Experimental Investigation on Self-Excited Thermoacoustic Instability in a Rijke Tube

Abstract

The experimental investigations into the thermoacoustic instability in a Rijke tube are presented. In order to capture the dynamics of the temperature, a single-ended tunable diode laser absorption spectroscopy (TDLAS) technique was developed, with a measurement rate of 5 kHz. The temperature was found to fluctuate periodically at a dominant frequency of 230 Hz, corresponding to the fundamental frequency of the Rijke tube used in the experiment. The flame chemiluminescence was detected by a high-speed camera to demonstrate flame response to thermoacoustic instability. It was evident that the flame front stretched regularly and had jagged edges. To quantitate the fluctuations of chemiluminescence intensity, the relative area was defined. According to the result, the intensity also oscillated at 230 Hz. Furthermore, the same feature was found in regard to pressure at the exit of the Rijke tube. Compared with temperature and chemiluminescent intensity, the pressure oscillations presented the most approximate standard waveform, as they suffered the least disruptions. The results indicated that the dominant frequencies of temperature, chemiluminescent intensity and pressure were consistent, in accordance with the fundamental frequency of the Rijke tube in the experiment. In addition, etalon effects on the TDLAS signals were mitigated efficiently by a lowpass filter.

1. Introduction

Thermoacoustic instability has drawn great attention in the design and operation of gas-turbine combustors [1]. It originates from the closed feedback between fluctuations of acoustic waves and heat release. The feedback typically consists of three steps depicted in Figure 1 [2]. Heat release rate fluctuations provide an energy source for acoustic waves, resulting in velocity and pressure oscillations propagating throughout combustors. Then, acoustic fluctuations stimulate the ratio of fuel to air variations, which contribute to further heat release rate fluctuations. Consequently, closed feedback is maintained. Under some conditions, the occurrence of thermoacoustic instability is inevitable and its resulting powerful pressure oscillations and significant temperature variations are potential dangerous for the structural strength and performance of gas turbines [3] and rocket engines [4]. Therefore, it necessitates comprehension of the essence of thermoacoustic instability to predict and control it.

Rijke tube, which is typically composed of an open-ended vertical or horizontal tube and a heat source inside it, has been widely employed to obtain the fundamental understanding of the dynamics in the thermoacoustic instability phenomena by numerous researchers. Bailey et al. [5] deduced the liner theory including driving and damping effects from the observation of two types of thermoacoustic instability. The predictions of the theory were shown to be in great agreement with experimental results. Hantschk et al. [6] simulated non-linear self-excited thermoacoustic instability numerically within different types of Rijke tubes. The oscillations of pressure were found to be stimulated at the tube’s natural frequencies. The obtained results were coincident with experiments. Li et al. [7] focused on characteristics of thermoacoustic behavior in Rijke-type combustors. It was found that oscillation amplitudes of pressure achieved the maximum in fuel-rich flames. Bittanti et al. [8] developed a model based on the physical analysis about heat release and gas dynamics to describe the thermoacoustic instability phenomenon in electrically heated Rijke tubes. For the purpose of predicting the amplitude and frequency of the thermoacoustic instability, Chatterjee et al. [9] captured the reaction flow physics in the Rijke tube by solving conservation equations numerically and compared the numerical and experimental results. Sujith et al. [10] reported the contributions of nonlinearity and non-normality to thermoacoustic instability in Rijke tubes. To acquire some understanding of the evolution of flow disturbances in initiating thermoacoustic instability, a generalized model of Rijke tubes was presented by Zhao et al. [11]. Furthermore, he developed a thermo-acoustic-piezo system based on Rijke-Zhao to extend the application for energy harvesting [12]. Matthew [13] combined the nonlinear governing equations with an optimized procedure to reveal the lowest initial energy which triggers self-sustained oscillations in a horizontal Rijke tube. Although the procedure is developed in a simple thermoacoustic system, it is still available for more complicated thermoacoustic models. Du et al. [14] studied the impacts of dispersion of heaters on Rijke tubes via altering heating property and the numerical simulation was performed to acquire temperature and flow field distribution with different heat sources. Zhang et al. [15] utilized the high-voltage microsecond plasma discharge for suppression of the thermoacoustic instability in Rijke tubes. More recently, Liu et al. [16] conducted experimental measurements upon the heat conduction transfer function during thermoacoustic instability in a horizonal Rijke tube. The findings demonstrated that the transfer function was influenced significantly by heater power as well as mean air flow velocity.

