Self-Excited Thermoacoustic Instability Behavior of a Hedge Premixed Combustion System with an Asymmetric Air/Fuel Supply or Combustion Condition

Abstract

Self-excited thermoacoustic instability (SETAI) is an undesirable and dangerous phenomenon in combustion systems. However, its control is difficult, thus greatly limiting the development of combustion technology. Our previous works clarified how the premixed chamber length (L_P) and equivalence ratio (φ) influence SETAI behavior in a symmetrical hedge premixed combustion system. On real-world sites, however, the supply structure or combustion condition in a multi-flame system could be asymmetric due to space limitations or combustion adjustment needs. This paper aims to clarify the SETAI behavior of a combustion system with an asymmetric supply structure or an asymmetric combustion condition. The results indicate that the sound pressure amplitude under strong oscillation can reach 160 dB, which is about 5% of the total pressure. The SETAI state under the asymmetric condition is determined by the coupling between the heat release oscillation and sound pressure oscillation on each side and their cooperation. The asymmetric supply structure leads to asynchronous heat release oscillations between the two sides; it may be that one promotes oscillation and that the other suppresses it, or that both have a promotion effect but with asynchronous action, thus partly canceling each other out to lower the system’s oscillation intensity. This brings an advantage for controlling SETAI, which can be achieved by only changing one side of the structure. The oscillation amplitude can be reduced by 80–90% by appropriately changing one L_P only by ~20%. Under an asymmetric combustion condition with φ differing between the two sides, the heat release oscillation on each side is dependent on the local φ but not the global φ. Consequently, SETAI can also be controlled by changing the distribution but maintaining a constant fuel feeding rate and φ. The concepts identified in this paper demonstrate that SETAI can be effectively controlled by adopting an asymmetric φ distribution or an asymmetric structure of the supply system. This provides a convenient SETAI control approach without affecting the equipment’s thermal performance.

1. Introduction

Self-excited thermoacoustic instability (SETAI) is a very undesirable phenomenon for gas- and liquid-fired equipment [1,2,3]. The accompanying strong vibration causes large noise, equipment damage, failure, and even personal safety accidents [4]. SETAI is more likely to occur in high-intensity combustion systems, where the released heat cannot be transferred away immediately, such as gas turbines, aero-engines, low-nitrogen boilers, and even compact gas water heaters.

While SETAI has been investigated for more than 200 years, this tough problem is still the main challenge limiting the development of combustion technology in practice. The famous Rayleigh criterion [5] proposed that the coupling between pressure oscillation and reaction heat release oscillation is the main cause of SETAI, and this coupling will encourage SETAI if the phase difference is less than 90°. This criterion has been widely confirmed and acknowledged by various scholars [1,2,6,7,8]. However, the oscillation phase difference is very microscopic, and it is virtually impossible to use operation parameters for SETAI prediction via the Rayleigh criterion.

SETAI behavior has been proven to be affected by the equipment’s structure and operation parameters, which provided the chance for SETAI control, but conflicting results have been observed with regard to the effect of the heat output [9,10,11], equivalent ratio (φ) [12,13,14], and system structure [15,16,17,18]. Due to insufficient knowledge, SETAI control is generally achieved via empirical or semi-empirical methods, mainly classified into two kinds of approaches. One is to increase the system’s acoustic damping [3,15,19,20], such as by adding quarter-wave tubes, half-wave tubes, Helmholtz resonators, perforated plates, and perforated liners. The other is to affect the coupling between the sound pressure oscillation and heat release oscillation to obtain an out-of-step oscillation by changing the combustion parameters or structure dimensions [4,9,11,13,20,21], or by adopting a flow jet [15,22,23]. These detailed control strategies always require a number of costly tentative retrofits of the combustion system, which also influence the combustion or thermal performance of the equipment. More unfortunately, when the operation condition is changed, the verified control approach could become invalid.

To accurately and conveniently control SETAI, it is necessary to clarify the mechanism by which the phase relation between the sound pressure oscillation and reaction heat release oscillation is affected during combustion. Fully relating this phase difference to the operation parameters and structure dimensions is the key to solving this tough problem. Our previous work demonstrated that the premixed supply duct length (L_P), fuel flow rate (V_F), and φ have a coupled effect that allows for the identification of the phase difference between pressure and heat release oscillation [24,25] by introducing the flow rate oscillation at the combustion chamber inlet. This flow rate oscillation is caused by the pressure oscillation, and their phase difference [8] is determined by the sound pressure mode shape within the premixed duct [2], which is further collaboratively influenced by L_P and V_F. However, this flow rate oscillation could cause heat release oscillation, and there is also a phase difference between them due to the time needed for the combustion reaction. Moreover, the heat release delay time relative to the flow rate oscillation (τ) [26] is mainly influenced by the combustion φ. In such a mode, these three parameters ultimately affect the coupling between sound pressure and heat release oscillation, thus determining SETAI behavior.

