Burner and Flame Transfer Matrices of Jet-Stabilized Flames: Influence of Jet Velocity and Fuel Properties

Nedden, Philipp zur; Eck, Mattias E. G.; Lückoff, Finn; Paschereit, Christian Oliver; Orchini, Alessandro

doi:10.1115/1.4066457

Abstract

To achieve the decarbonization of electrical power generation, gas turbines need to be upgraded to combust high-hydrogen content fuels reliably. One of the main challenges in this upgrade is the burner design. A promising burner concept for a high-hydrogen fuel mixture are jet burners, which are highly flashback resistant thanks to their high bulk velocity. Due to its nonacoustically compact extension and the presence of hydrogen in the fuel mixture, new challenges arise in assessing the (thermo)acoustic response of this burner design. A burner transfer matrix (BTM) and the flame transfer function (FTF) or transfer matrix (FTM) are typically measured with the multimicrophone method (MMM) to assess the performance of new burner types in relation to thermoacoustic stability. With the switch toward hydrogen, the fuel/air mixture is significantly altered in its properties regarding the speed of sound and density, which are of fundamental importance for acoustic waves propagation and their reconstruction via the MMM, as highlighted in recent work. In this work, we extend this discussion by studying the influence of the gas composition within the burner when measuring BTMs, and its indirect effect on the assessment of FTFs. Experimentally, we achieve this by adapting the preheating temperature during the measurement of the BTM with a nonreactive mixture in order to match the speed of sound of the hydrogen–air mixture required to flow in the burner under reactive conditions. Additionally, we present an analytical model for the jet burner transfer matrix, which is validated against the experimental data. Since the BTM is fundamental for the assessment of the FTM and FTF, the propagation of the error of changing fuel mixtures in the burner is evaluated. The influence of the variation in reactant composition of the BTM on the FTM assessment is noticeable, particularly in the gain of the FTFs. Furthermore, the influence of the total mass flow and, thus, the bulk flow velocity on the FTF is analyzed.

1 Introduction

To limit the human impact on the production of greenhouse gases, global electricity generation needs to be decarbonized. To achieve this target, a large share of energy must be produced from renewable sources. These sources are, however, intermittent. To guarantee a stable electrical grid, other on-demand power inputs are needed. This balancing power supply can be provided by stationary gas turbines. To future-proof stationary gas turbines, the use of carbon-free fuels is unavoidable. Therefore, burners for future gas turbines must be upgraded to be able to combust carbon-free fuels. Conventional gas turbine burner concepts, such as swirl burners, can reach a high level of hydrogen share, up to 60 vol. % [1]. This seemingly large amount of hydrogen in the fuel composition is, however, insufficient to significantly reduce the CO₂ emission levels of gas turbines [2]. Higher hydrogen shares, greater than 70%, must be burned. A possible method to increase the hydrogen share limit is to use a high amount of steam to stabilize the swirl combustor [3]. A promising new solution is to use lean premixed jet burners, which are highly flashback resistant thanks to their high bulk flow velocities [4].

Lean premixed combustion suffers from the drawback of being susceptible to thermoacoustic instabilities [5,6]. These instabilities may hinder the safe operation of the combustor and must be assessed in the early stages of a burner/combustor development. The flame transfer function (FTF) or flame transfer matrix (FTM) can be measured to assess the thermoacoustic properties of a newly developed combustor by means of the multimicrophone method (MMM) [7]. This method relies on the knowledge of the burner acoustics, assessed by the burner transfer matrix (BTM). The switch toward high-hydrogen content fuels poses new challenges in the assessment of the thermoacoustic response of a burner. First, the jet burner design is challenging for the BTM and FTM reconstructions due to the longer extension of the burner compared to traditional swirl burners. This leads to a higher influence of the propagation time through the burner in the reconstructed BTM. Also, the Mach number in the burner's mixing tube is significantly higher than in conventional swirl burners, which influences the shape of the BTM. Second, the hydrogen composition influences the reconstruction of the acoustic field. Due to the strong difference in density and speed of sound of hydrogen compared to air, even small amounts of H₂ significantly change the properties of the composition. If not accounted for, this results in a significant reconstruction error of the MMM, as shown by Blonde et al. [8].

In this work, we extend this discussion by studying the direct influence of the gas composition in the BTMs, and its indirect effect on the assessment of FTFs. A BTM is canonically measured at conditions where only (preheated) air is fed into the system. The acoustic properties of the burner, however, change when the gas flowing through the burner is a mixture of (preheated) air and fuel. This is particularly true for hydrogen-rich fuels because the thermodynamic properties (density, heat capacities) of hydrogen vary significantly compared to those of pure air. These changes influence the internal acoustic response of a burner, an effect that is beyond pure corrections of the MMM. The longer the burner, the larger the influence of the thermodynamic properties of the mixture on its acoustic response. This is particularly relevant for jet burners since they feature a relatively long tubular section required for flow-conditioning (see Fig. 1). In the following, we will first discuss the methodology changes required to appropriately consider the high hydrogen content to measure the required BTMs. Then, we will investigate the influence of the total mass flow, thus varying bulk flow velocity, on the FTF of the considered jet burner.

