Characterisation of three-port burner acoustics using a reduced-order model

Abstract

Combustion of steam-diluted fuels offers low emissions and fuel flexibility, but thermoacoustic instabilities remain a challenge. Burner acoustics play a crucial role in the prediction and control of such instabilities. This study reports an experimental investigation of the acoustic characteristics of a dual-swirl burner integrated in a humid combustor. In contrast to conventional two-port burners, this burner features three ports of acoustic flux: Two non-stiff inlets corresponding to air and steam-fuel mixture, and one outlet. A full characterisation of such a three-port is demanding for practical burners. Based on the assumption of acoustic compactness, a Reduced-order Model is proposed for the three-port Burner Transfer Matrix (BTM). The model treats the three-port as a combination of two compact two-ports with no interaction between the two inlets. It allows obtaining the reduced three-port BTM with only two linearly independent acoustic states, as is done for a two-port, but with additional microphones for the third port. First, the two-port characteristics of the burner were studied by blocking one of the two inlets at a time. The results substantiate the model assumptions for the three-port. However, the simplifications are valid for frequencies up to 400 Hz, beyond which some deviations are observed. The three-port shows higher resistive and reactive characteristics between the fuel inlet and the outlet ports, compared to the two-port case with the air inlet blocked. The ratio of flow momentum between the two inlets has only a mild effect on the BTM, whereas the speed of sound ratio exhibits a stronger influence.

Keywords

Burner transfer matrix dual-swirl fuel flexibility humid combustion non-stiff reduced-order model thermoacoustics three-port

Introduction

Global warming concerns and environmental protection goals are driving a worldwide shift from fossil fuels to renewable energy sources. The production of alternative fuels, their storage, and on-demand firing in gas turbines is deemed a solution to compensate for the intermittent generation of renewable electricity. Fuel candidates for the future generation of gas turbines include hydrogen, ammonia, and biogas. Nonetheless, natural gas will continue to hold a major share in the energy mix, at least in the near future, as combustion technologies have been developed around it, and it remains a fallback option. The uncertainty in fuel selection for next-generation systems and the time required to shift from natural gas to alternative fuel infrastructure necessitate the development of fuel-flexible energy systems.

Various techniques are employed to achieve fuel flexibility in gas turbines. These include but are not limited to: multi-jet flames, micromix burners, flameless combustion, exhaust gas recirculation, swirl control, and steam dilution.^1–5 Dilution of fuel by steam allows controlling the reactivity of the mixture, enabling the same combustor to fire highly reactive hydrogen or less reactive biogas by regulating the extent of steam dilution.⁶ Furthermore, the generation of an enhanced pool of radicals and reduced peak flame temperatures in the presence of steam leads to lower emissions.⁷ One such combustor has been recently developed at the Chair of Fluid Dynamics, TU Berlin. The combustor is based on swirl-stabilised flames in a steam environment. Fuel-steam mixture and air are injected through two different swirl stages in a dual-swirl configuration. The combustor is operated close to stoichiometric conditions. Its performance in terms of static stability, emissions, and fuel switching capability has been documented in Dybe et al.,⁸ demonstrating its high potential for gas turbine applications.

Another crucial requirement for the design of gas turbine burners is thermoacoustic stability. In these instabilities, the flame’s unsteady heat release dynamics couple with the acoustic modes of the combustion chamber.⁹ This leads to substantial pressure fluctuations that can significantly impair system performance, shorten the service life of the burner components, and, in severe cases, cause direct mechanical damage to the system. To mitigate thermoacoustic instabilities at the design stage of the gas turbine, both the acoustics of the combustion system as well as the flame’s response to acoustic perturbations have to be characterised over a wide range of fuels and operating conditions. The dual-swirl system presents a distinctive technical challenge in this regard due to the presence of two supply lines, which can both be subject to velocity fluctuations and cannot be considered acoustically stiff.

Stiff fuel injection is a common assumption when modelling burner acoustics. It implies that the fuel flow flux is steady and remains unaffected by acoustic fluctuations in the combustor. In this case, the acoustics of the burner is characterised by considering it as a two-port system with one inlet and one outlet. The corresponding description in frequency space is a Burner Transfer Matrix (BTM) relating the acoustic states upstream and downstream of the burner. However, this assumption is not always valid.¹⁰

In the dual-swirl humid combustor investigated in this study, the steam-fuel mixture as well as the air are fed through two low-impedance swirlers, none of which can be considered stiff. Therefore, the dual-swirl burner has to be considered as a three-port, with two inlet ports and one outlet port. Ignoring any of these inlets will reduce the system to a conventional two-port BTM. However, the results will not be transferable to another test or simulation setup since any change in the test rig geometry, configuration, or supply system will affect the measured BTM.¹¹

The acoustic characteristics of a three-port configuration have been studied before in the context of tee junctions in fluid piping systems and silencers. Karlsson et al. demonstrated that a 3 $\times$ 3 Scattering Matrix (SM) can be used to obtain rig-independent linear acoustic characteristics of a passive three-port.¹² In addition to the acoustic transfer across the three-port in no-flow condition, excitation of the side branch is possible due to a mean grazing flow resulting from the coupling of vorticity fluctuations at the unsteady shear layer and acoustic modes of the system.¹³ At low amplitudes, a flapping shear layer is observed, and at high amplitudes, discrete vortex-shedding occurs, which can have constructive or destructive interference with the acoustic field. Even a steady bias flow from the side branch can excite the acoustic modes of the tee-junction.¹⁴ These interactions appear as a source term in the acoustic transfer equations. As long as this flow-acoustic interaction is in the linear regime, the source term is captured by the measured SM of the three-port for a given bias-grazing flow condition.¹⁵ A similar approach can be employed for the three-port dual-swirl burner. The three-port can be characterised by using a 3 $\times$ 3 SM or Transfer Matrix (TM) that includes the influence of any flow-acoustic coupling inside the burner.

