Research on the Multi-Factor Critical Criterion for Acoustic Fire Extinguishment Based on the Flame-Fuel Cycle Model

1. Introduction

With the accelerating urbanization of human society, the threat of fires to urban safety and the lives of residents has become increasingly prominent. [1]. As a novel fire-extinguishing method that has attracted considerable attention in recent years, acoustic fire extinguishing, owing to its advantages of being non-contact, pollution-free and highly adaptable, exhibits great application potential in microgravity environments and fire protection for precision equipment, and its feasibility has been widely verified by experiments [2,3,4]. However, academic research on acoustic fire extinguishing still mainly focuses on investigating the influence of a single specific factor and often remains only at a qualitative level. Some studies, such as that by Gore et al. [5], suggest that the local flame temperature drop caused by pressure waves generated by acoustic waves is the main factor affecting the effectiveness of acoustic fire extinguishing, based on the ideal gas law. Research by Alexander, A. A. [2] considers the sound pressure level as the dominant indicator for extinguishing, and an increase in sound pressure level results in enhanced fire extinguishing effectiveness. Niegodajew P et al. [3] find that an increase in burner power leads to a significant rise in speaker extinction power, and this increase becomes more pronounced as the distance between the burner and the sound source increases. Shi X Q et al. [4] further emphasize that the alteration of the combustion environment due to sound pressure-induced airflow disturbances is the key factor in extinguishing. In contrast, Xiong, C. et al. [6] and Cliftmann, J. M. [7] argue that the mechanical movement of the speaker diaphragm, rather than the acoustic field itself, is the main cause of flame extinction, and attempt to define the flame extinction boundary using the critical displacement; however, their quantitative work exhibits significant shortcomings and fails to establish effective correlations with combustible material properties or flame dimensions. Consequently, it remains impossible to accurately calculate the required acoustic parameters for extinguishing based on fire source information. In response, Friedman, A. N. [8] attempts to establish critical criteria for acoustic fire extinction from a thermodynamic perspective by introducing the Nusselt number and the Spalding B number. However, his work only addresses liquid fuels and adopts liquid fuels with similar Prandtl numbers to neglect the influence of Prandtl number, resulting in the critical criterion model being applicable only to a very limited number of liquid fuels with close Prandtl numbers within his research scope. At the same time, the model severely lacks universality and practicality, as it fails to consider the significant impact of flame physical dimensions on the acoustic conditions required for extinction, as well as the theoretical applicability of relatively high-frequency acoustic waves.

In view of the above research gap in the quantitative critical extinction criteria considering multiple factors of acoustic fire extinguishing, this study addresses the limitations of Friedman’s flame-fuel cycle model by introducing correction terms for flame geometry and fuel properties, extending the theory from liquid fuels to gaseous premixed flames, and, combined with experimental data and log-linear regression analysis, establishes a generalized multi-factor extinction criterion that uniformly considers flame size, type, and fuel properties. Research indicates that the Prandtl number of gaseous fuels significantly influences the critical particle velocity for acoustic flame extinction. For gas diffusion and premixed flames ranging from 2.00 cm to 5.00 cm, the critical particle velocity for acoustic flame extinction exhibits an approximately proportional relationship with flame height. Furthermore, through an in-depth analysis of the critical particle velocity required for extinction efficiency under broadband acoustic fire suppression, this study further elucidates the dual influence of acoustic frequency on extinguishing efficacy. This research not only theoretically unifies the acoustic extinction criteria under the influence of multiple factors, but also provides key theoretical support for the development of high-efficiency acoustic fire-extinguishing equipment, which can predict the acoustic parameters required for fire extinguishing under different fuels, different flame sizes and different frequencies. Additionally, through in-depth analysis of dynamic behaviors such as flame necking and breakage in a wide frequency domain, this study also points out a new direction for subsequent research on the mechanism of acoustic fire extinguishing.