Fig. 1

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Schematic of the entire experimental test rig, comprising a combustion chamber and a burner enclosed between two duct sections allowing for the reconstruction of the acoustic fields with the MMM and the measurement of BTMs and FTMs

2 Experimental Setup

A generic piloted single hydrogen jet burner (POShyDON) is investigated in this study. The burner response is measured with two microphone arrays, one upstream and one downstream of the burner, each consisting of four microphones (G.R.A.S. 46 BP) that are nonuniformly spaced. The spacing between the first and the last microphone in both microphone arrays is approximately 300 mm. The microphones are mounted into cooled holders to enable a safe operation for the preheated conditions upstream and hot conditions due to the exhaust gas downstream of the burner element. The microphones are calibrated a priori with a pistonphone for absolute calibration and a calibration tube for relative phase calibration. Two sets of loudspeakers (B&C Speakers 18SW5-8) are mounted up- and downstream of the burner to provide two linearly independent forced acoustic fields needed for the assessment of the transfer matrices. The upstream set consists of four loudspeakers, whereas the downstream consists of two loudspeakers. We refer to Fig. 1 for a detailed description of the test rig.

Figure 2 shows a schematic view of the POShyDON burner, as shown in Ref. [9], which can be divided into three main components: an inlet nozzle (from A to C), a mixing tube (extending from C to D), and a dump plate at D. The tube length is chosen to be approximately ten times the diameter D of the pipe. This is done to align the flow and enhance the mixing of the reactants in a technically premixed configuration for the combustion (flow conditioning). For the safe ignition of the burner, a perfectly premixed pilot is used, which consists of eight jets in crossflow slanted toward the main jet center axis, displayed in the bottom left of Fig. 2. In this study, only perfectly premixed conditions are tested. For the perfectly premixed case, the fuel injection is situated further upstream of the upstream loudspeakers, indicated in the top inset of Fig. 1, guaranteeing a better level of spatial mixing of reactant. Using an orifice, the decoupling of the fuel supply from the acoustic forcing is achieved. A downstream anechoic termination is designed for each condition as in Ref. [10] to avoid the occurrence of thermoacoustic instabilities during the MMM measurements.

Fig. 2

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Detailed view of the POShyDON burner. Top: Cross section of the burner. Bottom left: Front view of the burner from the combustion chamber at the reference plane D. Bottom right: Feed channels for the pilot injection. The arrows display the flow direction of the premixed fuel and air. Capital letters indicate reference planes for area changes, and the numbers indicate propagation zones.

3 Methodology

The flame transfer matrix is measured to assess the flame response to acoustic fluctuations. It is typically measured with the multimicrophone method, which relates the pressure signals of two or more microphones to the propagation of planar acoustic waves [11,12]. To detect the downstream and upstream traveling planar acoustic waves,

\hat{f}

and

\hat{g}

⁠, respectively, the following relation is used:

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

(1)

with N the number of microphone signals available,

\bar{c}

the speed of sound,

\bar{u}

the velocity in the microphone section, and ω the angular frequency. When N > 2, the system of Eq. is over-defined and can be solved in a least squares sense. As discussed in Refs. [11,13], the (squared) error in the MMM due to noise in the pressure signals is proportional to

E^{2} = \frac{N}{N^{2} - | \sum_{j = 1}^{N} e^{i \frac{2 ω x_{j}}{c (1 - M^{2})}} |^{2}}

(2)

with x_j denoting the position of the microphone j and Ma the Mach number. This expression is used for the values assessed in this work, with the given microphones and the spacing for the up- and downstream microphone array. Figure 3 displays the values for both arrays. For the following evaluation, a threshold in the error assessment is chosen to be 25%, therefore frequencies below 110 Hz cannot be precisely measured with the given microphone arrays.

Fig. 3

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Leading error of the MMM reconstruction for the available experimental setup computed by Eq. (2), at the up- and downstream microphone array, for the reactive conditions presented in this paper. The error term is computed for all measured conditions, and the mean of these errors is plotted.

To determine the transfer matrix, two independent acoustic fields are required. They are provided by activating first only the upstream-mounted loudspeakers (i), and then only the downstream-mounted ones (ii). From the Riemann invariants reconstructed by the MMM, the scattering matrix (SM) can be calculated by solving

[\begin{matrix} {\hat{f}}_{d}^{(i)} & {\hat{f}}_{d}^{(i i)} \\ {\hat{g}}_{u}^{(i)} & {\hat{g}}_{u}^{(i i)} \end{matrix}] = \underset{SM}{\underset{︸}{[\begin{matrix} T_{d u} & R_{d d} \\ R_{u u} & T_{u d} \end{matrix}]}} [\begin{matrix} {\hat{f}}_{u}^{(i)} & {\hat{f}}_{u}^{(i i)} \\ {\hat{g}}_{d}^{(i)} & {\hat{g}}_{d}^{(i i)} \end{matrix}]

(3)

with _u and _d denoting the duct sections upstream and downstream of the burner, respectively. The diagonal elements T are the transmission coefficients of the planar waves through the burner, and R are the reflection elements on each side of the burner. The SM can be converted into a transfer matrix (TM), which relates the primitive variables of velocity and pressure fluctuations instead of relating the Riemann invariants f and g.

