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# A consistency test for the estimation of the Zeldovich number

Omar Dounia<sup>a ID,*</sup>, Nabiha Chaumeix<sup>b</sup>

## ARTICLE INFO

### Keywords:

Flame chemistry

Zeldovich number

Flame theory

Hydrogen chemistry

## ABSTRACT

The global activation energy Ea of a flame is an important parameter characterizing the thermal sensitivity of the reaction rate. Its estimation is of crucial importance in understanding the influence of diluents on flame propagation (Dounia, 2024), the sensitivity of flames to stretch via the Markstein number (Hamadi et al., 2024) and the potential onset of instabilities on flame surfaces (Berger et al., 2022), to cite a few examples. Usually, it is estimated from the sensitivity of the flame speed $s_L$ to small perturbations in burnt gases temperature $T_b$ using $Ea \approx -2R\partial(\ln(\rho_u s_L)) / \partial(1/T_b)$, where $\rho_u$ is the density of the unburnt gases. It is however often recognized that this estimate can strongly depend on how the perturbation in $T_b$ is obtained, either by perturbing the unburnt gases composition (Sun et al., 1999) (dilution) or temperature (Kumar and Sung, 2007), which constitutes a rather perturbing result. In this paper, we identify the origin of the discrepancies between these two methods and further highlight the large discrepancies between expected values of Ea and those predicted with the formula above, especially at moderate values of the Zeldovich number. A new formula is proposed to solve these two issues. Finally, we focus on the behavior of estimates of the Zeldovich number at the vicinity of kinetic limits for hydrogen flames and question the relevance of estimating the Zeldovich based on the sensitivity of integral parameters like the flame speed.

## 1. Introduction

One cornerstone of flame theory is the single-step formalism for the combustion process with a rate modeled by an Arrhenius law: $\dot{k}(T) \equiv AT^a \exp(-Ta/T)$. The activation temperature Ta plays a significant role in combustion theory as it dictates, to a large extent, the thermal sensitivity of flames. For most flames, Ta is large and the reaction zone is thin with respect to the remaining internal flame structure. This scale separation property is the reason for the success of large activation temperature asymptotic, under which Ta → ∞, in unveiling flame fundamental properties. The reader is referred to reference theoretical combustion textbooks [1,2] for an extensive body of theoretical works where Ta appears as a controlling parameter.

To understand different aspects of flame dynamics on the basis of these theoretical derivations, the question arises as to how to associate a global activation temperature Ta to a given flame. The usual way to do it is to take advantage of the theoretical scaling law for the mass flux per unit area $\dot{m} \equiv \rho_u s_L$ [3,4]:

$$ (\dot{m})^2 \propto \exp\left(-\frac{Ta}{T_b}\right) \Rightarrow Ta \approx -2 \frac{\partial \ln(\dot{m})}{\partial 1/T_b} \quad (1) $$

Ta is therefore computed by perturbing the burnt gas temperature of a flame. This is done either by: (dilution method [5]) perturbing the unburnt gas composition using a small amount of inert gas, usually $N_2$, where $Y_{N_2} = \pm 0.3\%$; (temperature method [6]) perturbing the unburnt gas temperature by $\pm 0.3\%$.

However, the estimation of the activation temperature Ta using Eq. (1) and the two methods described above show two important problems that are often overlooked or underreported in the literature:

**(P1) strong sensitivity of Ta to the perturbation method:** As pointed out in [7], the two methods described above can lead to values of Ta that are away from each other by a factor up to 2, which is a rather confusing result, one that even questions the relevance of the method. This also led to debates in the literature on which value most accurately describes the thermal sensitivity of flames [6–8].

**(P2) unrealistically high values of Ta close to cross-over:** Reported values of Ta for $H_2$/air flames at increasingly lean or rich conditions [3,9] show a “blow-up” of the values as the kinetic flammability limits are approached, defined by a burnt gas temperature $T_b$ equating the cross-over temperature $T_c$ at which the chain branching

<footnote><sup>a</sup> CERFACS, 42 Avenue Gaspard Coriolis, Cedex 1, 31057 Toulouse, France</footnote>

<footnote><sup>b</sup> CNRS-INSIS, I.C.A.R.E., 1C, Avenue de la recherche scientifique, cedex 2, 45071, Orleans, France</footnote>

<footnote>* Corresponding author.
E-mail address: dounia@cerfacs.fr (O. Dounia).</footnote>
production balances the chain consumption rates of the radical H [10]. This problem has already been pointed out by [11,12] without much attention.