2. Extension Study of the Multi-Factor Critical Criterion

This study adopts the flame-fuel cycle model proposed by Friedman, A. N. [8] as the core framework for acoustic flame extinction (i.e., the flame is displaced by acoustic disturbance away from the fuel bed, reducing the heat flux between the flame and the fuel, leading to extinction). The Spalding B-number and local Nusselt number are used to describe the stability of flame combustion and the disturbance caused by acoustic waves. However, since the study by Friedman, A. N. [8] employed a fixed-size burner to deliberately maintain identical flames and only used liquid fuel diffusion flames, its applicability to diffusion and premixed flames of different physical sizes and fuel states is severely underexplored. This limitation hinders the revelation of the significant influence of these factors on the acoustic flame extinction process and renders the theoretical model largely impractical. To address this, the present paper attempts to introduce additional factors not originally considered, such as flame height and combustion type, while simultaneously extending the theory to gaseous fuel flames.

2.1. Theoretical Extension

A model for the mass transfer Spalding B-number in gaseous fuel combustion is first constructed. The theoretical model developed by Quintiere [9] for small-scale flames is adopted, with its original formula as follows:

$$B={\displaystyle \frac{Y_{O_2,\infty }(\Delta h_c/r)-c_{p,\mathrm{air}}(T_b-T_{\infty })}{L}}$$

(1)

where $Y_{O_2,\infty }$ is the mass fraction of oxygen in the ambient air; $\Delta h_c$ denotes the heat of combustion per unit mass of fuel in $\mathrm{J}/\mathrm{k}\mathrm{g}$; r is the stoichiometric oxygen-to-fuel mass ratio; $c_{p,\mathrm{air}}$ represents the specific heat capacity of air at constant pressure in $\mathrm{J}/\mathrm{k}\mathrm{g}/\mathrm{K}$; $T_b$ is the boiling point of the fuel at one standard atmospheric pressure; $T_{\infty }$ is the ambient temperature in $\mathrm{K}$; and L is the latent heat of vaporization in $\mathrm{J}/\mathrm{k}\mathrm{g}$.

Since this formula is primarily designed for liquid fuels, considering the mass transfer driving force and heat loss resistance of gaseous fuels, the following modified B-number expression for gaseous fuel diffusion flames, denoted as Equation (2), is given:

$$B={\displaystyle \frac{Y_{O_2,\infty }(\Delta h_c/r)}{M}}$$

(2)

where M represents the combustion heat loss per unit mass of fuel in $\mathrm{J}/\mathrm{k}\mathrm{g}$. The ambient oxygen mass fraction is assumed as $Y_{O_2,\infty }=0.233$. For gaseous fuels, the latent heat of phase change is omitted since the fuel is already in the gaseous state.

Due to the flame displacement caused by the acoustic excitation, part of the fuel bed becomes directly exposed to the cold air under the action of the oscillatory airflow, leading to enhanced convective heat transfer between the unburned fuel and the cold air. To quantify this effect, a local Nusselt number $N{u}_{\xi }$ is employed to characterize the alteration in convective heat transfer intensity, thereby representing the perturbation imposed by the oscillatory particle motion on the flame-fuel feedback cycle. The Nusselt number is calculated as follows:

$$N{u}_{\xi }=cR{e}^{\gamma }P{r}^{\delta }$$

(3)

where c and $\delta$ are empirical parameters dependent on the ambient conditions; and $\gamma$ is taken as ${\displaystyle \frac{1}{3}}$ based on the findings of Friedman, A. N. [8]. $Pr$ and $Re$ denote the Prandtl and Reynolds numbers, respectively, defined in Equations (4) and (5).

$$Pr={\displaystyle \frac{c_p\mu }{\lambda }}$$

(4)

Here, $c_p$ is the specific heat capacity at constant pressure of the fluid (air, herein), in $\mathrm{J}/\mathrm{k}\mathrm{g}/\mathrm{K}$; $\mu$ is the dynamic viscosity, in $\mathrm{k}\mathrm{g}/\mathrm{m}/\mathrm{s}$; and $\lambda$ is the thermal conductivity, in $\mathrm{W}/\mathrm{m}/\mathrm{K}$.