However, previous measurements are mainly concerned with pressure or velocity. The instantaneous temperature measurements in the Rijke tube are rarely reported. The primary reason is that the time response of traditional tools for measuring temperature, such as thermocouple, is too slow, typically lower than 100 Hz, to capture the dynamics of temperature in thermoacoustic instability, which usually varies at the rate of hundreds of Hz. Temperature is a crucial parameter for characterizing the thermoacoustic instability in the Rijke tube. As such, the accurate and fast measurement of temperature is of considerable importance.

Numerous optical measurement techniques have emerged and have made abundant contributions, one of which is the tunable diode laser absorption spectroscopy (TDLAS). Benefiting from its high resolution, high accuracy, non-intrusiveness and fast response, TDLAS has been widely applied in numerous fields, such as combustion diagnostics [17,18], environmental monitoring [19], biomass energy [20] and even human health [21] and so forth [22,23].

In this paper, the instantaneous temperature of thermoacoustic instability in a Rijke tube was measured by a single-ended TDLAS technique. The measurement rate was 5 kHz. Meanwhile, the flame chemiluminescence intensity was recorded by a high-speed camera, demonstrating flame response to the thermoacoustic instability. Additionally, the pressure at the exit of the Rijke tube was monitored by a microphone. The analysis of frequency of temperature, flame response and pressure was presented. Furthermore, the etalon effects on TDLAS raw signals were mitigated by a lowpass filter to enhance measurement accuracy.

2. Theorical Background

2.1. Thermoacoustic Instability

The basic principle of thermoacoustic instability was first described qualitatively by Rayleigh [24]. The attenuation or amplification of pressure fluctuations due to heat release is dependent on whether their phases are in or out. The Rayleigh criterion can be formulated mathematically as [25].

$${\displaystyle \underset{V}{\int }{\displaystyle \underset{\tau }{\int }{q}^{\prime }(\overrightarrow{x},t)p^{\prime }(\overrightarrow{x},t)}}dtd\overrightarrow{x}>{\displaystyle \underset{V}{\int }{\displaystyle \underset{\tau }{\int }{\displaystyle \sum _i{L}_i}(\overrightarrow{x},t)}}dtd\overrightarrow{x}$$

(1)

where q′ represents heat variation, p′ pressure oscillation, L_i i-th wave energy dissipation process, V volume of interest, τ period of the oscillation. This inequality shows that during every cycle, only when the energy addition by heat release to acoustic field surpasses the total energy dissipation resulting from oscillation, will the thermoacoustic instability be excited and strengthened. It should be noted that the Rayleigh criterion provides the basic understanding of thermoacoustic instability, but it is difficult to obtain more valuable information only relying on the criterion.

The thermoacoustic modeling includes acoustic models, flame models and entropy wave models [26]. In the following, only the models concerning entropy waves are revised as they are significantly related to temperature.

Figure 2 presents the interaction between an unsteady heat source and acoustic waves in a Rijke tube. The incident flow with mean velocity, ${\overline{\mathrm{u}}}_1$, mean temperature, ${\overline{\mathrm{T}}}_1$, and acoustic wave p′₁ travels through region 1. When the flow encounters the flame, i.e., heat source, q′, the partial acoustic wave will be reflected back to region 1 and the rest transmits through the flame and propagates downstream in region 2. Meanwhile, the entropy wave, s′, emerges in region 2. The mean velocity and temperature in region 2 are represented as ${\overline{\mathrm{u}}}_2$, ${\overline{\mathrm{T}}}_2$, respectively. Fluid parameters can be divided into the summation of an average term $\overline{()}$ and fluctuating term ()′, e.g., T = $\overline{\mathrm{T}}$ + T′, for small perturbations. It has been verified that entropy fluctuations result from density and acoustic oscillations [27]:

$$s^{\prime }=c_v\frac{p^{\prime }}{\overline{p}}-c_p\frac{{\rho }^{\prime }}{\overline{\rho }}$$

(2)

where s is entropy, p pressure, $\rho$ density, c_v specific heat at constant volume and c_p specific heat at constant pressure. Combining Equation (2) with ideal gas relation, we can obtain

$$\frac{s^{\prime }}{c_p}=\frac{T^{\prime }}{\overline{T}}-(\frac{\gamma -1}{\gamma })\frac{p^{\prime }}{\overline{p}}$$

(3)

where $\gamma =c_p{/c}_v$.

In most combustion flows, the relative variation in pressure is much smaller than that of temperature. It means the last term on the right of Equation (3) can be ignored and therefore $s^{\prime }\approx c_p{T}^{\prime }/\overline{T}$, which states the fluctuations of entropy waves, can be indicated by temperature oscillations. As such, temperature is a critical parameter that has to be measured for a deeper insight into thermoacoustic instability.