When the flame is turned on, it is not possible to measure the flame transfer matrix directly, because the pressure transducers capture the combined acoustic response of the burner and flame, which we denote by flame-burner transfer matrix (FBTM). To acquire the FTM, first the BTM has to be determined and its contribution removed from the FBTM

FTM = FBTM \cdot {BTM}^{- 1}

(4)

In this process, one assumes that the BTM under reactive conditions is identical to that under nonreactive conditions. This is, however, not necessarily true, as it will be briefly discussed in the following.

For the BTM and FBTM measurements, the reference planes are selected as the burner inlet plane (A) for the upstream forcing measurement and as the burner exit plane (D) for the downstream forcing measurement. During the FBTM measurements, the acoustic forcing is regulated to have a constant amplitude (5% of the mean flow) at the burner outlet plane on the cold side. This amplitude is achieved by iteratively assessing the velocity fluctuations with the upstream microphone array for both forcing conditions as the forcing amplitude is varied. The BTM measured with the nonreactive mixture evaluates the velocity fluctuations at the burner exit.

3.1 Analytical Model of the Burner Transfer Matrix.

For the following investigation, an analytical model for the acoustic response of the POShyDON burner is constructed, comparable to the work of Beuth et al. [14], to assess how the key parameters influence the BTM. The model relates the acoustic fluctuations upstream and downstream of the jet burner via the burner transfer matrix

{[\begin{matrix} \tilde{p} \\ \hat{u} \end{matrix}]}_{d} = BTM {[\begin{matrix} \tilde{p} \\ \hat{u} \end{matrix}]}_{u}

(5)

with $\hat{p} / (\bar{ρ} \bar{c}) = \tilde{p}$ denoting the normalized pressure. The BTM model is composed of a chain of plane wave-propagation zones and area changes (see Fig. 2).

The propagation element of a straight duct with a constant cross section area and a uniform mean flow is expressed by

{TM}_{j} = \frac{1}{2} [\begin{matrix} e^{i \frac{ω L_{j}}{{\bar{c}}_{j} (1 - M_{j})}} + e^{- i \frac{ω L_{j}}{{\bar{c}}_{j} (1 + M_{j})}} & - e^{i \frac{ω L_{j}}{\bar{c_{j}} (1 - M_{j})}} + e^{- i \frac{ω L_{j}}{{\bar{c}}_{j} (1 + M_{j})}} \\ - e^{i \frac{ω L_{j}}{{\bar{c}}_{j} (1 - M_{j})}} + e^{- i \frac{ω L_{j}}{{\bar{c}}_{j} (1 + M_{j})}} & e^{i \frac{ω L_{j}}{{\bar{c}}_{j} (1 - M_{j})}} + e^{- i \frac{ω L_{j}}{{\bar{c}}_{j} (1 + M_{j})}} \end{matrix}]

(6)

where the index $_{j}$ denotes the section in which the propagation model is used, $L_{j}$ denotes the length of the element, and $M_{j}$ is the Mach number $M_{j} \equiv {\bar{u}}_{j} / {\bar{c}}_{j}$ ⁠. The parameters for this model are known from the burner geometry, the mass flow, and the chemical composition of the air/fuel blend. This model is used to propagate the acoustic wave in Secs. 1 and 2 of Fig. 2.

For the area change (contraction and expansion), the following model is used as in Ref. [15]:

J_{x} = [\begin{matrix} 1 & ζ_{x} M_{x} - i k L_{eff, x} \\ 0 & α \end{matrix}]

(7)

with $M_{x}$ denoting the upstream Mach number, the subscript $_{x}$ the reference plane (B-C and D in Fig. 1), α = A_us/A_ds the ratio of cross section areas, and $ζ_{x}$ the loss coefficient. $L_{eff, x}$ is the effective length of the area change and is estimated as described by Ref. [16]. The values for ζ are obtained by fitting them to the experimental values acquired. A simulated annealing algorithm, chosen due to its global optimization ability, performs this computation. The optimizer identifies the loss coefficients by minimizing the value of the L1 norm between the model and the experimental results for all experiments.

By concatenating wave propagation elements and area change elements, the model for the POShyDON burner is derived as follows:

BTM = J_{D} {TM}_{2} J_{B - C} {TM}_{1}

(8)

This expression relates the acoustics of the burner element from the inlet reference plane A to the outlet reference plane D. This expression makes clear that the presence of H₂ in the reactive mixture flowing through the burner changes the BTM, because it alters the value of the speed of sound, density and mean flow that enters in the propagation and area jump matrices. In the following, we propose a new procedure to explicitly account for these changes in the experimental assessment of the BTM.