For the Reynolds number, the characteristic length l is first defined as the root-mean-square (RMS) particle displacement per acoustic cycle, in $\mathrm{m}$. The expression for $Re$ can thus be transformed as:

$$Re={\displaystyle \frac{\rho U_{A}l}{\mu }}={\displaystyle \frac{U_{A}l}{\nu }}={\displaystyle \frac{U_A^2}{\nu \omega }}$$

(5)

where $\rho$ is the fluid density in $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$; $\nu$ is the kinematic viscosity in $\square \mathrm{m}/\mathrm{s}$; and $U_A$ is the RMS acoustic-induced particle velocity in $\mathrm{m}/\mathrm{s}$.

Considering the competitive relationship between the disturbance $\left(N{u}_{\xi }\right)$ and flame stability (represented by the mass transfer B-number), their ratio is adopted as a critical criterion for acoustic extinction, augmented with a flame size term $f(h,S)$ and a flame type factor k. A flame is predicted to extinguish when this ratio exceeds a critical value. The flame size term is empirically assumed to be separable as $f(h,S)={\displaystyle \frac{d}{h^{\alpha }g\left(S\right)}}$, where $g\left(S\right)$ is a function of the area S, d and $\alpha$ are empirical parameters, and we let $c^{\prime }=c·d$. For the factor k, a value of $k=1$ is assigned for diffusion flames by default; this parameter allows for the examination of potential differences in the extinction criteria between premixed and diffusion flames.

$${\Theta }_A=k{\displaystyle \frac{N{u}_{\xi }}{B}}f(h,S)=k{\displaystyle \frac{c^{\prime }{U}_A^{2\gamma }P{r}^{\delta }}{h^{\alpha }g\left(S\right){\left(\nu \omega \right)}^{\gamma }B}}$$

(6)

In a given experimental environment, ${\Theta }_A$ and $c^{\prime }$ should remain constant. Moreover, since the burner nozzle area S is controlled to be the same (i.e., using the same Bunsen burner with an unchanged nozzle size, ensuring the flame fully covers the nozzle), $g\left(S\right)$ is also constant. Equation (6) can thus be simplified to the following working critical extinction criterion ${\Theta }_A^{\prime }$ under the present experimental conditions:

$${\Theta }_A^{\prime }={\displaystyle \frac{{\Theta }_A}{c^{\prime }}}g\left(S\right)=k{\displaystyle \frac{U_A^{2\gamma }P{r}^{\delta }}{h^{\alpha }{\left(\nu \omega \right)}^{\gamma }B}}$$

(7)

It is noteworthy that Friedman, A.N. [8] employed a waveguide tube in their experiments to focus acoustic energy, which enabled flame extinction at a relatively low input power and possesses clear engineering application value. However, this device also alters the propagation mode of sound waves (tending towards plane wave/standing wave) and generates a significant accompanying flow, making the observed phenomena the result of “waveguide-constrained acoustic streaming coupling”. To investigate the fundamental physics of the interaction between the basic acoustic near-field and the flame without such complex boundary conditions, the present experiment does not employ a waveguide tube and directly examines the conditions in the acoustic near-field, aiming to reveal a more essential mechanism of acoustic fire extinguishment and provide a basis for the selection of technical approaches in different application scenarios. Although certain nonlinear acoustic effects are present under this configuration, the following conversion formula between sound pressure and acoustic-induced particle velocity [10] has been proven to remain valid under the experimental conditions of this work.

$$p=z{U}_{\mathbf{A}}$$

(8)

Here, p represents the sound pressure, in $\mathrm{Pa}$; and z denotes the acoustic impedance, in $\mathrm{Pa}\mathrm{s}/\mathrm{m}$.

All thermophysical data used in this experiment were obtained from the NIST Chemistry WebBook [11]. For properties at specific temperatures not available in the database, values were derived through methods such as interpolation, fitting and extrapolation, combined with theoretical models including the power-law model and the ideal gas law.

2.2. Experimental Analysis

2.2.1. Analytical Approach and Procedure

In the study by Friedman, A. N. [8], due to the selection of liquid fuels with similar Prandtl numbers, the influence of ${Pr}^{\delta }$ was considered approximately equal and thus directly neglected. Applying such an assumption to gaseous fuel diffusion and premixed flames is highly imprecise and would severely limit the theoretical applicability. Therefore, this study incorporates it into the discussion, and subsequent experiments demonstrate its significant impact on the critical extinction criterion for gaseous fuels. To simultaneously account for both the combustion type term k and the flame size term $f(h,S)$, the analytical approach and procedure for the experimental section of this paper are designed as follows. Through a well-conceived experimental design, additional factors are incorporated rationally and rigorously, leading to a substantially extended and more practically valuable multi-factor critical criterion for acoustic flame extinction.

