# On the Modelling of Differential Diffusion with Virtual Chemistry: An Application to Hydrogen Flames with Nitrogen Oxides

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

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## To cite this version:

Étienne Espada, Matthieu Préteseille, Tan-Phong Luu, Nasser Darabiha, Benoit Fiorina. On the Modelling of Differential Diffusion with Virtual Chemistry: An Application to Hydrogen Flames with Nitrogen Oxides. 13th Mediterranean Combustion Symposium (MSC2025), Jun 2025, Corfou, Greece. hal-05301954

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# ON THE MODELLING OF DIFFERENTIAL DIFFUSION WITH VIRTUAL CHEMISTRY: AN APPLICATION TO HYDROGEN FLAMES WITH NITROGEN OXIDES

E. Espada*, M. Préteseille*, T-P. Luu*, N. Darabiha* and B. Fiorina*

etienne.espada@centralesupelec.fr

*Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C, CentraleSupélec, 91190, Gif-sur-Yvette, France

## Abstract

Given the current imperatives of decarbonization, the accurate modelling of combustion processes is crucial for designing increasingly efficient combustion chambers. Decarbonized fuels, such as hydrogen, can be particularly challenging to simulate in turbulent flows, especially in lean regimes where turbulence-chemistry interactions are enhanced. These combustion regimes indeed play a pivotal role in lowering NOₓ pollutant emissions by reducing burnt gases temperature and suppressing major thermal chemical pathways [1]. Addressing these challenges and simplifying the underlying chemistry thus calls for the application of various kinetic mechanism reduction techniques. Among these strategies, virtual chemistry stands out as a compelling method, relying on the construction of a chemical scheme of virtual species and reactions with optimized thermochemical and kinetics properties to accurately reproduce selected flame characteristics [2]. The computation of a main mechanism is firstly performed by determining the optimal set of tabulated Arrhenius parameters and stoichiometric coefficients to capture reference flame structures for various equivalence ratios. The prediction of a specific pollutant is achieved through a satellite scheme, where chemical rates are similarly optimized and coupled with the main mechanism [2]. However, given its inherent assumption of a unity Lewis number transport model [2], the method faces limitations when applied to fuels like H₂ exhibiting complex differential diffusion effects. To address these issues, a new optimization strategy is proposed. First, the kinetic rates are optimized to retrieve chemical effects of unity Lewis flames. Then, the computation of optimal species Lewis numbers is achieved to account for complex diffusion effects. Relying on Cantera [3], this methodology is applied to optimize a H₂-Air virtual scheme with a NOₓ satellite mechanism. By decoupling the optimization of chemistry and diffusion, this approach succeeds to extend virtual chemistry capabilities to encompass complex transport phenomena whilst keeping a dramatically reduced number of species and reactions.

## Introduction

Amid growing concerns over climate change, the transition to alternative fuels is becoming increasingly popular. Hydrogen, relying on its seemingly simple chemistry and its large specific energy, is naturally one of the most promising candidates for decarbonized fuels. However, incorporating chemical pathways for pollutant prediction, such as NOₓ, substantially increases the complexity of chemical mechanisms. For instance, the detailed mechanism from Zhu et al. [4], specifically designed to model hydrogen and ammonia combustion with a detailed description of nitric oxides, includes 43 species and 312 reactions. Other low-carbon fuels, such as sustainable aviation fuels, involve even more complex chemistry, comprising thousands of reactions [5]. This participates to make simulations of industrial turbulent configurations computationally prohibitive. In addition to detailed chemistry, complex transport phenomena impact significantly the structure and dynamic of hydrogen flames. Indeed, accounting for differential diffusion is essential for accurately predicting fuel consumption and observing thermo-diffusive instabilities [6, 7].