3.2 Assessing the Flame Transfer Function From the Flame Transfer Matrix.

From a theoretical perspective, the Rankine–Hugoniot equations can be used to describe the jump conditions across an acoustically compact heat element, yielding an explicit expression for the FTM [12]

[\begin{matrix} {\tilde{p}}_{d} \\ {\hat{u}}_{d} \end{matrix}] = \underset{FTM}{\underset{︸}{[\begin{matrix} β & - β (T_{r} - 1) M_{1} (1 + FTF) \\ - (T_{r} - 1) γ M_{1} & 1 + (T_{r} - 1) FTF \end{matrix}]}} [\begin{matrix} {\tilde{p}}_{u} \\ {\hat{u}}_{u} \end{matrix}]

(9)

where β denotes the ratio of specific impedances, $β \equiv ({\bar{ρ}}_{u} {\bar{c}}_{u}) / ({\bar{ρ}}_{d} {\bar{c}}_{d})$ ⁠, γ is the ratio of specific heats, and $T_{r} = T_{d} / T_{u}$ denotes the temperature ratio across the flame. The FTM expression in (9) assumes low Mach numbers and neglects second order Mach number contributions. This is reasonable for our experiments since the Mach number in the burner tube reaches a maximum value of approximately M = 0.2. Also, Eq. (9) assumes a stiff fuel injection, resulting in no pressure fluctuation in the fuel supply [17]. This assumption is valid because we decoupled the fuel injection from the forcing via the use of an orifice. Last, to derive Eq. (9), it is assumed that the acoustic field does not interact with the turbulence field of the flame [13]. Following Ref. [8], in this study, T_d is estimated to be 90% of the adiabatic flame temperature, in order to account for heat loss due to the cooling and the radiation through the quartz combustion chamber. We note that this temperature is not the same temperature as in the downstream microphone duct, which is measured, since the temperature drops even more in this downstream section due to further heat losses effects.

The FTF relates the heat release rate fluctuations to the velocity fluctuations and is defined as

FTF (ω) = \frac{q^{'} (ω) / \bar{q} (ω)}{u^{'} (ω) / \bar{u} (ω)}

(10)

It can conveniently be extracted from the FTM without any measurement of the heat released by the flame. From the expression of the FTM₂₂ element (the bottom right element of the FTM), one obtains

FTF = \frac{{FTM}_{22} - 1}{T_{r} - 1}

(11)

Even if only one element of the FTM is used to reconstruct the FTF, the determination of the entire FTM is required to properly determine FTM₂₂, which is why two linearly independent acoustic forcings are required in the experimental procedure.

4 Results

In the following, we will first discuss the influence of the fuel composition on the BTM and how this effect changes the assessment of the FTM and, respectively, the FTF. Second, the influence of the air mass flow and, thus, the bulk velocity on the FTF is discussed.

4.1 Assessing Flame Transfer Function Measurements With High Hydrogen Concentration.

To investigate the effect of changing fuel compositions on the speed of sound and density, we vary the equivalence ratio $ϕ$ of the mixture composition and plot the relative errors of the gas properties compared to those of air. Figure 4 displays the variation in density and speed of sound, which are the two changing parameters for the assessment of $\bar{c} + \bar{u}$ ⁠, which can be found in Eq. (6) of the burner model. The reference condition is chosen differently for the educts and products of the combustion process. For the educts, air at room temperature is used as a reference; for the products, the air temperature is assumed to match the adiabatic flame temperature. This normalization generalizes the results for any given temperature since the values displayed rely solely on the gas properties of the mixture. In the case of a BTM measurement, both microphone segments see the gas properties of educts since no combustion is taking place. For the FBTM measurement, the upstream microphone section sees the educt gas properties, and the downstream section sees the product gas properties, which are obtained by using Cantera [18]. For the speed of sound plotted in Fig. 4, the gas properties of methane over $ϕ$ only slightly change for the educts and products, with a maximal error below 2%. On the other hand, the gas properties change significantly for the hydrogen mixtures as the hydrogen content is increased. The variation is higher for the educts than for the products due to the different chemical compositions of the combustion products. Focusing on the variation of the gas properties in the products, one can see that the deviation of the speed of sound is significantly higher for hydrogen. This is due to the fact that water vapor, the main product of H₂ combustion, has a speed of sound much higher than carbon dioxide, present in CH₄ combustion. The variation of the gas properties in the educts for the hydrogen combustion increases with the equivalence ratio and reaches values up to 19% for the speed of sound. Similar trends are observed for the density of the mixtures, with hydrogen reducing the density significantly, with a deviation of the educts of up to 29%. For methane, the density and speed of sound variations in comparison to the pure air case are minimal, which explains why these effects were never considered in detail in the past. However, for the hydrogen–air mixture, the density and the speed of sound significantly differ from the condition with pure air. They may induce significant errors in the BTM and FTM reconstructions.

Fig. 4

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Influence of the gas properties on the speed of sound and density for the reaction educts and products. A comparison between the influence of hydrogen and methane. The values are normalized to the reference conditions with only air. The temperature of the mixture is assumed to be $T_{in}$ = 23 °C is assumed as the input temperature. For the combustion products, the adiabatic flame temperature is used. The shaded areas denote the influence of a variation of $\pm 5 %$ of the corresponding temperature on the speed of sound and density. For simplicity reasons, the natural gas is assumed to be methane.