2.2. Tunable Diode Laser Absorption Spectroscopy

The absorption spectroscopy measurement technique is schematically shown in Figure 3. When a beam of light, whose center frequency is ν₀ and intensity is I₀, travels across a region of interest, its intensity will be attenuated to I_v due to the interaction between light and matter. The incident light, I₀, and the transmitted light, I_v are correlated by Beer-Lambert law:

$$I_v=I_{0,v}\exp \left(-k_{v}L\right)$$

(4)

where $k_v$ and L (cm) denote the absorption coefficient at frequency $v$ (cm⁻¹) and absorption length, respectively. The absorption coefficient is defined as

$$k_v=P_{total}\cdot X_i\cdot S\cdot \varphi (\nu )$$

(5)

where P_total (atm) represents total pressure, X_i mole concentration of absorption species, i, S (cm⁻² atm⁻¹) the linestrength of a specific absorption line, $\varphi (\nu )$ (cm) lineshape function, which can be normalized to unity, i.e., ${\displaystyle \int }\varphi (\nu )dv\equiv 1$, for every specific absorption line. Additionally, in most cases, the lineshape function can be accurately described by the Voigt function, which is the convolution of a Lorentzian function and a Gaussian function.

The linestrength of a specific absorption line at an arbitrary temperature is determined from partition function, low-state energy and the molecular reference temperature linestrength [28]:

$$\begin{array}{cc}\hfill S\left(T\right)=& S\left(T_0\right)\frac{Q\left(T_0\right)}{Q\left(T\right)}\left(\frac{T_0}{T}\right)exp\left[-\frac{hc{E}^{″}}{k}\left(\frac{1}{T}-\frac{1}{T_0}\right)\right]\hfill \\ \hfill & \times \left[1-exp\left(-\frac{hc{v}_0}{kT}\right)\right]{\left[1-exp\left(-\frac{hc{v}_0}{k{T}_0}\right)\right]}^{-1}\hfill \end{array}$$

(6)

where S(T₀) is the reference temperature linestrength, T₀ = 296 K, Q the partition function, E″(cm⁻¹) the lower-state energy and v₀ (cm⁻¹) the transition center frequency. The above mentioned parameters can be found in spectroscopic databases such as HITRAN [29]. h, c and k are Planck’s constant, speed of light and Boltzmann’s constant.

In TDLAS, the frequency of laser is tuned by current or temperature to scan across the overall or partial absorption profile of the targeted transitions, the scan rate ranging from a few of Hz to MHz. The absorption area, $A={{\displaystyle \int }}_0^L{{\displaystyle \int }}_{-\infty }^{+\infty }{k}_{v}dvdl$, is obtained when the transition is fully scanned across. Then, the gas temperature of interest is inferred from the dependence of linestrength on temperature by two-line thermometry [30].

$$\begin{array}{ll}\frac{A_1}{A_2}& =\frac{{\displaystyle {\int }_0^L{\displaystyle {\int }_{-\infty }^{+\infty }{k}_{v,1}dvdl}}}{{\displaystyle {\int }_0^L{\displaystyle {\int }_{-\infty }^{+\infty }{k}_{v,2}dvdl}}}=\frac{{\displaystyle {\int }_0^L{\displaystyle {\int }_{-\infty }^{+\infty }PX{S}_1(T){\varphi }_{v,1}(v)dvdl}}}{{\displaystyle {\int }_0^L{\displaystyle {\int }_{-\infty }^{+\infty }PX{S}_2(T){\varphi }_{v,2}(v)dvdl}}}\\ \hfill & =\frac{S_1(T)}{S_2(T)}=f(T)\end{array}$$

(7)

The ratio of absorption areas of two transitions is solely dependent on temperature.

3. Experimental Facility

Figure 4 demonstrates a schematic of the experiment facility employed to perform measurements of temperature, chemiluminescence and pressure in the Rijke tube during thermoacoustic instability. The setup consisted of a TDLAS system for temperature detection, a microphone for pressure detection, a high-speed camera for chemiluminescence detection, a Rijke tube with length of 800 mm and diameter of 30 mm and a Bunsen burner, whose exit diameter was 10 mm. Furthermore, a J-type thermocouple (not shown in the figure) in the Rijke tube was employed for temperature reference. The fundamental frequency of the Rijke tube employed in this experiment under standard conditions is 215 Hz, which is determined by $f=c/\left(2L_T\right)$ [31]. f is the fundamental frequency, c the sound speed, assuming to be 343 m/s and L_T the length of the Rijke tube. It is important to also mention that the fundamental frequency will be increased with the temperature rising.