This mechanism has been proven by many experiments conducted on a hedge premixed combustion system with a symmetrical air/fuel supply. With that symmetrical supply structure and combustion condition, the sound pressure distribution within the two sides of the premixed chamber is identical. Therefore, the flow rate oscillation is in phase between each side; thus, the resulting thermos-acoustic coupling effect synchronously promotes or suppresses the sound pressure oscillation. On real-world sites, however, the supply system could be asymmetric due to space limitations. The total V_F or φ cannot be changed freely, as they seriously affect the system’s heat output and combustion performance. In comparison, it is more feasible to change the combustion condition on each side under a constant total V_F and global φ by adopting an asymmetric combustion condition. Both the asymmetric supply structure and asymmetric combustion condition could lead the thermos-acoustic coupling to be out of step between each side, potentially changing the system’s SETAI state. However, there are very limited studies about how these asymmetric conditions influence SETAI behavior in a multi-flame combustion system. As such, this paper aims to explore SETAI behavior with an asymmetric supply structure or an asymmetric combustion condition via experiments. The influences of these asymmetric conditions are further discussed to provide theoretical guidance for controlling SETAI in practice.

2. Experimental Methodology

2.1. Experimental Systems and Measurement

The experimental system is depicted in Figure 1, whose primary components include a combustion chamber and two premixed chambers. The air was supplied by a compressor and the fuel (propane) was provided by gas cylinders. Their flow rates were regulated by mass flowmeters with a range of 5 and 50 L/min, respectively, for propane and air. The gases were mixed in the two premixed chambers and then entered into the combustion chamber after passing through a flame stabilizer. The combustion chamber was vertically placed and the two premixed chambers were arranged horizontally and connected to the combustion chamber.

The temperature distribution inside the combustion chamber was measured by placing 9 k-type thermocouples (with a maximum of 1300 K) along its vertical axis at the centerline. Seven dynamic pressure sensors (with a maximum ±10 kPa and 20 kHz for the measurement range and frequency range, respectively) were utilized to capture the pressures inside the combustion chamber and premixed chambers. During the experiment, the temperature within combustion chamber will continue to rise after ignition. After a duration of 20–30 min, the temperature field reaches a relatively steady distribution. Temperatures were manually recorded from those thermocouple indications, and the pressure signals were subsequently collected at a frequency of 10 kHz and recorded using a data acquisition system.

2.2. Data Analysis

The measured pressure variation with time (with frequency of 10 kHz) can visually illustrate the pressure oscillation conditions. In addition, the spectral analysis of these pressure signals was conducted using the Fast Fourier Transform (FFT) to identify the dominant frequency (f_M) and its amplitude of sound pressure for a quantitative analysis. Here, the root-mean-square pressure (P_RMS, Pa) was used to characterize the oscillation amplitude, as given by Equation (1), and the pressure signals collected at P₅ (combustion chamber bottom) were selected for representing the oscillation behavior of system.

$$P_{\mathrm{RMS}}=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{p}_i{}^2}}$$

(1)

where N is the number of samples and p_i is the measured pressure.

Alternatively, the sound pressure level (SPL, dB) was also used for a broader amplitude comparison, as given in Equation (2).

$$\mathrm{SPL}=20\lg \frac{P_{\mathrm{RMS}}/\sqrt{2}}{p_{\mathrm{ref}}}$$

(2)

where p_ref is the reference value (2 × 10⁻⁵ Pa, here).

Under sustained-oscillation conditions, a standing wave was formed within the whole system, and the mode shape is similar to a quarter wave within the combustion chamber. Consequently, the acoustic resonant frequency of combustion chamber (f) can be estimated using Equation (3), which can be used to analyze the system oscillation behavior. Here, the wave mode shape within combustion chamber is not perfectly a quarter wave, and thus f could differ slightly from the true oscillation frequency.

$$f=\frac{1}{4\times {\displaystyle \sum _i{h}_i/c_{\mathrm{mix},}{}_i}}$$

(3)

where h_i and c_mix,i are the height and acoustic speed of the mixture gas of the ith section of the combustion chamber (divided into nine sections along height, as shown in Figure 1), respectively, and the outlet correction [27] is included in h₉.

