. The reaction orders are defined differently depending on the nature of the reaction: for reversible reactions, they are set equal to the stoichiometric coefficients; for irreversible ones, reactants orders are treated as optimization variables [14].

The forward kinetic rate $k_f^r$ use the classical Arrhenius formulation:

$$k_f^r(T) = A^r T^{b^r} e^{-E_a^r/RT} \quad (6)$$

$A^r$, $b^r$ and $E_a^r$ being respectively the pre-exponential constant, temperature exponent and activation energy of reaction r. The backward kinetic rate $k_b^r$ of any reversible reaction is determined by its forward kinetic rate $k_f^r$ and its equilibrium constant $K_{eq}^r$ through:

$$k_b^r(T) = \frac{k_f^r(T)}{K_{eq}^r(T)} = \frac{k_f^r(T)}{e^{-\Delta G_r^\circ(T)/RT}} \quad (7)$$

where $\Delta G_r^\circ(T)$ is Gibbs free energy of reaction r at standard conditions.

Optimizing the virtual reaction model chemical rates involves determining, for each reaction r, the optimal pre-exponential factor $A^r$, temperature exponent $b^r$, activation energy $E_a^r$, and the reaction orders for irreversible reactions gathered in the following optimization vector $\boldsymbol{x}_{\text{Arrhenius}}$:

$$\boldsymbol{x}_{\text{Arrhenius}} = \left( \underbrace{(A^r, b^r, E_a^r) \dots \forall r \in [1..N_R]}_{\text{Arrhenius parameters}} , \underbrace{(n_F, \dots, n_{F_{NF}}, n_{Ox}, n_{I_1} \dots)}_{\text{Orders of reaction}} \right). \quad (8)$$

## 3.3. Transport properties

Transport properties and diffusion coefficients of virtual species are modeled using classical molecular transport laws derived from kinetic theory and molecular physics [19, 20]. For simplicity purposes, virtual species are treated as nonpolar and non-polarizable. Their viscosity $\mu_i$ and thermal conductivity $\lambda_i$ of a given species i are then expressed as follows [19]:

$$\mu_i = \frac{5}{16} \frac{\sqrt{\pi m_i k_B T}}{\pi \sigma_i^2 f_{ii}(\sigma_i, \epsilon_i)} \quad \lambda_i = \frac{25}{32} \sqrt{\frac{k_B T}{\pi m_i} \frac{C_v}{\sigma_i^2 A f_{ii}(\sigma_i, \epsilon_i)}} \quad (9)$$
$k_B$ being the Boltzmann constant, $T$ the temperature, $m_i$ the mass of molecule $i$, $(\sigma_i, \varepsilon_i)$ species' $i$ respective Lennard-Jones collision diameter and well-depth. For two molecules $i$ and $j$, $f_{ij}$ denotes a dimensionless reduced collision integral. Characterizing the collisions between the 2 molecules, this integral is expressed as a function of an intermolecular potential. For nonpolar and non-polarizable molecules, this potential simplifies into the Lennard-Jones potential $\varphi_{ij}$ [20]:

$$ \varphi_{ij}(r) = 4\sqrt{\varepsilon_i\varepsilon_j} \left[ \left( \frac{\sigma_i + \sigma_j}{2r} \right)^{12} - \left( \frac{\sigma_i + \sigma_j}{2r} \right)^6 \right] \quad (10) $$

which is only dependent of $(\sigma_i, \sigma_j, \varepsilon_i, \varepsilon_j)$. Lastly, $C_v$ symbolizes the molar heat capacity whereas $\mathcal{A}$ represents Avogadro's number. Knowing $\mu_i$ & $\lambda_i$ of each species $i$ composing the gas mixture, mixture laws are finally used to compute mixture viscosity $\bar{\mu}$ and thermal conductivity $\bar{\lambda}$ [12, 13, 20].