Figure 1. Flowchart of the experimental analysis approach.

Figure 1. Flowchart of the experimental analysis approach.

2.2.2. Fitting and Validation of the Flame Height Exponent $\alpha$ for Diffusion Flames

Based on the mathematical characteristics of Equation (7) derived from the theoretical analysis, it can be fitted as follows to first obtain the empirical flame height parameter $\alpha$.

Considering experiments conducted with the same fuel under identical experimental conditions, ${\Theta }_A^{\prime }$ and ${\displaystyle \frac{{Pr}^{\delta }}{B}}$ remain constant. The flame type factor k is defined as $k=1$ for diffusion flames, leading to Equation (9).

$${\displaystyle \frac{{\left(\frac{U_A^2}{\nu \omega }\right)}^{\gamma }}{h^{\alpha }}}=C$$

(9)

where C is a constant. Taking the logarithm of the above equation and rearranging terms yields Equation (10).

$$\gamma ln{\displaystyle \frac{U_A^2}{\nu \omega }}=\alpha lnh+lnC$$

(10)

After rearrangement, Equation (11) is obtained.

$$ln{\displaystyle \frac{U_A^2}{\nu \omega }}={\displaystyle \frac{\alpha }{\gamma }}lnh+{\displaystyle \frac{lnC}{\gamma }}$$

(11)

By treating $ln(U_A^2/\left(\nu \omega \right))$ as a linear function of $lnh$, the coefficient $\alpha /\gamma$ can be fitted using the least squares method based on experimental data.

Following the above fitting principle, MATLAB was used to fit the experimental data (covering different flame heights and acoustic frequencies). The calibration was performed using experimental data from butane gas diffusion flames. The average value from multiple fitting results was taken, yielding $\alpha =0.6868$, and the fitting plot is shown in Figure 2.

As shown in Figure 2, the measured and fitted values of the critical acoustic particle velocity for extinction agree well for butane gas diffusion flames when $\alpha =0.6868$ (the slope of the fitted line in Figure 7 is 2.0603, which corresponds to $\alpha /\gamma$, allowing the calculation of $\alpha$).

Furthermore, to verify whether this value is applicable to other fuels, experiments were repeated using methane and propane gas fuels, substituting $\alpha =0.6868$ for validation, as shown in Figure 3.

It can be seen from Figure 3 that this value exhibits good applicability, with MREs all below 0.03. Therefore, $\alpha =0.6868$ is determined as the empirical flame height parameter for the experimental environment in this study.

It is worth noting that the value of $\alpha$ is close to the exponent ${\displaystyle \frac{2}{3}}$ for $U_A$, suggesting a possible proportional relationship between ${\displaystyle \frac{U_A}{{\omega }^{0.5}}}$ and the flame height h. Consequently, at the same frequency, there may exist a proportional relationship between the acoustic-induced particle velocity $U_A$ and the flame height h. To verify this hypothesis, a zero-intercept linear fit was performed for the independent variable h and dependent variable ${\displaystyle \frac{U_A}{{\omega }^{0.5}}}$ based on experimental data from butane diffusion flames, and the residual vs. fitted value scatter plot and Q-Q plot were drawn, as shown in Figure 4, Figure 5 and Figure 6.

As shown in Figure 4, Figure 5 and Figure 6, the experimental data points (blue) closely distribute around the fitted line (red). The residuals are randomly scattered around the $y=0$ line without apparent systematic trends, and they approximately follow a normal distribution. The above analysis indicates a significant proportional relationship between ${\displaystyle \frac{U_A}{{\omega }^{0.5}}}$ and the flame height h. The same method was applied to analyze experimental data for methane and propane, both diffusion and premixed flames. The metrics of the zero-intercept linear regression models are summarized in Table 1.