This work demonstrates a method of characterising burners with non-stiff fuel injection as acoustic three-ports. With this method, a dual-swirl burner used in a steam-diluted fuel-flexible combustor is acoustically characterised. The BTM is formulated based on a Reduced-order Model (ROM) for the acoustics of compact burners. This reduced TM is experimentally measured using microphone arrays in the ducts connected to two of the ports and two additional acoustic pressure probes in the plenum connected to the third port. The assumptions made are assessed, and the acoustic transfer characteristics of the burner are investigated for different flow conditions.

Methodology

Experimental setup

The combustion system under investigation is a prototype dual-swirl burner for humid combustion, adapted for laboratory experiments in the thermoacoustics test rig at TU Berlin. A schematic of the test rig is shown in Figure 1, which also depicts how the dual-swirler is integrated in the test rig. Each swirling stage is fed by an individual flow line. The upstream swirling stage is designed to be fed with a steam-fuel mixture. In the test rig, it protrudes upstream into the fuel plenum. The downstream swirling stage is fed through a radial air plenum and is used to supply combustion air. The air is preheated to 573 K to mimic the compressor discharge conditions in the gas turbine. Four radially arranged stainless steel hoses supply the radial plenum with the preheated air. In order to achieve a homogeneous inflow around the circumference, a perforated plate is installed in the plenum as indicated in Section A-A in Figure 1. The streams fed to the two swirlers experience strong mixing in the mixing tube due to a high shear between the two swirling flows. After passing through the mixing tube, the mixture flows into the combustion chamber and subsequently through the exhaust tube into the lab ventilation unit. An orifice is mounted at the end of the exhaust tube to control the acoustic boundary condition of the test rig.

Figure 1.

Schematic representation of the thermoacoustic test rig and the dual-swirl burner.

Mimicking engine-relevant speeds of sound and velocities

The measurements presented in this work were intended to characterise the acoustic transfer characteristics of the dual-swirl burner, which can later be used to measure the acoustic response of the flame. In order to be able to transfer the methodology and results to reacting tests, the operating conditions were selected to mimic the relevant reacting cases. We expect the burner acoustics to be largely governed by: (a) the speed of sound in the different burner sections, and (b) the momentum ratio between the two swirlers. Thus, the operating conditions were selected to match these quantities between the non-reacting tests and reacting conditions.

The air line, which feeds the downstream swirler, allows for the direct selection of mass flow and temperature values from the reacting operating range. In contrast, the use of steam in the fuel plenum was avoided in order to facilitate the acoustic measurements with water-cooled microphones, which are designed to operate only in dry environments. Instead, the fuel plenum was supplied with preheated air. In order to achieve speed of sound values similar to those of superheated steam-fuel mixtures, helium was added to the preheated air, as depicted in Figure 2. It shows the speed of sound in a mixture of air and helium at different compositions and temperatures as a colour map. The speed of sound in a steam-hydrogen mixture at the engine-relevant temperature of 550 K is also shown with solid lines for different mass fractions of hydrogen.

Figure 2.

Speed of sound in helium-air mixture for different helium mass fractions $Y_{He}$ and temperatures. Solid lines represent the speed of sound in a steam-hydrogen mixture at 550 K for different hydrogen mass fractions $Y_{H_{2}}$ .

The necessity of matching the speed of sound for accurate BTM measurements has been demonstrated before.^16,17 The use of helium to reproduce realistic speed of sound values for high-hydrogen fuels has also been reported earlier.¹⁸ The required temperature for matching the speed of sound of the steam-fuel mixture reduces significantly with the addition of helium in the air-helium mixture. For example, without the addition of helium, mimicking the steam-fuel speed of sound with 5% of hydrogen would require preheating air to more than 800 K. This requirement becomes increasingly stringent for increasing hydrogen content and cannot be met by the available laboratory hardware. Therefore, helium addition was utilised. The speed of sound in the air plenum remains unchanged since it is always maintained at the engine-relevant temperature of 573 K. In order to reproduce a specific operating point corresponding to the reactive condition, the helium-air ratio was adapted to the intended speed of sound in the fuel plenum. The absolute mass flow through the fuel swirler was then selected so that the momentum ratio between the two swirling stages corresponds to the reacting condition. The mixture speed of sound downstream of the burner depends upon the speed of sound and the flow rates of the two streams. A schematic indicating the exemplary distribution of the speed of sound assumed for this study is depicted in Figure 3. Complete mixing is assumed at the end of the mixing tube. Note that the values are not shown for the mixing regions within the burner and the mixing tube, where strong concentration gradients are expected.

Figure 3.

Exemplary speed of sound distribution assumed and quantities measured at reference locations.

Acoustic measurement setup

To characterise the acoustic transmission properties of the dual-swirler, a number of microphones were used. The upstream fuel plenum, which feeds the first swirler stage, accommodated an array of five microphones installed in water-cooled holders at different axial positions. Their signals were analysed using the Multi-Microphone Method (MMM), to identify the upstream and downstream acoustic waves in the duct. These were then combined to obtain acoustic velocity and pressure at reference position ‘ $f$ ’, which is located at the inlet of the fuel swirler, as shown in Figure 3. To monitor the acoustics in the exhaust tube, another array of water-cooled microphones was used. By means of the MMM, the recorded pressure signals were processed to quantify velocity and pressure variations at the burner exit, reference position ‘ $m$ ’. Thermocouples were provided in each duct to estimate the speed of sound and gas density. Details of the wave reconstruction using the MMM are explained in the next sub-section.