Therefore, in the following, we will focus on the influence of hydrogen on the assessment of the BTM and FTM. First, the influence of the difference in speed of sound and density on the BTM is investigated. We use the previously introduced model and compare it to measured data, considering variations in the properties of the mixture propagating through the burner. Second, we investigate the influence of the BTM on the FTM by evaluating the same flame conditions with differing BTMs. For these investigations, we will show two different conditions. The conditions represent the mixture required for the burner to operate at a thermal power of 100 kW with an air mass flow of 160 kg/h. This is done for hydrogen (H₂), and a reference condition without fuel further denoted as “Air.”

The resulting thermodynamic and combustion parameters for each of the mixtures considered are displayed in Table 1. For a precise assessment of the BTM, the change in the speed of sound of the reactive mixture needs to be accounted for. It is, however, not possible to measure a BTM while injecting a reactive fuel composition without combusting it, due to safety and environmental concerns. To evaluate experimentally a BTM with the properties of the fuel/air mixture, we adapt the speed of sound of pure air by increasing the preheating temperature $T_{in}$ ⁠. These values and the corresponding speed of sound and density are denoted with a subscript $_{c}$ in Table 1. The preheating temperature is increased to match the speed of sound c. This procedure does not generally guarantee that the density of the heated air matches the one of the fuel/air mixture. For the condition investigated in this configuration, this difference is, however, only 1%.

Table 1

Density and speed of sound values for varying fuel types for a flame with constant thermal power $P_{th}$ = 100 kW and an air mass flow of 160 kg/h

Type	Fuel	${\dot{m}}_{fuel}$ (kg/h)	$\bar{ρ}$ (kg/m³)	$\bar{c}$ (m/s)	$T_{in}$ (°C)	$T_{ad}$ (°C)	$ϕ$
FTM	Air + H₂	3	0.9731	382.1246	18	1637	0.642
BTM	Air	—	1.21259	342.1928	18	—	—
BTM	${Air}_{c, H_{2}}$	—	0.9692	382.1246	91	—	—

In the first row, the reference values are displayed for the conditions with pure air. The values with the subscript $_{c}$ are corrected to match the speed of sound of the fuel/air mixture with pure air by changing the preheating temperature $T_{in}$ ⁠.

4.1.1 Influence of Fuel Variation on the Assessment of the Burner Transfer Matrix.

In this section, the burner acoustics are characterized. The derived model in the previous section depends on the Mach number. The Mach number itself depends on the density and the speed of sound—thus, the fuel composition. The comparison between the BTM obtained experimentally and the model is displayed in Fig. 5, where Fig. 5(a) shows the absolute values of the BTM and Fig. 5(b) the phase values. Because all experiments are performed with air, the preheating air temperature of the experiment was adjusted to adjust the speed of sound of the experiment to that of the model with fuel.

Fig. 5

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Influence of the fuel composition on the values of the modeled and experimentally measured BTM. The mass flowrate is adapted accordingly to the total mass flowrate of the FTM: (a) absolute values and (b) phase values.

All elements of the BTM obtained from the theoretical model match the experimental values, with some minor differences. In the BTM₁₁ element, which relates the upstream pressure fluctuations to the downstream pressure fluctuations, the results of the experiments and the models match well. At low frequency, however, the model and the experimental results do not fit as closely, which can be due to the MMM reconstruction uncertainties discussed previously. The BTM₁₂ element of the model is in good agreement with the measured results for both conditions, resulting in relatively high gain values compared to the other BTM elements. This element relates the downstream pressure fluctuations ${\tilde{p}}_{d}$ to the upstream velocity fluctuations ${\hat{u}}_{u}$ ⁠. Therefore, the upstream velocity fluctuation has a high gain on the downstream pressure fluctuations. The BTM₂₁ element, which relates the downstream velocity fluctuations to the upstream pressure fluctuations, has a low gain compared to the other elements, indicating that the influence of the upstream pressure fluctuations on the downstream velocity fluctuations is damped through the burner. The gain results slightly differ at low frequencies and at frequencies above 400 Hz, whereas the phase predicted by the model compares very well with the experimental results in the entire frequency range. Last, for the BTM₂₂ element, which relates upstream to downstream velocity fluctuations, the gain varies around a value of about two, corresponding to the ratio between the downstream and upstream cross section areas, $α = 2.01$ in Eq. (7). The phase shows a good agreement between the experimental and analytical results.

In the BTM₂₁ and BTM₂₂, in the frequency region between 100 Hz and 350 Hz, the experimental results show fluctuations, which are not predicted by the model. From a look at the SM, it is seen that these fluctuations are represented only in the element representing the reflection at the burner exit $| R_{dd} |$ ⁠, suggesting that the fluctuations in this range are induced by the burner front plate. In Fig. 6, the gain of the corresponding SM element is shown; since the phase values are not as strongly affected, they are not shown. The burner plate has eight jets in crossflow, which are depicted in Fig. 2, and a further cavity for the ignition torch. In Fig. 2 in the bottom right, a detailed view of the feed channels for the pilot injection is shown, consisting of additional cavities. In order to study the influence of these cavities in the burner front plate, the cavities are closed using tape. This can only be done for cold measurements such as the BTM ones, since the pilot flame and ignition torch are required for a stable ignition of the flame.

