Numerical Investigation on the Head‑on Quenching (HoQ) of Laminar Premixed Lean to Stoichiometric Ammonia–Hydrogen‑Air Flames

The Head-on Quenching (HoQ) of laminar premixed ammonia–hydrogen-air flames under lean to stoichiometric condition is numerical investigated. Detailed chemistry including 34 reactive species and detailed multi-component transport model including thermal diffusion (Soret effect) are applied. The quenching distance is considered as a representative quantity for the HoQ process, and the influence of different system parameters on it has been investigated. These parameters involve fuel/air equivalence ratios, hydrogen content in gas mixture and pressure. It was found that an increase of quenching distance can be caused by a lower hydrogen addition and a leaner mixture condition. Furthermore, it was found that, regardless of the gas mixture, the quenching distance decreases monotonically with increasing pressure, obeying a power function with the exponent − 0.7. Moreover, numerical results show a relation between the quenching Peclet number and the dimension-less wall heat flux normalized by the flame power. Additionally, sensitivities of quenching distances with respect to the transport model, considering the heat loss in the wall and the chemical kinetics are studied.


Introduction
In most engineering applications such as gas turbines and internal combustion engines, flame quenching due to flame-wall interaction (FWI) is an important topic, since it involves numerous physico-chemical processes such as formation of pollutants or the flame stability (Potter Jr 1960;Daniel 1957;Ferguson and Keck 1977;Bruneaux et al. 1996;Lai et al. 2018;Salimath et al. 2020).Due to its importance, the flame-wall interaction has been intensively studied from numerical or experimental aspects for different configurations of FWI (Poinsot and Veynante 2005), namely the (i) Head-on Quenching (HoQ), where the flame propagates towards the wall and the flame front is parallel to the wall; (ii) Side-Wall Quenching (SWQ), where the flame front is perpendicular to the wall; and (iii) tube quenching, where the flame propagates inside the tubes and can quench in case of tubes with sufficient small diameter.In the last decades, the FWI investigation has been performed for various fuels such as hydrogen (Gruber et al. 2010;Mari et al. 2016;Yenerdag et al. 2017), hydrocarbon (Häber and Suntz 2018;Andrae et al. 2002;Popp et al. 1996;Baigmohammadi et al. 2020;Kosaka et al. 2020) and Biofuels (Kosaka et al. 2020;Kaddar et al. 2022;Wan et al. 2019).
Ammonia (NH 3 ) has been considered as next generation fuel because it can be produced by using renewable energy sources such as solar and wind energy (Valera-Medina et al. 2018, 2021), with which the goal of net zero-carbon emission can be achieved.While the combustion processes of ammonia-based fuels such as auto-ignition (Li et al. 2019;He et al. 2019;Dai et al. 2020) and flame propagation (Mei et al. 2019;da Rocha et al. 2019;Jing et al. 2021;Chi et al. 2022) have been intensively studied in recent years, limited literature involving its interaction with solid wall, to author's knowledge, can be found.In Okafor et al. (2021) the influence of wall heat loss on the premixed ammonia-air swirling flames interacting with the combustor wall has been experimental studied by employing Planar Laser-Induced Fluorescence (PLIF) imaging of OH radicals and Fourier Transform Infrared (FTIR) spectrometry, and it is found that large heat loss in the wall results in the the flame quenching and large amount of unburnt NH 3 emission.In Wei et al. (2021) the stabilization characteristics of NH 3 -air premixed flame near blow-off limit of NH3/air flame numerical studied using the Large Eddy Simulation (LES), and it is found that the wall heat loss showed greater influence on NH 3 -air flame stabilization.In Yu et al. (2023) the steady laminar stagnation flow NH 3 -H 2 -air premixed flame was considered and the thermal stresses in the plane wall induced by the combustion processes have been investigated, and it is found that the solid wall would probably undergo plastical deformation under certain conditions such as high strain rates (corresponding to high flow velocity) and high pressures.Furthermore, the larger the heat conductivity of the wall has, the more stable is the flame against extinction.
Our objective in this work is to explore the flame-wall interaction of unstrained premixed NH 3 -H 2 -air flames and the quenching behavior, which is, to author's knowledge, less investigated.The Head-on Quenching (HoQ) process will be considered, because this is a nice canonical configuration which can be described by only one spatial coordinate, and the corresponding governing equations for mass, species, momentum and energy can be simplified to one-dimensional formulation.Furthermore, the numerical simulation can be performed by using detailed transport models and detailed chemical kinetics, without increasing the computational cost significantly.Nevertheless, the flow patterns in real combustion applications are very complex and some phenomenon such as shear flow effect and preferential diffusion effect in different directions may add additional effect on the flame, but such a model is beyond the scope of this paper.This work aims at studying the idealized one-dimensional laminar flame model, and give the first insight into the HoQ process of the ammonia system.
In the present work, the focus will be on the hydrogen-enriched ammonia-air mixture, because the addition of hydrogen can improve the combustion efficiency and stability to overcome the low reactivity and easy distinguish of pure ammonia-air system (Valera-Medina et al. 2021, 2018).Furthermore, we will focus on the lean to stoichiometric combustion.Such combustion is of great interesting, because various groups are now considering the use of ammonia for replacement in current gas turbines which operate under Dry Low NOx (DLN) conditions that fall into the lean regime with single-stage combustion chambers.Khateeb et al. (2021) conducted the first studies showing that ammonia could be burned under lean conditions close to operational gas turbine equivalence ratios.The results showed that NO tends to decrease with equivalence ratio, whilst stability was somehow degraded.The work opened further research for high-pressure analyses of these lean ammonia/hydrogen blends (Khateeb 2020).However, further analyses performed by Mashruk et al. (2022) showed that those conditions of operation would be detrimental to the production of N 2 O, as the heat losses of these lean flames promote reactions such as NO + NH → N 2 O + H at the flame front, hence generating this powerful greenhouse gas (i.e.300 times more damaging than CO 2 ).Nevertheless, the potential of using lean combustion conditions to ensure minor retrofitting of current gas turbine units is economically attractive, thus requiring further research to decrease heat losses whilst understanding the reaction mechanisms behind the interaction of radicals such as NH (mainly produced via OH pools) under these conditions.
The main result parts will be divided into two parts: • In the first part, the quenching distance will be discussed.The influence of the system parameters such as the mixture compositions (incl.equivalent ratios and hydrogen content) and the system pressures on the quenching distance will be investigated.• In the second part, the results will focus on the influence of different numerical models such as the molecular transport models, considering the heat loss in the wall and reaction mechanisms.It will be discussed how the predicted quenching distances vary with different numerical models.

