Flame Transfer Functions of the Lean Burnout Zone of an RQL Combustion Chamber - Dynamic Response to Primary Zone and Mixing Port Velocity Fluctuations

Abstract

This article describes an experimental investigation of thermoacoustic flame transfer functions of the lean burnout zone of an rich–quench–lean combustion chamber. With spatial separation of the rich primary from the lean secondary dilution zone, the latter is independently examined. The multi-microphone-method was employed to characterize the combustor acoustic velocity response to acoustic forcing coming from the primary zone and the mixing ports. The lean secondary zone is then treated as a pure acoustic 3-port network element connected to a 2-port Rankine-Hugoniot flame element. Focusing only on heat release fluctuations due to velocity fluctuations, the former are described by two linear superimposed flame transfer functions as a function of the velocity fluctuations coming from the primary zone and the mixing ports, respectively. Based on a non-reacting and a reacting measurement the two flame transfer functions could be extracted from the experimental data. Within this research, flame transfer functions from the new acoustical approach are presented and compared with ones measured using chemiluminescence and a photomultiplier tube. It is found that the inverse diffusion flame in the burnout zone reacts to velocity fluctuations from the primary zone in the low frequency range and a clear low pass behavior is observed. The mixing port velocity fluctuations create a more broadband response. In the presented cases, the flame transfer functions calculated from chemiluminescence match those from the acoustic method very well.

Keywords

Thermoacoustics low frequency rich–quench–lean mixing zone flame transfer function

Introduction

Ambitious and stringent goals in the aviation industry have been set by the Advisory Council for Aeronautical Research in Europe (ACARE) to reduce ${CO}_{2}$ by $74 %$ and ${NO}_{x}$ emissions by $90 %$ by 2050 compared to the year 2000.¹ In recent years, the technological development has reduced jet engines fuel burn per passenger kilometer by nearly $70 %$ . Almost $50 %$ of the fuel reduction is achieved by improvement of jet engine cycle efficiency (e.g. increasing overall pressure ratio, bypass ratio, turbine inlet temperature, etc.).² Several low-emission combustion systems have been developed and are still the focus of many research studies. Within these, the RQL (rich–quench–lean) combustor is an effective method to reduce ${NO}_{x}$ emissions. In an RQL combustor, the combustion process begins in the primary fuel rich zone close to the injector, followed by air quenching, and fuel lean secondary zone where the combustion process is completed. In this way, combustion in regions close to stoichiometric conditions, where ${NO}_{x}$ production is expected to be highest, are minimized.

The ability of RQL combustors to achieve low emissions were evaluated by Holdeman³ and Rosfjord et al.⁴ They investigated an RQL combustion chamber with a cylindrical flame tube and various quench jet-in-crossflow (JIC) configurations to reduce emissions. The results showed that ${CO}_{2}$ and ${NO}_{x}$ emissions were very sensitive to the performance of the JIC quenching and mixing zone and less sensitive to the performance of the used injector. Multiple experimental and numerical studies focused on the flow-field and the mixing and reaction processes in the secondary zone. It was found that the momentum-flux ratio (J) as well as orifice geometry and spacing are crucial parameters influencing the mixing, combustion and emission processes.³

