Experiments and Modelling on the E ff ect of an Adjustable Boundary on Thermoacoustic Stability

Audrey Blondé 1

CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich Sonneggstrasse 3, Zürich 8092, Switzerland

Bruno Schuermans 2

CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich Sonneggstrasse 3, Zürich 8092, Switzerland

Nicolas Noiray 3

CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich Sonneggstrasse 3, Zürich 8092, Switzerland

ABSTRACT Predicting the existence of thermoacoustic instabilities is a key step in the design of modern gas turbines and the choice of their operating conditions. The stability of a combustion system crucially depends on the acoustic boundary conditions. In order to systematically investigate the influence of these boundary conditions, a test facility with variable inlet and outlet geometry has been developed. Cold flow tests confirmed that the acoustic terminations allow for a change of the reflection coe ffi cient from close to -1 (open end) to 0 (anechoic) to 0.8 (almost close end) over a large frequency range. In the present work, we present the design of an adjustable exit boundary enabling a change in the thermoacoustic stability without modifying the flame operating conditions. Experiments have been conducted in a turbulent axial atmospheric combustor. The acoustic reflection coe ffi cients of the exit boundary are measured for di ff erent boundary geometries and the impact of these geometries on the flame stability is assessed. A parametrized model is derived and reproduces the experiments well.

1. INTRODUCTION

Thermoacoustic instabilities constitute a major issue in gas turbine industry. Oscillations of pressure generate these instabilities when acoustic pressure and heat release are in phase [1]. These oscillations can lead to severe mechanical damage. Predicting their existence is therefore a key step in the design of modern gas turbines. The acoustic boundary conditions of a combustor crucially impact the stability of the systems. As shown in [2], the growth rate of an instability is highly sensitive

1 ablonde@ethz.ch

2 bschuermans@ethz.ch

3 noirayn@ethz.ch

ante. noize 21-24 AUGUST SCOTTISH EVENT CAMPUS O ? . GLASGOW

to the outlet boundary condition. Several studies mention tunable acoustic boundary conditions to ensure thermoacoustic stability [3–5] but no further design rule is provided. In the suppression of thermoacoustic instabilities, Helmholtz dampers [6] or acoustic liners [7] can be used. Their designs are well-described. However, these devices are often e ffi cient for a small range of frequencies. Boundary conditions of a test rig can be tuned by adjusting the phase between loudspeakers [4]. Due to space constraints, this is not a viable option in gas turbines, yet. An area contraction can also be employed to suppress instabilities [8]. Actually, using the theory of Bechert on sound absorption [9], a properly designed orifice placed at the end of a pipe is a non-reflecting termination at a specific Mach number. For a given flow velocity, there is therefore an optimum area of the boundary orifice such that most of the acoustic energy is dissipated into vortex shedding. The Mach number dependence of quasi-steady responses of orifice is further demonstrated in [10]. This dependence is also presented in [11] where acoustic reflection coe ffi cients have been measured at various Mach numbers for a fixed nozzle geometry. This concept is used in the present study to design a tunable acoustic boundary condition. The present work presents the design of an adjustable boundary that enables the change of thermoacoustic stability. A 3D printed prototype has first been tested in cold conditions. This variable boundary has then been manufactured and tested in combustion conditions. In this paper, the experimental setups are first outlined. Second, the theoretical background is presented, highlighting the post-processing of acoustic data and the modelling of the boundary acoustic response. Subsequently, results are presented. Concluding remarks close the paper.

