# Low-order modeling of the efficiency of flame ignition by Nanosecond Repetitively Pulsed discharges

Stéphane Q.E. Wang, Yacine Bechane, Nasser Darabiha, Benoît Fiorina

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Stéphane Q.E. Wang, Yacine Bechane, Nasser Darabiha, Benoît Fiorina. Low-order modeling of the efficiency of flame ignition by Nanosecond Repetitively Pulsed discharges. 11th European Combustion Meeting, Apr 2023, Rouen, France. hal-05109875

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Submitted on 12 Jun 2025

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CentraleSupélec

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université
PARIS-SACLAY

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European Research Council
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# Low-order modeling of the efficiency of flame ignition by Nanosecond Repetitively Pulsed discharges

S. Q. E. Wang, Y. Bechane, N. Darabiha, B. Fiorina

Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C - France

## Plasma-Assisted Combustion / Ignition

Plasma-Assisted Combustion by
**Nanosecond Repetitively Pulsed (NRP) discharges** can:

* Stabilize lean flames [1]

* Extend flammability domain [1]

* Promote ignition [2]

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

NRP discharges **enhance the chemical activity** [3] :

* Increasing the gas temperature,

* Producing active radicals, such as atomic O.

The repetitive nature will promote the **accumulation** of these thermal and chemical effects.

The ultra-fast heating induces **hydrodynamic effects** [4] limiting the interelectrode residence time.

The efficiency of the NRP discharges to ignite the
mixture is driven by:

* the deposited energy per pulse $E_p$

* the pulse repetition frequency $f_p$

<figure><img src="image_11.png" /></figure>

How do the parameters $E_p$ and $f_p$ impact the ignition process?
What parameters result in an **efficient Plasma-Assisted Ignition**?

LES can capture these **complex interactions** between plasma, turbulence and combustion [3].
However, the **CPU cost is very expensive**. Not feasible for parametric studies.

## Perfectly-Stirred Reactor modelling

Low-order PSR model developed in [3] and implemented in REGATH 0D solver [5]

<figure><img src="image_12.png" /></figure>

GRI-Mech 3.0 detailed scheme for CH₄-Air combustion
(53 species and 325 reactions)

$$ \frac{dY_k}{dt} = \frac{1}{\tau_{res}} (Y_k^0 - Y_k) + \left[ \frac{W_k \dot{\omega}_k^c}{\rho} + \frac{W_k \dot{\omega}_k^p}{\rho} \right] + \text{PSR} $$

$$ \frac{dh}{dt} = \frac{1}{\tau_{res}} (h^0 - h) + \left[ \frac{\dot{E}_{chem}^p}{\rho} + \frac{\dot{E}_{heat}^p}{\rho} \right] + \text{NRP} $$

$$ \frac{de_{vib}}{dt} = \frac{1}{\tau_{res}} (e_{vib}^0 - e_{vib}) + \left[ \frac{\dot{E}_{vib}^p}{\rho} - \frac{\dot{R}_{VT}^p}{\rho} \right] + \text{Recirculating zone} $$

Residence time $\tau_{res}$

Ultrafast phenomena
of NRP discharges

$$ \frac{\dot{R}_{VT}^p}{\rho} $$

Slow phenomena
of NRP discharges

## Phenomenological NRP discharge model

<figure><img src="image_13.png" /></figure>

$$ \dot{R}_{VT}^p = \rho \frac{e_{vib} - e_{vib}^{eq}(T)}{\tau_{VT}} $$

$$ \dot{E}_{vib} = (1 - \alpha) \dot{E}^p $$

$$ \dot{E}_{heat}^p = \alpha \dot{E}^p - \dot{E}_{chem}^p $$

$$ \dot{E}_{chem}^p = \eta \frac{Y_{O_2}}{Y_{O_2}^0} \left(1 - \frac{e_{O_2}}{e_O}\right) \dot{E}^p $$

$$ \begin{aligned} \dot{\omega}_O^p &= \eta \frac{Y_{O_2}}{Y_{O_2}^0} \frac{\dot{E}^p}{e_O W_O} \\ \dot{\omega}_{O_2}^p &= -\frac{W_O}{W_{O_2}} \dot{\omega}_O^p \\ \dot{\omega}_k^p &= 0 \quad \text{if } k \neq \{O, O_2\} \end{aligned} $$

Castela et al.
(2016) [6]

## Acknowledgments & References

This work was partially funded by:

* the **PASTEC** project of ANR (Grant no. ANR-16-CE22-0005),

* the **GREENBLUE** project of ERC (Grant no. 101021538).

Performed using HPC resources from the "Mésocentre"
of CentraleSupélec, ENS Paris-Saclay and Uni. Paris-Saclay
supported by CNRS and Région Île-de-France.

[1] G. Pilla et al., *IEEE Trans. Plasma Sci.* **34** (2006)

[2] Y. Ju and W. Sun, *Prog. Energy Combust. Sci.* **48** (2015)

[3] Y. Bechane and B. Fiorina, *Proc. Combust. Inst.* **38** (2021)

[4] M. Castela et al., *Proc. Combust. Inst.* **36** (2017)

[5] N. Darabiha, *Combust. Sci. Technol.* **86** (1992)

[6] M. Castela et al., *Combust. Flame.* **166** (2016)

For a given $E_p$ or $f_p$: $\uparrow \tau_{res} \rightarrow \uparrow n_p$.

* When $\tau_{res} \rightarrow \tau_{res}^{crit}$, $n_p \rightarrow +\infty$. No successful ignition for $\tau_{res} < \tau_{res}^{crit}$.

* When $\uparrow E_p$ or $\uparrow f_p$, $\uparrow \tau_{res}^{crit}$. Except for some conditions, where ignition is followed by an extinction, thus $\uparrow \tau_{res}^{crit}$.

For a given $f_p$: $\uparrow E_p \rightarrow \uparrow n_p \rightarrow \begin{cases} \uparrow E_p^{tot} & (\text{for } \tau_{res} < 0.4 \text{ ms}) \\ \uparrow E_p^{tot} & (\text{for } \tau_{res} > 0.7 \text{ ms}) \end{cases}$

For a given $E_p$: $\uparrow f_p \rightarrow \uparrow n_p$ and $\uparrow E_p^{tot}$ (for $\tau_{res} < 2$ ms)
Otherwise, $n_p$ and $E_p^{tot}$ do not vary much.

## Observations & Conclusions

As for the ignition delay time:

$\uparrow E_p$ or $\uparrow f_p \rightarrow \uparrow \tau_{ign}$ (for almost all $\tau_{res}$)

Except for some $\tau_{res}$, where chemical activity is sufficient to not apply another pulse.

It is more energy-efficient to ignite the mixture with:

* high $E_p$ and high $f_p$ (for low $\tau_{res}$).

* low $E_p$ and high $f_p$ (for high $\tau_{res}$).

There is a residence time $\tau_{res}^{crit}$ below which ignition failed.