Low emission solutions for combustion chambers are limited by the occurrence of thermoacoustic instabilities, which reduce the operational flexibility. Like all high power density combustors RQL combustion is prone to thermoacoustic instabilities, which are the result of dynamic interactions between acoustic oscillations, flow and unsteady heat release in the combustion system. Therefore, the prediction of the thermoacoustic stability of the engine combustor is an important step in the design process. The understanding of the instability mechanisms may allow for greater fuel and operational flexibility. Eckstein et al.⁵ was one the first to study the dynamic behavior of RQL combustors. The main objective was the low-frequency phenomenon called “rumble” or “growl,” which is observed in aero engines during idle and sub-idle conditions. The research was focused on the response of a non premixed spray flame in an RQL combustor under the influence of primary air forcing and on the entropy-wave interaction with the combustor exit nozzle. Most of the findings were related to the spray atomization and evaporation process. Missing the possibility of downstream acoustic excitation (only available in later studies e.g. Bade et al.,⁶ Stadlmair and Sattelmayer,⁷ de Rosa et al.⁸) the quantitative determination of the systems thermoacoustic response, for example, captured as flame-transfer-matrix (FTM) or flame-transfer-function (FTF) was not possible. The more recent results from Cai et al.⁹ support the conjecture that the thermoacoustic behavior of the lean combustion zone plays a decisive role for the overall combustion dynamics of RQL burners. Cai et al.⁹ was able to identify different instabilities depending on the operational mode of his RQL combustor segment. Using natural gas as fuel and operating under ambient pressure Cai et al.⁹ used chemiluminescence (CL) heat release images and dynamic pressure measurements inside the chamber. As a result they showed a correlation between the acoustic emissions during the RQL operation mode and the interaction of the fuel-rich mixture with the secondary air jets, but their dynamics could not be quantified in an FTF.

In the authors’ previous research a novel test rig was presented to measure and characterize the flame dynamics of the lean secondary zone in an RQL combustion chamber.¹⁰ Separating the rich primary zone and the lean dilution zone allows the independent investigation of the secondary zone and its contribution to the whole thermoacoustic system. With an experiment ensuring acoustically stiff mixing ports, flame dynamics in form of FTFs representing the response to primary zone velocity fluctuations were determined based on the 2-port Rankine-Hugoniot (RH) methodology.

In this article, the extension of the 2-port RH technique to a 3-port topology is presented, which results in two separated FTFs. They characterize the response of the lean secondary zone to acoustic fluctuations from either the primary zone and the secondary air velocity fluctuations. For better understanding of the considered fluctuations and the dynamic processes in the lean secondary zone the RQL topology is introduced in terms of acoustic network system elements. Based on the network system theory a pure acoustic approach is given and a formulation of the flame heat release depending on two linear superimposed FTFs is derived. In the following the experimental setup and the data processing is described. Then multiple FTFs depending on the velocity fluctuations upstream and from the mixing ports are presented and discussed.

The lean secondary zone as a thermoacoustic 3-port

The network element technique (e.g. Munjal,¹¹ and Schuermans et al.¹²) is a well known tool for the identification and description of the thermoacoustic properties of combustion systems. In such a network, the thermoacoustic element describes the functional relations between the acoustic variables, for example, acoustic pressure $p^{'}$ and acoustic velocity $u^{'}$ , at its ports based on the conservation equations and element specific transfer functions representing, for example, acoustic losses of ducts, junctions and injectors or the dynamic flame response. In many simple cases these elements consist of two inputs and can be referred as 2-port systems. Nevertheless, the approach is not restricted in the amount of inputs and $N - ports$ ( $N > 2$ ) are possible as described by Lavrentjev and Åbom.¹³ Any linear time-invariant $N$ -port can be described by relating the input state variables to the corresponding output variables.

By selecting the reference location of the ports as shown in Figure 1 on the upper left, the compact RQL combustor can be considered as a 2-port system (Ⓐ-Ⓐ). In this arrangement, the flame dynamics of the primary and secondary zones are inseparable. To identify the individual parts of flame dynamics the zones of the RQL combustor are separated as shown in Figure 1 on the upper right. The secondary zone of the separated RQL forms a $N$ -port system with $N = 3$ connecting the flow from the primary zone, the secondary mixing air flow and the burnout zone (Ⓒ-Ⓒ). This system can be modeled as a network consisting of a 3-port with an adjoining 2-port flame, as shown in the lower part of Figure 1, where the 3-port represents the acoustic system behavior and the flame element describes the heat release. State ① represents the upstream $(\cdot)_{u}$ state coming from the primary zone towards the secondary zone, ② represents the state of the mixing air added through the side mixing ports $(\cdot)_{s}$ , ③ is the virtual intermediate $(\cdot)_{i}$ coupling state, and ④ represents the state downstream $(\cdot)_{d}$ of the lean secondary zone.