Binary diffusion coefficients $D_{ij}$, representing how any virtual species $i$ diffuses related to any species $j$, are also expressed following [19]:

$$ \forall (i, j), \quad D_{ij} = \frac{3 \sqrt{2\pi k_B^3 T^3 (m_i + m_j) / (m_i m_j)}}{4 p\pi (\sigma_i + \sigma_j)^2 g_{ij} (\sigma_i, \sigma_j, \varepsilon_i, \varepsilon_j)} \quad (11) $$

$p$ being the pressure and $g_{ij}$ a second collision integral function of species' Lennard-Jones collision diameters $(\sigma_i, \sigma_j)$ and well-depths $(\varepsilon_i, \varepsilon_j)$. Depending on the transport model used for combustion computations, Eq. (11) or simplified equations are employed to calculate species diffusion coefficients [12, 13, 20].

Resulting from Eqs. (9, 10, 11), once the atomic composition and thermodynamic properties of a virtual species are known, its transport behavior is fully characterized by its Lennard-Jones collision diameter and potential well-depth. As the transport properties of real species are preserved and only those of virtual species are subject to optimization, the associated transport optimization vector $\mathbf{x}_{\text{Transport}}$ is given by:

$$ \mathbf{x}_{\text{Transport}} = \left( (\sigma_i, \varepsilon_i) \dots \forall i \in [1..N_P^v + N_I + N_F] \right). \quad (12) $$

The final optimization vector $\mathbf{x}$ characterizing the virtual reaction model therefore reads:

$$ \mathbf{x} = \left( \mathbf{x}_{\text{Scheme}}, \mathbf{x}_{\text{Arrhenius}}, \mathbf{x}_{\text{Transport}} \right). \quad (13) $$

### 3.4. Virtual scheme optimization

The next optimization stage dedicated to retrieve combustion kinetics is illustrated in Fig. (2) and follows three steps:

#### Step A

The numbers of desired pyrolysis fragments and intermediates species, $N_F$ and $N_I$, are first specified by the user to define the architecture of the virtual mechanism (Mech.1). An initial population for the optimized vector $\mathbf{x}$ is then randomly chosen.
<figure><img src="image_2.png" /><figcaption>**Figure 2.** Optimization protocol for virtual reaction models.</figcaption></figure>

If pyrolysis is considered, the atomic composition of each pyrolysis fragment $F_f$ is computed to ensure elemental conservation during fuel F decomposition.

### Step B

The stoichiometric coefficients of the $N_P$ virtual products ($\alpha'_{r,P_k}$) $k \in [1..N_P]$ and ($\alpha''_{r,P_k}$) $k \in [1..N_P]$ are also deduced from elemental mass conservation. Kinetic rate constants and transport properties are updated by $\boldsymbol{x}_{\text{Arrhenius}}$ and $\boldsymbol{x}_{\text{Transport}}$, respectively.

### Step C

One-dimensional (1D) laminar flame archetypes are computed using the virtual mechanism for a prescribed set of operating conditions $\psi_c$. A set of targeted quantities is compared against reference data, obtained from detailed chemistry flame archetypes under identical conditions. The agreement between virtual and reference results is quantified using a fitness function $f$. As long as the fitness function does not satisfy a predefined tolerance, the optimization algorithm proceeds with another iteration, returning to Step A.

## 4. Kinetic optimization results

Optimizations were conducted using the CMA-ES evolutionary algorithm [21] with an initial standard deviation $\sigma_0 = 0.2$ as for the thermodynamics optimization step. The population size was however increased to 100 individuals to account for the growing number of optimization variables.
### 4.1. $H_2-Air$ combustion

#### 4.1.1. Virtual mechanism

The reference flame database is computed using the detailed mechanism from Zhu & al. [15], which encompasses 43 species and 312 reactions. Molecular diffusion is modeled with the multicomponent model [20] with the Soret effect included, as suggested in [22].

The following two schemes are considered:

**Mechanism V-0/0-H₂**

$$ 
\begin{aligned}
H_2 + \frac{1}{2} O_2 &\longrightarrow \alpha''_{1,P_1} P_1 + \alpha''_{1,H_2O} H_2O \\
H_2 + \alpha'_{2,P_1} P_1 &\rightleftharpoons \alpha''_{2,H_2O} H_2O \\
O_2 + \alpha'_{3,H_2O} H_2O &\rightleftharpoons \alpha''_{3,P_1} P_1
\end{aligned}
$$