Numerical Method and Model Configuration
To study the FWI numerically, the in-house code INSFLA (Maas and Warnatz 1988) has been used.This code solves for the spatio-temporal evolution of the system of PDEs for the given initial and boundary conditions using the method of lines.INSFLA has the advantage that a non-uniform grid is used and is automatically adapted according to the instantaneous flame structure.Furthermore, it involves an automatic adaptive time stepping with order and stepsize control, such that significant computational time can be saved without loosing accuracy.The code INSFLA has already been validated for the FWI including differential diffusion and surface reactions for different combustion systems such as methane/ air and iso-octane/air mixtures (Strassacker et al. 2018(Strassacker et al. , 2019(Strassacker et al. , 2021)).
In the numerical calculation, a detailed molecular transport model, the multi-component diffusion model including the thermal diffusion (Soret effect: molecular diffusion due to temperature gradient), is applied.The multi-component diffusion model takes into account the differential diffusion caused by different species, and the diffusion coefficient of each species into the mixture depends on the instantaneous mixture composition.The formulation for the diffusion fluxes j i,g is written as Hirschfelder et al. (1964): where x i is the mole fraction of i− th species.D M i represents the mixture-averaged diffusion coefficient, and D T i the thermal diffusion coefficient for the Soret effect.Both D M i and D T i are calculated according to Ref. Hirschfelder et al. (1964).Note that more detailed models such as Maxwell-Stefan model (Bothe 2011) are available, but it has been shown that the multi-component diffusion model and the mixture-averaged diffusion model give similar prediction for the flame quenching at a cold wall (Ganter et al. 2017;Zirwes et al. 2021).Moreover, according to Popp and Baum (1997), the Dufour effect for the head-on quenching is negligible.Furthermore, the thermal conductivity of gas mixture is calculated via where i is the heat conductivity of i-th species determined by the Chapman-Enskog solu- tion of kinetic gas theory (Kee et al. 2005).
The Head-on Quenching (HoQ) configuration studied in the present work is illustrated in Fig. 1.The laminar flat flame, which is considered as initial condition, is propagating to a cold wall and quenches there due to heat loss to the cold wall.The wall is located at x = 0 m, and the unburnt gas (cold premixed gas) has the same temperature as the cold wall temperature: T ub = T w = 300 K. Since the wall temperature is sufficiently low, the impact of the surface reactions is negligible small and the wall can be approximately considered as inert here Popp and Baum (1997).For this reason, the boundary condition for the species diffusion flux j i,x at the wall can be formulated as (1) where j M i,x is the mass diffusion determined by the Fick's law and j T i,x the thermal diffusion describing the Soret effect (Hirschfelder et al. 1964) At the right domain, there exists hot burnt gas and the gradient of temperature T and mass fractions of all species w i are set to be zero: where Ω is the length of the whole computational domain.
Moreover, the pressure p is considered to be constant.
In Table 1 we list all the relevant boundary and initial conditions applied in our numerical simulation, as described in this section.