Because of its complex geometry, the radial air plenum could not be equipped with a microphone array. In order to characterise the acoustics in this part of the burner, two Waveguide (WG) systems were installed. They measured the acoustic pressure inside and outside the perforated plate at reference positions ‘ $a_{1}$ ’ and ‘ $a_{2}$ ’, respectively. These WGs work as infinite line pressure probes.¹⁹ Each WG consists of a metal tube that ends at the measurement position so that the acoustic pressure waves propagate into the tube. The acoustic pressure was recorded by a microphone that is mounted flush with the wall of the tube, around 100 mm outside of the hot components of the test rig. In order to avoid acoustic reflections, the opposite side of the tube was connected to a long extension hose in which the waves dissipate. To protect the microphone from thermal damage, the WGs were purged with cold air from the other end. The time delay and the attenuation of the WG systems were calibrated against the other microphones before the experiments. A thermocouple was used to estimate the speed of sound and density in the air plenum.

Overall, this setup provided the measurement of acoustic pressure fluctuations at all reference positions and additional measurement of velocity fluctuations at reference positions ‘ $f$ ’ and ‘ $m$ ’, as summarised in Figure 3. To acoustically force the combustion system, a set of loudspeakers was installed both upstream and downstream, as shown in Figure 1. The loudspeaker housings were purged with cold air to prevent ingress of hot gases and thermal damage to the speakers.

Processing of microphone signals

The microphone signals were measured in Volts by analogue input modules from National Instruments. In order to convert them to meaningful acoustic pressure signals, all microphones were calibrated against a common reference microphone by installing them in an impedance tube at the same cross-section and exciting over a range of frequencies.²⁰ This provided the relative calibration amongst the microphones, which is necessary for the multi-microphone method discussed in the next section. An absolute calibration of the reference microphone using a piston-phone provided the Volts-to-Pascals conversion.²¹ To capture only the frequency component coherent to the acoustic forcing, the Cross Power Spectral Density was evaluated for each microphone against the forcing signal. This was then normalised by the Auto Power Spectral Density of the forcing signal to finally obtain the acoustic pressure.

Multi-microphone method

The Multi-Microphone Method (MMM) allows the detection of plane acoustic waves from pressure signals measured by multiple microphones at different axial locations in a straight duct.^22,23 For frequencies below the cut-on frequency of the first transverse mode, the response of the acoustic pressure and velocity to harmonic forcing can be expressed as a linear combination of two Riemann invariants ( $f$ and $g$ ), which represent downstream and upstream travelling waves:

\tilde{p} = \frac{p (x)}{ρ c} = \underset{f (x)}{\underset{⏟}{\hat{f} e^{- i ω (\frac{x}{c + \bar{u}})}}} + \underset{g (x)}{\underset{⏟}{\hat{g} e^{+ i ω (\frac{x}{c - \bar{u}})}}},

(1)

u (x) = f (x) - g (x),

(2)

where

\tilde{p}

is the acoustic pressure

p

normalised by the characteristic acoustic impedance

ρ c

u

is the acoustic velocity,

ρ

the density,

c

the speed of sound,

\bar{u}

the mean flow velocity, and

ω

the angular frequency. The travelling waves

f (x)

and

g (x)

at different axial positions

x_{i}

and

x_{j}

are related through

f (x_{j}) = f (x_{i}) \exp (- i ω \frac{x_{j} - x_{i}}{c + \bar{u}})

and

g (x_{j}) = g (x_{i}) \exp (+ i ω \frac{x_{j} - x_{i}}{c - \bar{u}})

. At a reference position

x = 0

, the values for

\hat{f}

and

\hat{g}

can thus be determined from the pressure signals of a microphone array with

N

microphones at positions

x_{1}, \dots x_{N}

by solving the following equation for the Riemann invariants:

[\begin{matrix} {\tilde{p}}_{1} \\ ⋮ \\ {\tilde{p}}_{N} \end{matrix}] = [\begin{matrix} e^{- i ω \frac{x_{1}}{c + u}} & e^{i ω \frac{x_{1}}{c - u}} \\ ⋮ & ⋮ \\ e^{- i ω \frac{x_{N}}{c + u}} & e^{i ω \frac{x_{N}}{c - u}} \end{matrix}] [\begin{matrix} \hat{f} \\ \hat{g} \end{matrix}] .

(3)

For

N > 2

, Equation (3) is over-determined. The Least Squares Method (LSM) is applied to identify

\hat{f}

and

\hat{g}

at the reference position, which can then be used in Equations (1), (2) to obtain the velocity and pressure fluctuations at other positions.

BTM modelling of a compact two-port burner

The acoustic response of a burner is generally characterised through its SM, by assuming that only planar waves propagate through the burner. This holds true provided that the considered frequencies are low so that the burner can be considered to be acoustically compact. Under this assumption, the SM relates the Riemann invariants travelling into the burner to those travelling out of the burner, as schematised in Figure 4. For a two-port burner, the elements of the SM can be measured experimentally by applying the MMM in the ducts connected to the two ports. Having to identify the four unknown elements of the SM from two equations requires collecting experimental data for two linearly independent acoustic states. This is achieved by forcing the system acoustically from the Upstream (US) and the Downstream (DS) side of the burner, one at a time. The SM represents a causal relation of the acoustic state across the burner, which is often translated into a non-causal relation between the acoustic velocity and pressure states across the burner by using the relations in Equations (1), (2). This results in defining the BTM between the inlet ‘ $a$ ’ and outlet ‘ $m$ ’, namely $T^{a m}$ , as:

[\begin{matrix} {\tilde{p}}_{m} \\ u_{m} \end{matrix}] = \underset{T^{a m}}{\underset{⏟}{[\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix}]}} [\begin{matrix} {\tilde{p}}_{a} \\ u_{a} \end{matrix}] .

(4)

Figure 4.

Top: Acoustic scattering matrix for a classic burner, represented as an acoustic two-port with one inlet (‘ $a$ ’) and one outlet (‘ $m$ ’). Bottom: Acoustic scattering matrix for a three-port burner with two non-stiff flow inlets (‘ $a$ ’, ‘ $f$ ’) and one outlet (‘ $m$ ’).