**Mechanism V-0/1-H₂**

$$ 
\begin{aligned}
H_2 + \frac{1}{2} O_2 &\longrightarrow \alpha''_{1,I} I + \alpha''_{1,P_1} P_1 + \alpha''_{1,H_2O} H_2O \\
I + \alpha'_{2,P_1} P_1 &\longrightarrow \alpha''_{2,H_2O} H_2O \\
H_2 + \alpha'_{3,P_1} P_1 &\rightleftharpoons \alpha''_{3,H_2O} H_2O \\
O_2 + \alpha'_{4,H_2O} H_2O &\rightleftharpoons \alpha''_{4,P_1} P_1
\end{aligned}
$$

Scheme **V-0/0-H₂** does not include an intermediate virtual species I while scheme **V-0/1-H₂** does. Naturally, pyrolysis is not considered in hydrogen combustion. For both mechanisms, combustion products are respectively $H_2O$ and $P_1$, $P_1$ being the optimized virtual product from thermodynamics optimization.

These two $H_2-Air$ virtual schemes are optimized to minimize the following fitness function $f_{H_2}$:

$$ f_{H_2} = f^T + f^{s_L} + 0.1 f^{\lambda}. \qquad (14) $$

The first fitness term, $f^T$, is designed to accurately replicate the flame structure by means of the temperature profile [7]:

$$ f^T = \sum_{c=1}^{N_c} \frac{\|T^v(x) - T^d(x)\|_{\mathcal{L}_2}(\psi_c)}{\|T^d(x)\|_{\mathcal{L}_2}(\psi_c)} \qquad (15) $$

$\|\cdot\|_{\mathcal{L}_2}$ being the $\mathcal{L}_2$ norm operator.

The second cost function $f^{s_L}$ in Eq. (14) is dedicated to capture the correct laminar flame velocity $s_L$ over the selected range of operating conditions [7]:

$$ f^{s_L} = \sum_{c=1}^{N_c} \left| \frac{s_L^v - s_L^d}{s_L^d} \right| (\psi_c). \qquad (16) $$

A third fitness term $f^{\lambda}$ is introduced to recover the thermal conductivity $\lambda$ in the
flame front:

$$f^{\lambda} = \sum_{c=1}^{N_c} \frac{\|\lambda^v(x) - \lambda^d(x)\|_{\mathcal{L}_2}(\psi_c)}{\|\lambda^d(x)\|_{\mathcal{L}_2}(\psi_c)} \quad (17)$$

The properties of the reference and virtual schemes, as well as targeted optimization conditions, are summarized in Table (2).

<table><caption>**Table 2.** Virtual scheme architectures and detailed mechanism comparison for H₂-Air combustion. *φ*: Equivalence ratio, *T*₀: Fresh gas temperature, *p*: Pressure.</caption><thead><tr><th>Scheme</th><th>Species</th><th>Reactions</th><th>Transport model</th><th>Target conditions</th></tr></thead><tbody><tr><td>Zhu & al. [15]</td><td>43</td><td>312</td><td>multicomponent + Soret</td><td>/</td></tr><tr><td>V-0/0 - H₂</td><td>5</td><td>3</td><td>multicomponent + Soret</td><td>1D laminar premixed flames, φ ∈ [0.4, 4.0], T₀ = 300K, p = 1 atm</td></tr><tr><td>V-0/1 - H₂</td><td>6</td><td>4</td><td>multicomponent + Soret</td><td>1D laminar premixed flames, φ ∈ [0.4, 4.0], T₀ = 300K, p = 1 atm</td></tr></tbody></table>

### 4.1.2. Results on H₂-Air laminar flames

For all H₂ laminar flame configurations, the fresh gases temperature is T₀ = 300K, the pressure p = 1 atm, and the multicomponent transport model is considered with Soret effect. Flame initialization with virtual models is slightly modified to enforce the absence of intermediate species in burnt gases.

Fig. (3) compares the laminar flame speed *s*<sub>L</sub> predicted by the reference mechanism with those obtained from the two optimized reaction models **V-0/0-H₂** & **V-0/1-H₂**.

<figure><img src="image_3.png" /><figcaption>**Figure 3.** Laminar flame velocity *s*<sub>L</sub> for H₂-Air combustion over an equivalence ratio range of φ ∈ [0.4, 4.0]. Comparison of two virtual scheme architectures **V-0/0-H₂** & **V-0/1-H₂** with the reference kinetic mechanism from Zhu & al. [15].</figcaption></figure>

The **V-0/0-H₂** model fails to accurately capture the flame speed in rich combustion regimes, primarily due to the absence of the intermediate species I, which plays a key role in controlling fresh gases consumption speed. In contrast, the **V-0/1-H₂** model achieves close agreement with the reference chemical scheme, with a relative error *f*<sub>sL</sub> ≈ 2.3%.