2. EXPERIMENTAL SETUP

2.1. Cold Measurements These experiments were conducted in a non-reactive test rig at atmospheric conditions. The experimental test rig is sketched in Figure 1. The test stand features an anechoic inlet in which the

Loudspeaker

Microphone Exhaust Nozzle

Anechoic Termination

Figure 1: Configuration of the impedance tube for cold measurements.

air stream is injected and straightened. Six targeted air mass flow rates are considered in this work. They are summarized in Table 1. At the targeted mass flow ˙ m 5 = 100 g / s , a fluctuation of ± 3% of the air mass flow is observed and explained by the air compressor limitations. The correct and actual mass flow rate is accounted for in the modelling. An impedance tube of cross section 62x62 mm 2 is attached to the anechoic termination. One BEYMA SW1600Nd loudspeaker and six 1 / 4” microphones (GRAS Type 46BD-FV) are placed on the tube. The loudspeaker generates acoustic forcing at discrete frequencies between 50Hz and 500Hz

Variable Air Mass Flow Rate (g / s)

˙ m 1 0

˙ m 2 10

˙ m 3 55.6

˙ m 4 70

˙ m 5 100

˙ m 6 140

Table 1: Targeted Air Mass Flow Rates.

to measure the reflection coe ffi cient of the boundary. The pressure fluctuations are captured by the microphones. At the downstream boundary of the test rig, a 3D printed exhaust nozzle is attached. It consists in an orifice in which a conical piston can be moved. By moving the piston, the cross section of the flow passage and therefore the Mach number at the orifice are changed. In what follows, the acoustic element {orifice + piston} will be referred as the exhaust nozzle. Five discrete positions of the piston are considered in this work (see Table 2).

Piston Position Opening Area (mm 2 )

1 595

2 882

3 1201

4 1484

5 1780

Table 2: Discrete positions of the piston and the corresponding opening areas.

2.2. Combustion Test Rig The experiments in hot conditions were conducted in a modular axial laboratory-scale combustor at atmospheric conditions. The experimental test rig is sketched in Figure 2. The test stand features

Microphone

Exhaust Nozzle

Air

Hot Gases

Natural Gas + H 2

Figure 2: Configuration of the atmospheric combustion test rig.

a plenum and a combustion chamber of 62x62 mm 2 cross section. It is terminated by an adjustable exhaust nozzle which design is inspired from the 3D printed version presented in the previous section. Its position is varied throughout the experiment such that the opening area varies. Three positions are considered and the corresponding opening areas are presented in Table 3. Interchangeable walls are

Piston Position Opening Area (mm 2 )

1 900

2 1400

3 1600

Table 3: Discrete positions of the piston and the corresponding opening areas in combustion conditions.

mounted on 250mm long water-cooled aluminium modules. These walls are either water-cooled aluminium plates (to support microphones, pressure sensors or the igniter) or quartz windows (to allow optical access). The plenum features an acoustically sti ff air injection. In the present study, the combustor is equipped with a swirled burner, is operated at 42.6kW and equivalence ratio of 0.8, and fueled with natural gas and hydrogen in technically premixed conditions. The power fraction of hydrogen is 12%. One 1 / 4” water-cooled microphone (GRAS Type 46BD-FV) is placed in the combustion chamber to measure the pressure fluctations and deduce the thermoacoustic stability state of the combustor.

3. THEORETICAL BACKGROUND

3.1. Multi-Microphone Method The reflection coe ffi cient R of the exhaust nozzle can be obtained from the pressure signals of N microphones using the multi-microphone method [3]. In this method, the acoustic pressure at the axial location x is expressed as the solution of the one-dimensional wave equation with mean flow :

ˆ p ( ω, x )

c x 1 + M + ˆ g ( ω ) e i ω

ρ c = ˆ f ( ω ) e − i ω

c x 1 − M (1)

Here, ˆ p denotes the complex amplitude of the Fourier transform of the acoustic pressure. ρ and c stand for density and speed of sound. The Riemann invariants ˆ f and ˆ g are integration constants obtained from initial and boundary conditions. ω is the angular frequency and M the flow Mach number. For brevity, the substitution ˆ ψ = ˆ p / ρ c is made. The pressure signals of the N microphones can then be expressed with respect to the Riemann invariants as: 

c x k . x k stands for the axial position of the microphone k with respect to the reference position characterizing the position of the acoustic element. The Riemann invariants ˆ f , ˆ g , solution of the over-determined system in Eq.2, are obtained using least-square inversion. The complex reflection coe ffi cient ˆ R of an acoustic element relates the incoming acoustic wave to the acoustic wave reflected by the acoustic element and is expressed as:

ˆ R = ˆ g

ˆ f (3)

It can then be obtained from the multi-microphone method.