These two problems raise serious concerns regarding the estimation of several key parameters controlling different aspects of flame dynamics: flame thermal mitigation [13], stretch effects on flames [8], onset of instabilities on flame surfaces [9], classification of regimes of flame acceleration [14], deflagration to detonation transition [15–17], to cite a few examples. This paper proposes to delve into problems (P1) and (P2) to clarify their origin and propose ways to circumvent them. Only hydrogen flames are considered in this paper for two reasons: (1) their strong stretch sensitivity and tendency to initiate markedly unstable fronts, which makes the estimation of the activation temperature Ta a crucial element for their understanding; (2) the ease of some theoretical derivations that will be shown in this paper that build upon recently derived analytically reduced chemistry for hydrogen combustion by Sanchez, Williams and co-workers [10]. Sections 2 and 3 delve into problem (P1) and most of the arguments provided therein are valid for arbitrary flames. Section 4, on the other hand, focuses on hydrogen combustion and problem (P2).

## 2. A consistency test for the estimation of Ta

We first propose a simple test to assess the accuracy of Eq. (1). Take a simple irreversible single-step chemistry for hydrogen combustion:

$$2 \mathrm{H}_2 + \mathrm{O}_2 \xrightarrow{\dot{\omega}} 2 \mathrm{H}_2\mathrm{O} \qquad (2)$$

$$\text{where } \dot{\omega} = C_{\mathrm{H}_2}^{n_{\mathrm{H}_2}} C_{\mathrm{O}_2}^{n_{\mathrm{O}_2}} A T^{\alpha} \exp \left( -\frac{\mathrm{Ta}^0}{T} \right) \qquad (3)$$

The idea here is to solve for a premixed 1D flame with reaction rate (3) and imposed value of the activation temperature $Ta^0$. The accuracy of Eq. (1) can then be assessed by evaluating the error between the numerical value Ta, obtained with Eq. (1), and $Ta^0$. We stress that an accurate estimation of $Ta^0$ in this simple test is a necessary condition that any estimate of the activation temperature must satisfy before being applied to characterize the global thermal sensitivity of detailed chemistry flames. Emphasis is put here on the non-dimensional counterpart of Ta, namely the Zeldovich number $\beta \equiv \mathrm{Ta}(T_b - T_u)/T_b^2$.

To fix the other parameters and ensure consistency with properties of hydrogen combustion, the global reaction order $n_r$ is computed from the pressure dependence of the mass flux obtained with the UCSD mechanism [18] ($n_r/2 = \partial \ln(\dot{m})/\partial \ln(P)$) and imposed s.t. $n_{H_2} = n_{O_2} = n_r/2$. The pre-exponential factor A is imposed such that the flame speed of the detailed chemistry is recovered for all values of $Ta^0$. Therefore, both values of $n_r$ and A depend on the equivalence ratio $\phi$. The values of $n_r$ are provided in the supplementary material for completeness. In Eq. (3), $\alpha=0$ here for simplicity (note however that all the derivations in later sections of this manuscript are valid for arbitrary values of $\alpha$). Finally, non-unity Lewis numbers are accounted for and are evaluated from a detailed-chemistry flame profile at the burnt gases state: $Le_{H_2} = 0.34$, $Le_{O_2} = 1.2$, $Le_{H_2O} = 0.88$, $Le_{N_2} = 1.37$. $T_u$ is fixed here at 300 K.

Fig. 1 displays the relative error between the imposed value of Zeldovich number $\beta^0$ and the value $\beta$ estimated based on Eq. (1) with both the dilution [5] and temperature [6] methods. First, Fig. 1 confirms that the temperature method systematically leads to higher over-predictions of the Zeldovich number compared to the dilution method (with values almost two times the latter). Regarding the dilution method, while it exhibits satisfactory accuracy (error below the 10% threshold shown in gray) for high values of $\beta^0$, it also leads to serious over-predictions for intermediate to low values of $\beta^0$, namely $\beta^0 < 10$, which correspond to a range of $\beta^0$ of interest for [3,9]. Finally, a considerable scatter of the results is observed for each method and seems to positively correlate with the burnt gas temperature $T_b$. These observations question the relevance of estimates for $\beta$ based on Eq. (1). The origin of these discrepancies is clarified in the following section.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Relative error associated to the estimation of the Zeldovich number based on Eq. (1) for different imposed values $\beta^0$. Plus signs and circles indicate errors associated to the temperature [6] and dilution [5] methods respectively.</figcaption></figure>