Temperature, thermal conductivity and real product H₂O mass fraction profiles predicted with **V-0/1-H₂** on freely-propagating laminar flames are shown in Fig. (4) for lean (φ = 0.4), stoichiometric (φ = 1.0), rich (φ = 2.0), and very rich (φ = 4.0) combustion regimes.
<figure><img src="image_4.png" /><figcaption>(a) Temperature T(x).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Thermal conductivity λ(x).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) H₂O mass fraction Y_H₂O(x).</figcaption></figure>

<caption>**Figure 4.** (a) Temperature, (b) thermal conductivity and (c) real product, H₂O, mass fraction profiles of H₂-Air 1D laminar premixed flames for various equivalence ratio *φ*. Comparison of the virtual scheme **V-0/1-H₂** with the reference kinetic mechanism from Zhu & al. [15].</caption>
The virtual reaction model V-0/1-H₂ captures both reference temperature profiles and the global flame thickness. Fig. (4) reveals a more pronounced discrepancy of thermal conductivity in the flame front, particularly near stoichiometric conditions. This behavior is consistent with the weighted formulation of the fitness function $f_{H_2}$, in which reproducing $\lambda$ was given lower priority compared to temperature and laminar flame velocity (see Eq. (14)). Nonetheless, the results remain satisfactory, as $\lambda$ is still reasonably well captured, indicating that the virtual scheme retains good predictive capability for thermal diffusion. Finally, Fig. (4) compares real combustion products mass fractions predicted by the reference and virtual mechanisms. Interestingly, despite not being included in the definition of the fitness function $f_{H_2}$, the production of virtual scheme's single real combustion product H₂O is well captured in the flame front across all equivalence ratios. Its concentration in the burnt gases is almost perfectly modelled, reflecting the effective recovery of the reference equilibrium states.

The virtual reaction model is then validated as *a posteriori* using 1D burner-stabilized flames, a challenging canonical non-adiabatic configuration for simplified chemistry techniques [23, 24]. A set of 1D H₂-Air burner-stabilized simulations are performed with CANTERA for lean ($\phi = 0.6$), stoichiometric ($\phi = 1$), and rich ($\phi = 2$) combustion scenarios. For each case, fresh gas mass fluxes $\dot{m}$ are adapted to cover a spectrum of enthalpy defects $\Delta h$ from low to high heat losses ($\Delta h \in [10\%, 40\%]$). As shown in Fig. (5), the H₂-Air virtual mechanism naturally reproduces non-adiabatic flame structures. Predicted equilibrium temperatures $T_{eq}$ closely match reference solutions: the largest deviation, observed at $\phi = 2$ with a heat loss ratio $\Delta h = 20\%$, remains below 4%.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Temperature profiles $T(x)$ of 1D H₂-Air burner-stabilized flames in (a) lean, (b) stoichiometric and (c) rich combustion regimes for various heat loss ratios $\Delta h$. Comparison of the virtual scheme **V-0/1-H₂** with the reference kinetic mechanism from Zhu & al. [15].</figcaption></figure>

The virtual mechanism optimized on 1D premixed flames is ultimately tested as *a posteriori* on 1D laminar counterflow diffusion flamelets. A set of counterflow diffusion flames is simulated using CANTERA for various strain rates. The strain rate $\dot{\phi}$ is calculated following the definition proposed by Aguerre [25]. Fig. (6) presents the temperature profiles, revealing excellent overall agreement between the two reaction mechanisms across all strain rates. For each strain rate, the flame structure is closely reproduced, with only a slight overestimation of the peak temperature $T_{max}$. The error $\epsilon_{T_{max}}$, averaged on these 4 flames, is $\epsilon_{T_{max}} \approx 3.3\%$. The analysis is extended to assess the virtual scheme's capability to predict non-premixed combustion: Fig. (7) presents
<figure><img src="image_8.png" /><figcaption>**Figure 6.** 1D H₂–Air counterflow diffusion flames for various strain rates *a*. Comparison of the virtual scheme **V-0/1-H₂** with the reference kinetic mechanism from Zhu & al. [15].</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Dependence of maximum temperature $T_{max}$ on strain rate *a* for 1D H₂–Air counterflow diffusion flames. Comparison of the virtual scheme **V-0/1-H₂** with the reference kinetic mechanism from Zhu & al. [15].</figcaption></figure>