From a modelling perspective, the explicit expressions of the elements of the BTM stem from the conservation of mass and momentum across the burner.²⁴ For acoustically compact swirl burners, a commonly employed model for the elements of $T^{a m}$ is the so-called $L$ – $ζ$ model. In this model, the acoustic mass flux is conserved, an effective length $L$ accounts for the inertial effects associated with flow acceleration and/or deceleration, and a pressure loss coefficient $ζ$ accounts for the flow resistance through the burner.²⁵ With this model, the BTM reads:

[\begin{matrix} {\tilde{p}}_{m} \\ u_{m} \end{matrix}] = \underset{T^{a m}}{\underset{⏟}{[\begin{matrix} β_{a} & T_{12} \\ 0 & α_{a} \end{matrix}]}} [\begin{matrix} {\tilde{p}}_{a} \\ u_{a} \end{matrix}],

(5)

where

β_{a} \equiv (ρ_{a} c_{a}) / (ρ_{m} c_{m})

α_{a} \equiv (ρ_{a} A_{a}) / (ρ_{m} A_{m})

, and

T_{12} \equiv (- ζ M a - \frac{i ω}{c} L_{})

, with

A

for effective area and

M a

being the Mach number. This non-causal representation is, however, not unique. Rather than relating the acoustic states across the burner, one can equivalently express the BTM as matrix

B^{a m}

which relates the acoustic velocities at the two ports of the burner to the acoustic pressures at the two ports. By rearranging Equation (5) one obtains:

u_{a} = \frac{1}{T_{12}} ({\tilde{p}}_{m} - β_{a} {\tilde{p}}_{a}),

(6)

u_{m} = \frac{α_{a}}{T_{12}} ({\tilde{p}}_{m} - β_{a} {\tilde{p}}_{a}),

(7)

which can be written in the matrix form as:

[\begin{matrix} u_{a} \\ u_{m} \end{matrix}] = \underset{B^{a m}}{\underset{⏟}{[\begin{matrix} - β_{a} / T_{12} & 1 / T_{12} \\ - α_{a} β_{a} / T_{12} & α_{a} / T_{12} \end{matrix}]}} [\begin{matrix} {\tilde{p}}_{a} \\ {\tilde{p}}_{m} \end{matrix}] .

(8)

This form has the advantage that it can be generalised to burners with multiple ports, as discussed in the next section. It emphasises that, provided the $T_{21}$ element of the matrix $T^{a m}$ in Equation (4) vanishes, the acoustic velocity is a function of the pressure drop $Δ \tilde{p} = {\tilde{p}}_{m} - β_{a} {\tilde{p}}_{a}$ . If no temperature or composition changes occur, then $β_{a} = 1$ .

BTM modelling of a compact three-port burner

The burner considered in this work has the peculiarity of having two non-stiff inlet ports and one outlet port. To characterise the SM of this burner, one needs to relate the three Riemann invariants travelling into the burner to the three Riemann invariants travelling out of the burner (see Figure 4). The full characterisation of the resulting 3 $\times$ 3 SM requires identifying the nine unknown elements from a set of three linearly independent acoustic states. This would require using the MMM in the three flow paths, one for each inlet/outlet. Moreover, it requires the ability to force acoustically each flow path of the burner, one at a time, including the two inlets and one outlet. This more complex experimental configuration was not feasible for the current dual-swirl burner design. Therefore, the experimental characterisation of the burner required model reduction based on simplifying assumptions. First, by transforming the Riemann invariants into acoustic pressures and velocities, one can convert the three-port SM into a 3 $\times$ 3 BTM, whose coefficients need to be characterised. The classic representation of the BTM, namely $T$ in Equation (4), which relates the acoustic states at the inlet to those at the outlet, cannot be generalised to this burner, and it is instead more natural to relate the acoustic velocities to the acoustic pressures at the three ports of the burner via matrix $B^{f a m}$ , in analogy to Equation (8). We recall that in single-swirl two-port burners that can be approximated using an $L$ – $ζ$ model, the acoustic velocity at the inlet is a function of the pressure drop. Extending this result to both inlet ports of the dual-swirl burner, one finds:

u_{f} = B_{13} ({\tilde{p}}_{m} - β_{f} {\tilde{p}}_{f}),

(9)

u_{a} = B_{23} ({\tilde{p}}_{m} - β_{a} {\tilde{p}}_{a}),

(10)

where

B_{13}

and

B_{23}

are frequency-dependent functions that need to be characterised. Imposing conservation of acoustic mass flux for a compact burner and applying the expressions from Equations (9) and (10) yields:

\begin{matrix} u_{m} & = \frac{ρ_{f} A_{f}}{ρ_{m} A_{m}} u_{f} + \frac{ρ_{a} A_{a}}{ρ_{m} A_{m}} u_{a}, \\ \Rightarrow u_{m} & = - β_{f} α_{f} B_{13} {\tilde{p}}_{f} - β_{a} α_{a} B_{23} {\tilde{p}}_{a} \\ + (α_{f} B_{13} + α_{a} B_{23}) {\tilde{p}}_{m}, \end{matrix}

(11)

By combining Equations (9), (10), and (11), the three-port BTM for the dual-swirl burner $B^{f a m}$ can be modelled as:

[\begin{matrix} u_{f} \\ u_{a} \\ u_{m} \end{matrix}] = \underset{B^{f a m}}{\underset{⏟}{[\begin{matrix} - β_{f} B_{13} & 0 & B_{13} \\ 0 & - β_{a} B_{23} & B_{23} \\ - β_{f} α_{f} B_{13} & - β_{a} α_{a} B_{23} & α_{f} B_{13} + α_{a} B_{23} \end{matrix}]}} [\begin{matrix} {\tilde{p}}_{f} \\ {\tilde{p}}_{a} \\ {\tilde{p}}_{m} \end{matrix}] .