## 3. Clarifying the origin of the discrepancies observed with the estimate (1) of Ta

### 3.1. The flame speed under the reactive-diffusive approximation

To identify the origin of the discrepancies highlighted in Fig. 1, it is important to recall the main parameters controlling the flame speed and how they may be impacted by perturbations in burnt gas temperature in the formula (1). Only the main steps are recalled here, all the details can be found in [1,19]. Under high activation temperature, the thin reaction zone can be approximated as a reactive-diffusive layer (neglecting convection) confined in the temperature range [$T_c, T_b$] ($T_c$ is a-priori unknown) and is governed by:

$$\frac{\rho D_T}{v_k W_k Le_k} \frac{d^2 Y_k}{dx^2} = \dot{\omega} \quad \text{for } k = \{\mathrm{H}_2, \mathrm{O}_2\} \qquad (4)$$

$$\rho c_p D_T \frac{d^2 T}{dx^2} = -q\dot{\omega} \qquad (5)$$

where $v_k, W_k, Le_k$ and $Y_k$ are the stoichiometric coefficient, the molecular weight, the Lewis number and the mass fraction of species $k$. $\rho, c_p$ and $q \equiv -v_{H_2O} A H_f^0 (H_2O)$ are the density, the heat capacity and the heat release per unit of mass respectively. $c_p$ and $\rho D_T$ are assumed constant for simplicity and evaluated at the burnt gases. Multiplying Eq. (5) by $dT/dx$ and integrating it from $T_c$ to $T_b$ yields an expression for the heat flux from the reaction zone, which can be matched with the solution in the convective-diffusive preheat zone. This yields the following expression for the mass flux provided that $T_c/T_b - 1 \ll 1$ (consistent with the high activation temperature assumption):

$$\dot{m}^2 \approx \frac{2(v_{\mathrm{H}_2} W_{\mathrm{H}_2})^2}{(Y_{\mathrm{H}_2,u} - Y_{\mathrm{H}_2,b})^2} \left(\frac{\rho D_T}{q/c_p}\right) \int_{T_c}^{T_b} \dot{\omega} dT \qquad (6)$$

In most flames, $\alpha - n_r = O(1)$ and the thermal sensitivity of the chemical source term in the region [$T_c, T_b$] is essentially governed by the exponential term, the term $\rho n_r^\alpha T^\alpha$ being approximated by their values at the burnt gases. This approximation is valid to the lowest order in the limit of high activation temperature, and the integral in the right hand side of Eq. (6) can then be simplified to [1,2]:

$$\int_{T_c}^{T_b} \dot{\omega} dT \approx \frac{A \rho_b^{n_r} T_b^{\alpha} \exp(-Ta/T_b)}{W_{\mathrm{H}_2}^{n_{\mathrm{H}_2}} W_{\mathrm{O}_2}^{n_{\mathrm{O}_2}}} \Omega_T \qquad (7)$$

where:

$$\Omega_T = \int_{T_c}^{T_b} Y_{\mathrm{H}_2}^{n_{\mathrm{H}_2}} Y_{\mathrm{O}_2}^{n_{\mathrm{O}_2}} \exp \left[ \frac{\mathrm{Ta}(T - T_b)}{T_b^2} \right] dT \qquad (8)$$

The evaluation of $\Omega_T$ necessitates relating $Y_{\mathrm{H}_2}$ and $Y_{\mathrm{O}_2}$ to temperature. This is done by linearly combining Eqs. (4) and (5) to eliminate the
chemical source term and integrating it twice from $T_c$ to $T_b$ yielding:

$$ \frac{Y_k - Y_{k,b}}{\nu_k W_k} = \frac{Le_k}{q/c_p} (T_b - T), \quad k = \{\text{H}_2, \text{O}_2\} \qquad (9) $$