the variation of peak temperature $T_{max}$ with strain rate. As expected, virtual chemistry exhibit a consistent overestimation of $T_{max}$ across the entire strain range. The relatively small overprediction of typically $\epsilon_{T_{max}} \approx 2.5\%$ in the low to moderate strain rates ($a \in [10, 1000]s^{-1}$), tends to grow with increasing *a*, resulting in a pronounced overestimation of the quenching limit. These discrepancies likely stem from differences in the dominant chemical pathways near extinction, compared to those in premixed flames spanning various equivalence ratios.

## 4.2. $nC_{12}H_{26}$–Air

### 4.2.1. Virtual mechanism

The method is further challenged on the complex chemistry of $nC_{12}H_{26}$ combustion, a major component of Sustainable Aviation Fuel (SAF). The 413-species C0–C16 CRECK model [16], designed for heavy hydrocarbon combustion, is retained to build the reference database of 1D premixed flames. Transport processes are accounted with
the *mixture-averaged* model [20].

Two competing virtual mechanism designs are explored:

### Mechanism V-0/1-nC₁₂H₂₆

$$n\text{C}_{12}\text{H}_{26} + \frac{37}{2}\text{O}_2 \longrightarrow \alpha''_{1,I}\text{I} + \alpha''_{1,P_1}\text{P}_1 + \alpha''_{1,H_2O}\text{H}_2\text{O} + \alpha''_{1,\text{CO}_2}\text{CO}_2$$

$$\text{I} + \alpha'_{2,P_1}\text{P}_1 \longrightarrow \alpha''_{2,H_2O}\text{H}_2\text{O} + \alpha''_{2,\text{CO}_2}\text{CO}_2$$

$$n\text{C}_{12}\text{H}_{26} + \alpha'_{3,\text{CO}_2}\text{CO}_2 \rightleftharpoons \alpha''_{3,H_2O}\text{H}_2\text{O} + \alpha''_{3,P_1}\text{P}_1$$

$$\text{O}_2 + \alpha'_{4,P_1}\text{P}_1 \rightleftharpoons \alpha''_{4,H_2O}\text{H}_2\text{O} + \alpha''_{4,\text{CO}_2}\text{CO}_2$$

### Mechanism V-1/0-nC₁₂H₂₆

$$n\text{C}_{12}\text{H}_{26} \longrightarrow \alpha''_{1,\text{pyro}}\text{F}_1$$

$$\alpha''_{1,\text{pyro}}\text{F}_1 + \frac{37}{2}\text{O}_2 \longrightarrow \alpha''_{1,P_1}\text{P}_1 + \alpha''_{1,H_2O}\text{H}_2\text{O} + \alpha''_{1,\text{CO}_2}\text{CO}_2$$

$$\text{F}_1 + \alpha'_{3,\text{CO}_2}\text{CO}_2 \longrightarrow \alpha''_{3,H_2O}\text{H}_2\text{O} + \alpha''_{3,P_1}\text{P}_1$$

$$\text{O}_2 + \alpha'_{4,P_1}\text{P}_1 \rightleftharpoons \alpha''_{4,H_2O}\text{H}_2\text{O} + \alpha''_{4,\text{CO}_2}\text{CO}_2$$

The **V-0/1-nC₁₂H₂₆** adopts the same architecture as **V-0/1-H₂** for Dodecane, whereas **V-1/0-nC₁₂H₂₆** features a pyrolysis-dedicated structure. Combustion products are CO₂, H₂O and the optimized P'₁ from thermodynamics optimization (cf. Table (1)). Both virtual schemes are composed of 7 species and 4 reactions. Characteristics of the detailed and virtual kinetic mechanisms, as well as their associated target conditions are specified in Table (3). The single-objective fitness function *f*<sub>nC<sub>12</sub>H<sub>26</sub></sub> is defined analogously to *f*<sub>H₂</sub>, expressed in Eq. (14).