3.2. L- ζ Model of the Exhaust Nozzle In the present work, a simple model is developed to reproduce the acoustic impedance of the exhaust nozzle. The acoustic impedance ˆ Z of an acoustic element is the ratio of the acoustic pressure ˆ p to the acoustic velocity ˆ u at the location of the element: ˆ Z = ˆ p

ˆ u (4)

As described in [9], part of the acoustic energy is transformed into vorticity at the exhaust nozzle. In practice, these e ff ects are di ffi cult to capture. Therefore, in the present model, they are simply modeled through a pressure loss coe ffi cient ζ . The loss coe ffi cient ζ can be expressed as:

ζ = 2 P s

ρ U 2 n (5)

where P s , ρ and U n are the static pressure in the test rig, the air density and the flow velocity in the exhaust nozzle, respectively. A first estimate ζ exp of the pressure loss coe ffi cient is obtained experimentally from static pressure measurements at di ff erent air mass flow rates. ζ exp is function of the cross section in the exhaust nozzle. A correction factor d is introduced as a model parameter such that ζ = d ζ exp . Combining the unsteady Bernoulli equation with the mass conservation equation and the aforementioned pressure loss term, it follows in frequency domain:

ˆ Z = ζ M n A rig A nozzle + i ω L ef f

c A rig A nozzle (6)

where M n , A rig and A nozzle denote the Mach number in the exhaust nozzle, the cross section of the test rig and the reference cross section of the exhaust nozzle, respectively. L ef f , referred as an "end correction", accounts for the mass of air moving due to pressure waves at the exhaust nozzle location. As shown in [12], this end correction is a function of the cross section in the exhaust nozzle. In the present model, the end correction is also function of the Mach number M n in the exhaust nozzle. The parameters d and L ef f of the model are fitted onto the experimental data using a genetic algorithm based optimization. Finally, the acoustic reflection coe ffi cient ˆ R of the exhaust nozzle is retrieved from the acoustic impedance ˆ Z using Eq. 7.

ˆ R = ˆ Z − 1 ˆ Z + 1 (7)

4. RESULTS

4.1. Cold Conditions Two variations of operating conditions are presented here:

1. The air mass flow rate in the impedance tube is kept constant and the piston position is changed.

2. The piston position is kept constant and the air mass flow rate is varied.

Constant Mass Flow Rate The reflection coe ffi cient of the exhaust nozzle is first measured for a mass flow rate ˙ m 5 = 100 g / s and five di ff erent piston positions (see Table 2). The amplitude and phase of the measured reflection coe ffi cients are presented in Figure 3. The symbols represent the measurements and the solid lines represent the fitted model. Due to the presence of resonances in the impedance test tube between 100 Hz and 150 Hz, the experimental results for this range of frequencies are discarded from the fitting routine. This range is highlighted by the dashed red lines in Figure 3.

The results depicted in Figure 3 first underline the broad range of the amplitude of the reflection coe ffi cient that we can obtain with the present design. Additionally, the simple model presented in Section 2 reproduces well the change of amplitude and phase of the reflection coe ffi cients. For these conditions, the values of the Mach number M n at the exhaust nozzle, the end correction L ef f and the loss coe ffi cient ζ for the five di ff erent piston positions are summarized in Table 4.

1

Pos.1 Pos.2 Pos.3 Pos.4 Pos.5

0.8

0.6

| R | ∠ R

0.4

0.2

0

π

π/ 2

0

− π/ 2

− π

100 200 300 400 500 50 150 250 350 450

Frequency [Hz]

Figure 3: Reflection Coe ffi cients of the Exhaust Nozzle for a constant air mass flow rate and five di ff erent piston positions.