Gas Mixture
In this work, the H 2 -enriched ammonia-air (NH 3 -H 2 -air) mixtures are considered.The addition of H 2 in the gas mixtures is described by the mole fraction of H 2 in the fuel: where Φ is the fuel/air equivalence ratio.For = 0 we have pure ammonia-air mixture, and for = 1 pure hydrogen-air mixture.Although the HoQ behavior of the NH 3 -H 2 -air mixture is the focus of this work, pure NH 3 -air mixture and pure H 2 -air mixture are also considered as references to compare the performance.
For the mixture compositions, we focus on varying H 2 content with 0 ≤ ≤ 0.4 , as such level of H 2 addition keeps ammonia-based features whilst being more representative to future industrial systems working on NH 3 -H 2 blends (Valera-Medina et al. 2018, 2021).
Furthermore, the fuel/air equivalence ratios are selected to be 0.6 ≤ Φ ≤ 1.0 .Since, as mentioned in the introduction section, various groups are now considering the use of ammonia for replacement in current gas turbines which operate under Dry Low NOx (DLN) conditions that fall into the lean regime with single-stage combustion chambers.

Effective Lewis Number of Unburnt Gas Mixture
A very important property of ammonia with different levels of hydrogen additions is the significant differences on molecular transport, because hydrogen is much lighter than ammonia and thus a faster diffusion.Therefore, it is helpful to check the Lewis number of gas mixture Le eff .Note that there are various ways to determine the Le eff such as volume-based, diffusionbased and heat-release based formulation.However we are not focusing on investigating the correctness of various formulations in this work, but on using it to check the differential diffusion effect caused by hydrogen addition, we use the simplest way, the volume-based formulation, to calculate the Le eff .Some reviews and comparison of different formulations can be (3)  • Calculate the Lewis number Le NH 3 and Le H 2 of spcies NH 3 and H 2 via where a mix ub , mix ub , mix ub and c mix p,ub are the thermal diffusivity, heat conductivity, density and isobaric specific heat capacity of unburnt NH 3 -H 2 -air gas mixture.D i,ub is the molecular diffusivity of species i in unburnt gas mixture.
• Calculate the Le eff based on the mole fractions in fuel via (Ichikawa et al. 2015) Figure 2 compares the Lewis number of fuel Le eff of gas mixture with different levels of hydrogen addition for fuel/air equivalence ratios Φ at T ub = 300 K and p = 1 bar.We observe that the effect of hydrogen addition in gas mixture on the Le eff is much more significant than of the fuel/air equivalence ratios Φ .The more the hydrogen is added into gas mixture, the smaller is the Le eff .