The measured temperatures were used together with properties of mixture gas to estimate the c_mix,i, as shown in Equation (4).

$$c_{\mathrm{mix},i}=\sqrt{\mathrm{R}\frac{{\gamma }_{\mathrm{mix},i}{T}_i}{M_{\mathrm{mix}}}}$$

(4)

where $M_{\mathrm{mix}}$ is the molar mass of mixture (ignore the change in combustion chamber), ${\gamma }_{\mathrm{mix},i}$ is the adiabatic exponent of the mixture in the ith section, and $T_i$ is the temperature measured from the ith thermocouple.

In addition, the acoustic wavelength (λ_P) within the premixed chamber can be calculated by the ratio of c_mix within the premixed chamber to the oscillation frequency (f or f_M). This λ_P is crucial for determining the pressure mode shape before the combustion chamber inlet, and thus influences the oscillation behavior through the Rayleigh criterion [24,25], as is also discussed in Section 4.1.

2.3. Set of Cases

This study primarily investigated the SETAI behavior with an asymmetric gas supply structure or under asymmetric combustion conditions. The situations under symmetrical conditions were first examined as comparison basis, as shown by the detailed experimental cases in

Table 1. (a) Cases under symmetric supply structure and combustion condition. (b) Cases under asymmetric air/fuel supply structure. (c) Cases under asymmetric supply system with two premixed chambers differing by a half wavelength. (d) Cases under asymmetric air/fuel supply structure and asymmetric combustion condition. (e) Cases under asymmetric combustion condition (L₁ = L₂ = 1.85 m).

(a)
Cases	LP(m)			φ
1–12	0.55, 0.65, 0.86, 1.06, 1.25, 1.46, 1.65, 1.85, 2.06, 2.26, 2.46, 2.66			0.9
13–27	0.55, 0.65, 0.86, 1.06, 1.25, 1.46, 1.65, 1.85, 2.06, 2.26, 2.46, 2.66, 2.86, 3.06, 3.26			0.7
(b)
Cases	L1(m)	L2(m)		φ
28–38	2.06	0.55, 0.65, 0.86, 1.06, 1.25, 1.46, 1.65, 1.85, 2.26, 2.46, 2.66		0.9
39–48	1.65	0.55, 0.65, 0.86, 1.06, 1.25, 1.46, 1.85, 2.26, 2.46, 2.66		0.9
49–62	2.66	0.55, 0.65, 0.86, 1.06, 1.25, 1.46, 1.65, 1.85, 2.06, 2.26, 2.46, 2.86, 3.06, 3.26		0.7
(c)
Cases	L1(m)	L2(m)		φ
63	3.87	3.87		0.9
64	1.65	3.87		0.9
65	2.86	0.55		0.7
(d)
Cases	Premixed chamber 1		Premixed chamber 2
	L1(m)	φ1	L1(m)	φ2
66	2.66	0.9	1.65	0.7
67	2.66	0.7	1.65	0.9
(e)
Cases	φ1		φ2
68–71	0.7		0.8, 0.9, 1.1, 1.3
72–73	0.9		1.1, 1.3

a.

Previous work demonstrated that L_P and V_F could affect the sound pressure distributions to determine the state of SETAI, and the influence from changing L_P is more pronounced. Therefore, experiments under asymmetric gas supply structure with L_P differing on two sides were conducted, as shown by the cases in Table 1b (L₁ and L₂ represent the length of left and right premixed chamber, respectively).

In addition, previous work also demonstrated that increasing L_P by a half λ_P can result in similar oscillation conditions. With this, two additional cases were conducted to further clarify the mechanism of the asymmetric supply system influence on SETAI behavior, as shown by cases 64 and 65 in Table 1c.

On the other hand, φ also has a great impact on SETAI behavior, and thus two groups of experiments were conducted under asymmetric combustion conditions with φ differing on two sides, as shown in Table 1d,e (φ₁ and φ₂ are φ at left and right premixed chamber, respectively). All the cases in the present experiment were conducted under a constant V_F of 0.924 L/min, equally divided on both sides.

2.4. Uncertainty Analysis

In the present experiments, the measurement error of thermocouple and pressure sensor was ±50 Pa and ±6 K, respectively. The error of pressure sensor could cause an uncertainty of less than 2.5% for the oscillation amplitude under SETAI condition. Moreover, the error of thermocouple could introduce an uncertainty of 1 m/s for the sound speed, and then the resulting error for f was less than 0.5%. In addition, the measurement error of flowmeter was ±0.0174 L/min in propane flow rate and ±0.1006 L/min for air flow rate. These errors could contribute to a maximum error of 2.8% for φ.