The reduction of this computational cost can principally comes from multiple chemistry reduction techniques classically used in the combustion community, such as tabulated chemistry,
reduced chemistry or optimized chemistry. Tabulated chemistry consists in tabulating thermo-
chemical variables as a function of coordinates that are chosen to recover the thermochemical
space manifold [8, 9]. By tabulating the chemistry as a function of the progress variable $Y_c$
and the mixture fraction $Z$, the number of transported scalars is significantly reduced - limited
to the selected coordinates - while also eliminating issues related to chemical stiffness. This
approach, however, faces significant challenges when dealing with combustion processes in-
volving multiple chemical timescales, such as those observed in $NO_x$ chemistry [1]. In the
framework of tabulated chemistry, differential diffusion is typically addressed by correcting the
mixture fraction in the direction normal to the flow [10]. This approach has been further ex-
tended to capture tangential effects, such as curvature, in thermo-diffusively unstable flames
[11]. However, these improvements in precision come at the cost of increased computational
complexity, as they necessitate augmenting the manifold dimensions by introducing one or two
additional scalar transport variables to account for preferential diffusion effects. An alterna-
tive is to develop optimized global schemes. For instance in the virtual chemistry method [2],
a chemical mechanism is built from scratch using virtual species and reactions whose thermo-
chemical coefficients and kinetic rates are optimized on 1D flamelets to retrieve the macroscopic
properties of the flame structure - such as the heat release rate $\dot{\omega}_T$ or laminar flame speed $s_L^0$ -
and the production of pollutants. Specifically designed to leverage an optimally calibrated set
of parameters, virtual chemistry enables to accurately capture the combustion behavior of a fuel
across its entire flammability range with a remarkable accuracy using a minimum number of
reactions and species. It already demonstrated its effectiveness in modeling methane and ethy-
lene combustion, as well as increasingly complex pollutants like $NO_x$ [12] and soot [13].

While virtual chemistry was originally developed in the context of hydrocarbon combus-
tion - where differential diffusion is less critical - this study proposes a simplified approach
to incorporate such effects. This work thereby extends the capabilities of virtual chemistry
by integrating transport property optimization, offering a more robust framework for capturing
complex transport phenomena.

This report is structured as follows: The first section introduces the virtual chemistry method-
ology applied to the *main* scheme, along with the resulting flame structure predictions. By
optimizing the Lewis numbers of individual species, the proposed method effectively accounts
for differential diffusion without significantly increasing computational overhead. The second
section focuses on optimizing the *satellite* NO scheme and coupling it to the main to accurately
capture pollutant mass fractions. Finally, the report concludes with a discussion of key findings
and future perspectives.

## Main virtual scheme for flame structure prediction

### *Method overview*

Virtual chemistry aims to simplify chemical mechanisms by optimizing a minimal set of vir-
tual species and reactions to capture reference flame properties. The reference flame database
is a collection of 1D flamelets and 0D reactors built from detailed chemistry and complex trans-
port. Hereinafter, the reference mechanism used will be the CRECK [14], from which carbon
species and carbon-involving reactions have been removed. This choice is motivated by Ca-
purso et al. [15] study, which showed the reliability of CRECK mechanism to capture flame
key properties as well as nitrogen oxides. Dealing with virtual scheme's structure, Cailler et al.
[2] highlighted that the following two-step mechanism reproduces the flame consumption speed
and the thermal flame structure over the flammability limit range:
$$ 
\begin{aligned}
& \sum_{i=1}^{3} \alpha'_{i} \mathcal{R}_{i} \longrightarrow \alpha_{I} \mathcal{I} + \sum_{i=1}^{3} \alpha''_{i} \mathcal{R}_{i} \qquad (\mathcal{R}_{1}) \\
& \alpha_{I} \mathcal{I} + \sum_{i=1}^{3} \alpha''_{i} \mathcal{R}_{i} \longrightarrow \sum_{i=1}^{3} \alpha''_{i} \mathcal{R}_{i} + \sum_{j=1}^{N_{P}} \beta''_{j} \mathcal{P}_{j} \qquad (\mathcal{R}_{2})
\end{aligned}
$$

where $\mathcal{R}_i$ & $\mathcal{P}_j$ respectively represent real species i and virtual product $P_j$. The real species are denoted as F, Ox, and D for the corresponding Fuel, Oxidizer, and Diluent. Each virtual species is characterized by its optimized molar mass and set of NASA7 thermodynamic coefficients, and is transported analogously to real species in any thermochemical solver. Species source terms are closed using reactions' traditional Arrhenius formulation. $\alpha_i$ & $\beta_j$ denotes stoichiometric coefficients associated with real species i and virtual species $P_j$, respectively. Superscripts ' & " respectively symbolizes a variable in reactants' or products' sides. In the present study, we consider $\mathcal{R} = \{F, Ox, D\}$ and $\mathcal{P} = \{P_1, P_2, \dots, P_{N_P}\}$. The chemical reaction connecting fresh and burnt gases are expressed. The total number of species involved is therefore $N_S = N_P + 3$.