Piston Position 1 2 3 4 5

Mach number M n 0.43 0.3 0.23 0.19 0.15

Loss Coef. ζ 1.60 1.92 2.12 2.26 2.39

End Correction L ef f (cm) 1.03 1.53 1.94 2.31 2.73

Table 4: Mach number in the Exhaust Nozzle, End Correction and Loss Coe ffi cient for the five di ff erent piston positions and an air mass flow rate of 100 g / s.

Constant Piston Position The reflection coe ffi cient of the exhaust nozzle is then measured for a constant piston position (Pos. 2 in Table 2) and six di ff erent air mass flow rates (see Table 1). Similarly, due to resonances in the rig, the experimental results between 100 Hz and 150 Hz are discarded from the modelling. Results are presented in Figure 4. The amplitude of the measured reflection coe ffi cient ranges from 0.1 to almost 1 in absolute value. The maximum amplitude corresponds to the air mass flow rate of 0 g / s. At this mass flow, according to [9], there is no vortex shed and therefore no dissipation of acoustic energy. In this condition, the exhaust nozzle acts as a closed end. Furthermore, Figure 4 highlights the capability of the L- ζ model to reproduce the experiments. For these conditions, as the cross section area in the exhaust nozzle is constant, the pressure loss coe ffi cient ζ is constant and equals 1.92. The values of the Mach number M n at the exhaust nozzle and the end correction for the six di ff erent mass flow rates are summarized in Table 5.

1.2

˙ m 1 ˙ m 2 ˙ m 3 ˙ m 4 ˙ m 5 ˙ m 6

1

0.8

| R | ∠ R

0.6

0.4

0.2

0

π

π/ 2

0

− π/ 2

− π

100 200 300 400 500 50 150 250 350 450

Frequency [Hz]

Figure 4: Reflection Coe ffi cients of the Exhaust Nozzle for a constant piston position and six di ff erent air mass flow rates.

Air Mass Flow Rate (g / s) 0 10 55.6 70 100 140

Mach number M n 0 0.029 0.16 0.22 0.3 0.38

Loss Coef. ζ 1.92

End Correction L ef f (cm) 1.84 1.81 1.67 1.61 1.53 1.44

Table 5: Mach number in the Exhaust Nozzle, End Correction and Loss Coe ffi cient for the six di ff erent air mass flow rates.

Existence of an Optimum Figure 5 depicts the minimum amplitude of the measured reflection coe ffi cients over the entire range of frequencies (50 to 500 Hz) as function of the Mach number in the nozzle for the two variations. For each variation, it exists an optimal Mach number for which the minimum amplitude is the closest to zero (nearly anechoic termination). This is in agreement with the figure 5 in [9]. Figure 5 further underlines the broad range of reflection coe ffi cient amplitudes that can be obtained with the present design of the rig boundary.

1

constant ˙ m constant Pos

0.8

0.6

| R min |

0.4

0.2

00

0.2 0.1 0.3

0.4 0.5

M n [-]

Figure 5: Minimum Amplitude of the Reflection Coe ffi cient as function of the Mach number in the nozzle.

4.2. Combustion Conditions Figure 6 shows the power spectral density of the acoustic pressure measured by the microphone located in the combustion chamber. Three positions of the exhaust nozzle are considered (see Table 3), leading to three di ff erent cross section areas and therefore to three di ff erent acoustic coupling between the flame and the combustor boundary. The spectrum for the first position of the piston is flat over the entire range of frequencies meaning that the system is thermoacoustically stable. However, the opening of the area leads to a sharpening of the spectrum peak around 250 Hz (see brown and yellow curves in Figure 6). The change of the outlet boundary condition brings the combustor into a thermoacoustically unstable state. This highlights the capability of the designed exhaust nozzle to change the thermoacoustic stability of the combustor, without changing the combustion conditions.