3. Results

3.1. Oscillation Condition under Asymmetric Supply Structure

Figure 2 presents a comparison of sound pressure oscillation between an asymmetric supply structure (case 35) and two symmetric systems (cases 8 and 9). In case 35, L₁ is the same as the premixed chamber length in case 8, and L₂ is the same as the premixed chamber length in case 9. Both the symmetric structures exhibit strong oscillation, with frequencies of 70.4 and 72.6 Hz, respectively. In case 35, the sound pressures also demonstrate a strong oscillation, with its frequency spectrum showing only one peak at 72.4 Hz, rather than two separate oscillations at 70.4 and 72.6 Hz.

Figure 3 displays the oscillation characteristics of another asymmetric condition (case 32) in comparison to its corresponding symmetric systems (case 9 and 5). The frequency spectrum of case 32 exhibits a bimodal feature, similar to case 5. Additionally, an important observation is that the amplitude is lower for the asymmetric system compared to the two symmetric systems. Both Figure 2 and Figure 3 support the notion that the sound pressure oscillation behavior of an asymmetric system does not arise from the linear superposition of the two corresponding symmetric systems, and its frequency differs from either.

3.2. Oscillation Variation with Changing Length of One Side of Premixed Chamber

The dependence of oscillation on the length of one side of the premixed chamber was investigated while keeping the length of the other side fixed (cases 28–62 in Table 1b). The variation in sound pressure amplitude and frequency with L₂ was compared to the corresponding symmetric system (L₁ changing equally with L₂), as shown in Figure 4, Figure 5 and Figure 6. When L₁ was fixed at 2.06 m, the combustion successively underwent non-, intermittent, and continuous oscillation as L₂ increased from 0.55 to 2.06 m. Subsequently, it returned to non-oscillation at 2.26–2.46 m, initiating another cycle. The oscillation dependence on L₂ was similar to that on L_P under the symmetric supply systems [25]. Furthermore, for all conditions of continuous oscillation, the P_RMS under the asymmetric supply systems was consistently lower than that of the corresponding symmetric systems. The experiments conducted under φ = 0.7 and L₁ = 2.66 m also exhibited the same trend, as shown in Figure 5. The most intense oscillation for the asymmetric supply system, with L₁ fixed at 2.06 and 1.65 m, occurred in case 35 and 45, respectively, both with L₂ = 1.85 m. The P_RMS values were 30% and 42% lower than those of the symmetric supply system (L₁ = L₂ = 2.06 m), respectively.

Figure 6 reveals a few minor differences compared to Figure 4 and Figure 5. In case 45 (L₁ = 1.65 m, L₂ = 1.85 m) and case 63 (L₁ = 1.65 m, L₂ = 1.45 m), the oscillation intensity fell between those of the two corresponding symmetric systems. An intermittent oscillation occurs under case 60 (L₁ = 1.65 m, L₂ = 0.85 m), while both corresponding symmetric systems displayed sustained oscillation. The combination of 1.65 m (sustained oscillation in symmetric systems) and 2.26 m (non-oscillation in symmetric systems) resulted in sustained oscillation. These apparently irregular relationships indicate that the oscillation characteristic under asymmetric supply structures is not straightforward.

3.3. Oscillation Condition under Asymmetric Combustion Condition

Figure 7 illustrates the comparison of sound pressure oscillation and its frequency spectrum between two cases (case 66 and 67) with asymmetric supply structure and asymmetric combustion conditions. While for the supply system structure, the total V_F and global φ remain the same in both cases and case 67 exhibits strong oscillation, case 66 only displays combustion noise. The P_RMS under case 66 is approximately 1/10 of that under case 67.

4. Discussion

4.1. SETAI Behavior under Asymmetric Supply Structure

Previous investigations have shown that L_P and φ collectively influence the interaction between sound pressure oscillation and heat release oscillation, determining whether SETAI can be encouraged [9,25]. Specifically, when L_P/λ_P complies with Equation (5), the mode shape within the premixed chamber corresponds to a quarter to a half wave. In this scenario, a pressure node will exist upstream of the flame stabilizer, as the far end (inlet) of the premixed chamber acts as a pressure antinode due to its near-closed acoustic boundary condition. It should be noted that there is a potential error regarding the far end of the premixed duct not completely adhering to a closed boundary condition. Thus, when the instantaneous pressure at the flame stabilizer exceeds its average value, the pressure will increase in the flow direction. This pressure difference between the two sides of the flame stabilizer will reduce the mixture flow rate, causing a flow rate oscillation that is phase-led by 90° relative to the pressure oscillation. In contrast, the flow rate oscillation will be phase-lagged by 90° relative to the pressure oscillation when Equation (6) is satisfied. For combustion with φ = 0.9, τ satisfies Equation (7) (where T_M is the oscillation period). This indicates that the phase lag of heat release oscillation is less than 180° relative to the flow rate oscillation. In this case, the phase difference between sound pressure oscillation and heat release oscillation would be less than 90°, promoting the oscillation as long as Equation (5) is also satisfied. When Equations (6) and (7) are simultaneously satisfied, however, the sound pressure oscillation and heat release oscillation are out of phase, resulting in a negative Rayleigh criterion that suppresses oscillation. In the combustion of φ = 0.7, however, Equation (8) is satisfied, and therefore the condition of L_P/λ_P satisfying Equation (6) can result in an oscillation promotion, while Equation (5) results in oscillation suppression. In a symmetric supply structure, both sides have the same effect of either promoting or suppressing oscillation. In an asymmetric supply system, a different L_P value may lead to the interaction between heat release oscillation and sound pressure oscillation promoting each other on one side but suppressing each other on the other side.