Substituting Eq. (9) into Eq. (8) yields:

$$ \Omega_T = \frac{(q/c_p) L_{\text{H}_2}^{n_{\text{H}_2}} L_{\text{O}_2}^{n_{\text{O}_2}}}{[\beta/(Y_{\text{H}_2,u} - Y_{\text{H}_2,b})]^{n_r+1}} G(\xi_c) \qquad (10) $$

where $L_k \equiv \nu_k W_k Le_k$, and $\xi_c \equiv \text{Ta}(T_b - T_c)/T_b^2$ and $G$ is the non-dimensional counterpart of $\Omega_T$:

$$ G(\xi_c) = \int_0^{\xi_c} (\xi + a_{\text{H}_2})^{n_{\text{H}_2}} (\xi + a_{\text{O}_2})^{n_{\text{O}_2}} e^{-\xi} d\xi \qquad (11) $$

with $a_k \equiv (\beta/Le_k)(Y_{k,b}/(Y_{k,u} - Y_{k,b}))$. Under either lean or rich conditions, we have $a_{H_2} = 0$ or $a_{O_2} = 0$ respectively. Moreover, to the leading order in the limit $\beta \to \infty$, we have $G(\xi_c) \to G(\infty)$. Under lean conditions, the following expression for the mass flux is obtained:

$$ \dot{m}^2 \approx C_{\dot{m}} \frac{(\rho D_T) Y_{\text{H}_2,u}^{n_r-1} G(\infty)}{\beta^{n_r+1} \rho_b^{-n_r} T_b^{-\alpha}} \exp(-\frac{T_a}{T_b}) \qquad (12) $$

$C_{\dot{m}}$ is a constant which depends on the Lewis numbers $Le_{H_2}$ and $Le_{O_2}$, its expression is not of importance for this study. Eq. (12) for rich mixtures is obtained by simply replacing $Y_{H_2,u}$ by $Y_{H_2,u} - Y_{H_2,b}$. We recover, in Eq. (12), the scaling at the basis of Eq. (1), namely $\dot{m}^2 \propto \exp(-Ta/T_b)$. However, Eq. (12) shows that many terms are actually impacted by a perturbation in burnt gases temperature $T_b$, the contribution of which is omitted in the estimate Eq. (1), which may be the origin of the discrepancies observed in Fig. 1.

### 3.2. Recovering the missing terms in Eq. (1)

Eq. (12) allows to derive a new estimate for $Ta$:

$$ Ta \approx -2 \frac{\partial \ln(\dot{m})}{\partial 1/T_b} - \mathcal{E}(T_b, n_r) \qquad (13) $$

with:

$$ \mathcal{E} = T_b \left( 1 + \alpha - (n_r + 1) \frac{T_u}{T_b - T_u} \right) + T_b^2 \left[ \frac{\partial \rho D_T / \partial T_b}{\rho D_T} - (n_r + 1) \frac{(\partial T_b / \partial T_u)^{-1}}{T_b - T_u} + (n_r - 1) \frac{(\partial T_b / \partial Y_{H_2,u})^{-1}}{Y_{H_2,u}} \right] \qquad (14) $$

The details of the derivation are provided in the supplementary material. Note that the term $G(\infty)$ in Eq. (12) is treated as a constant and is neglected in Eq. (13) (approximations for $G(\infty)$ can be found in [1,19]). $\mathcal{E}$ represents the error introduced when computing $Ta$ using Eq. (1). Its expression provides an explanation for the discrepancies observed in Fig. 1: (i) the error $\mathcal{E}$ increases with increasing $T_b$, thereby explaining the large scatter observed for each method in Fig. 1 and its positive correlation with $T_b$, the term in $T_b^2$ accounting for the blow up of the errors for low values of $Ta$; (ii) the two last terms in Eq. (13) are specific to the method used to introduce the perturbation in $T_b$, namely the dilution and temperature method respectively, and have different scaling with the reaction order $(n_r - 1$ for dilution and $n_r + 1$ for temperature), thereby explaining the larger errors observed with the latter.

Fig. 2 displays the relative error between the imposed value of Zeldovich number $\beta^0$ and the value $\beta$ estimated based on the new expression (13). Fig. 2 shows that using Eq. (13) as an estimate for $Ta$ leads to drastically increased accuracy for a larger range of values of $\beta^0$, the dilution method yielding errors below the 10% threshold down to low values of $\beta^0$. Eq. (13) also solves problem (P1) leading to values that are independent of the method used for values of $\beta^0 \ge 5$. Discrepancies with the temperature method are still observed for lower

<figure><img src="image_4.png" /><figcaption>Fig. 2. Relative error associated to the estimation of the Zeldovich number based on Eq. (13) (neglecting the term $G(\infty)$) for different imposed values $\beta^0$. Plus signs and circles indicate errors associated to the temperature [6] and dilution [5] methods respectively.</figcaption></figure>

values of $\beta^0$ and are due to the treatment of integral term $G(\infty)$ as a constant despite its greater sensitivity to temperature perturbations ($T_u$).