From the above model metrics, it can be seen that under the experimental conditions of this study, a significant proportional relationship exists between ${\displaystyle \frac{U_A}{{\omega }^{0.5}}}$ and the flame height h for all tested fuels and flame types. It should be noted that the flame heights and frequencies investigated in this work are relatively limited. The above conclusion is valid only within the studied data range and experimental environment. In the subsequent model calculations, $\alpha =0.6868$ is still used as the empirical flame height parameter to enhance the model’s universality. Nonetheless, this finding remains noteworthy.

2.3. Determination of the Prandtl Number Exponent $\delta$ and the Critical Extinction Criterion ${\Theta }_A^{\prime }$ for Diffusion Flames

After determining $\alpha =0.6868$, the determination and validation of the Prandtl number exponent $\delta$ are further considered. From Equation (7), if the experimental environment remains unchanged, ${\Theta }_A^{\prime }$ should be constant for different fuels, and the values of B and Pr are determined under the same experimental conditions. For a given value of $\delta$, the average of ${\Theta }_A^{\prime }$ calculated within a specific fuel is taken as the ${\Theta }_A^{\prime }$ value for that fuel under this $\delta$. Using this method, the functions of ${\Theta }_A^{\prime }$ versus $\delta$ for the three fuels are obtained. Since ${\Theta }_A^{\prime }$ should remain the same across different fuels, the $\delta$ value that minimizes the coefficient of variation of ${\Theta }_A^{\prime }$ is selected, simultaneously yielding ${\Theta }_A^{\prime }$, as illustrated in Figure 7.

Figure 7. Determination of $\delta$ and ${\Theta }_A^{\prime }$ ($\delta =4.4356$; ${\Theta }_A^{\prime }=0.0817$).

Figure 7. Determination of $\delta$ and ${\Theta }_A^{\prime }$ ($\delta =4.4356$; ${\Theta }_A^{\prime }=0.0817$).

It is important to note that the derived Prandtl number exponent $\delta$ is positive and relatively large in magnitude. This indicates that an increase in Pr significantly reduces the critical particle velocity for extinction. This is because a decrease in thermal diffusivity makes it more difficult for the heat of the cooled unburned gas, after the flame is displaced from the fuel bed by acoustic disturbance, to be promptly replenished from the flame zone. Consequently, the flame-fuel cycle becomes more susceptible to disruption. Furthermore, the influence of this thermal diffusivity is found to be extremely significant.

2.3.1. Validation of the Critical Extinction Criterion Model for Diffusion Flames

At this stage, the theoretical critical acoustic particle velocity for extinction can be calculated for diffusion flames. Selecting data where different fuels share the same flame height or acoustic frequency, the universality of the critical extinction criterion model under different experimental conditions was validated. It was found that the model is applicable to all tested experimental conditions, as shown in Figure 8.

2.3.2. Generalization of Empirical Parameters for Premixed Flames and Determination of Flame Type Term k

For premixed flames, the calculation method for the mass transfer number B is first modified. Since in premixed flames the fuel and air are pre-mixed, the limiting factor for their burning rate differs from that of diffusion flames, which is limited by the air diffusion rate. Consequently, the terms $Y_{O_2,\infty }$ and r in Equation (2) for the mass transfer number of gaseous fuel diffusion flames are removed. The equivalence ratio $\varphi$ (defined as the ratio of the theoretical air required for complete combustion to the actual air supplied) is used to describe the calorific value per unit mass of the premixed gas. Simultaneously, considering that unrealistically large mass transfer numbers may occur when the equivalence ratio is small, a combustion efficiency factor $\eta \left(\varphi \right)$ is introduced, as shown in Equation (12).

$$B^{\prime }=\eta \left(\varphi \right){\displaystyle \frac{\Delta h_c/\varphi }{M}}$$

(12)

For the critical extinction criterion ${\Theta }_A^{\prime }$ of gaseous fuel premixed flames, $B^{\prime }$ is substituted for calculation, and different values of the flame type term k are considered. Since the equivalence ratio was controlled at $\varphi \approx 1$ in the experiments, $\eta \left(\varphi \right)=1$ is assumed by default. The determination of $\eta \left(\varphi \right)$ for a wider range of equivalence ratios will be specifically addressed in future research (note: the fitting of the height term discussed later does not involve the calculation of B; the description of $B^{\prime }$ for premixed flames is presented here for narrative convenience).