Figure 1.

Upper: Comparison of combustion chamber topology between compact and separated arrangement in terms of a network element approach; lower: MISO (multiple input–single output) approach for the secondary zone as a network system of a $3 \times 3$ scattering matrix and a $2 \times 2$ flame element.

Karlsson and Åbom¹⁴ and Holmberg et al.¹⁵ introduced a technique, where a 3-port can be evaluated with a $3 \times 3$ scattering matrix (SM) representing the reflection and transmission coefficients of the system $R$ and $T$ for the selected state variables. With the application to thermoacoustics in RQL combustors in mind it is convenient to introduce the acoustic Riemann invariants $f = 1 / 2 (p^{'} / \bar{ρ} c + u^{'})$ and $g = 1 / 2 (p^{'} / \bar{ρ} c - u^{'})$ as state variables to describe the network elements in terms of harmonic acoustic disturbances that propagate in co-flow and counter-flow direction. This leads to the following representation of a 3-port, where subscripts $(\cdot)_{u, s, i}$ refer to the corresponding ports of the disturbances at the system boundary, see lower part of Figure 1

\begin{aligned} (\begin{matrix} g_{u} \\ g_{s} \\ f_{i} \end{matrix}) = \underset{{S M}_{fg, 3 \times 3}}{\underset{⏟}{(\begin{matrix} R_{u} & T_{s, u} & T_{i, u} \\ T_{u, s} & R_{s} & T_{i, s} \\ T_{u, i} & T_{s, i} & R_{i} \end{matrix})}} (\begin{matrix} f_{u} \\ f_{s} \\ g_{i} \end{matrix}) \end{aligned}

(1)

Adding the RH 2-port type element into the network system results in a compact arrangement with coupling intermediate state variables

f_{i}

and

g_{i}

for the

{S M}_{f g, 3 \times 3}

and the

2 \times 2

flame element. The RH jump conditions are one-dimensional (1D) integral conditions describing mass, momentum, and energy conservation across a discontinuity—here a flame.¹⁶ Using the speed of sound

c^{2} = γ p / ρ

, the Mach number

M = u / c

, the Riemann invariants

f

and

g

and applying the jump condition to the presented case, leads to

\begin{aligned} (\begin{matrix} f_{d} \\ g_{d} \end{matrix}) = & \frac{1}{2} (\begin{matrix} B - A γ M_{i} + 1 \\ B + A γ M_{i} - 1 \end{matrix} \\ \begin{matrix} B - A γ M_{i} - 1 & A {\bar{u}}_{i} \\ B + A γ M_{i} + 1 & - A {\bar{u}}_{i} \end{matrix}) (\begin{matrix} f_{i} \\ g_{i} \\ {\dot{Q}}^{'} / \bar{\dot{Q}} \end{matrix}) \\ A ≜ (\frac{T_{d}}{T_{i}} - 1) B ≜ \frac{(ρ c)_{i}}{(ρ c)_{d}} \end{aligned}

(2)

as the formulation for the flame transfer matrix in

f

and

g

-notation. The subscript “i” denotes variables upstream and “d” denotes quantities downstream the flame. The downstream acoustic quantities are related to the upstream fluctuations through mean flow quantities (

T_{i}, T_{d}, {\bar{u}}_{i}

) and heat release fluctuations

{\dot{Q}}^{'}

. Following the well established method for 2-port elements (e.g. Bade et al.,⁶ Stadlmair and Sattelmayer,⁷ de Rosa et al.,⁸ Kaufmann et al.¹⁷) the dynamic flame response is often quantified with a FTF. In the case of velocity coupled flame response, the definition of the FTF is given as the normalized heat release fluctuation