Flame Thickness of the Unstrained Premixed Flame
Later, the quenching Peclet number will be introduced where the quenching distance is normalized by the flame thickness (see below).Therefore, the flame thickness for the considered combustion system will be shortly discussed.According to Bychkov and Liberman (2000) the flame thickness f can be evaluated using the dimensional analysis ( 5) x NH 3

Fig. 2 effective Lewis number
Le eff versus fuel/air equivalence ratios Φ for gas mixture with different hydrogen addition at T ub = 300 K and p = 1 bar where a ub is the thermal diffusivity of the unburnt premixed gas mixture, and s f the laminar flame speed of unstrained premixed flame.Note that although there are also many other definitions for the flame thickness such as the thermal flame thickness estimated by considering the temperature gradient ( f =(T b − T ub )∕(dT∕dx) max ), they will not be consid- ered here because the tendency of flame thickness depending on the fuel/air equivalent ratios Φ and the hydrogen content does not alter by considering other definition for flame thickness.
In Fig. 3 the flame thickness f is plotted against fuel/air equivalence ratios Φ for gas mixtures with different hydrogen addition at T ub = 300 K and p = 1 bar.It is observed that the flame thickness decreases with increasing hydrogen content and fuel/air equivalent ratios in regime 0.6 ≤ Φ ≤ 1.0 , which is also confirmed in e.g.Li et al. (2014).

Chemical Mechanism
For the numerical simulation, the Li-2019 detailed chemical mechanism is applied (Li et al. 2019).This mechanism is originally designed for NH 3 -H 2 -CH 4 air combustion system, which has been intensively validated and documented for the ignition delay times (IDT), laminar burning velocities (LBV), and speciation (Li et al. 2019).To avoid unnecessary computational cost for inert gas such as Ar and He and carbon-related species, these species are removed and the remaining mechanism has 34 species and 252 reactions.

Quenching Process and Quenching Distance
Figure 4 shows one representative quenching process.In the first phase, the flame propagates towards the wall.Since the flame is still far from the wall, the cold wall has no influence on the flame structure (here black and purple solid lines).During the second stage, flame approaches the wall and the cold wall begins to affect the flame structure.The closer the flame approaches the wall, the larger is the heat loss towards the wall.When the heat loss becomes sufficiently large, the heat release inside the flame is not large enough to support the flame propagation and thus flame starts to quench.
According to the Fourier's law, the heat loss to the cold wall ̇qw can be calculated as: where is the heat conductivity of gas mixture.The flame quenches, after the heat loss ̇qw has reached its maximum.And the corresponding quenching distance d q is the dis- tance between the cold wall and the location of the maximum of the total heat release rate (HRR), which is defined as: where ̇ i and h i are the net production rate and enthalpy for species i.
Figure 5 shows representative temperature (T), heat release rate (HRR) and species mole fractions profiles at the time of quenching.It is interesting to mention that the left figure shows non-zero HRR at the wall ( x = 0 ), which is consistent with the observation in Dabireau et al. (2003), Palulli et al. (2019), Owston et al. (2007).As discussed there, such non-zero HRR at the wall is attributed to the low-temperature induced radical recombination reactions, which can contribute over 50% to the total HRR at the wall.
(8 Furthermore, as studied in Poinsot et al. (1993), Vosen et al. (1985), Boust et al. (2007), Bruneaux et al. (1996) by means of the analytical solution for the HoQ premixed flames, two important dimensionless quantities can be introduced, which are helpful for the analysis of quenching processes for different fuel types and are correlated with each other.They are the quenching Peclet number Pe q and the normalized wall heat loss : • The quenching Peclet number is defined as where d q is the quenching distance and f is the flame thickness (c.f.Eq. 7).Vari- ous works (Poinsot et al. 1993;Boust et al. 2007;Popp and Baum 1997;Chauvy et al. 2010;Guiberti et al. 2020) have already confirmed both numerical and experimental, that the quenching Peclet number Pe q is in the order of three ( Pe q ≈ 3.0 ) for methane- air flames.However, this quenching Peclet number varies noticeable with fuel types.In Rißmann et al. (2017) they showed that for C 2 H 4 Pe q ≈ 5.5 , and Pe q ≈ 1.4 or 1.7 for H 2 flame (Dabireau et al. 2003;Gruber et al. 2010).Moreover, in Lai and Chakraborty (2016) it is stated based on the DNS analysis that smaller Lewis numbers could lead to higher quenching Peclet numbers.
• The normalized wall heat loss is defined as describing the wall heat loss at the quenching moment max( ̇qw ) normalized by the so- called "flame power" Q Σ which can be determined as Boust et al. (2007), Poinsot et al. (1993), Bruneaux et al. (1996) The analytical thermal formulation for quenching of transient laminar flames at cold wall in Boust et al. (2007) shows that and Pe q are correlated in a simple relation approxi- mately as: More details can be found in Boust et al. (2007).One must, however, emphasizes at this stage that this simple relation is based on the pure thermal transport analysis, and the influence of species transport on the quenching processes is not considered in this analytical thermal formualtion.Later we will investigate the relation between both quantities for NH 3 -H 2 -air system.