$$\frac{2n+1}{4}<\frac{L_{\mathrm{P}}}{{\lambda }_{\mathrm{P}}}<\frac{2n+2}{4}$$

(5)

$$\frac{2n}{4}<\frac{L_{\mathrm{P}}}{{\lambda }_{\mathrm{P}}}<\frac{2n+1}{4}$$

(6)

$$\frac{2m{T}_{\mathrm{M}}}{2}<\tau <\frac{\left(2m+1\right)T_{\mathrm{M}}}{2}$$

(7)

$$\frac{\left(2m+1\right)T_{\mathrm{M}}}{2}<\tau <\frac{\left(2m+2\right)T_{\mathrm{M}}}{2}$$

(8)

The ratio of L_P to λ_P was calculated on each side to analyze the mechanism of encouraging or suppressing oscillation under an asymmetric supply structure, combining the results with those obtained from measuring P_RMS, as shown in

Table 2. Ratio of L_P to λ_P on each side and the P_RMS (L₁ = 2.06 m, φ₁ = φ₂ = 0.9).

Case	fM (Hz)	Premixed Chamber 1		Premixed Chamber 2			PRMS (Pa)
		L1/λP	Effects on Oscillation	L2 (m)	L2/λP	Effects on Oscillation
28	91.4	0.570	Suppression	0.55	0.152	Suppression	146
29	69.6	0.434	Promotion	0.65	0.137	Suppression	1591
30	72.0	0.449	Promotion	0.86	0.187	Suppression	1112
31	73.2	0.456	Promotion	1.06	0.235	Suppression	848
32	73.8	0.460	Promotion	1.25	0.279	Promotion	899
33	74.4	0.464	Promotion	1.46	0.329	Promotion	1289
34	73.6	0.459	Promotion	1.65	0.368	Promotion	1780
35	72.4	0.451	Promotion	1.85	0.405	Promotion	2857
36	87.2	0.544	Suppression	2.26	0.596	Suppression	362
37	94.0	0.586	Suppression	2.46	0.700	Suppression	324
38	71.6	0.446	Promotion	2.66	0.576	Suppression	1352
9 (Symmetric)	70.4	0.439	Promotion	2.06	0.439	Promotion	4108

,

Table 3. Ratio of L_P to λ_P on each side and the P_RMS (L₁ = 1.65 m, φ₁ = φ₂ = 0.9).

Case	fM (Hz)	Premixed Chamber 1		Premixed Chamber 2			PRMS (Pa)
		L1/λP	Effects on Oscillation	L2 (m)	L2/λP	Effects on Oscillation
39	88.4	0.441	Promotion	0.55	0.147	Suppression	123
40	80.6	0.403	Promotion	0.65	0.159	Suppression	597
41	73.4	0.367	Promotion	0.86	0.191	Suppression	972
42	74.2	0.371	Promotion	1.06	0.238	Suppression	927
43	83.0	0.414	Promotion	1.25	0.314	Promotion	1194
44	79.8	0.399	Promotion	1.46	0.353	Promotion	1922
45	75.2	0.376	Promotion	1.85	0.421	Promotion	2366
46	74.0	0.370	Promotion	2.26	0.506	Suppression	1013
47	81.2	0.406	Promotion	2.46	0.605	Suppression	374
48	76.8	0.384	Promotion	2.66	0.618	Suppression	867
7 (Symmetric)	75.2	0.376	Promotion	1.65	0.376	Promotion	2176

and

Table 4. Ratio of L_P to λ_P on each side and the P_RMS (L₁ = 2.66 m, φ₁ = φ₂ = 0.7).

