# On the Critical Condition for Flame Acceleration in Hydrogen-Based Mixtures

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## Abstract

**:**

## 1. Introduction

## 2. Problem Setup

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the problem setup. Channel width equal to H; in most of calculations H = 10 mm, R = 1 mm.

**Figure 2.**Results of the convergence test for 11% hydrogen in air combustion. The curve with signs shows the flame thickness’s dependence on the linear size of the numerical cell. The dashed line shows the exact solution estimated according to Richardson’s technique. Numerals show the sizes of the numerical cell and the error value in percent.

**Figure 3.**Evolution of the flame in channel: (

**a**) H = 40 mm, 29.5 % of hydrogen in mixture; (

**b**) H = 10 mm, 29.5 %; (

**c**) H = 10 mm, 17.0 %; (

**d**) H = 10 mm, 11.0 %. Flame position is shown with the use of temperature iso-lines T = 1000 K. Time intervals between subsequent instants ${u}_{f}\Delta t/H\approx 0.025$ (400 $\mathsf{\mu}$s (

**a**), 100 $\mathsf{\mu}$s (

**b**), 500 $\mathsf{\mu}$s (

**c**), 3750 $\mathsf{\mu}$s (

**d**)).

**Figure 4.**Time histories of flame speeds (

**a**,

**b**) and maximal pressure in channel with H = 10 mm filled with different mixtures containing 11 % (red), 12 % (green), and 17 % (black) hydrogen. Highlighted regions correspond to the early stage of flame acceleration (I), stage of flame deceleration (II), and further flame development either with acceleration or propagation with a constant speed. Flame speed (D) and time (t) are presented in SI (

**a**,

**c**) and in dimensionless values (

**b**). Pressure (p) is presented in atm units. Dashed lines in frame (b) show the values of $D=\Theta {u}_{f}$, while $D/{u}_{f}=1$ corresponds to the value $D={u}_{f}$.

**Figure 5.**Time scales characterizing the energy release inside the reaction zone (${\tau}_{b}$) and heat losses due to heat transfer (${\tau}_{\chi}$) depending on the nitrogen content in the oxidizer (O${}_{2}$/N${}_{2}$). Hydrogen content in all the mixtures is 11 %. Values with index “ref” correspond to the hydrogen–air mixture ($\left[{\mathrm{N}}_{2}\right]$ = 79 %).

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**MDPI and ACS Style**

Kiverin, A.; Tyurnin, A.; Yakovenko, I. On the Critical Condition for Flame Acceleration in Hydrogen-Based Mixtures. *Materials* **2023**, *16*, 2813.
https://doi.org/10.3390/ma16072813

**AMA Style**

Kiverin A, Tyurnin A, Yakovenko I. On the Critical Condition for Flame Acceleration in Hydrogen-Based Mixtures. *Materials*. 2023; 16(7):2813.
https://doi.org/10.3390/ma16072813

**Chicago/Turabian Style**

Kiverin, Alexey, Alexey Tyurnin, and Ivan Yakovenko. 2023. "On the Critical Condition for Flame Acceleration in Hydrogen-Based Mixtures" *Materials* 16, no. 7: 2813.
https://doi.org/10.3390/ma16072813