Results: Phenomenology Discussion on Quenching Distance
In this section, we will focus on the effect of three different system parameters on the quenching distance d q , namely the fuel/air equivalence ratio 0.6 ≤ Φ ≤ 1.0 , the pressure 1 bar ≤ p ≤ 20 bar and the hydrogen addition in the mixture 0 ≤ ≤ 0.4 .Figure 6 shows (10) the dependence of d q on these three system parameters, which will be discussed in the following.

Influence of Mixture Composition
As shown in Fig. 6, for all pressures and fuel/air equivalence ratios, the quenching distances d q decreases with increasing hydrogen content in the gas mixture monotonically.
According to the numerical simulation involving total heat release rate at various hydrogen addition in Li et al. (2019), the more hydrogen added in the mixture, the higher heat release rate has the mixture.Therefore, flames of mixture with higher hydrogen contents are capable of lossing more heat to the cold wall.In this sense, the higher the hydrogen added into the mixture, the closer the flame reaches the cold wall and remains stable against quenching.This effect of hydrogen addition in the ammonia system is also consistent with the behavior in methane-hydrogen systems (Guo et al. 2022).Moreover, Fig. 6 further shows that the quenching distances d q decreases with increas- ing fuel/air equivalence ratios monotonically in range of lean to stoichiometric conditions for all pressures, which is also consistent to the analysis that the flame speed reaches its maximum in the stoichiometric condition and thus the d q its minimum.

Influence of Pressure
In the following, the influence of pressure on the quenching distance will be discussed, as shown in Fig. 7 where different fuel/air equivalence ratios and hydrogen addition in mixtures are considered.To clarify the pressure-dependence, a least-square extrapolation fit of a power function d q ∼ p a is also included, which is also used in e.g.Refs.Suckart et al.  (2017), Boust et al. (2007).
It is mainly observed here that the quenching distance decreases monotonically with increasing pressures for all cases.Such dependence has also been confirmed by experimental and analytical investigation in other literature focusing on methane-air flame with d q ∼ p −0.56 (Westbrook et al. 1981), gasoline flame with d q ∼ p −0.55 (Suckart et al. 2017)  and methanol flame with d q ∼ p −0.88 (Westbrook et al. 1981).This trend is attributed to the fact, as discussed in Refs.Chu et al. (2019), Boust et al. (2007), Colson et al. (2016), Westbrook et al. (1981), that a higher pressure increases the chemical reaction rate in terms of higher collision rate of reactive molecules, resulting in a growth of heat release rate (HRR).Thus, flame can get closer to the wall and more resistant to quenching.
Based on the numerical prediction, the correlation between quenching distance d q and pressure p by using the power function reads: Fig. 6 Quenching distance d q (mm) over fuel/air equivalence ratios Φ for different hydrogen additions and different pressures In Zhang et al. (2022), it is suggested that the laminar flame speed of ammonia-hydrogenair system S u also follows a power function of pressure as S u ∼ p −0.32 .Therefore a correla- tion between d q ⋅ S u and the pressure has the form which is consistent with the correlation form of d q ⋅ S u ∼ p −1.0 for both methane-and meth- anol-air system as shown by Westbrook et al. (1981).