10 6

Pos. 1 Pos. 2 Pos. 3

10 4

| S pp |

10 2

10 0

0 100 200 300 400 500 600

Frequency [Hz]

Figure 6: Power Spectral Density of the acoustic pressure p measured in the combustion chamber for three di ff erent piston positions.

The experimental and theoretical estimations of the reflection coe ffi cients for these three combustion conditions will be performed and presented in future work.

5. CONCLUSION

A new design of an adjustable boundary has been presented in this work. It consists of an orifice in which a conical piston can be moved to change the opening area. Experiments in cold conditions have shown the ability of this simple design to generate a broad range of acoustic responses, from closed end to anechoic terminations. A simple model, based on a pressure loss coe ffi cient and an end correction, reproduces the experiments well. This model will enable the prediction of acoustic responses at conditions that have not been experimentally measured. Finally, the impact of this adjustable boundary on thermoacoustic stability has been assessed experimentally. Future work will include the modelling of the acoustic response of the exhaust nozzle in combustion conditions and the prediction of thermoacoustic stability of the combustor for various nozzle positions using an acoustic network.

ACKNOWLEDGEMENTS

This work is funded by the ETH Foundation. They are gratefully acknowledged for their support.

REFERENCES

[1] Lord Rayleigh. The explanation of certain acoustical phenomena. Roy. Inst. Proc. , 8:536–542, 1878. [2] Alexander Avdonin, Stefan Jaensch, Camilo F. Silva, Matic ˇ Cešnovar, and Wolfgang Polifke. Uncertainty quantification and sensitivity analysis of thermoacoustic stability with non-intrusive polynomial chaos expansion. Combustion and Flame , 189:300–310, 2018. [3] Bruno Schuermans. Modelling and Control of Thermoacoustic Instabilities . Thesis, EPFL, 2003. [4] Christian Oliver Paschereit, Ephraim Gutmark, and Wolfgang Weisenstein. Control of thermoacoustic instabilities and emissions in an industrial-type gas-turbine combustor. In Symposium (International) on Combustion , volume 27, pages 1817–1824. Elsevier, 1998. [5] Bernhard ´ Cosi´c, Ste ff en Terhaar, Jonas P Moeck, and Christian Oliver Paschereit. Response of a swirl-stabilized flame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes. Combustion and Flame , 162(4):1046–1062, 2015. [6] Zhiguo Zhang, Dan Zhao, Nuomin Han, Shuhui Wang, and Junwei Li. Control of combustion instability with a tunable helmholtz resonator. Aerospace Science and Technology , 41:55–62, 2015. [7] Liangliang Xu, Jianyi Zheng, Guoqing Wang, Zhenzhen Feng, Xiaojing Tian, Lei Li, and Fei Qi. Investigation on the intrinsic thermoacoustic instability of a lean-premixed swirl combustor with an acoustic liner. Proceedings of the Combustion Institute , 38(4):6095–6103, 2021. [8] K. T. Kim, J. G. Lee, B. D. Quay, and D. A. Santavicca. Spatially distributed flame transfer functions for predicting combustion dynamics in lean premixed gas turbine combustors. Combustion and Flame , 157(9):1718–1730, 2010. [9] W.D. Bechert. Sound Absorption Shedding Caused By Vorticity Shedding, Demonstrated With a Jet Flow. Journal of Sound and Vibration , 70(3):389–405, 1980. [10] P. Durrieu, G. Hofmans, G. Ajello, R. Boot, Y. Aurégan, A. Hirschberg, and M. C. A. M. Peters. Quasisteady aero-acoustic response of orifices. The Journal of the Acoustical Society of America , 110(4):1859–1872, 2001. [11] Markus Weilenmann and Nicolas Noiray. Experiments on sound reflection and production by choked nozzle flows subject to acoustic and entropy waves. Journal of Sound and Vibration , 492:115799, 2021. [12] George B Thurston and John K Wood. End corrections for a concentric circular orifice in a circular tube. The Journal of the Acoustical Society of America , 25(5):861–863, 1953.