(12)

It is important to emphasise that, apart from conserving acoustic mass flux, this ROM assumes no interaction between the two inlet streams. Therefore, the pressure drop across each stream is assumed to be independent of the other stream, leading to $B_{12} \equiv B_{21} \equiv 0$ . At the merger point of the two inlet streams into the outlet stream, pressure uniformity is assumed. If this merger occurs close to the burner exit, the streams are expected to behave independently. Furthermore, the model assumes that only one dimensional acoustic modes exist in the radial air plenum within the frequency range of interest, i.e. 100–600 Hz. This was verified by an acoustic simulation using COMSOL, which is beyond the scope of this paper.

An additional complication was caused by the fact that the experimental setup did not allow for a measurement of the acoustic velocity $u_{a}$ at the second inlet port, due to the lack of MMM capabilities at this position. To overcome this obstacle, the acoustic velocity in the air plenum was estimated from the acoustic pressure drop across the perforated plate. By employing admittance $H_{a}$ in analogy to the definition of the Rayleigh conductivity,²⁶ the acoustic velocity can be obtained from the acoustic pressures measured upstream and downstream of the perforated plate, see Figure 3, as:

u_{a} = H_{a} ({\tilde{p}}_{a 1} - {\tilde{p}}_{a 2}) = \frac{({\tilde{p}}_{a 1} - {\tilde{p}}_{a 2})}{- ζ_{p e r} M a - \frac{i ω}{c} L_{p e r}},

(13)

where

1 / H_{a} \equiv (- ζ_{p e r} M a - \frac{i ω}{c} L_{p e r})

is an approximation of the perforated plate impedance based on

L

–

ζ

model.

ζ_{p e r}

represents the pressure loss due to skin friction and hydrodynamic effects, e.g. flow separation. As a first approximation, the pressure loss coefficient for steady flow through an orifice of finite thickness was used. A value of

ζ_{p e r} = 1.72

was estimated for 5 mm holes in the 3 mm thick perforated plate with a porosity of 12% from handbook data²⁷ and was used throughout the study. The effective length accounting for the acoustic reactance due to flow inertia was estimated to be

L_{p e r} = 5

mm based on an empirical model for two dimensional discontinuities,^28,29 including a length correction for the contraction at the upstream and expansion at the downstream side. Note that Equation (13) provides the acoustic velocity outside the perforated ring, at location ‘

a_{2}

’, which is considered to be the burner air inlet ‘

a

’. Furthermore, the equation assumes that the WGs in the air plenum are very close to the perforated plate and hence measure the acoustic pressure directly upstream and downstream of the plate; no change in cross-sectional area is considered between the WGs and the perforated plate.

This impedance model allows for estimating the acoustic velocity in the air plenum from the two pressure readings at locations ‘ $a_{1}$ ’ and ‘ $a_{2}$ ’, such that all three acoustic pressures and velocities can be derived from the measurements.

Results and discussion

In order to better understand the acoustic properties of the burner, it was first characterised by blocking one of the two inlet ports. This not only allowed us to study the behaviour of the burner in a conventional manner and compare it with the model for the two-ports but also helped in validating the assumptions made in deriving Equation (12).

Two-port BTM (air plenum blocked)

First, the air swirler was replaced with a dummy part having a solid wall instead of slots, so that the burner becomes a single-swirler connecting the fuel plenum to the exhaust duct. MMM was employed to resolve the acoustic field upstream and downstream of the burner for the two acoustic states resulting from US and DS acoustic forcing. Hence, the full 2 $\times$ 2 BTM $T^{f m}$ could be determined between the ports ‘ $f$ ’ and ‘ $m$ ’, similar to Equation (4).

For such a two-port burner, the $L$ – $ζ$ model can be expected to approximate the acoustic characteristics reasonably well if the assumption of acoustic compactness holds. This implies that the element $T_{11}$ of $T^{f m}$ should be approximately constant with respect to frequency, with a value close to the specific impedance ratio $β_{f} = (ρ_{f} c_{f}) / (ρ_{m} c_{m})$ whereas $T_{21} \approx 0$ . The amplitude and phase of the measured BTM $T^{f m}$ elements $T_{11}$ and $T_{12}$ depend on the distance between the two reference positions upstream and downstream of the burner, selected for MMM processing. To identify the distance which produces the best results, the BTM was evaluated for a range of lengths and a value of 80 mm was chosen for which the condition $T_{11} \equiv (ρ_{f} c_{f}) / (ρ_{m} c_{m})$ is almost true. This is shown in Figure 10 in the Appendix. The same burner length was assumed throughout the study.

Note that for the two-port burner, $β_{f}$ is close to 1 at low preheating temperatures when the heat loss is not significant. A variation in the He-Air mixture composition, preheating temperature, and flow rate was considered to produce different values of speed of sound and flow momentum through the burner. The results are shown in Figure 5.

Figure 5.

Two-port BTM (air plenum blocked) for varying flow momentum and speed of sound. Dashed horizontal lines for the amplitude of $T_{11}$ and $T_{22}$ represent $β_{f}$ and $α_{f}$ , respectively.

The flow momentum, increasing from dark to light blue hues, has a mild influence on the BTM. The amplitude of $T_{12}$ and $T_{22}$ elements of $T^{f m}$ increases slightly with increasing momentum. In contrast, an increase in the speed of sound, increasing from dark to light green hues, decreases the amplitude of the $T_{12}$ element and increases the amplitude of the $T_{22}$ element considerably. This effect is more pronounced at higher frequencies. For all considered operating conditions, one can verify that $T_{11} \approx 1$ , that $T_{21} \approx 0$ , and that the $T_{12}$ element increases almost linearly with the frequency, as per Equation (5). However, the element $T_{22}$ is neither constant nor equal to $α_{f} = (ρ_{f} A_{f}) / (ρ_{m} A_{m})$ , shown by dashed black lines. Amplitude for this element decreases almost linearly with frequency until 450 Hz, after which it starts to increase again. This implies that the burner is otherwise compact but shows some compliance. Corresponding to an approximately constant amplitude of $T_{11}$ , the phase of this element almost remains constant throughout the frequency range. Whereas, the phase of $T_{12}$ shows a slight increase with the frequency. The phase of the element $T_{21}$ is not a very informative quantity because the amplitude of this element is close to zero. The phase changes significantly only for the $T_{22}$ element when changing the speed of sound. This effect is noticeable for higher frequencies beyond 350 Hz.