In the original virtual chemistry formulation [2], a unity Lewis number is assumed to represent the molecular diffusion coefficients of virtual species. As a result, physical phenomena associated with differential diffusion, such as local demixing or intrinsic flame instabilities, cannot be captured. We therefore propose to extend virtual chemistry capabilities to account for complex diffusion following a natural optimization protocol summed up in Figure (1). Firstly, virtual products molar mass $M_k$, stoichiometric coefficients $\beta_k''$ and NASA7 parameters are optimized to retrieve equilibrium mixture's thermodynamic quantities of interest and ensure mass conservation. Secondly, the optimal reaction rates of mechanism (1) are determined to accurately capture the chemical effects connecting real fresh gases with the previously optimized burnt gases. Given the intrinsic coupling between chemical kinetics and diffusion in flames [16], these rates are optimized on flames computed under unity Lewis assumption to exclude the influence of complex diffusion. Finally, the impact of diffusion is incorporated by optimizing species Lewis numbers while freezing combustion chemistry variables, ensuring the reproduction of reference flame structures derived from advanced transport models. More details on each optimization step will be given in the following subsections.

<figure><img src="image_3.png" /><figcaption>Figure 1: Proposed optimization procedure of main virtual chemical mechanism.</figcaption></figure>
The objective of this first optimization concerns thermodynamic coefficients of virtual products. This aims to return the optimal set of parameters to match reference mixture's equilibrium thermochemical states for a given range of equivalence ratio. This set of equivalence ratio is predefined to cover the entire flammability domain of the selected fuel F. $N_c$ different equivalence ratio are noted ($\phi_c$)$_{c \in [1, N_c]}$.

As traditionally made with virtual chemistry [2], virtual products ($P_k$)$_{k \in [1, N_P]}$ are defined with a molar mass $M_k$ and a set of NASA7 coefficients from the classical parametrization of [17]. For a given virtual species k, its heat capacity $C_{p,k}$, enthalpy $H_k$ and entropy $S_k$ are conveniently derived from its NASA7 coefficients according to the subsequent temperature polynomials:

$$ 
\begin{aligned}
\forall k \in [1, N_p], \quad \frac{C_{p,k}(T)}{R} &= a_{1,k} + a_{2,k}T + a_{3,k}T^2 + a_{4,k}T^3 + a_{5,k}T^4 \\
\frac{H_k(T)}{RT} &= a_{1,k} + \frac{a_{2,k}}{2}T + \frac{a_{3,k}}{3}T^2 + \frac{a_{4,k}}{4}T^3 + \frac{a_{5,k}}{5}T^4 + \frac{a_{6,k}}{T} \\
\frac{S_k(T)}{R} &= a_{1,k}\ln(T) + a_{2,k}T + \frac{a_{3,k}}{2}T^2 + \frac{a_{4,k}}{3}T^3 + \frac{a_{5,k}}{4}T^4 + a_{7,k}
\end{aligned}
\quad (2) $$

As demonstrated by Cailler et al. [2], optimizing temperature-dependent parameters alone within the virtual chemistry framework is insufficient to fully capture the range of thermochemical equilibrium states of detailed mechanisms across varying equivalence ratios $\phi_c$. To address this limitation, a tabulation approach for $\beta_k''(\phi_c)$ is introduced, ensuring burnt gases mandatory diversity to capture equilibrium states for our set of equivalence ratio. Within this formalism, Cailler et al. [2] empirically showed that $N_P = 4$ was, independently of the fuel, the optimal number of virtual products. This assumption will naturally be maintained throughout the remainder of this study.

The thermodynamic optimization vector $\boldsymbol{x}_T$ is therefore defined according to Equation (3)

$$ \boldsymbol{x}_T = \left( \underbrace{(a_{1,k}, a_{2,k}, a_{3,k}, a_{4,k}, a_{5,k}, a_{6,k}, a_{7,k})}_{\text{Thermodynamics coefficients}}, \underbrace{(\beta_k''(\phi_c))}_{\substack{\text{Tabulated} \\ \text{stoichiometric} \\ \text{coefficients}}}, \underbrace{M_k \dots \forall k \in [1, N_p]}_{\substack{\text{molar} \\ \text{mass}}} \right) \quad (3) $$

A fitness function $f_T$ is built to recover mixture's equilibrium thermodynamic quantities through its minimization. As proposed in the classical virtual chemistry formalism [2, 12, 13], targeted mixture properties are gas mean molecular mass $\bar{M}$ and mean thermodynamic coefficients $\bar{a}_i = \sum_{k=1}^{N_S} a_{i,k} Y_k$, $\forall i \in [1, 7]$. These coefficients indeed express the temperature-dependency of mixture's heat capacity, enthalpy and entropy. The notations $|_v$ and $|_d$, as well as the subscripts v and d, are adopted to differentiate quantities related to virtual and detailed schemes, respectively.