Fig. 6

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Influence of the cavity at the burner front plate, the comparison is taken without preheating and an air mass flow of 220 kg/h. The condition without cavities is acquired with the burner front plate taped to exclude the influence of the cavities of the pilot injection and the ignition torch.

In Fig. 6, it can be seen that, with the closed cavities, the fluctuations in the reflection coefficient at the burner plate disappear. This confirms the assumption that the cavities of the burner plate induce a small frequency-dependent variation in the reflection coefficient. which can be translated to small oscillations in the BTM₂₁ and BTM₂₂ elements which are; however, not critical for assessing the BTM, since the values are repeatable. However, the strong assumption taken for the assessment of the FTM that the BTM does not change at hot conditions is even more questionable for the cavities at the burner plate due to their proximity to the hot gases of the combustion.

Overall, the influence of the hydrogen share on the BTM is noticeable in the model as well as in the experimental results, this is especially the case for the gains of the BTM₁₁ and BTM₁₂ elements. The influence on the gains of the BTM₂₁ and BTM₂₂ elements is lower compared to the previous two elements although, a sudden variation of the conditions adapted to the hydrogen speed of sound can still be noticed. The phase values for all elements are not as strongly affected by the adaption to hydrogen conditions.

Our results show that hydrogen's influence on the BTM is significant for reconstructing the downstream pressure fluctuations, and should not be neglected. The strong influence of the speed of sound on the BTM is especially the case for the burner investigated in this study; due to its significant length, the propagation time through the burner is altered with differing sound speeds (see Eq. (6)). In the following, we will investigate the influence of the BTM on the FTM and FTF.

4.1.2 Influence of an Error in the Burner Transfer Matrix on the Flame Transfer Matrix.

To assess an FTM, only an FBTM can be directly measured, from which the previously acquired BTM is “subtracted.” The change of the gas properties of the BTM does change the influence on the FTM in a nontrivial way. Therefore, in the following, the FTM is displayed and evaluated once with the BTM measured with air and once with the BTM adapted to the gas properties of the hydrogen and air mixture. The results are shown in Fig. 7, with Fig. 7(a) displaying the absolute values and Fig. 7(b) the phase.

Fig. 7

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Absolute values of the single elements of the FTM for hydrogen combustion (see Table 1) with the perfectly premixed injector. The color corresponds to the previously displayed BTM: (a) absolute values and (b) phase values.

Generally, it can be stated that the influence of the variation of the BTM has a marginal influence on the phase values for all elements of the FTM. For the absolute values of the FTM, the influence of the BTM is noticeable in all elements. The result of $| {FTM}_{11} |$ should correspond to the specific impendence ratio β, as per the Rankin-Hugoniot equation with a stiff fuel injector and low Mach numbers as stated in Eq. (9). In Fig. 7(a), it can be seen that if the BTM is accounted for the hydrogen conditions, this assumption is matched closer for frequencies below 300 Hz.

The gain of the FTM₁₂ element results in high values compared to the other elements of the FTM. Nevertheless, if hydrogen is not accounted for in the BTM, the results are even larger, with up to twice the magnitude. The BTM values were noticeably higher by one order of magnitude in the BTM measurements for the element linking the upstream velocity fluctuation to the downstream pressure fluctuation. The large gain observed in the FTM₁₂ element are due to a change in the BTM in reactive conditions due to the high temperature combustion gases.² One may account for this effect, which however was found to have little influence on the FTM₂₂ element and the FTFs, and this correction is therefore not considered in this study.

The gain of the FTM₂₁ element has low values, consistently with the theoretical predictions of Eq. (9). However, an increase in the gain can be seen in the low and high-frequency limits. In the following section, we will extract the FTF from the FTM₂₂ element. Since the computation is a division by constant value, the influence of the BTM on the FTF is analogous to that of the BTM on the FTM₂₂ element.

4.1.3 Extracting an Flame Transfer Function From the Flame Transfer Matrix.

An FTF can be extracted from the FTM by using Eq. (11). The results for the perfectly premixed conditions are displayed in Fig. 8. We will discuss the influence of the BTM on the assessment of the FTF.

Fig. 8

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FTF values derived from the FTM for perfectly premixed conditions. The color denotes the BTM used for the evaluation of the FTM. The operational conditions are noted in Table 1.