To verify the applicability of the empirical parameters to premixed flames, the flame height term $\alpha$ is first generalized. The fitted values calculated using $\alpha$ are compared with the actual experimental data for acoustic extinction of methane and propane premixed flames, as shown in Figure 9.

From Figure 9, it can be seen that the diffusion flame height term $\alpha$ also shows strong applicability to premixed flames under the experimental conditions. Therefore, by substituting the empirical parameters obtained from the diffusion flame study along with the critical particle velocity data for gaseous fuel premixed flame extinction, the flame type terms for methane and propane gas premixed flames at $\varphi \approx 1$ are derived as $k_M=3.7975$ and $k_P=2.8123$, respectively. The k values for premixed flames calculated using the premixed flame experimental data and the empirical parameters are shown in Figure 10.

When the flame type and equivalence ratio remain constant, the k value is relatively stable across different frequencies. This indirectly reflects the universality of the empirical parameters determined under diffusion flame conditions when applied to premixed flames. It should be noted that Figure 10 is a summary of k values calculated separately under various factors such as different flame heights and acoustic frequencies, aiming to verify the applicability of the empirical parameters to different experimental conditions and the stability of the k value.

From Figure 10, it is evident that the k values for methane and propane premixed flames are not identical. This is primarily because the original flame-fuel cycle model was designed for diffusion flames rather than premixed flames. There are significant differences between the two in terms of burning rate and flame structure, and premixed flames of different fuels possess fundamentally different physicochemical characteristics. Nevertheless, this experiment still demonstrates that the empirical parameters determined in the diffusion flame extinction criterion study can be applied to premixed flames as well. Moreover, for premixed flames of the same fuel, the k value can be considered identical. This marks an important step towards unifying the acoustic extinction mechanisms for diffusion and premixed flames.

2.3.3. Determination of the Critical Extinction Criterion ${\Theta }_A^{\prime }$

After completing the above work, the ${\Theta }_A^{\prime }$ values for different gaseous fuels and combustion types are plotted as shown in Figure 11.

As shown in Figure 11, the critical extinction criterion ${\Theta }_A^{\prime }$ values for different fuels and flame types exhibit a stable and concentrated distribution. The value ${\Theta }_A^{\prime }=0.0817$ is adopted as the critical extinction criterion for gaseous fuel flames. It is noteworthy that this value was determined alongside $\delta$ in the diffusion flame experiments, and the determination process did not involve premixed flames. Nevertheless, after theoretical correction for premixed flames, this value is applicable to both diffusion flames and premixed flames at $\varphi \approx 1$ (where $B^{\prime }$ should be substituted for premixed flames).

Finally, the generalized quantitative critical extinction criterion formula for acoustic flame extinction based on the flame-fuel cycle model is as follows:

$$0.0817=k{\displaystyle \frac{U_A^{2\gamma }{Pr}^{\delta }}{h^{\alpha }{\left(\nu \omega \right)}^{\gamma }B}}=k{\displaystyle \frac{U_A^{2/3}{Pr}^{4.4356}}{h^{0.6868}{\left(\nu \omega \right)}^{1/3}B}}$$

(13)

where the flame type term $k=1$ for diffusion flames; for methane and propane gas premixed flames at $\varphi \approx 1$, the values are $k_M=3.7975$ and $k_P=2.8123$, respectively.

2.3.4. Comparison of Critical Particle Velocities between Premixed and Diffusion Flames

Furthermore, a comparison is made of the critical particle velocities required for acoustic extinction between methane and propane diffusion flames and premixed flames at $\varphi \approx 1$ under the same flame height $h=3.00\mathrm{c}\mathrm{m}$ and identical acoustic frequency, as shown in Figure 12.