Quenching Peclet Number Pe q and Normalized Wall Heat Loss '
As mentioned above, the Quenching Peclet number Pe q normalized wall heat loss are two important quantities in the description of the transient quenching processes, which takes into account different gas mixtures, and a simple relation is proposed based on thermal transport analysis.In this part, the relation between quenching Peclet number and the normalized wall heat loss introduced in Sect. 3 for gas mixtures with different hydrogen additions are presented in Fig. 8.The aim of this part is to investigate whether a similar simple relation is also valid for ammonia-hydrogen systems with different hydrogen contents.
We observe from Fig. 8 clearly that the quenching Peclet numbers Pe q vary significantly for different gas mixtures, and are different from those of the hydrocarbon system with values around 3.0-5.5 (Poinsot et al. 1993;Boust et al. 2007;Popp and Baum 1997;Chauvy et al. 2010;Guiberti et al. 2020;Rißmann et al. 2017;Ganter et al. 2017).However, a similar relation proposed in Boust et al. (2007) can be observed, namely an increase of Pe q leads to decreasing normalized wall heat loss and vice versa.Furthermore, an empirical extrapolation fit shows a correlation which is close to the theoretical formulation derived in Boust et al. (2007) (c.f. Eq. 13).However, it should be emphasized here that the simple relation proposed in Boust et al. (2007) is only based on the thermal transport analysis, and the complexity of molecular ( 14) 1.54 Pe q − 1.07 , Fig. 7 Quenching distances versus pressure for gas mixture with different hydrogen addition and fuel/air equivalence ratios Φ=0.8 and Φ=1.0 .Both axes in logarithmic scales diffusion transport is not taken into account, which leads to a difference in the correlation between and Pe q .However, this result shows that the usage of normalized wall heat loss is helpful for the study of quenching process for the NH 3 -H 2 -air combustion system, since they include the influence of different gas mixtures on the combustion properties in terms of e.g. the fuel mass fraction, enthalpy of reaction and laminar flame speeds.

Results: Sensitivity of the Simulated Quenching Distance
Due to the lack of experimental measurement, the discussion above is rather quantitative.In this results section, we focus on to check which physical and chemical models are important in the prediction of quenching distances.

Transport Model
All the results in Sect. 4 were calculated using the detailed multi-component transport model including the thermal diffusion.In this part, it is interesting to see whether the transport model could have impact on the quenching distances.In order to answer this question, we use other two different transport models: (1) multi-component transport model without thermal diffusion; (2) unity Lewis number model (equal diffusivity for all species).
In the unity Lewis number model, all the species have the same diffusion coefficient which is computed as: where , and p are the heat conductivity, density and isobaric heat capacity of the gas mixture.It should be mentioned here that the heat conductivity and viscosity of the gas mixture are still computed in a detailed way without any simplification.
Figure 9 shows two representative results of predicted quenching distances over fuel/air equivalence ratios by using different diffusion models for pure NH 3 -air flames ( = 0 ) and NH 3 -H 2 -air flames with 20 and 40% hydrogen addition ( = 0.2 and (17) Fig. 8 Relation between quenching Peclet number Pe q and normalized wall heat loss for gas mixture with different hydrogen additions under p = 1 bar = 0.4 ).It is observed that the predicted quenching distances are more significantly affected by the diffusion models with increasing hydrogen contents, which is also expected.For pure NH 3 -air flames whose effective Lewis number of fuel is around unity ( Le eff ≈ 1 , c.f. Fig. 2), the deviation between using very detailed transport model (circle solid lines) and using the unity Lewis number model (diamond point lines) is maximum around at Φ=0.6 .However, for NH 3 -H 2 -air flames with 20% hydrogen addi- tion ( = 0.2 ), the use of unity Lewis number model predicts almost 3 times smaller compared to those using very detailed transport model (multicomponent transport + Soret effect), and the discrepancy increases further if the hydrogen content is further increased (almost 7 times smaller for Φ=0.6 ).The large difference caused by using the unity Lewis number model has also been confirmed in the hydrocarbon-air system such as shown in Zirwes et al. (2021); Ganter et al. (2017).
The Soret effect, on the other hand, plays a smaller role in the prediction of the quenching distance, compared to those predicted by the unity lewis diffusion model.This is also consistent with Popp and Baum (1997).However, it should be mentioned that the Soret effect can still alter the prediction of quenching distance, because as discussed in Kovaleva et al. (2023) the Soret effect accelerates the diffusion of H 2 towards the flame front and shifts the position of the temperature profile towards the lean conditions.This is also consistent with the observation in Wang et al. (2021), Liang et al.