The absolute values of the FTF show an evident influence of the BTM on the FTF. More precisely, the values of the FTF that do not account for the hydrogen share in the BTM measurements result in an overestimation of the gain values. The general slope of the function remains comparable, except for high frequencies where the slope of the uncorrected FTF increases. This can be due to the BTM, the slope of the elements relating to the downstream pressure fluctuations (BTM₁₁ and BTM₁₂) do show a significant difference at frequencies above 400 Hz. As in the FTM measurement, the phase values are not affected notably by the influence of hydrogen on the BTM. For a better understanding, the absolute error between the corrected and uncorrected FTFs is computed as

ϵ = | \frac{{FTF}_{Air}}{{FTF}_{{Air}_{c, H_{2}}}} - 1 |

(12)

Figure 9 displays the error assessed by Eq. (12) over the measured frequency range. The error created by assessing the BTM at differing ambient parameters induces a positive error over the entire frequency range. The mean error created by the uncorrected ambient parameters in the BTM is 42%, showing that if the influence of the BTM is not accounted for, the absolute value of the FTF is overestimated on the whole measured frequency range. Especially at higher frequencies, the error increases, with its highest error at the highest measured frequency of 504 Hz, with an error of 120%. This is because the main difference between the differing BTMs measured under the two differing conditions is to be found especially above 400 Hz. The lowest error of 11% is at the local minimum in the FTF at 170 Hz. This should be due to the lower absolute differences in the BTMs at these conditions.

Fig. 9

View large Download slide

Error of the FTF if the influence of hydrogen is not accounted for in the assessment of the BTM

This comparison demonstrates an influence of the correction of the gas properties of the BTMs by nearly halving the gain values of the FTF. A further discussion of the FTF is considered in the next paragraph to address the influence of the bulk velocity on the flame response.

4.2 Influence of Mass Flux for Pure Hydrogen on the Flame Transfer Function.

In this section, we further investigate the behavior of the FTF while varying the bulk velocity. We choose to keep the thermal power constant and vary the air mass flow. The parameters for this variation are displayed in Table 2.

Table 2

For all measurements, a thermal power of $P_{th}$ of 100 kW is taken, and the air mass flow is adapted, leading to a variation in equivalence ratio and thus adiabatic flame temperature

Type	${\dot{m}}_{air}$ (kg/h)	${\dot{m}}_{fuel}$ (kg/h)	Re	$v_{jet}$ (m/s)	l_f (D)	$T_{in}$ (°C)	$T_{ad}$ (°C)	$ϕ$
FTM	160	3	92416	52.1	3.03	18	1641	0.6418
FTM	200	3	114846	62.4	3.39	18	1397	0.5134
FTM	220	3	126068	67.5	3.72	18	1300	0.4667

For the cases listed in Table 2, the Reynolds number, the bulk jet velocity

v_{jet}

⁠, and the flame length f_l expressed in D are computed. The Reynolds number is computed as

R e = v D / ν

⁠, with ν being the kinematic viscosity. The flame length is taken from chemiluminescence images of

{OH}^{*}

⁠. The length is assessed by using the method of the center of gravity on the Abel-transformed mean images. These images have been taken with a slightly different burner configuration, equipped with technical fuel injection [9]. However, it is assumed that the mean field is not strongly affected by the fuel injection due to the long mixing length for hydrogen and air for the technical configuration of 8D. As proposed by Ref. [19], we scale the acquired FTF with the Strouhal number defined as

S t_{i} = \frac{f L_{i}}{v_{jet}}

(13)

with L_i denoting a characteristic length. In the following, we will use the flame length, l_f as characteristic length, as proposed by Ref. [19].

The acquired FTFs shown in Fig. 10 are all measured using a complete assessment of the FTM, where, for each case, a separate BTM is required to account for the varying influence of hydrogen. The FTF is shown in the first row as a function of the dimensional frequency. The results show that for the measured condition, the gain over the measured frequency range are close to one; for the lowest measured frequencies, an increase in the gain can be observed. We can only speculate about the trend at low frequencies since, with our microphone setup, we are not able to perform accurate measurements below 110 Hz.

Fig. 10

View large Download slide

FTF values derived from the FTM for the differing mass flows. The operational conditions are noted in Table 2. All results are acquired with the previously introduced correction of the BTM to hydrogen. The first row shows the unscaled results. The second row shows the scaled results by the Strouhal number St with the length of the flame.

The phase value converges to zero, for all three conditions. This convergence is expected for a perfectly premixed configuration, as shown by Ref. [17]. All three configurations show two gain peaks in the range between 200 Hz and 500 Hz. The peaks of the ${\dot{m}}_{air}$ = 200 kg/h and ${\dot{m}}_{air}$ = 220 kg/h match quite well, which may be due to the smaller difference in bulk flow velocity compared to the ${\dot{m}}_{air}$ = 160 kg/h case. The phase value for all three configurations shows a similar shape, which consists of an initial constant negative slope up to 220 Hz, followed by a flattening of the phase slip at higher frequencies, with further local maxima at the position of the minima in the absolute values.

With increasing flow rates, both the jet velocity U and the flame length l_f are higher. In terms of time-lag that perturbations take to travel along the flame, these two effects counter each other, since the time-lag $τ = l_{f} / U$ ⁠, and the time delay may remain (approximately) the same as we vary the mass flowrate. This is indeed shown in the phases plot of Fig. 10, where the slope of the FTFs for the three operating conditions are approximately the same.