Case	fM (Hz)	Premixed Chamber 1		Premixed Chamber 2			PRMS (Pa)
		L1/λP	Effects on Oscillation	L2 (m)	L2/λP	Effects on Oscillation
49	72.6	0.582	Promotion	0.55	0.120	Promotion	1866
50	72.8	0.584	Promotion	0.65	0.143	Promotion	1632
51	77.2	0.619	Promotion	0.86	0.200	Promotion	1054
52	79.6	0.638	Promotion	1.06	0.254	Suppression	832
53	81.0	0.650	Promotion	1.25	0.305	Suppression	1054
54	79.4	0.637	Promotion	1.46	0.349	Suppression	1387
55	52.0	0.417	Promotion	1.65	0.259	Suppression	181
56	78.6	0.630	Promotion	1.85	0.438	Suppression	174
57	78.4	0.629	Promotion	2.06	0.487	Suppression	169
58	76.4	0.613	Promotion	2.26	0.521	Promotion	180
59	75.4	0.605	Promotion	2.46	0.559	Promotion	177
60	72.4	0.581	Promotion	2.86	0.626	Promotion	2062
61	75.6	0.606	Promotion	3.06	0.697	Promotion	1083
62	80.0	0.642	Promotion	3.26	0.786	Suppression	1042
24 (Symmetric)	70.2	0.563	Promotion	2.66	0.563	Promotion	3367

. In asymmetric supply systems, the heat release oscillation is probably in a different phase between the two sides, generating two sources (OS₁ and OS₂, respectively) with different effects on pressure oscillation. Taking the results in Table 2 as an example, in the symmetric supply system with L₁ = L₂ = 2.06 m, L_P/λ_P is 0.439. Under this condition, both OS₁ and OS₂ promote oscillation, resulting in a strong oscillation with P_RMS as high as 4108 Pa. However, when L₂ is changed, OS₂ may suppress the oscillation, as observed in cases 29, 30, 31, and 38, resulting in an overall weak oscillation. Furthermore, it is notable that although OS₁ still has a strong promotion effect in cases 28, 36, and 37, these conditions only exhibit combustion noise without obvious oscillation.

The results in Table 3 and Table 4 also demonstrate the same regularity, where most cases show that the whole system can only be pushed into a strong oscillation when OS₂ has a promotion effect. An exceptional circumstance occurs in cases 58 and 59, where both OS₁ and OS₂ have a promotion effect, but the whole system does not exhibit oscillation. This may be because the sound pressure antinode is too close to the stabilizer [24], resulting in a relatively weak flow rate oscillation and muting the thermo-acoustic interaction. These observations indicate that, for an asymmetric supply structure, a necessary but not sufficient condition for strong oscillation is that both L₁ and L₂ satisfy oscillation promotion.

Table 5. Phase of sound pressure fluctuation at 7 measuring points (L₁ = 2.06 m, φ₁ = φ₂ = 0.9).

Case	L2 (m)	Phase of Sound Pressure Fluctuation (°)							PRMS (Pa)
		P1	P2	P3	P4	P5	P6	P7
28	0.55	174.2	−6.0	−17.7	−20.9	0	1.5	2.7	146
29	0.65	−168.8	5.6	−20.2	−22.6	0	2.5	1.9	1591
30	0.86	−171.4	4.3	−39.6	−44.7	0	1.9	1.9	1112
31	1.06	−172.8	3.5	−61.1	−72.3	0	1.6	1.8	848
32	1.25	−173.2	2.8	−47.6	−103.1	0	1.8	1.6	899
33	1.46	−174.4	1.3	35.6	−132.9	0	1.7	1.3	1289
34	1.65	−173.4	1.5	26.2	−148.0	0	2.0	1.3	1780
35	1.85	−169.8	3.5	20.2	−156.1	0	3.1	1.5	2857
36	2.26	176.0	−4.4	−7.6	169.7	0	0.7	1.9	362
37	2.46	170.9	−8.4	−29.0	141.3	0	1.0	1.7	324
38	2.66	−169.4	4.0	−3.5	174.6	0	3.5	1.9	1352
9 (Symmetric)	2.06	−162.4	9.3	17.6	−160	0	7.3	3.1	4108

lists the phase of sound pressure oscillation at seven measurement points under various cases. These data clearly demonstrate that the sound pressure differs within the two premixed chambers in an asymmetric supply structure. In particular, the pressure oscillation phase at the close end (P₂ and P₃) is very close in the symmetric supply structure but differs significantly in some asymmetric supply structure cases. This difference in pressure oscillation phase between P₂ and P₃ implies the asynchronized heat release oscillation between two inlets of the combustion chamber. In the symmetric supply system, the flow rate oscillation at the combustion chamber inlet is almost synchronous between the two sides, resulting in synchronous heat release oscillation. Therefore, the coupling between pressure and heat release oscillation is also the same between the two sides, and OS₁ and OS₂ have the same effect on the system’s SETAI.