Two-port BTM (fuel plenum blocked)

Similarly, a conventional two-port BTM was measured between the air port ‘ $a$ ’ and the burner outlet port ‘ $m$ ’ at the end of the mixing tube. The acoustic pressure and velocity at the burner outlet were available from MMM. Two WGs in the air plenum provided a direct measurement of acoustic pressure and an indirect measurement of acoustic velocity using Equation (13). The upstream fuel plenum inlet to the burner was blocked by installing a dummy with solid walls instead of swirler slots. Thus, the acoustic forcing with the upstream loudspeakers was not possible. Therefore, only one independent acoustic state could be measured, which is not sufficient to obtain all the elements of the 2 $\times$ 2 BTM $T^{a m}$ . To proceed, it was assumed that the $T_{11}$ and $T_{21}$ elements of $T^{a m}$ are equal to the ratio of the characteristic impedances and zero, respectively. These assumptions are similar to the results shown in Figure 5 for the air blocked case. The remaining elements of the BTM were then calculated using only the downstream forcing.

The results for the estimated elements $T_{12}$ and $T_{22}$ of the BTM $T^{a m}$ are shown for different flow momenta through the air plenum in Figure 6. Similar to the air blocked case, the element $T_{12}$ approximately increases linearly at low frequencies up to 300 Hz. However, it plateaus at a higher frequency close to a value of 1. The $T_{22}$ element shows very different behaviour compared to the air blocked case. The dashed line for the $T_{22}$ amplitude indicates the area-density ratio $α_{a} = (ρ_{a} A_{a}) / (ρ_{m} A_{m})$ . Both the elements show a slight increase in amplitude with increasing flow momentum, except for the frequency range 250–300 Hz. The phase of both elements remains fairly constant with respect to frequency, except for a sudden increase near 160 Hz corresponding to a slight dip in the amplitudes.

Figure 6.

Two-port BTM (fuel plenum blocked) for varying flow momentum. The dashed line for $T_{22}$ amplitude represents $α_{a}$ .

Three-port BTM (both plena open)

The tests by blocking the air and fuel plena allowed us to demonstrate that the assumptions applicable to a compact burner approximately hold true for the dual-swirl burner. These simplifications offered the possibility to apply a ROM for the 3 $\times$ 3 BTM of the three-port, namely $B^{f a m}$ . Experiments were conducted with both plena open in order to evaluate this BTM, see Figure 7 for the amplitude and phase of the selected elements of the matrix $B^{f a m}$ . The first and the second rows of $B^{f a m}$ were solved individually for $B_{13}$ and $B_{23}$ , respectively, using both US and DS forcings. Two acoustic states generated, corresponding to two different acoustic excitations, provided a validation of the compactness assumption in Equations (9), (10). The element $B_{13}$ shows little deviation between the two forcing cases. The deviation increases at frequencies above 400 Hz. The amplitude of the element $B_{23}$ also shows a very good match up to 400 Hz, but the phase for the two forcing cases shows significant deviation. Note that the y-axis scale for the $B_{23}$ amplitude is 10 times that of the other elements. This does not affect the phase plots. Both the elements show amplitude values fairly constant in the 200–500 Hz range and increasing in the very low and very high frequencies. The amplitude of $B_{23}$ is 6–7 times more than that of $B_{13}$ . The amplitude of $B_{23}$ also shows two distinct peaks at frequencies corresponding to 150 and 450 Hz. They can be explained by a very low acoustic pressure difference between the air inlet and burner outlet at these frequencies, while the acoustic pressure drop across the perforated plate and hence the acoustic velocity at ‘ $a$ ’ remains considerable. This may be an artefact of high resonance in the air plenum, leading to non-linear behaviour and requires further investigation. The phase of $B_{13}$ remains fairly constant throughout the frequency range, whereas the phase of $B_{23}$ shows significant variations at lower frequencies.

Figure 7.

Three-port BTM (both plena open) with assumptions. Validation using upstream and downstream acoustic forcing states.

The elements of the third row are, in principle, a linear combination of $B_{13}$ and $B_{23}$ , see Equation (11). This holds the assumption that the quantities $α_{f}$ and $α_{a}$ are constant with respect to frequency and equal to the corresponding density-area ratios. However, the results for the air blocked case in Figure 5 show otherwise. Hence, the burner shows some compliance, and the quantities $α_{f}$ and $α_{a}$ have to be treated as frequency-dependent variables. Therefore, the third row of the matrix $B$ was solved for the two unknowns, namely $C_{31} \equiv α_{f} B_{13}$ and $C_{32} \equiv α_{a} B_{23}$ , see Equation (12). $B_{33}$ is a linear combination of $B_{31}$ and $B_{32}$ . Two equations corresponding to two acoustic states generated by US and DS forcings were solved for the two unknowns. These two variables then allowed the calculation of all three elements in the third row. The amplitude and phase of the elements $B_{31}$ , $B_{32}$ , and $B_{33}$ remain relatively flat with respect to frequency. Exceptions include some deviations in the frequency range of 100–200 Hz. The amplitude of $B_{31}$ is considerably lower than the amplitude of $B_{32}$ , indicating a minor contribution of fuel inlet to the burner outlet mass flux.