Based on previous remarks, one can define two distinct fitness functions. To account for variables' different orders of magnitude, a fitness based on relative differences is adopted. Thus, writing

$$ f_T^{\bar{M}}(\boldsymbol{x}_T) = \left| \frac{\sum_{k=1}^{N_S} M_k Y_k \bigg|_v - \sum_{k=1}^{N_S} M_k Y_k \bigg|_d}{\sum_{k=1}^{N_S} M_k Y_k \bigg|_d} \right| \quad \& \quad f_T^{a_{i,k}}(\boldsymbol{x}_T) = \sum_{i=1}^{7} \left| \frac{\sum_{k=1}^{N_S} a_{i,k} Y_k \bigg|_v - \sum_{k=1}^{N_S} a_{i,k} Y_k \bigg|_d}{\sum_{k=1}^{N_S} a_{i,k} Y_k \bigg|_d} \right| \quad (4) $$

one can define the final single-objective fitness function for thermodynamics $f_T$ coupling Equa-
tions (4) into

$$f_{\tau}(\mathbf{x}_{\tau}) = \sum_{i=1}^{N_c} \left( f_{\tau}^{\overline{M}} + f_{\tau}^{a_{i,k}^{v}} \right) \qquad (5)$$

This is finally the minimization of $f_{\tau}$ that ensures an accurate description of mixture's equilibrium thermodynamic states. The accuracy of the optimisation is measured through the retrieval of detailed scheme equilibrium temperature, as illustrated in Figure (2).

<figure><img src="image_4.png" /><figcaption>**Figure 2:** Comparison of detailed and virtual equilibrium temperatures $T_{eq}$ in function of the equivalence ratio $\phi$ at atmospheric pressure.</figcaption></figure>

### Chemistry optimization

Relying on previously optimized virtual products $P_k$ to accurately retrieve burnt gases mixture properties, this second optimization aims to capture flame front chemical dynamics throughout the optimization of virtual mechanism (1) reaction rates. The proposed strategy significantly differs from the traditional virtual chemistry approach, which optimizes Arrhenius parameters to retrieve flamelets with complex transport models without retrieving the local demixing caused by differential diffusion effects. Furthermore, it makes the capture of thermo-diffusive instabilities impossible. Conversely, the objective is to optimize Arrhenius rates to replicate reference pure chemical effects through the decoupling of chemistry and transport, before optimizing species transport properties. Consequently, a unity Lewis assumption is first undertaken to simply consider, for each chemical species of detailed and virtual schemes, an equal diffusion coefficient $\mathcal{D}$ being mixture's thermal diffusion coefficient $\mathcal{D}_{th}$. It follows

$$\mathcal{D} = \mathcal{D}_{th} = \frac{\lambda}{\rho C_p} = \frac{\mu}{\rho \text{Pr}} \qquad (6)$$

$\lambda$, $\rho$, $C_p$, $\mu$ & Pr being respectively mixture's thermal conductivity, density, heat capacity, dynamic viscosity and Prandtl number. The proper modelling of mixture's thermal conductivity $\lambda$ and viscosity $\mu$ is carried out through the fitting of detailed profiles using power laws

$$\mu = \mu_0 \left( \frac{T}{T_0} \right)^{\beta} \quad \text{and} \quad \lambda = \frac{\mu C_p}{\text{Pr}} = \frac{\mu_0 C_p}{\text{Pr}} \left( \frac{T}{T_0} \right)^{\beta} \qquad (7)$$

This technique aims to find the best tabulated sets of parameters ($\mu_0$, Pr, $\beta$) for the chosen fresh gas temperature $T_0$ in function of the equivalence ratio $\phi_c$, thereby improving the procedure proposed by Cailler et al. [2]. Rate constants of irreversible reactions $R_1$ and $R_2$ are simply
expressed using Arrhenius laws

$$ 
\begin{aligned}
k_{\mathcal{R}_1} &= A_{\mathcal{R}_1}(\phi_c) \exp\left(\frac{-E_{a,\mathcal{R}_1}}{RT}\right) [\text{F}]^{n_{\mathcal{R}_1}^\text{F}} [\text{Ox}]^{n_{\mathcal{R}_1}^\text{Ox}} \\
k_{\mathcal{R}_2} &= A_{\mathcal{R}_2} \exp\left(\frac{-E_{a,\mathcal{R}_2}}{RT}\right) [\text{I}]^{n_{\mathcal{R}_2}^\text{I}}(\phi_c)
\end{aligned} 
\quad (8) $$