In Figure 12, it can be clearly observed that, under otherwise identical experimental conditions, the critical particle velocity required to extinguish premixed flames with $\varphi \approx 1$ is significantly higher than that for diffusion flames. It is important to note that substituting the previously determined premixed flame type term $k>1$ into Equation (7) does not imply a decrease in the critical particle velocity. The primary reason for the increased critical particle velocity for premixed flames here is the significant increase in the mass transfer number B and the kinematic viscosity of air $\nu$. The magnitude of increase in these two factors exceeds the actual increase in the critical particle velocity observed in the premixed flame experiments. The condition $k>1$ precisely reflects the more macroscopic adjustment role of the flame type term in calculating the critical particle velocity for different flame combustion types.

3. Discussion and Implications of Derived Phenomena

3.1. Dual Effect of Acoustic Frequency on Extinction Effectiveness

According to the above theoretical derivation and experimental results, it can be seen that a higher acoustic frequency requires a larger critical particle velocity $U_A$ for extinction. However, data analysis from Figure 13 on the influence of different acoustic frequencies on extinction time reveals that, under the condition of the same particle velocity where extinction is achievable, acoustic waves in the 6085$\mathrm{Hz}$ range lead to shorter extinction times.

This reveals the dual effect of acoustic frequency on extinction effectiveness. A possible explanation for this phenomenon is the existence of a Stokes viscous boundary layer of certain thickness on the flame surface. Within a certain range, a higher acoustic frequency results in a thinner boundary layer, which in turn requires a larger velocity gradient for the acoustic wave to penetrate [12]. When extinction is achievable, for the same root-mean-square particle velocity, acoustic waves at 6085$\mathrm{Hz}$, possibly due to a resonant coupling effect, amplify flame instability, consequently leading to the fastest extinction speed and higher extinction efficiency. This phenomenon holds significant implications for the specific selection of acoustic frequency in different application scenarios and for revealing the fundamental principle of acoustic wave extinction. It should be a major focus of future research. Here, only a preliminary discussion and conjecture are provided.

It is important to note that the optimal frequency range for the fastest flame extinction here is considerably influenced by flame type and fuel state. The conclusions drawn in this study are representative primarily for gaseous hydrocarbon fuels, especially low-carbon alkanes.

3.2. Acoustically Induced Flame Necking and Fracture

Observations from high-speed camera experiments reveal that, due to acoustic interference during upward development, diffusion flames exhibit localized significant thinning, which subsequently leads to partial flame fracture. We term this phenomenon “acoustically induced flame necking and fracture.” To investigate this phenomenon in depth, while keeping other conditions constant, a comparison of the single-cycle flame necking and fracture behavior was conducted for propane diffusion flames with heights of 3.00$\mathrm{c}$$\mathrm{m}$, 4.00 $\mathrm{c}$$\mathrm{m}$, and 5.00 $\mathrm{c}$$\mathrm{m}$ under a 30 $\mathrm{Hz}$ acoustic frequency, as shown in Figure 14.

From the above phenomena, it can be clearly observed that as the flame height continuously increases, the volume of the flame separated from the main body correspondingly grows. A possible reason for this phenomenon is that the acoustically induced oscillatory flow excites flame instabilities (such as the Kelvin-Helmholtz instability), leading to strong shear and stretching effects on the flame surface, and generating vortex structures that cause flame detachment, thereby resulting in flame necking and fracture. This phenomenon is highly instructive for exploring the mechanism of acoustic flame extinction from the perspectives of flame stability and fluid dynamics. It contributes to a deeper understanding of the intrinsic influence and action process of acoustically induced oscillatory flow on flame combustion and can serve as a new starting point for future research endeavors.

3.3. Anomalous Increase in Critical Particle Velocity for Extinction at and above 90 Hz

It was observed that in the aforementioned experiments, the critical particle velocity for extinguishing methane and propane diffusion flames exhibited a trend where the actual increase exceeded the theoretical value as the acoustic frequency increased. Although the deviation of the theoretical value remains within an acceptable range for the primary frequency range of acoustic extinction at and below 80 $\mathrm{Hz}$, this suggests that the flame-fuel cycle model proposed by Friedman, A. N. [8] has a certain applicable scope. Experiments were conducted to compare the theoretical and experimental values for diffusion and premixed flames of methane and propane gaseous fuels under a broader range of acoustic frequency conditions.