When the measurements are scaled by the Strouhal number based on the flame length, $S t_{f_{l}}$ ⁠, the frequency positions of the gain peaks above 200 Hz match well for all three configurations. This indicates that these peaks are characterized by the flame itself, e.g., by the flame front and the shear layer of the turbulent jet. Similar observations can be made for the phase. The influence on the FTF measurements for differing bulk velocities and equivalence ratios results in an intriguing observation where the richer equivalence ratio case has a higher overall gain for frequencies above 200 Hz, indicating that the interaction of the shear layer with the flame front results in a higher thermoacoustic gain for higher equivalence ratios. Instead, the FTF trend below 200 Hz results in an inverse order where, with increasing ${\dot{m}}_{air}$ ⁠, the gain is increased. For the assessment of pure hydrogen flames, the low-frequency regime seems to be prone to feedback of the forcing velocity and the heat release rate.

5 Conclusion

In this study, we discuss potential error sources when measuring the acoustic response of hydrogen flames instead of methane/natural gas ones. First, we discuss the influence of the fuel composition on the BTM, which is used to obtain an FTM. The BTM must be measured with air (or alternatively, with an inert gas like helium). Therefore, to assess the influence of the H₂ mixture's properties on the measurements, we proposed to measure a BTM at an elevated temperature to correct for the influence of the speed of sound induced by the high hydrogen share. We investigated this influence with an analytical model of the presented burner which shows good agreement with the experimental data. We found that accounting for the changes in gas composition due to the hydrogen results in slightly differing values of the BTM, especially for the BTM₁₁ and BTM₁₂ elements at high frequency.

Moreover, we show the influence of varying gas composition on the speed of sound and, consequently, on the assent of the BTM. Our results show that if the change in properties of density and speed of sound is accounted for in the assessment of the BTM, this changes the results of the measured FTM distinctly. When an FTF is extracted from the FTM, we show that the differing BTMs have little influence on the phase value of the FTF. Contrarily, accounting for hydrogen in the assessment of the BTM results in a large deviation of the gain values for the corresponding FTF. Overall, the gain is overestimated if the influence of hydrogen is not accounted for.

Second, we investigate the influence of the ${\dot{m}}_{air}$ on the FTF, resulting in overall gain values close to one. In the low-frequency limit the gain rises for all three conditions. However, a low-frequency convergence limit cannot be reliably deducted since our experimental setup does not allow for accurate measurements below 110 Hz. The trend of the phase at low frequencies suggest a convergence toward zero, consistently with the literature. Applying a Strouhal number scaling for the absolute value of the FTFs, considering the flame length as a characteristic length, collapses the gain peaks around 250 Hz for the assessed FTFs. This finding indicating that these frequency-dependent variations in gain are due to an interplay of the flame front with the shear layer of the jet flame.

Acknowledgment

The authors would like to thank Bruno Schuermans for fruitful discussions on the BTMs, and Thorsten Dessin, Andy Göhrs, and David Lück for their technical support. Furthermore, the authors would like to thank Katharina Schmidt and Harout Nenejian for their support during the measurements.

Funding Data

European Research Council (ERC) funding under the Advanced Grant HYPOTHESIs (Grant No. 101019937; Funder ID: 10.13039/501100000781).
the Bundesministerium für Wirtschaft und Klimschutz (BMWK) (Grant No. 03EE5144; Funder ID: 10.13039/501100006360).
FVV e.V. (Grant No. 601421; Funder ID: 10.13039/501100003162).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Footnotes

2

Private communication with Bruno Schuermans.

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Burner and Flame Transfer Matrices of Jet-Stabilized Flames: Influence of Jet Velocity and Fuel Properties

Abstract

1 Introduction

2 Experimental Setup

3 Methodology

3.1 Analytical Model of the Burner Transfer Matrix.

3.2 Assessing the Flame Transfer Function From the Flame Transfer Matrix.

4 Results

4.1 Assessing Flame Transfer Function Measurements With High Hydrogen Concentration.

4.1.1 Influence of Fuel Variation on the Assessment of the Burner Transfer Matrix.

4.1.2 Influence of an Error in the Burner Transfer Matrix on the Flame Transfer Matrix.

4.1.3 Extracting an Flame Transfer Function From the Flame Transfer Matrix.

4.2 Influence of Mass Flux for Pure Hydrogen on the Flame Transfer Function.

5 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

Footnotes

References

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Burner and Flame Transfer Matrices of Jet-Stabilized Flames: Influence of Jet Velocity and Fuel Properties

Abstract

1 Introduction

2 Experimental Setup

3 Methodology

3.1 Analytical Model of the Burner Transfer Matrix.

3.2 Assessing the Flame Transfer Function From the Flame Transfer Matrix.

4 Results

4.1 Assessing Flame Transfer Function Measurements With High Hydrogen Concentration.

4.1.1 Influence of Fuel Variation on the Assessment of the Burner Transfer Matrix.

4.1.2 Influence of an Error in the Burner Transfer Matrix on the Flame Transfer Matrix.

4.1.3 Extracting an Flame Transfer Function From the Flame Transfer Matrix.

4.2 Influence of Mass Flux for Pure Hydrogen on the Flame Transfer Function.

5 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

Footnotes

References

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