It is therefore recommended to avoid estimating the Zeldovich number on the basis of Eq. (1) and replacing it with Eq. (13) using the dilution method (neglecting the term in $G(\infty)$ for simplicity). Note that, in the diffusive term, it is understood that $\rho D_T$ is evaluated at the burnt gases as already mentioned in Section 3.1.

With the conclusions above in mind, it is worth comparing the predictions of Eqs. (1) and (13) on hydrogen flames computed with detailed chemistry. The UCSD is used here. Multicomponent transport and Sorret effect are accounted for. Fig. 3 confirms that, away from stoichiometry, the discrepancies obtained between the dilution and the temperature methods are completely depleted with Eq. (13) highlighting the importance of including the additional terms in Eq. (13). Around stoichiometry, inconsistencies between the two methods are still visible and are due to the omission of the term in $G(\infty)$ in Eq. (13) which introduces prediction errors which scale with $T_b^2$ and are therefore important in the region of equivalence ratios close to unity. The omission of this same term is also responsible for the non-monotonic behavior systematically observed close to stoichiometry (local peak at $\phi = 1$). This behavior was initially incorrectly attributed in [20] to the invalidity of the unitary reactant approximation close to $\phi = 1$ (assuming a largely deficient reactant), since the approximation is not used here. Careful consideration of this reaction-integral term $G$ is therefore necessary to remove these remaining inconsistencies. Nevertheless, we have already shown above that the dilution method is less impacted by this term and the estimates of $\beta$ using Eq. (13) with the dilution method should be considered as the reference for future studies. Note that Eq. (13) yields values of $\beta$ which are systematically lower than those provided by Eq. (1), which could have strong implications for many studies in which $\beta$ plays a central role.

## 4. Analytical expression of the Zeldovich number of H₂/air flames

### 4.1. The QSS-based single-step mechanism for H₂/air flames [12,21,22]

In this part of the paper, we seek an analytical expression for the Zeldovich number of H₂/air flames observed in Fig. 3, which is able to account for the “blow-up” of the Zeldovich number at the vicinity of the kinetic flammability limits. In this section, we will focus on the lean-limit but the same approach can be extended to the rich-limit as well. To explore analytically the behavior of $\beta$ as the lean limit is approached, we need a simplified mechanism which retains as much information as possible on the thermal sensitivity of hydrogen
<figure><img src="image_5.png" /><figcaption>Fig. 3. Zeldovich number of hydrogen/air flames using Eq. (1) (dotted lines) and Eq. (13) (solid lines). Red and black lines correspond to the temperature and dilution methods. The UCSD mechanism is used here. The term in G(∞) is neglected for convenience in Eq. (13).</figcaption></figure>

combustion predicted by detailed chemistry. The solution is provided by the works of Galisteo et al. [12,21,22]. They first show that only a reduced set of nine elementary reactions are actually controlling hydrogen combustion, 4 of which are recalled here for their importance later in the paper: 3 shuffle reactions H + O₂ ⇌ OH + O, H₂ + O ⇌ OH + H, H₂ + OH ⇌ H₂O + H and 1 recombination reaction H + O₂ + M ⇌ HO₂ + M. The expression for the rate of each of these reactions can be found in [12,21]. They then derived a one-step mechanism by assuming Quasi-steady state (QSS) for all intermediates. Its has the same from as Eq. (2) but with a chemical source term which reads: ω₄f = CₕC₉C₄k₄f, kₕ and Cₘₕ indicate the reaction rate and effective third-body concentration of reaction r. Cₘⱼ is evaluated taking into account Chaperon efficiencies. The concentration of the radical H is given by its QSS approximation:

$$C_H = \frac{1}{GH} \frac{k_{2f} k_{3f} C_{H_2}^2}{k_{1b} k_{4f} C_{M_4} C_{O_2}} \left( \frac{k_{1f}}{\alpha k_{4f} C_{M_4}} - 1 \right) \quad (15)$$

where G and H are parameters which depend on the rate of the reactions of the nine-step mechanism other than reactions 1-4 above. Their full expression can be found in [22] and will not be important in the following. Finally, the term between parenthesis in Eq. (15) allows to define a crossover temperature T_c:

$$\frac{k_{1f}(T_c)}{\alpha k_{4f}(T_c)C_{M_4}} = 1 \quad (16)$$

Near the lean limit φᵢ, defined by T_b = T_c, a simplified expression is provided for ω₄f:

$$\dot{\omega}_{4f} \underset{\phi \to \phi_l}{\longrightarrow} C_{H_2}^2 \dot{K}(T) \quad (17)$$

$$\dot{K}(T) = \left( \frac{k_{1f}}{k_{4f} C_{M_4}} - 1 \right) \frac{k_{2f} k_{3f} / k_{1b}}{1 + \frac{k_{3b} C_{H_2O}}{k_{4f} C_{M_4} C_{O_2}}} \quad (18)$$

Fig. 4 shows that evaluating the flame speed of lean H₂/air flames using Eq. (6) with the chemical source term Eq. (17) leads to predictions in remarkable agreement with that of the UCSD mechanism. Note that the better results observed here compared to [12] are due to careful considerations for the evaluation of the concentration of O₂ and H₂O replacing their values at equilibrium (as done in [12]) by their evolution in the diffusive-reaction layer, i.e. Yₖ - Yₖ,ₙ = vₖWₖLeₖ(Tₖ - T)/(q/cₚ), similarly to Eq. (9).

The rate coefficient K associated to the chemical source term ω₄f can be further simplified as already proposed in [21]. Under the

<figure><img src="image_6.png" /><figcaption>Fig. 4. Unstrained laminar flame speed for H₂/air flames obtained with: (black line) Cantera [23] and the UCSD mechanism with multicomponent transport and Sorret effects accounted for; (blue line) Cantera and the one-step mechanism ω₄f with (15) and mixture-averaged transport; (red line) Eq. (6) with ω₄f and (15); (dashed red line) Eq. (6) with the lean-limit one-step mechanism (17); (green solid and dashed lines) analytical expression (23) with G' given by (25) and (31) respectively.</figcaption></figure>

condition k₃bCₗH₂O ≫ k₄fCₘ₄C₉O₂, valid only sufficiently away from T_c, K̇ reduces to:

$$\dot{K} \approx \frac{k_{2f} k_{3f} k_{4f} C_{O_2}}{k_{1b} k_{3b} C_{H_2O}} \left( \frac{k_{1f}}{k_{4f}} - 1 \right) \quad (19)$$

Injection Eq. (16) and neglecting the term C₉O₂/CₗH₂O and replacing its value by 1/4 (this is done to simplify the analytical derivations while ensuring acceptable accuracy for sₗ predictions in Fig. 4, the exact value is of no consequence for the formulas for the Zeldovich number provided in Section 4.2) yields:

$$\dot{K} \approx A^* T^{\alpha^*} \exp \left( -\frac{T^{\alpha^*}}{T} \right) (1 - E_c) \quad (20)$$

$$E_c = \exp \left( - \frac{T_{ac}}{T_c} \left[ \frac{T - T_c}{T} \right] \right) \quad (21)$$

with α* ≈ 2.3, Ta* ≈ 4141, Ta_c ≈ 8587 and A* ≈ 0.8 [S.I.]. The main difference between the rate coefficient (20) and the classical generalized Arrhenius law in (3) resides in the presence of the crossover term 1 - E_c which makes the reaction rate vanish at the vicinity of T_c. Sufficiently away from the lean limit, a generalized Arrhenius law is retrieved:

$$\dot{K} \approx A^* T^{\alpha^*} \exp \left( -\frac{T^{\alpha^*}}{T} \right) \quad (22)$$

It is shown in Fig. 5 that the rate coefficient K̇ can be reasonably approximated by Eq. (20), except at the vicinity of the burnt gases for mixtures close to stoichiometry. Fig. 5 confirms that the generalized Arrhenius law Eq. (22) is a crude approximation of K̇, especially close to T_c as already pointed out by [11,21].