For asymmetric supply systems, it is important to note that although OS₁ and OS₂ still experience the same pressure oscillation within the combustion chamber, their heat release oscillation may not be synchronized. Specifically, during certain periods within an oscillation cycle, the heat release oscillation may increase sound pressure on one side but have a reducing effect on the other side. In this case, there is a possibility that either OS₁ or OS₂ may individually encourage SETAI, but their promotional effects may partly cancel each other out. Consequently, the oscillation in an asymmetric supply system is generally weaker than in the two corresponding symmetric systems, as reflected in Table 2, Table 3 and Table 4 and Figure 4, Figure 5 and Figure 6. The non-synchronous sound pressure oscillation between the two premixed chambers is the reason for the reduced oscillation under the asymmetric supply structure, and thus the system SETAI could potentially be controlled simply by altering the length of one premixed chamber.

Figure 8 compares the sound pressure oscillation in case 64 with the two corresponding symmetric systems. The oscillation condition in case 64 (L₁= 1.65 m and L₂ = 3.87 m) is almost the same as in the two symmetric supply structures, in terms of both frequency and amplitude. This is unlike the phenomenon observed in Figure 2 or Figure 3. Case 64 was specially designed such that one premixed chamber is longer than the other by half λ_P. This implies that the pressure mode shape at the upstream position of the flame stabilizer is the same between the two sides, despite having different L_P. Under the same combustion conditions, a synchronous heat release oscillation was obtained between OS1 and OS2, resulting in pressure oscillation behavior that is as strong as that in the corresponding symmetric system. The same results were observed under the condition of φ = 0.7, as shown in Figure 9.

The oscillation mechanism of these two cases was further investigated by analyzing the corresponding L_P/λ_P and the sound pressure oscillation phase at each measurement point, as shown in

Table 6. Ratio of L_P to λ_P on each side and the P_RMS under cases 64 and 65.

Case	φ1 = φ2	fM (Hz)	Premixed Chamber 1		Premixed Chamber 2		PRMS (Pa)
			L1/λP	Effects on Oscillation	L2/λP	Effects on Oscillation
64	0.9	75.4	0.377	Promotion	0.883	Promotion	2145
65	0.7	71.2	0.616	Promotion	0.118	Promotion	2198

and

Table 7. Phase of sound pressure fluctuation at 7 measuring points under case 64 and 65.

Case	Phase of Sound Pressure Fluctuation (°)							PRMS (Pa)
	P1	P2	P3	P4	P5	P6	P7
64	−147.5	24.0	25.0	35.0	0.0	3.1	1.6	2145
65	162.2	−17.6	−18.8	−20.5	0.0	2.5	1.6	2198

, respectively. In each case, both OS₁ and OS₂ satisfy the requirements for prompting oscillation. Furthermore, what is significant is that the sound pressure oscillation is almost identical between P₂ and P₃, indicating an in-phase waveform of sound pressure that drives a synchronous flow rate oscillation between the two sides. With this and the same φ, the pressure oscillation promotion from OS₁ and OS₂ is synchronous, and hence their combined result is the same as that of the symmetric supply system. This further confirms that the system oscillation characteristic is determined by the effects on each side and their cooperation. The opposite phase between P₁ and P₄ is attributed to an additional half wave formed within the longer premixed chamber [18].

4.2. SETAI Behavior under Asymmetric Combustion Condition

Table 8. Ratio of L_P to λ_P on each side and the P_RMS under asymmetric combustion condition of cases 66 and 67.

Case	fM (Hz)	Premixed Chamber 1			Premixed Chamber 2			PRMS (Pa)
		L1/λP	φ1	Effects on Oscillation	L2/λP	φ1	Effects on Oscillation
66	85.0	0.637	0.9	Suppression	0.397	0.7	Suppression	436
67	79.4	0.684	0.7	Promotion	0.423	0.9	Promotion	1947

presents the ratio of L_P to λ_P on each side and the measured P_RMS for two different asymmetric combustion conditions, namely cases 66 and 67. In case 67, both OS₁ and OS₂ promote oscillation, but neither satisfies the requirement for promoting oscillation in case 66. This observation aligns with the findings illustrated in Figure 7, and it supports the notion that the occurrence of oscillation in an asymmetric condition is not determined by the overall φ but is dependent on the specific condition (combination of L_P/λ_P and φ) on each side. Under an asymmetric combustion condition where φ varies between the two sides, the value of τ relies on the local φ, rather than the global φ. This is logical since the heat release at each flame should be controlled by the local φ, rather than being influenced by the total φ [24,26]. Consequently, the heat release oscillation differs between the two sides, even when there is an identical flow rate oscillation. This discrepancy leads to an unsynchronized effect on the pressure experienced on both sides.