where $k_{\mathcal{R}_1}$ & $k_{\mathcal{R}_2}$ respectively represent $\mathcal{R}_1$ & $\mathcal{R}_2$ rate constants. ($A_{\mathcal{R}_i}$, $E_{a,\mathcal{R}_i}$) symbolizes the couple of Arrhenius parameters in the reaction $i$. [X] is the concentration of species X and $n_{\mathcal{R}_i}^X$ denotes the order of reaction of species X in reaction $\mathcal{R}_i$. Similarly as in the thermodynamics optimization, tabulated variables are introduced - here $A_{\mathcal{R}_1}$ and $n_{\mathcal{R}_2}^I$ - as new degrees of freedom to improve the capture of flame front dynamics [2]. As demonstrated in the literature, this method successfully replicates reference temperature profiles of various hydrocarbon flames across a range of equivalence ratios [2, 12, 13]. However, in hydrogen flames, this conclusion weakens, as the chemical rates involved exceed those of hydrocarbon combustion kinetics by an order of magnitude. To tackle these challenges and accurately capture the thin flame fronts in H₂-Air combustion, the optimization of intermediate's NASA7 thermodynamic coefficients is proposed to better align with flame front kinetics. This approach indeed yields an optimized temperature-dependent set of seven coefficients that directly influence the enthalpy $H_I(T)$. This parametrizes in return the local enthalpy of the mixture, and consequently, its temperature $T$ and heat release rate $\dot{\omega}_T$.

The chemistry optimization vector $\mathbf{x}_C$ is therefore divided in two major terms :

$$ \mathbf{x}_C = \left[ \mathbf{x}_C^k, \mathbf{x}_C^I \right] \quad (9) $$

where $\mathbf{x}_C^k$ represents the optimized Arrhenius parameters, whereas $\mathbf{x}_C^I$ symbolizes intermediate's thermodynamic coefficients. The definition of each vector is respectively given by

$$ \mathbf{x}_C^k = \left( \underbrace{(A_{\mathcal{R}_1}(\phi_c), n_{\mathcal{R}_2}^I)}_{\text{Tabulated}} , E_{a,\mathcal{R}_1}, E_{a,\mathcal{R}_2}, n_{\mathcal{R}_1}^\text{F}, n_{\mathcal{R}_1}^\text{Ox} \right) \quad \text{and} \quad \mathbf{x}_C^I = \left( \underbrace{(a_1^I, a_2^I, a_3^I, a_4^I, a_5^I, a_6^I, a_7^I)}_{\text{Intermediate thermodynamics coefficients}} \right) \quad (10) $$

As in the thermodynamics optimization subsection, the cost function $f_C$ is formulated as a single-objective fitness function encompassing multiple quantities to minimize. While originally designed to reduce discrepancies between detailed and virtual scheme temperature profiles, it is now redefined to focus on minimizing errors in the profiles of the heat release rate $\dot{\omega}_T(x)$ and the heat capacity $C_p(x)$. A major improvement of this strategy lies in the capability to accurately predict sensitive variables in hydrogen flames, such as the flame thickness $\delta_F$, relying on capturing similarly sensitive variables physically representative of combustion. The weighted fitness function finally writes, using norm $\mathcal{L}_2$,

$$ f_C(\mathbf{x}_C) = \sum_{i=1}^{N_c} \left[ w_{\dot{\omega}_T} \sqrt{\frac{\int_{-\infty}^{+\infty} (\dot{\omega}_T|_v - \dot{\omega}_T|_d)^2 dx}{\int_{-\infty}^{+\infty} \dot{\omega}_T|_d dx}} + w_{C_p} \sqrt{\frac{\int_{-\infty}^{+\infty} (C_p|_v - C_p|_d)^2 dx}{\int_{-\infty}^{+\infty} C_p^2|_d dx}} \right. \\ \left. + w_{s_L} \left| \frac{s_L^0|_v - s_L^0|_d}{s_L^0|_d} \right| \right] \quad (11) $$

where the set of weights ($w_{\dot{\omega}_T}$, $w_{C_p}$, $w_{s_L}$) is assigned to balance the contribution of each fitness term in Equation (11). Results of the chemistry optimization are presented in Figure (3), which displays temperature and heat release rate profiles across lean, stoichiometric, and rich combus-
tion regimes. Optimized scheme is compared against 1D laminar premixed flames computed with detailed chemistry under a unity Lewis number transport model. The flamelet computation is done at atmospheric pressure and for a fresh gas injection of 300K. The optimization shows almost perfect agreement with the reference database, allowing the flame thickness and temperature gradient within the flame front to be accurately captured.