The TDLAS system works as follows: a function generator (Rigol 4102, Rigol, Beijing, China) generated sawtooth waveforms to drive two diode laser controllers (Arroyo 6305, Arroyo), which enable the output of two DFB diode lasers’ (NEL, NTT, Tokoyo, Japan) wavelengths around 7444.35 cm⁻¹ and 7185.59 cm⁻¹, respectively, corresponding to two absorption transitions of H₂O. The time division multiplexing strategy was employed, and the repetition rate was 5 kHz to capture the fluctuations of temperature. The lights coming from the two DFB lasers were combined and then divided into two beams by a fiber splitter. One was directed through a Fabry–Perot interferometer (Thorlabs SA200-12B, Thorlabs, Newton, MA, USA) and detected by a photodetector (Thorlabs PDA20CS2, Thorlabs, Newton, MA, USA) to obtain the wavelength of lights. The other was directed to a customized fiber bundle. The fiber bundle contained a single mode fiber (9 μm core diameter) for transmitting the laser light, which was surrounded by six multimode fibers (105 μm core diameter) for collecting reflected lights and pitching it onto a photodetector. The fiber-coupled beam was collimated to a beam diameter of ~2 mm by a fiber collimator (Thorlabs F220APC-1310, Thorlabs, Newton, MA, USA) and passed through the region of interest. Subsequently, it was reflected by a flat mirror and coupled into multimode fibers by the collimator. Ultimately, the transmitted light was directed onto another photodetector. Signals from each photodetector were recorded by a 4-channel, 16-bit DAQ card (NI 9223, National Instruments, Austin, TX, USA) with a sample rate of 1 MS/s. Except for the measurement area, the laser lights travelled inside optical fibers.

The heat source was provided by a Bunsen burner with butane as fuel, with a volume flow rate of 200 mL/min. The exit of the Bunsen burner was located 200 mm away from one end of the Rijke tube, corresponding to a quarter of tuber length to enable the occurrence of thermoacoustic instability. The fuel and air were premixed before entering the burner. The chemiluminescence of flame was recorded by a high-speed camera (Fastcam SA-Z 2100K, Photron, Tokoyo, Japan) with 3000 fps to demonstrate the flame response to thermoacoustic instability. The pressure at the exit of the Rijke tube was monitored by a microphone to capture the variation in pressure with a measurement rate of 3 kHz.

4. Results and Discussions

4.1. Etalon Effects

Etalon effects originate from optical length differences when light travels through surfaces in an optical system, which can distort the true signals. In this experiment, cautions have been taken for laser alignment to avoid etalon effects as much as possible. Nevertheless, there are still some fringes induced by etalon effects remaining. Figure 5 shows partial measurement signals and the blue line represents the raw signal. It is obvious that there exist high-frequency interferences distorting the true signal. For the high-frequency etalon effects, it is suggested to mitigate them by averaging the raw signals [32]. This method is effective but has to sacrifice the measurement rate. In this work, the lowpass filter was conducted on the raw signals to relieve the effects and the cut-off frequency was set as 120 kHz weighing up the measurement accuracy and the removal of etalon effects. The pink line represents the filtered data, and it is clearly observed the filtered data is much smoother than the raw one. Figure 5b shows the increment in every equal interval sampling in the green dot square region in Figure 5a. Theoretically, the increment should be the same value, coinciding with the slope of sawtooth waveforms. However, the increment varies due to etalon effects and other noises. As the bule line in Figure 5b shows, the increment in raw data ranges from 0.032 to 0.058 V and its standard deviation, σ, is 0.008 V. In contrast, that of filtered data is in range of 0.045 to 0.050 V and 1 − σ is only 0.0018 V, which is less than a quarter of the former. It is proved that the etalon effects on the signals are relieved efficiently and the measurement signals suffer less distortions.

4.2. Temperature

The temperature fluctuations above flame, 30 mm from the Bunsen burner exit, during the thermoacoustic instability were captured by the single-ended TDLAS, with a measurement rate of 5 kHz. Figure 6 demonstrates the typical absorption spectrum of two H₂O transitions of interest centered at 7185.59 cm⁻¹ and 7444.35 cm⁻¹. The red dots represent measured absorption signals, while the bule lines are the Voigt profile best-fitting to the measured signals. Both fitting residuals of H₂O transitions are within 0.4%. The absorption areas of these two transitions, A₁ and A₂, can be deduced from the best-fitting curves. Then, the temperature is determined by using Equation (7). Figure 7 illustrates the measured temperature time histories during thermoacoustic instability with single-ended TDLAS and a J-type thermocouple in 1 s. It is evident that the temperature measured by TDLAS fluctuates from 1000 K to 1300 K in Figure 7a. The measurement by the thermocouple is stable at 1153 K due to its low response, making it fail to capture the dynamics of temperature. Figure 7b presents the zoom-in view of the green square region in Figure 7a. It can be found that the temperature fluctuates periodically, which is one of the main features of entropy waves.