Fig. 9
Quenching distances versus fuel/air equivalence ratio for gas mixture with = 0 , 0.2 and 0.4 at p = 1 bar using different diffusion models (2013) that the Soret effect becomes especially important for hydrogen and high hydrogen contained combustion system.

Conjugate Heat Loss in the Wall
Till now, all the calculations above are based on the assumption that the wall temperature at x = 0 in Fig. 1 was fixed.In this subsection, the mathematical model introduced in Sect. 2 will be coupled with conjugate heat loss in the wall.This means that both sides at x = 0 in Fig. 1 share the same contact temperature and the heat transfer And the back-side of the wall is assumed to be fixed at 300 K.The wall is modeled to be stainless steel with a thickness of 2 mm, following the investigation on the influence of conjugate heat transfer for the side-wall quenching flame in Zirwes et al. (2021).All other material properties such as heat conductivity, heat capacity and density are also identical to those in Zirwes et al. (2021).
Figure 10 compares the quenching distances predicted by fixed wall temperature (solid symbol lines) and by considering conjugate heat loss in the wall (dashed symbol lines).We notice that there are almost no visible differences between both cases.In other words, the effect of considering the conjugate heat loss in the wall on the quenching distances is negligible.Near the quenching point, the wall temperature at x = 0 reaches its maximum value of around 310 K, which is an increase of about only 10 K. Consequently, the quenching distance is slightly changed within 0.5%.This can be attributed to fact that the time of the flame quenching is much shorter than the time-scale of the heat conductivity in the wall, so that the wall temperature has no sufficient time to react.Such conclusion is consistent with the observation in Zirwes et al. (2021), Häber and Suntz (2018) as well, where the methane-air flame quenching process is considered.However, it should be mentioned that the heat loss in the wall can affect the flame structures and quenching distances in other cases such as side-wall quenching configuration reported in Zhang et al. (2022) or (18) T wall (x = 0 − )=T flame (x = 0 + ) (19) j q,wall (x = 0 − )=T q,flame (x = 0 + ).turbulent flame-wall interaction reported in e.g.Zhao et al. (2018), Gruber et al. (2010) or use of other materials with lower heat conductivities (Kai et al. 2020).

Reaction Mechanisms
In this section, we focus on the influence of the chemical kinetics on the predicted quenching distance.Before performing the sensitivity analysis, it is interesting to investigate how large difference on the quenching distances exist, if different detailed chemistries are used.To show this, two more detailed chemical mechanisms, the Shrestha-2021 mechanism (Shrestha et al. 2021) and the Otomo-2018 (UT-LCS) mechanism (Otomo et al. 2018) (see Table 2), are used for the prediction of quenching distance for flame of gas mixture NH 3 :H 2 = 0.8:0.2 with different fuel/air equivalence ratios at two pressures, as shown in Fig. 11.It is observed that the leaner the mixture is, the larger is the difference of predicted quenching distance for both pressures.
Note that there are also many other mechanisms such as Nakamura et al. (2017), Stagni et al. (2020) and Gotama et al. (2022).However, our aim here is only to state that there exist large differences using different chemical kinetics.Thus it is important to know which elementary reactions are especially important for the accuracy of the predicted quenching distance.If there are experiment measurements in the future, it is then recommended to test every chemical mechanism for its ability in the prediction of quenching distance.In order to answer this question, the sensitivity analysis is a helpful tool to quantify which elementary reactions are the key reactions.
The relative sensitivity S rel (d q ) used in this work describes the normalized sensitivity of d q with respect to the reaction rate constant of one elementary reaction as: Figure 12 shows the sensitivity of the quenching distance S rel (d q ) (red bars) with respect to the reaction rate constants based on the Li-2019 mechanism (Li et al. 2019).Note that these values are shown in −S rel (d q ) for convenient readability.Together shown in this figure is the sensitivity of the laminar burning velocity S rel (LBV) (blue bars), which is a common topic in other literature (c.f.da Rocha et al. 2019;Shrestha et al. 2021;Xiao et al. 2017).Several important observations can be addressed here: • The S rel (d q ) has always an opposite sign of the S rel (LBV) .In other words, an increase of LBV leads to a decrease of d q , and vice versa.This is because, as discussed above, that an increase of LBV means a higher heat release rate in the flame, and consequently more stable against quenching due to heat loss.Therefore, the d q decreases with increasing LBVs.• The LBV and the d q are sensitive to the similar elementary reactions.Among them, the most sensitive elementary reaction is H + O 2 = OH + O.This indicates that an accurate prediction of LBV is prerequisite for an accurate prediction of d q . (20) S rel = k r d q ⋅ d q k r (evaluated by finite difference) d q (k r ) .
Fig. 12 sensitivity of both laminar burning velocity (blue bars) and the quenching distance (red bars) with respect to the reaction rates.Conditions: gas mixture with NH 3 :H 2 = 0.8:0.2;fuel/ air equivalent ratio Φ=0.8 .p = 1 bar.Chemical mechanism (Li et al. 2019) It should be mentioned that although this sensitivity analysis is based on the Li-2019 mechanism, these two observations also hold for the other two mechanisms considered (not shown here).In Sect. 4.2 it is shown that the correlation between quenching distance d q and pressure p follows a power function Eq. ( 14) by using the Li-2019 mechanism (Li et al. 2019).In Fig. 13 such correlation is also checked for other two mechanisms listed in Table 2.And it is observed that the power function of Eq. ( 14) is independent on the mechanisms we used.