It can be hypothesized that altering φ on each side, even with a consistent total φ, may impact the oscillation characteristic of the system, similar to adopting an asymmetric structure for air/fuel supply. To validate this hypothesis, experiments were conducted under symmetric supply structures (L₁ = L₂ = 1.85 m) but with asymmetric combustion conditions, as detailed in Table 1e. Figure 10 illustrates the observed oscillation characteristic, while

Table 9. Ratio of L_P to λ_P on each side and the P_RMS under asymmetric combustion condition of case 68–73.

Case	fM (Hz)	Premixed Chamber 1			Premixed Chamber 2			PRMS (Pa)
		L1/λP	φ1	Effects on Oscillation	L2/λP	φ1	Effects on Oscillation
68	81.0	0.452	0.7	Suppression	0.453	0.8	Suppression	316
69	81.4	0.454	0.7	Suppression	0.456	0.9	Promotion	731
70	82.2	0.458	0.7	Suppression	0.462	1.1	Promotion	163
71	81.8	0.456	0.7	Suppression	0.462	1.3	Promotion	159
72	70.6	0.395	0.9	Promotion	0.397	1.1	Promotion	3120
73	70.4	0.394	0.9	Promotion	0.397	1.3	Promotion	2947

analyzes the respective effects of OS₁ and OS₂. Previous experiments have demonstrated that a φ range of 0.7–0.8 and 0.9–1.4, respectively, needs to be combined with an L_p/λ_P range of (2n/4, (2n + 1)/4) and (n/4, 2n/4) in order to generate a promotion effect. Among cases 68–73, where L_P is 1.85 m, φ values of 0.9, 1.1, and 1.3 satisfy this requirement. Consequently, both sides have a promotion effect in cases 72 and 73, which is consistent with the observations in Figure 10 and Table 7, where strong oscillation occurs under these two cases. In cases where OS₂ promotes oscillation but OS₁ suppresses it, weak oscillation was observed in case 69, while non-oscillation occurs in cases 70 and 71.

In the case of an asymmetric supply structure with a symmetric combustion condition, τ on two sides is identical, but the sound pressure waveform is different, resulting in distinct effects on system oscillation. In comparison, the situation for an asymmetric combustion condition differs in approach but yields similar results. Consequently, SETAI can also be controlled by generating different effects through the asymmetric φ distribution between the two sides. For instance, in case 72, where the global φ is 0.91, a strong oscillation is observed under the symmetric combustion condition (φ₁ = φ₂ = 0.9) with the same supply system. In case 71, however, the asymmetric combustion condition of φ₁ = 0.7 and φ₂ = 1.3 suppresses the oscillation, without modifying the total heat release or structure. This finding holds significant importance for controlling SETAI in actual combustion systems.

5. Conclusions

This paper investigates the SETAI behavior under a hedge premixed combustion condition, focusing on the influence of the asymmetric structure of the air/fuel supply system and the asymmetric combustion condition. To achieve this, experiments were conducted with different L_P or different φ on each side. The main conclusions are as follows:

(1)

The SETAI characteristic under an asymmetric condition is not simply the combination of the states under the two corresponding symmetric conditions. Instead, it is determined by the condition on each side and the cooperation between them in influencing sound pressure oscillation.

(2)

An asymmetric supply structure has different effects on two sides, potentially promoting oscillation on one side while suppressing it on the other side. The system with an asymmetric supply structure can generate strong oscillation only when the coupling between heat release oscillation and sound pressure oscillation on both sides promotes oscillation. Even so, the promotion effect may be asynchronous between the two sides, partly cancelling each other out and reducing the system’s oscillation amplitude. Adopting an asymmetric supply structure allows for control of SETAI by only changing the length of one side of the premixed chamber, which is more convenient than a symmetric structure distribution.

(3)

Under an asymmetric combustion condition with different φ on two sides, the oscillation of heat release on each side depends on the local φ, not the global φ. This causes different effects on two sides due to the different phase relation between heat release and sound pressure oscillation. The concepts identified in this paper demonstrate that SETAI can be effectively controlled by adopting an asymmetric φ distribution or an asymmetric structure of the supply system. This provides the potential to control SETAI without changing the fuel feeding rate, global φ, and other main combustion parameters, thus preserving the equipment’s thermal performance.

Further investigation into how asynchronous heat release oscillations are coupled to influence pressure can provide a better understanding of SETAI behavior under asymmetric conditions. In addition, the dependence of SETAI behavior on the supply structure before premixing will be investigated to develop a better asynchronous control technology for SETAI in multi-flame combustion systems.