<figure><img src="image_5.png" /><figcaption>**Figure 3:** Comparison of detailed and virtual 1D premixed flame temperature $T(x)$ and heat release rate profiles $\dot{\omega}_T(x)$ as a function of the equivalence ratio $\phi$. Black solid lines (-) correspond to the Unity-Lewis detailed flame, whereas red dashed lines (---) represent the flame predicted by the main virtual mechanism.</figcaption></figure>

## Transport optimization

Building on a previously optimized frozen chemistry, this final stage focuses on identifying the optimal set of species' Lewis numbers to accurately reproduce hydrogen's differential diffusion phenomena. The critical role of complex transport and the Soret effect in capturing the physics of laminar hydrogen flames - particularly the laminar flame speed $s_L^0$ - is well-established [18]. Accordingly, reference flames are computed using a multicomponent transport model with the Soret effect included. As switching to this transport model alters mixture's reference thermal conductivity $\lambda$ and viscosity $\mu$, power laws in Equation (7) are recalibrated based on multicomponent reference flame profiles.

To simplify the optimization process, unity Lewis numbers are imposed to unity for the oxidizer Ox and the diluent D. As a result, the diffusion optimization vector $x_D$ is defined as follows:

$$x_D = \left( Le_F, Le_I, \underbrace{(Le_{P_1}, \dots, Le_{P_{N_P}})}_{\substack{\text{Virtual products} \\ \text{Lewis numbers}}} \right) \qquad (12)$$

The weighted fitness function $f_D$ is constructed similarly to the chemistry optimization section. However, due to the comparison of two distinct transport models - multicomponent with Soret effect for the detailed scheme and a much simpler variable Lewis number model for the vir-
tual one - it is important to note that sensitive variables, such as heat release rate $\dot{\omega}_T$, can not be finely reproduced. As a result, $f_D$ is formulated to optimize both laminar flame speed and macroscopic temperature profile by determining the optimal set of species Lewis numbers. The resulting virtual scheme is compared to detailed chemistry, using a *multicomponent* transport model including the Soret effect. To highlight the improvements brought by the optimized species Lewis numbers, the newly calculated laminar flame speed is plotted alongside the flame speed obtained after chemistry optimization (which assumes equidiffusive properties). As shown in Figure (4), the Lewis optimization provides an almost perfect match for the flame speed. Three equivalence ratios, $\phi_c$, are selected to assess the accuracy of the two-step mechanism in predicting the temperature profile. The results demonstrate that the Lewis optimization accurately captures both the steep temperature gradients at the flame front and the heat diffusion away from the reactive zone.

<figure><img src="image_6.png" /><figcaption>**Figure 4:** - Top: Laminar flame speed $s_L^0$ as a function of the equivalence ratio $\phi$ for simplified *Unity-Lewis* and complex transport models.</figcaption></figure>

Black solid (-) and dashed (--) lines: Detailed mechanism with *multicomponent* and *Unity-Lewis* transport models respectively. Red dots (●) and triangle (▲) symbols: Virtual mechanism with optimized species Lewis numbers and *Unity-Lewis* transport model.

- Bottom: Comparison of detailed (-) and virtual (--) temperature profiles $T(x)$ for lean, stoichiometric & rich combustion regimes at $\phi = 0.3$ (Left), $\phi = 1.1$ (Middle) & $\phi = 4.1$ (Right) respectively. 1D flames are calculated at atmospheric pressure and a temperature injection at 300 K

## Satellite scheme for pollutant prediction

### Satellite scheme structure

Complementing the primary H₂-Air virtual scheme designed to describe the flame structure, the satellite NO virtual scheme focuses on accurately capturing pollutant formation across its diverse timescales. Inspired of Maio et al. [12], the following 4-step mechanism is proposed
$$ \begin{aligned} \alpha_F F + \alpha_{Ox} Ox & \longrightarrow \alpha_1^1 V_1 + \alpha_2^1 V_2 + \alpha_3^1 V_3 & (\mathcal{R}_1^{NO}) \\ V_1 & \longrightarrow \alpha_{NO}^2 NO + \alpha_2^2 V_2 & (\mathcal{R}_2^{NO}) \\ V_3 & \longrightarrow \alpha_{NO}^3 NO & (\mathcal{R}_3^{NO}) \\ V_2 & \longleftrightarrow \alpha_{NO}^4 NO & (\mathcal{R}_4^{NO}) \end{aligned} \qquad (13) $$

Thermal and prompt pathways are captured by reactions ($\mathcal{R}_2^{NO}$) and ($\mathcal{R}_3^{NO}$) respectively, whilst ($\mathcal{R}_4^{NO}$) ensures the recovery of NO equilibrium mass fraction. The first reaction, designed to be consistent with the main virtual mechanism (1), represents fuel and oxidizer consumption. Consequently, ($\mathcal{R}_1^{NO}$) kinetic rates, reactants' stoichiometric coefficients and reaction orders are directly imposed from reaction ($\mathcal{R}_1$) [2, 12].