## 4.2. An explicit formula for the Zeldovich number of lean H₂/air flames

The steps to get an expression for ṁ based on the chemical source term (18) with the rate (20) are exactly the same as in Section 3.1 and yield:

$$\dot{m}^2 \approx C_m^* \frac{(\rho D_T) Y_{H_2, u} G(\xi_c)}{(\beta^*)^3 \rho_b^{-2} T_b^{-\alpha^*}} \exp(-\frac{T_{ac}^*}{T_b}) \quad (23)$$

with G(ξ_c) given by:

$$G(\xi_c) = \int_0^{\xi_c} \xi^2 e^{-\xi} (1 - e^{-\gamma\xi}) d\xi \quad (24)$$
<figure><img src="image_7.png" /><figcaption>Fig. 5. The rate coefficient $\dot{K}$ according to Eq. (18) (colored dashed lines), Eq. (20) (colored solid lines) for two values of equivalence ratio $\phi$. Black dashed line indicates the Arrhenius approximation Eq. (22).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Comparison between numerical and theoretical predictions of the Zeldovich number of hydrogen/air flames. (black line) Eq. (13) using the UCSD mechanism and the dilution method. (Solid green line) analytical prediction using Eq. (28) with $G'$ and $G$ given by Eq. (25) and (29) respectively. (Dashed green line) analytical prediction using Eq. (28) with $G'$ and $G$ given by Eq. (31) and (32). black dotted line indicates the lean kinetic limit $\phi_l$.</figcaption></figure>

and $\gamma \equiv (T_b/T_c)^2(\text{Ta}_c/\text{Ta}^*)$ controls the crossover term. $G(\xi_c)$ can be computed analytically:

$$G(\xi_c) = 2 - \left( \xi_c^2 + 2\xi_c + 2 - 2\Gamma(\xi_c) \right) e^{-\xi_c} \quad (25)$$

with $\Gamma$ given by:

$$\Gamma(\xi_c) = \frac{e^{-(\gamma-1)\xi_c} - \frac{\xi_c^2}{2}(\gamma-1)^2 + \xi_c(\gamma-1) - 1}{(\gamma-1)^3} \quad (26)$$

Fig. 4 shows that evaluating the flame speed using Eq. (23) with $G$ given by (25) provides a reasonable prediction of the flame speed for mixtures up to stoichiometry.

Using Eq. (23) to evaluate the term $-2d \ln(\dot{m})/d(1/T_b)$ in Eq. (13), it is also possible to compute analytically the estimate of the effective activation temperature, the error term $\mathcal{E}$ vanishing:

$$\text{Ta} = \text{Ta}^* \left[ 1 + \left( \frac{2T_c - T_b}{T_b} \right) \frac{dG(\xi_c)/d\xi_c}{G(\xi_c)} \right] \quad (27)$$

$$= \text{Ta}^* \left[ 1 + \left( \frac{2T_c - T_b}{T_b} \right) \frac{G'(\xi_c)}{G(\xi_c)} \right] \quad (28)$$

where $G'(\xi_c)$ reads:

$$G'(\xi_c) = -2\gamma\Gamma(\xi_c) \quad (29)$$

It is important to note that the presence of the term $G'/G$ is only due to the strong assumption made in Eq. (13) that the integral term $G(\xi_c)$ is negligible and not impacted by perturbations in $T_b$. Therefore, Eq. (28) highlights the impact of neglecting $G(\xi_c)$ on the accuracy of the estimate $\text{Ta}$ with respect to the expected value $\text{Ta}^*$.

Fig. 6 compares the predictions of Eq. (28) to the numerical prediction obtained for the UCSD mechanism and Eq. (13) with the dilution method. The evolution of the Zeldovich number with equivalence ratio for lean H₂/air flames is reasonably reproduced by Eq. (28). More importantly, Eq. (28) shows that the Zeldovich number computed with the thermal sensitivity of $\dot{m}$ becomes singular as the lean limit is approached ($\phi \to \phi_l \Leftrightarrow \xi_c \to 0$):

$$\text{Ta} \underset{\phi \to \phi_l}{\longrightarrow} \text{Ta}^* \left[ 1 + O\left( \frac{1}{\xi_c} \right) \right] \quad (30)$$

thereby explaining the blowup of the Zeldovich number close to $\phi_l$.