<table><caption>**Table 3.** Virtual scheme architectures and detailed mechanism comparison for nC₁₂H₂₆-Air combustion. *φ*: Equivalence ratio, *T₀*: Fresh gas temperature, *p*: Pressure.</caption><thead><tr><th>Scheme</th><th>Species</th><th>Reactions</th><th>Transport model</th><th>Target conditions</th></tr></thead><tbody><tr><td>CRECK [16]</td><td>413</td><td>14,922</td><td>mixture-averaged</td><td>/</td></tr><tr><td>V-0/1-nC<sub>12</sub>H<sub>26</sub></td><td>7</td><td>4</td><td>mixture-averaged</td><td>1D laminar premixed flames, φ ∈ [0.7, 1.4], T<sub>0</sub> = 400K, p = 1 atm</td></tr><tr><td>V-1/0-nC<sub>12</sub>H<sub>26</sub></td><td>7</td><td>4</td><td>mixture-averaged</td><td>1D laminar premixed flames, φ ∈ [0.7, 1.4], T<sub>0</sub> = 400K, p = 1 atm</td></tr></tbody></table>

### 4.2.2. Results on 1D premixed nC₁₂H₂₆-Air laminar flames

As mentioned in Table (3), 1D freely-propagating flames are computed with a fresh gas temperature *T*₀ = 400K, pressure *p* = 1 atm, and transport processes are calculated using the *mixture-averaged* model.

Fig. (8) compares the laminar flame speed *s*<sub>L</sub> predicted by the CRECK reference mechanism with those obtained from the two optimized reaction models **V-0/1-nC₁₂H₂₆** & **V-1/0-nC₁₂H₂₆**. The classical model with a single intermediate I, **V-0/1-nC₁₂H₂₆**, largely fails to capture *s*<sub>L</sub> across the entire flammability range. By contrast, **V-1/0-nC₁₂H₂₆**, which includes a single pyrolysis fragment F₁, matches the reference laminar flame speed for all equivalence ratios, with a relative error *f*<sub>sL</sub> ≈ 2.6%. This difference is explained by the underlying chemistry. In particu-
<figure><img src="image_10.png" /><figcaption>(a) Temperature $T(x)$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b) Heat release rate $\dot{\omega}_T(x)$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(c) CO₂ & H₂O mass fractions $Y_k(x)$.</figcaption></figure>

**Figure 9.** (a) Temperature, (b) heat release rate and (c) real products mass fraction profiles of 1D $nC_{12}H_{26}$-Air laminar premixed flames for various equivalence ratio $\phi$. Comparison of the virtual scheme V-1/0-nC₁₂H₂₆ with the reference kinetic mechanism from CRECK [16].
<table><caption>**Table 4.** Chemical stiffness $\tau$ of H₂-Air and nC₁₂H₂₆-Air virtual schemes and their associated detailed mechanisms.</caption><thead><tr><th rowspan="2">Scheme</th><th colspan="3">H₂-Air combustion</th></tr><tr><th>Detailed: Zhu et. al. [15]</th><th>Virtual</th><th>Ratio</th></tr></thead><tbody><tr><td>Chemical stiffness τ</td><td>2.75 × 10<sup>-9</sup>s</td><td>2.27 × 10<sup>-7</sup>s</td><td>| ×83</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">Scheme</th><th colspan="3">nC<sub>12</sub>H<sub>26</sub>-Air combustion</th></tr><tr><th>Detailed: CRECK [16]</th><th>Virtual</th><th>Ratio</th></tr></thead><tbody><tr><td>Chemical stiffness τ</td><td>1.88 × 10<sup>-10</sup>s</td><td>5.15 × 10<sup>-7</sup>s</td><td>| ×2745</td></tr></tbody></table>

These results underscore the effectiveness of virtual chemistry in reducing not only the number of species and reactions, but also the underlying chemical stiffness: Virtual schemes consistently exhibit significantly lower chemical stiffness than their detailed counterparts. For H₂-Air combustion, the virtual mechanism is measured 83 times less stiff than the detailed one. This stiffness reduction is even more impressive for nC₁₂H₂₆-Air, since reducing the mechanism from over 10⁴ reactions to only four results in a dramatic stiffness decrease exceeding three orders of magnitude.