The measured temperature uncertainty comes primarily from the uncertainty in the reference linestrength and Voigt fitting routine [33]. The uncertainty in the reference linestrength of 7185.59 cm⁻¹ and 7444.35 cm⁻¹ is less than 2% and 5%, respectively [29]. On the other hand, the residuals of the best fitting to absorption signals of these two transitions are no more than 0.4%. Therefore, the measurement uncertainty is about 5.41%.

In this work, both amplitude and frequency of temperature fluctuations are given equal attention. To find out the domain frequency of temperature, the short-time Fourier transform was performed on the measured temperature. Figure 8 displays the spectrogram of the temperature illustrating the frequency as a function of time. There exist a strong frequency at 230 Hz during the whole measurement corresponding to the fundamental frequency of the Rijke tube. Second harmonics of the frequency also appear. Furthermore, some frequencies with low amplitudes occur occasionally. Additionally, the frequency distribution explains why the temperature time series deviates from sinuous wave severely, as show in Figure 7b. This indicates that the local gas temperature suffers intricate perturbations, such as convection, within the inner wall of the Rijke tube.

4.3. Flame Response and Pressure

The flame can stretch regularly and the intensity of chemiluminescence changes greatly during thermoacoustic instability. Figure 9 presents the flame response to thermoacoustic instability in about one period. What is clearly seen is that the flame front has jagged edges due to the perturbations of pressure and velocity. The intensity of chemiluminescence fluctuated periodically and was most apparent at the exit of the burner. At t = 0 ms, the intensity was strongest and then began to grow weaker, reaching the bottom at around 1.98 ms. Subsequently, the strength underwent enhancement and peaked at approximately 4.29 ms. Immediately after that, the intensity declined and the “decline and ascent” process continued to repeat.

In order to describe the chemiluminescent intensity quantitively, only the intensity exceeding a certain threshold is reserved via binarization. The relative area is defined as the reserved intensity spatial distribution area within a fixed visual area (shown in the last subfigure of Figure 9). Figure 10a illuminates the fluctuations of a relative area. It is observed the area ranges from 1% to 4% periodically. The dominant frequency is near 230 Hz, as well as the appearance of the second and higher harmonics seen in Figure 10c. This suggests that the chemiluminescent intensity oscillates at a specific frequency.

The measurements of pressure at the exit of the Rijke tube are shown in Figure 10b. As we can see, the pressure varies from −20 Pa to 20 Pa, which approximates a standard sinuous waveform. This may result from the fact that no other perturbations were exerted on the acoustic propagation except the unsteady heat release. The dominant frequency of pressure coincides with that of relative area.

5. Conclusions and Outlook

An experimental investigation into the thermoacoustic instability of a Rijke tube was presented. To capture the fast dynamics of temperature, a single-ended TDLAS technique was designed and employed. The TDLAS measurement rate was 5 kHz, fast enough to capture temperature fluctuations. The etalon effects encountered during TDLAS measurements were mitigated efficiently by a lowpass filter. The measured temperature oscilliated from 1000 K to 1300 K at the dominant frequency of 230 Hz, which corresponds to the fundamental frequency of the Rijke tube. However, the temperature variations severely deviated the sinuous waveform. This indicates that the entropy waves produced from the unsteady heat release suffer complicated disturbances while propagating downstream. One of the main disturbances might be the heat convection between local gas and the inner wall of the Rijke tube.

The measurements of flame chemiluminescence demonstrate that the flame front has jagged edges and stretches regularly. Chemiluminescence intensity also fluctuated periodically at 230 Hz. Furthermore, the pressure at the exit of Rijke tube was also found to oscillate at the same frequency. Additionally, pressure oscillated the most standard sinuous waveform as it underwent the least perturbations in the Rijke tube compared with temperature and chemiluminescence. The dominant frequencies of temperature, chemiluminescence and pressure are consistent with each other, all agreeing with the fundamental frequency of the Rijke tube in the experiment.

In addition, the successful application of a single-ended TDLAS technique extends the measurement area to where the optical windows are very limited. Only one optical window is enough to employ this optical measurement technique. In future work, the single-ended TDLAS sensor will be improved to be accustomed to a more hostile environment, such as aeroengine combustors or industrial gas turbines, to perform measurements.