In order to better understand the acoustic behaviour of the three-port, the $1 / B_{13}$ element of $B^{f a m}$ was compared with the element $T_{12}$ of the two-port BTM $T^{f m}$ for the case in which the air plenum was blocked. The three-port and the two-port cases were chosen such that they have similar momentum and speed of sound in the fuel plenum. The real and imaginary parts for the mentioned elements are plotted in Figure 8.

Figure 8.

Comparison of $T_{12}$ element of the two-port BTM (air plenum blocked) with 1/ $B_{13}$ element of the three-port BTM (both plena open). Dashed lines represent the $L$ – $ζ$ model.

Although both elements represent the impedance for the passage between location ‘ $f$ ’ and ‘ $m$ ’, one should not expect them to be identical, since the acoustic flux balances are altered when opening the second inlet. But it is reasonable to expect similarities. The results indeed show that when the air plenum is open, the impedance through the fuel swirler changes, but remains qualitatively similar up to 400 Hz. To quantify the change, the resistive component represented by the real part was fitted against $- M a_{s w i r l e r} ζ$ , with $M a_{s w i r l e r}$ being the Mach number through the fuel swirler. The entire frequency range was considered for the case when the air plenum was blocked, but only the frequency range in which the real part of $1 / B_{13}$ is approximately constant (100–400 Hz) was considered for the three-port case. A higher resistance through the fuel swirler is estimated when the air swirler is open compared to when it is closed. Similarly, the reactance represented by the imaginary part was fitted against $- (ω / c) L$ , considering the same frequency ranges as before. The two-port air blocked case shows an almost linear decrease in the imaginary part with increasing frequency, which is in line with the $L$ – $ζ$ model. The three-port case follows an $L$ – $ζ$ type of model only in the low-frequency range up to 400 Hz, beyond which it shows significant deviations. The large values of the resistive ( $ζ$ ) and reactive ( $L$ ) parameters indicate that the burner imposes a strong impedance on the acoustic field. The effective length shows non-physically high value which suggests that the simple $L$ – $ζ$ model may be insufficient, and a more advanced model with additional parameters could better capture the burner’s acoustic behaviour.

The influence of air-to-fuel momentum and speed of sound ratios on the three-port burner acoustics was assessed by varying the flow through the air plenum, and the flow as well as the speed of sound in the fuel plenum. The results for the amplitude and phase of the TM $B^{f a m}$ are depicted in Figure 9. Note that the first two rows were solved here for $B_{13}$ and $B_{23}$ using LSM employing both the acoustic states. An increase in the air-to-fuel momentum ratio (dark to light blue hues) results in a slight decrease in the amplitude of $B_{13}$ . It implies that the acoustic pressure drop between the fuel inlet and burner outlet results in a reduced fuel inlet velocity as the flow rate through the air plenum increases. A similar argument can be made for the amplitudes of $B_{23}$ for frequencies above 300 Hz. The other elements also show a slight change with the flow momentum ratio. The variation in the air-to-fuel speed of sound ratio shows a stronger impact on the elements $B_{13}$ and $B_{31}$ . Increasing the speed of sound ratio (dark to light green hues) increases the amplitude of $B_{13}$ and the phase of $B_{31}$ . The amplitudes of the elements $B_{23}$ , $B_{32}$ , and $B_{33}$ also show a slight increase with increasing speed of sound ratio at frequencies above 400 Hz.

Figure 9.

Three-port BTM (both plena open) with assumptions. Influence of air-to-fuel momentum and speed of sound ratios.

Conclusions and outlook

This study highlights the importance of treating burners with non-stiff fuel injection as three-port systems to achieve rig-independent BTMs, which are essential for accurate thermoacoustic modelling in practical combustion systems. Unlike the conventional two-port approach, the three-port definition accounts for the influence of multiple inlet flows, enabling a more comprehensive understanding of the acoustic behaviour of complex burners. Although a full three-port definition is necessary for obtaining rig-independent BTMs, it was discussed how the assumption of acoustic compactness, valid at low frequencies, can be used to simplify the experimental characterisation of such systems. These simplifications led to a ROM for the three-port BTM that was obtained by generating only two independent acoustic states with MMM in upstream and downstream ducts, and two extra microphones in the radial air plenum connected to the third port. The presented measurements demonstrate that the $L$ – $ζ$ approach is valid for this burner only for the low frequency range up to 400 Hz. Moreover, a change in momentum ratio between the two inlets has little effect on the BTM, whereas a change in speed of sound ratio has a significant influence.

Further improvement of the analysis is possible by experimentally determining the impedance for the perforated plate in the air plenum. This would allow a better estimation of the acoustic velocity from the two acoustic pressures and enable the derivation of a more meaningful BTM. Moreover, generating three independent acoustic states, e.g. by changing the boundary condition in the air plenum in addition to the US and DS forcings, can be exploited to obtain the full three-port BTM without any assumption.

Footnotes

ORCID iDs

Muhammad Yasir

Johann Moritz Reumschüssel

Christian Oliver Paschereit

Alessandro Orchini

Funding

The research was funded by the EU HORIZON-JTI-CLEANH2 project ACHIEVE under grant agreement number 101137955. The project is supported by the Clean Hydrogen Partnership and its members. Project website: .

Declaration of conflicting interests

The authors declare no potential conflicts of interest with respect to research, authorship, and publication of this article.

Appendix A: Burner length for acoustic calculations Since the BTM depends upon the length of the burner assumed for the wave reconstruction using MMM, it was evaluated for a number of predefined lengths. The results are shown in Figure 10 for increasing burner length (dark to light red hues), indicating that the element $T_{11}$ of the conventional BTM $T^{f m}$ is a strong function of this length. The elements $T_{21}$ and $T_{22}$ are almost unchanged and therefore not shown. A burner length of 80 mm meets the condition that the amplitude of $T_{11}$ is constant with respect to frequency and equal to the ratio of the characteristic impedances i.e. $β_{f}$ , shown by the dashed black line. This holds true for all the operating conditions tested. Therefore, a burner length of 80 mm was assumed for the entire analysis.