The same result can be obtained even when the rate coefficient does not exhibit a crossover term but the reaction rate is put to zero below $T_c$. This is the case when $\dot{K}$ is given by the generalized Arrhenius law Eq. (22) and is used in Eq. (28). This case is recovered in Eq. (24) as the limit $\gamma \to \infty$, where the cross over term $1 - \exp(-\gamma\xi)$ reduces to unity everywhere except at the boundary $\xi = \xi_c$. In this limit, $G$ and $G'$ reduce to:

$$G(\xi_c) \underset{\gamma \to \infty}{\longrightarrow} 2 - (\xi_c^2 + 2\xi_c + 2)e^{-\xi_c} \quad (31)$$

$$G'(\xi_c) \underset{\gamma \to \infty}{\longrightarrow} \xi_c^2 e^{-\xi_c} \quad (32)$$

yielding the same behavior Eq. (30) close to the singularity at $\phi \to \phi_l$.

Fig. 6 shows that, for $\phi > 0.7$, Eq. (28) gives the same value whether finite or infinite $\gamma$ is used, since large values $\gamma$ are reached there ($\gamma \propto (T_b/T_c)^2$). This implies that this range of equivalence ratios is outside the region of influence of the crossover term, which explains the plateau observed for the Zeldovich number with UCSD in this region. What is more important is that Eq. (28) also predicts a singularity close to crossover even when the reaction rate is of Arrhenius type everywhere in the flame profile above crossover, i.e. for $\gamma \to \infty$ representing the rate (22). This suggests that the high values of $\beta$ generally associated to very lean H₂/air flames is only a consequence of the omission of the integral parameter $G'(\xi_c)$ and its vanishing behavior close to the kinetic limits $\xi_c \to 0$. To the authors, this questions the meaning of the Zeldovich number computed based on methods that rely on $\dot{m}$, when the vanishing integral rate $G$ is not taken into account. Of course, accounting for $G$ is not straightforward in practice since it relies on an a-priori knowledge of the single-step form of the reaction rate close to the kinetic limits. This potential inconsistency advocates for estimates of the Zeldovich number based on a local analysis of the thermal sensitivity of the flame profile, in a manner similar to [24].

It is important to stress that the blowup of the estimate $\text{Ta}$ close to the kinetic limits does indeed indicate enhanced thermal sensitivity of these flames as already pointed out by [21], but can hardly be interpreted as an exploding Zeldovich number, at least in the sense that it is not equivalent by any means to the activation energy in the exponential term $\exp(-\text{Ta}/T)$ of the reaction rate, hence cannot be used as a value for $\beta$ to inject in asymptotic results of combustion theory [1].

## 5. Conclusion

The paper has identified serious inconsistencies in the classical estimation of the Zeldovich number for flames, namely using Eq. (1). A new formula is proposed, which accounts for previously missing terms. It also explicitly identifies the origin of the strong dependency of the activation energy to the perturbation method used to evaluate it. The second part of the paper questions the meaning of the Zeldovich number of H₂/air flames at the vicinity of the kinetic limits, showing that the blow up of the estimate of $\beta$ is a consequence of the omission of a vanishing reaction integral term. An explicit formula for this estimate is also derived explaining the origin of the singularity. Future efforts will focus on unifying results from local and integral estimates for $\beta$ proposed in the literature (see [24] for example of an estimate of $\beta$ based on an inner layer temperature).
## Novelty and significance statement

This research is a fundamental contribution to the estimation of the global Zeldovich number $\beta$ for flames, a parameter of crucial importance in analyzing and understanding flame dynamics. First, discrepancies associated to the classical estimate for $\beta$, recognized but not explained in the literature, are formally shown to originate from the omission of the thermal sensitivity of additional terms in the mass flux $\dot{m}$ formula. Second, a new formula, accounting for these missing terms and solving these discrepancies, is proposed and shows to be more accurate over a wider range of $\beta$ through a consistency test specifically derived to assess the validity of estimates for $\beta$. Finally, the paper demonstrates analytically that any estimate of $\beta$ based on $\dot{m}$ is singular close to the kinetic limits if the vanishing reaction rate at this limit is not properly accounted for, which questions the validity of the high-values of $\beta$ computed in the literature near these limits and their use to predict flame dynamics on the basis of classical asymptotic results.

## CRediT authorship contribution statement

**Omar Dounia:** Writing – original draft, Conceptualization, Methodology, Investigation.
**Nabiha Chaumeix:** Conceptualization, Methodology, Investigation.

## 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

OD would like to thank Jonathan Wirtz for initially having pointed out a potential problem with estimates of Ta obtained with Eq. (1).

## Appendix A. Supplementary data

For the sake of completeness, many details of the derivations presented in this manuscript have been included in the Supplementary Material.

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.proci.2025.105879.

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