### Chemical rates optimization for NOx prediction

The remaining reaction rates ought to be optimized using the so-called Arrhenius formulation for irreversible reactions ($\mathcal{R}_2^{NO}$) & ($\mathcal{R}_3^{NO}$)

$$ \forall i \in [2, 3], \quad k_{\mathcal{R}_i^{NO}} = A_{\mathcal{R}_i^{NO}}(\phi_c) \exp\left(\frac{-E_{a, \mathcal{R}_i^{NO}}}{RT}\right) [V_k]^{n_{\mathcal{R}_i^{NO}}} \qquad (14) $$

and equilibrium reaction ($\mathcal{R}_4^{NO}$)

$$ k_{\mathcal{R}_4^{NO}} = A_{\mathcal{R}_4^{NO}}(\phi_c) \exp\left(\frac{-E_{a, \mathcal{R}_4^{NO}}}{RT}\right) \left([V_2] - \frac{[NO]}{K_c}\right) \qquad (15) $$

The equilibrium constant $K_c$ is obtained at chemical equilibrium following

$$ K_c = \frac{[NO]^{eq}}{[V_2]^{eq}} \qquad (16) $$

The equilibrium concentration of NO being determined from the detailed scheme, $K_c$ becomes solely dependent on the equilibrium concentration of $V_2$, which is known a priori. With the concentrations of F, Ox, and D provided by the main mechanism, and considering that $V_1$ and $V_3$ are completely consumed in reactions ($\mathcal{R}_2^{NO}$) and ($\mathcal{R}_3^{NO}$), the species balance at equilibrium gives

$$ [V_2]^{eq} = 1 - [F]^{eq} - [Ox]^{eq} - [D]^{eq} - [NO]^{eq} \qquad (17) $$

A one-way coupling is established; the main mechanism feeds the NO satellite's without any feedback. Since the flame thermochemical properties are predicted by the main mechanism, only the species conservation equations of reactions (13) are solved [2]. Therefore, thermochemical properties of species $V_i$ are not relevant and do not require any optimization. For each reaction ($\mathcal{R}_1^{NO} - \mathcal{R}_4^{NO}$) (13), stoichiometric coefficients are partially optimized and imposed to verify mass conservation principle.

This leads to the expression of the satellite optimization vector $\boldsymbol{x}_{NO}$ accordingly

$$ \boldsymbol{x}_{NO} = \underbrace{(W_{V_1}, W_{V_2}, W_{V_3})}_{\substack{\text{Virtual products} \\ \text{molar mass}}} \underbrace{(\alpha_1^1(\phi_c), \alpha_2^1(\phi_c), \alpha_{NO}^2(\phi_c))}_{\substack{\text{Tabulated} \\ \text{stoichiometric coefficients}}}, \left( A_{\mathcal{R}_2^{NO}}(\phi_c), E_{a, \mathcal{R}_2^{NO}}, n_{\mathcal{R}_2^{NO}}^{V_1} \right), \\ \left( A_{\mathcal{R}_3^{NO}}(\phi_c), E_{a, \mathcal{R}_3^{NO}}(\phi_c), n_{\mathcal{R}_3^{NO}}^{V_3} \right), \left( A_{\mathcal{R}_4^{NO}}(\phi_c), E_{a, \mathcal{R}_4^{NO}}(\phi_c) \right) \qquad (18) $$
Particularly remarkable in (18), the activation energy $E_{a,R_3^{NO}}(\phi_c)$ is optimized for each equivalence ratio. This strategy is proposed since this reaction accounts for NO thermal pathways, especially activated in high temperature hydrogen flames [1, 19]. The optimization seeks to replicate the NO mass fraction profile for every equivalence ratio [12]. Accordingly, a first proposed fitness would be calculated as

$$f_{\text{NO}}(\mathbf{x}_{\text{NO}}) = \sum_{i=1}^{N_c} \left[ \sqrt{\frac{\int_{-\infty}^{+\infty} (Y_{\text{NO}}|_v - Y_{\text{NO}}|_d)^2 dx}{\sqrt{\int_{-\infty}^{+\infty} Y_{\text{NO}}^2|_d dx}}} \right] \quad (19)$$