Appendix B: Nomenclature

BTM: Burner Transfer Matrix

DS: Downstream

LSM: Least Squares Method

MMM: Multi-Microphone Method

ROM: Reduced-order Model

SM: Scattering Matrix

TM: Transfer Matrix

US: Upstream

WG: Waveguide

References

Haque

Nemitallah

Abdelhafez

, et al. Review of fuel/oxidizer-flexible combustion in gas turbines. Energy Fuels 2020; 34: 10459–10485.

Perpignan

Gangoli Rao

Roekaerts

. Flameless combustion and its potential towards gas turbines. Prog Energy Combust Sci 2018; 69: 28–62.

Beuth

Reumschüssel

Saldern

, et al. Thermoacoustic characterization of a premixed multi jet burner for hydrogen and natural gas combustion. J Eng Gas Turbines Power April 2024; 146(4): 041007. DOI: 10.1115/1.4063692

Guethe

de la Cruz García

Burdet

. Flue gas recirculation in gas turbine: Investigation of combustion reactivity and NOx emission, 2009, pp.179–191. ASME Turbo Expo. DOI: 10.1115/GT2009-59221.

Eck

MEG

Nedden

von Saldern

JGR

, et al. Experimental design validation of a swirl-stabilized burner with fluidically variable swirl number. J Eng Gas Turbines Power 2024; 147: 041017.

Kuhn

Terhaar

Reichel

, et al. Design and assessment of a fuel-flexible low emission combustor for dry and steam-diluted conditions. In: ASME Turbo Expo 2015, 2015, p.V04BT04A024. DOI: 10.1115/GT2015-43375.

Stathopoulos

Kuhn

Wendler

, et al. Emissions of a wet premixed flame of natural gas and a mixture with hydrogen at high pressure. J Eng Gas Turbines Power 2016; 139: 041507.

Dybe

Yasir

Güthe

, et al. On the demonstration of a humid combustion system performing flexible fuel-switch from pure hydrogen to natural gas with ultra-low nox emissions. J Eng Gas Turbines Power 2023; 146: 051018.

Culick

. Unsteady Motions in Combustion Chambers for Propulsion Systems. California Institute of Technology: NATO Research & Technology Organisation, 2006.

10.

Polifke

Lawn

. On the low-frequency limit of flame transfer functions. Combust Flame 2007; 151: 437–451.

11.

Huber

Polifke

. Dynamics of practical premixed flames, part II: Identification and interpretation of CFD data. Int J Spray Combust Dynam 2009; 1: 229–249.

12.

Karlsson

Åbom

. Aeroacoustics of T-junctions—an experimental investigation. J Sound Vib 2010; 329: 1793–1808.

13.

Bauerheim

Boujo

Noiray

. Numerical analysis of the linear and nonlinear vortex-sound interaction in a T-junction. In: AIAA AVIATION 2020 FORUM. DOI: 10.2514/6.2020-2569.

14.

Belfroid

SPC

Schiferli

Peters

RMCAM

, et al. Flow-induced pulsations caused by split flow in a T-branch connection. In: ASME 2006 pressure vessels and piping/ICPVT-11 conference, 2008, pp.583–589. DOI: 10.1115/PVP2006-ICPVT-11-93884.

15.

Karlsson

Åbom

. Quasi-steady model of the acoustic scattering properties of a T-junction. J Sound Vib 2011; 330: 5131–5137.

16.

Blondé

Schuermans

Pandey

, et al. Effect of hydrogen enrichment on transfer matrices of fully and technically premixed swirled flames. J Eng Gas Turbines Power 2023; 145: 121009.

17.

Zur Nedden

Eck

MEG

Lückoff

, et al. Burner and flame transfer matrices of jet stabilized flames: Influence of jet velocity and fuel properties. In: ASME Turbo Expo 2024, 2024, p.V03AT04A048. DOI: 10.1115/GT2024-124934.

18.

Beuth

Reumschüssel

von Saldern

, et al. Experimental study of the dynamics of lean premixed hydrogen flames in a multi jet combustor. J Eng Gas Turbines Power 2024; 147: 031013.

19.

Englund

Richards

. The infinite line pressure probe. NASA Tech Memorandum 1985; 19850055553: 24(2, 19).

20.

Fischer

. Hybride, thermoakustische Charakterisierung von Drallbrennern. PhD, Technischen Universität München, Germany, 2004.

21.

Alemela

. Measurement and scaling of acoustic transfer matrices of premixed swirl flames. PhD, Technischen Universität München, Germany, 2009.

22.

Schuermans

. Modeling and Control of Thermoacoustic Instabilities. PhD Thesis, École Polytechnique Fédérale de Lausanne (EPFL), 2003.

23.

Jang

. On the multiple microphone method for measuring in-duct acoustic properties in the presence of mean flow. J Acoust Soc Am 1998; 103: 1520–1526.

24.

Polifke

van der Hoek

Verhaar

. Everything you always wanted to know about f and g. Technical Report of ABB Research, 1997.

25.

Bellucci

Schuermans

Nowak

, et al. Thermoacoustic modeling of a gas turbine combustor equipped with acoustic dampers. J Turbomach 2005; 127: 372–379.

26.

Luong

Howe

McGowan

. On the Rayleigh conductivity of a bias-flow aperture. J Fluids Struct 2005; 21: 769–778.

27.

Idelchik

. Handbook of Hydraulic Resistance. 4 ed. New York: Begell House, 2007.

28.

Nesterov

. Experimental study of the conductance of a circular hole, in a partition across a tube. USSR Acad Sci 1941; 31: 879–882.

29.

Jaouen

Chevillotte

. Length correction of 2D discontinuities or perforations at large wavelengths and for linear acoustics. Acta Acust united Ac 2018; 104: 243–250.