Conclusions
The numerical investigation on the head-on quenching (HoQ) process of NH 3 -H 2 -air flames is performed.The gas mixture is considered in the lean to stoichiometric range due to its pratical application in gas turbines which operate Dry Low NOx (DLN).Focusing on the quenching distance d q , the main phenomenology findings of the present work are: • The quenching distances decrease with increasing H 2 addition and increasing fuel/air equivalent ratios in range of lean to stoichiometric conditions for all pressures; • The quenching distances decrease with increasing pressure, following the power function d q ∼ p −0.7 ; • An increasing quenching Peclet number Pe q results in decreasing normalized wall heat loss , and an empirical extrapolation fit as a rational function has been suggested.
Furthermore, models and model parameters have been altered to study the sensitivity of the quenching distances.It is found that: • The molecular transport model plays an important role, especially when the hydrogen contents are increased in the gas mixtures; • The consideration of the conjugate heat loss in the wall during the quenching processes has negligible effect on the predicted quenching distance with a change of maximum 0.5% for the considered system; Fig. 13 Quenching distances versus pressure for gas mixture with 20% hydrogen addition ( = 0.2 ) and fuel/air equivalence ratios Φ=0.8 for three detailed chemical kinetics listed in Table 2.Both axes in logarithmic scales • The predicted quenching distance is sensitive to several key elementary reactions such as H + O 2 = OH + H and NH 2 + NO = N 2 + H 2 O.And these key elementary reactions, which are significant for the accurate prediction of the quenching distances, are similar to those for the laminar burning velocity (LBV).This suggests that the accurate prediction of LBV is a prerequisite for an accurate prediction of quenching distance; • although different detailed chemical mechanisms can provide different values of quenching distances, their dependence on pressure obeys the power function d q ∼ p −0.7 .

Fig. 3
Fig. 3 Flame thickness f versus fuel/air equivalence ratios Φ for gas mixture with different hydrogen addition at T ub = 300 K and p = 1 bar

Fig. 4 Fig. 5
Fig. 4 Representative quenching process.Temperature profile for different times

Fig. 10
Fig. 10Quenching distances versus fuel/air equivalence ratios for different hydrogen additions by considering fixed wall temperatures (solid symbol lines) and conjugate heat loss in the wall (dashed symbol lines)

Table 1
Boundary conditions for the numerical simulation +TemperatureT ub = T w = 300 K Species diffusion flux j i,x = 0 (c.f.Eq. 3) Species mass fraction w i = w i,ub Pressure p right boundary x =Ω Species mass fraction

Table 2 Three
Fig.11Quenching distances d q over fuel/air equivalence ratios Φ at two pressures for gas mixture with NH 3 :H 2 = 0.8:0.2.Three different detailed chemical mechanisms are used, as listed in Table2