By simultaneously reducing chemical stiffness and the number of species and reactions, virtual chemistry significantly lowers computational cost, compared to detailed mechanisms. To illustrate this gain, the CPU time required for simulating 1D premixed flames is measured. The raw computational time *t* is averaged over the number of Newton iterations achieved by the solver and the number of grid points. This provides a normalized metric *cpu*\* closely tied to chemical stiffness and species/reactions numbers. Results are presented in Table (5).

<table><caption>**Table 5.** Computational time/grid point/Newton iteration *cpu*\* (on Apple M1) for 1D premixed flame computations with H₂-Air and nC₁₂H₂₆-Air virtual schemes and their associated detailed mechanisms.</caption><thead><tr><th rowspan="2">Scheme</th><th colspan="3">H₂-Air combustion</th></tr><tr><th>Detailed: Zhu et. al. [15]</th><th>Virtual</th><th>Ratio</th></tr></thead><tbody><tr><td>Computational time cpu\*</td><td>3.51 × 10<sup>-3</sup>s</td><td>2.39 × 10<sup>-5</sup>s</td><td>| ×147</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">Scheme</th><th colspan="3">nC<sub>12</sub>H<sub>26</sub>-Air combustion</th></tr><tr><th>Detailed: CRECK [16]</th><th>Virtual</th><th>Ratio</th></tr></thead><tbody><tr><td>Computational time cpu\*</td><td>2.01s</td><td>2.37 × 10<sup>-5</sup>s</td><td>| ×85000</td></tr></tbody></table>

Virtual mechanisms largely outperform detailed reaction models in terms of efficiency. While computing a premixed flame with the H₂-Air virtual scheme is already ≈ 150 times faster than its corresponding detailed counterpart, the gain becomes striking with nDodecane, reaching a factor of 85,000.

## 5. Conclusions

The standardized virtual chemistry formalism introduced by Préteseille & al. [14] has been extended to capture both the complex combustion chemistry and transport processes of laminar flames. A general optimization procedure has been proposed through the simultaneous optimization of virtual scheme's structure, reaction rates, and virtual species transport properties on reference 1D flamelets. The virtual mechanisms,
structured analogously to any CANTERA/CHEMKIN detailed mechanism, are able
to incorporate user-defined real species such as H₂O or CO₂, accommodate any number
of intermediate species, and introduce an original dedicated architecture for pyrolyzing
fuels.

Applied to model H₂-Air flames, the resulting optimized 6-species/4-reaction mech-
anism reproduced reference premixed flame structures computed with detailed chem-
istry and differential diffusion with excellent agreement. *A posteriori* tests achieved
on burner-stabilized and counterflow flames also highlighted virtual mechanism's abil-
ity to capture non-adiabatic and non-premixed combustion regimes. The method was
further validated on the complex nC₁₂H₂₆-Air combustion, where the introduced
pyrolysis-dedicated 7-species/4-reaction mechanism significantly outperformed stan-
dard virtual scheme architectures. The highly-reduced mechanism faithfully described
the reference premixed flame structures and laminar flame speeds computed with a
full mechanism comprising 413 species and 14,922 reactions.

Beyond drastically reducing the number of species and reactions, the virtual chem-
istry strategy also proved highly effective in lowering stiffness by 2-3 orders of magni-
tude, leading to dramatic computational speed-ups in flame simulations ranging from
a factor of 147 for H₂ to 85,000 for Dodecane.

These results underscore the ability of these new, standardized, non-tabulated,
highly-reduced virtual mechanisms to model combustion with both accuracy and effi-
ciency, while enabling seamless integration into any reactive flow solver. Future work
aims to incorporate pollutant prediction, such as nitrogen oxide and soot formation,
within this standardized framework.

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 support from the MONTHY project funded by the French National Research Agency (ANR) under France 2030 program and reference ANR 22-PEHY-0014. A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions. Authors also acknowledge support from ArianeGroup and the HESTIA project which received funding from the European Union Horizon Europe research and Innovation program under Grant Agreement N°101056865. This research was conducted using HPC resources from IDRIS, TGCC, and CINES under the allocation A0132B10253 by GENCI (Grand Équipement National de Calcul Intensif).

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