As shown in Figure 15, Figure 16, Figure 17 and Figure 18, the growth rate of the critical particle velocity for extinction at and above 90 $\mathrm{Hz}$ accelerates significantly with increasing frequency, deviating completely from the theoretical model values. It should be noted that the research work of Friedman, A. N. [8] was concentrated within the very narrow frequency range of 3050 $\mathrm{Hz}$, and existing studies on acoustic flame extinction [3,5,6,7] have also mostly focused on the extremely low-frequency environment of 3080 $\mathrm{Hz}$. The analysis of this phenomenon—the significant increase in critical particle velocity for extinction above 90 $\mathrm{Hz}$—not only provides important assistance in complementing the applicable scope of the flame-fuel cycle model proposed by Friedman, A. N. [8], but also holds significant supplementary and heuristic value for investigating the mechanisms of acoustic flame extinction in the relatively high-frequency range.

4. Conclusion

Addressing the research gap in the multi-factor quantitative critical extinction criterion for acoustic flame extinction, this study adopts the acoustic flame extinction model centered on the flame-fuel cycle proposed by Friedman, A.N. [8]. First, its physical logic chain was revised and supplemented. Subsequently, by constructing the Spalding B-number for gaseous fuel combustion and introducing parameters such as the flame height term, flame size term $f(h,S)$, and combustion type term k, the original theory was substantially extended to incorporate flame size, flame type, and fuel state. Through a rigorous and well-designed experimental framework involving fitting, validation, generalization, and re-validation, we overcame the limitation in [8]—where variations in $\delta$ had to be neglected to derive the critical criterion ${\Theta }_A$, restricting its applicability to fuels with similar Prandtl numbers—and incorporated additional key factors influencing the critical acoustic particle velocity for extinction into a generalized acoustic extinction criterion model. Through multiple sets of experiments conducted under various conditions, the flame height exponent $\alpha =0.6868$, the Prandtl number exponent $\delta =4.4356$, and the flame type terms $k_M=3.7975$ and $k_P=2.8123$ for stoichiometric $\varphi \approx 1$ methane and propane premixed flames were determined. Ultimately, the critical extinction criterion for gaseous fuel flames, ${\Theta }_A^{\prime }=0.0817$, was established, successfully constructing a quantitative formula for calculating the critical acoustic particle velocity under the influence of multiple factors such as flame size and flame type for gaseous fuels. This generalized acoustic extinction criterion model is not merely an explanatory formula for acoustic flame extinction but also a “computational tool” capable of predicting the acoustic parameters required for extinction across different fuels, flame sizes, and frequencies. However, it must be noted that the specific functional form of the flame area term $g\left(S\right)$ within the model has not yet been experimentally validated. Future research should determine the functional form of $g\left(S\right)$ by varying the fuel bed dimensions, thereby extending the model to more general combustion scenarios. Finally, other significant derivative phenomena observed in the experimental study—such as the dual effect of acoustic frequency and acoustically induced flame necking and fracture—which may further elucidate the fundamental mechanisms of acoustic flame extinction, were discussed. Additionally, the applicable scope of the flame-fuel cycle model proposed by Friedman, A. N. [8] was addressed, and the current gap in research on acoustic flame extinction at frequencies of 90 $\mathrm{Hz}$ and above was supplemented. Through the discussion of these numerous highly instructive derivative phenomena, this work provides a clear direction and a solid foundation for subsequent research.

CRediT Authorship Contribution Statement:

Guo Mengze: Conceptualization, Methodology, Validation, Investigation, Resources, Data Curation, Writing - Original Draft, Writing - Review & Editing, Project administration. Zhang Jieming: Methodology, Software, Validation, Formal analysis, Data Curation. Guo Yuning: Writing - Review & Editing, Supervision, Resources. Pan Jiani: Software, Writing - Original Draft, Visualization. Zhao Shumin: Writing - review & editing, Writing - original draft, Supervision, Methodology, Investigation. Deng Qinghua: Resources, Funding Acquisition, Supervision. Fang Aiping: Resources, Supervision, Funding Acquisition. Cao Wen: Writing - Review & Editing, Supervision, Resources, Project Administration, Funding Acquisition. 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.