Nevertheless, due to $NO_x$ production multiple timescales, an alternative optimization strategy is employed. The cost function is defined in two distinct regions, each targeting a specific pathway. The first region, located at the flame front, captures the fast $NO_x$ formation timescales, where the objective is to recover the steep NO profile characteristic of the prompt pathway. The second region, situated downstream of the flame front, is dominated by the slow thermal pathway, where slow and equilibrium reactions prevail. To distinguish these two areas, the following criterion is used, inspired by the work of Maio et al. [12]

$$\eta^+ = \frac{\left| \frac{d^2 Y_{\text{NO}}}{dx^2} \right|}{\left| \frac{d^2 Y_{\text{NO}}}{dx^2} \right|_{+\infty}} \quad (20)$$

with a separating threshold $\eta^+ = 0.003$ identified in [12]. Let $x_p$ be the spatial location where $\eta^+ = 0.003$, the overall weighted fitness $f_{\text{NO}}$ finally becomes

$$f_{\text{NO}}(\mathbf{x}_{\text{NO}}) = \sum_{i=1}^{N_c} \left[ W \sqrt{\frac{\int_{-\infty}^{x_p} (Y_{\text{NO}}|_v - Y_{\text{NO}}|_d)^2 dx}{\int_{-\infty}^{x_p} Y_{\text{NO}}^2|_d dx}} + (1-W) \sqrt{\frac{\int_{x_p}^{+\infty} (Y_{\text{NO}}|_v - Y_{\text{NO}}|_d)^2 dx}{\int_{x_p}^{+\infty} Y_{\text{NO}}^2|_d dx}} \right] \quad (21)$$

with $W$ the weighting factor empirically chosen as $W = 0.7$. The 4-step satellite mechanism is evaluated against detailed chemistry employing *multicomponent* transport in typical 1D premixed flame simulations. To ensure the satellite mechanism accurately represents the fast NO formation pathways, a particular focus is placed on nitrogen oxides production within the flame front. The results, presented in Figure (5), demonstrate that the satellite mechanism effectively captures both prompt and thermal NO formation pathways, achieving excellent agreement with detailed chemistry. The mass fraction at the domain's boundary is accurately reproduced, confirming the suitability of the equilibrium reaction $R_4^{NO}$ within the satellite scheme (13).

## Conclusions & perspectives

The virtual chemistry framework has been extended to hydrogen-air combustion to account for differential diffusion effects and accurately capture the different timescales of nitrogen oxides formation. This extension was achieved by decomposing the optimization methodology into three distinct procedures: thermodynamics, chemistry, and transport. Validation of the optimized virtual H₂-Air scheme was performed using 1D premixed flames. Results demonstrate excellent agreement with detailed chemistry and complex transport reference flames, thereby validating the proposed approach. Specifically, the optimized Lewis number successfully reproduces the enhanced consumption speed and provides an accurate description of the flame front structure predicted by *multicomponent* detailed schemes.

Furthermore, a new virtual satellite scheme dedicated for NO, inspired by Maio et al. [12], has been coupled to the previously optimized hydrogen-air scheme. The mechanism achieves
<figure><img src="image_7.png" /><figcaption>**Figure 5:** - Top: Comparison of detailed (-) and virtual (--) NO mass fraction profiles $Y_{NO}(x)$ for lean, stoichiometric & rich combustion regimes at $\phi = 0.4$ (Left), $\phi = 1.0$ (Middle) & $\phi = 2.0$ (Right) respectively.</figcaption></figure>

- Bottom: Zoom, for each equivalence ratio $\phi$, in the flame front defined following the $\eta^+$ criterion.

near-perfect agreement with detailed chemistry, capturing both prompt and thermal NO pathways. The equilibrium reaction incorporated in the scheme ensures as expected to recover the pollutant concentration in the post-flame region.

Overall, the virtual H₂-air scheme effectively captures both flame dynamics and pollutant formation while dramatically reducing the computational complexity, requiring only 11 species and 6 reactions. Future work will focus on validating the proposed scheme in counterflow and partially premixed flame configurations, as well as in 2D thermodiffusive unstable flames, such as those investigated by Lapenna et al. [6].

## Acknowledgements

The authors acknowledge support from the MONTHY project funded by the French National Research Agency (ANR) under France 2030 program and reference ANR-PEHY-0014 and 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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