1 Introduction
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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.
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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.
2 Mathematical Modeling
2.1 Numerical Method and Model Configuration
Boundary conditions | ||
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Left boundary \(x=0^+\) | Temperature | \(T_\text {ub} = T_w = 300\) K |
Species diffusion flux | \(j_{i,x} = 0\) (c.f. Eq. 3) | |
Species mass fraction | \(w_i = w_{i,\text {ub}}\) | |
Pressure | p | |
right boundary \(x=\Omega \) | Species mass fraction | \(\frac{\partial w_i}{\partial x} = 0\) |
Temperature | \(\frac{\partial T}{\partial x} = 0\) | |
Pressure | p |
Initial condition for \(t=0\) | ||
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Unstrained premixed flame under adiabatic condition |
2.2 Combustion System
2.2.1 Gas Mixture
2.2.2 Effective Lewis Number of Unburnt Gas Mixture
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Calculate the Lewis number \(Le_{\text {NH}_3}\) and \(Le_{\text {H}_2}\) of spcies NH\(_3\) and H\(_2\) viawhere \(a^\text {mix}_\text {ub}\), \(\lambda ^\text {mix}_\text {ub}\), \(\rho ^\text {mix}_\text {ub}\) and \(c^\text {mix}_\text {p,ub}\) are the thermal diffusivity, heat conductivity, density and isobaric specific heat capacity of unburnt NH\(_3\)–H\(_2\)-air gas mixture. \(D_{i,\text {ub}}\) is the molecular diffusivity of species i in unburnt gas mixture.$$\begin{aligned} Le_i = \frac{a^\text {mix}_\text {ub}}{D_{i,\text {ub}}} = \frac{\lambda ^\text {mix}_\text {ub}}{\rho ^\text {mix}_\text {ub} \cdot c^\text {mix}_\text {p,ub} \cdot D_{i,\text {ub}}}, \end{aligned}$$(5)
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Calculate the \(Le_\text {eff}\) based on the mole fractions in fuel via (Ichikawa et al. 2015)$$\begin{aligned} Le_{\text {eff}} = \frac{x_{\text {NH}_3}+x_{\text {H}_2}+x_{\text {O}_2}}{ \frac{x_{\text {NH}_3}}{ Le_{\text {NH}_3}} + \frac{x_{\text {H}_2}}{ Le_{\text {H}_2}} + \frac{x_{\text {O}_2}}{ Le_{\text {O}_2}}}. \end{aligned}$$(6)
2.2.3 Flame Thickness of the Unstrained Premixed Flame
2.3 Chemical Mechanism
3 Quenching Process and Quenching Distance
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The quenching Peclet number is defined aswhere \(d_\text {q}\) is the quenching distance and \(\delta _f\) is the flame thickness (c.f. Eq. 7). Various 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 \(Pe_\text {q}\) is in the order of three (\(Pe_\text {q} \approx 3.0\)) 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\(_4\) \(Pe_\text {q} \approx 5.5\), and \(Pe_\text {q} \approx 1.4\) 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.$$\begin{aligned} Pe_\text {q} = \frac{d_\text {q}}{\delta _f}, \end{aligned}$$(10)
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The normalized wall heat loss \(\varphi \) is defined asdescribing the wall heat loss at the quenching moment \(\text {max}(\dot{q}_w)\) normalized by the so-called “flame power” \(Q_\Sigma \) which can be determined as Boust et al. (2007), Poinsot et al. (1993), Bruneaux et al. (1996)$$\begin{aligned} \varphi = \frac{\text {max}(\dot{q}_w)}{Q_\Sigma }, \end{aligned}$$(11)$$\begin{aligned} Q_\Sigma = \rho _\text {ub} c_p s_f (T_\text {b} - T_\text {ub}) \end{aligned}$$(12)
4 Results: Phenomenology Discussion on Quenching Distance
4.1 Influence of Mixture Composition
4.2 Influence of Pressure
4.3 Quenching Peclet Number \(Pe_\text {q}\) and Normalized Wall Heat Loss \(\varphi \)
5 Results: Sensitivity of the Simulated Quenching Distance
5.1 Transport Model
5.2 Conjugate Heat Loss in the Wall
5.3 Reaction Mechanisms
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The \(S^\text {rel}(d_\text {q})\) has always an opposite sign of the \(S^\text {rel}(\text {LBV})\). In other words, an increase of LBV leads to a decrease of \(d_\text {q}\), 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 \(d_\text {q}\) decreases with increasing LBVs.
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The LBV and the \(d_\text {q}\) 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 \(d_\text {q}\).
6 Conclusions
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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;
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The quenching distances decrease with increasing pressure, following the power function \(d_\text {q} \sim p^{-0.7} \);
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An increasing quenching Peclet number \(Pe_\text {q}\) results in decreasing normalized wall heat loss \(\varphi \), and an empirical extrapolation fit as a rational function has been suggested.
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The molecular transport model plays an important role, especially when the hydrogen contents are increased in the gas mixtures;
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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;
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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;
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although different detailed chemical mechanisms can provide different values of quenching distances, their dependence on pressure obeys the power function \(d_\text {q} \sim p^{-0.7} \).