{\dot{Q}}^{'} / \bar{\dot{Q}}

divided by a normalized reference velocity fluctuation

u_{r e f}^{'} / {\bar{u}}_{r e f}

. As two flows enter the lean secondary zone both are considered to contribute to the heat release fluctuations. The secondary zone can then be seen as a MISO (multiple input–single output) network system. It is assumed that the individual contributions of the two velocity fluctuations

u_{u}^{'}

and

u_{s}^{'}

, on the unsteady heat release rate can be linearly superposed and quantified by individual FTFs (see also Innocenti et al.¹⁸, Bothien et al.,¹⁹ and Gant et al.²⁰). For the linear case, the heat release rate fluctuations can be expressed as two SISO (single input–single output) FTFs

\frac{{\dot{Q}}^{'}}{\bar{\dot{Q}}} = {FTF}_{u, M} \frac{u_{u}^{'}}{{\bar{u}}_{u}} + {FTF}_{s, M} \frac{u_{s}^{'}}{{\bar{u}}_{s}}

(3)

= {FTF}_{u, M} \frac{f_{u} - g_{u}}{{\bar{u}}_{u}} + {FTF}_{s, M} \frac{f_{s} - g_{s}}{{\bar{u}}_{s}}

(4)

with

{FTF}_{u, M}

and

{FTF}_{s, M}

as the complex valued flame-transfer-functions of each input. This MISO approach is particularly useful to separate the physical mechanisms adding to the systems behavior. Inserting equation (4) into (2) and eliminating

f_{i}

and

g_{i}

and by substituting the resulting formulation into equation (1) leads to a

3 \times 3

matrix (called

{BFSM}_{MISO}

) including the reflection- and transmission coefficients, temperatures before and after the flame and the desired FTFs. The detailed procedure how the coefficients are determined and the measurements are performed is given in the measurement methods section.

Experimental setup

The experiments were performed on the RQL test rig shown in Figure 2. The test rig is operated at atmospheric pressure in a thermal power range of $45 - 80 kW$ . Its design is discussed by March et al.²¹ and more details and additional information on the current configuration can be found in Renner et al.¹⁰. Natural gas is used as fuel in the first stage. The preheated primary air flow enters the test rig with a constant inlet temperature of $T_{p r e}^{p z} = 473 K$ , measured with a Type-K thermocouple. The two combustion chambers, the primary on the left and the secondary on the right have the same cross-sectional area, are optically accessible and connected by the upstream excitation element and the transition duct. The latter is needed to reconstruct the 1D acoustic field. The preheated secondary air with $T_{p r e}^{s z} = 473 K$ enters the test rig through the two side ducts and the mixing hole section (see upper left of Figure 2) mounted perpendicular to the secondary zone. Inserts allow variation of the hole geometry and different patterns can be realized. In the work reported here the mixing jet configuration is shown in the lower left of Figure 2 and geometrical parameters are specified in Table 1. With two loudspeakers on the sides, acoustic excitation of the mixing air is possible. The end duct connects the secondary zone with the downstream excitation element and ends in the exhaust. A perforated plate at the end of the test rig provides a low acoustic reflection coefficient. For acoustic measurements, the test rig is equipped with 12 water-cooled Synotech PCB 106B piezoelectric dynamic pressure transducer located in the transition, side and end ducts. In each duct three dynamic pressure transducers are mounted at non-equidistant spacing to ensure high quality wavefield reconstruction with the multi-microphone-method (MMM).^22–25 In addition, a photomultiplier tube (PMT) is employed for CL measurements. The PMT is mounted above the lean secondary zone and views the frame through the upper quartz glass window. A narrow bandpass filter with a wavelength around $308 nm$ full width at half maximum of $10 nm$ is used to detect emissions in the ${OH}^{*}$ CL band. Dynamic pressure and the PMT-data are recorded at a sample rate of $f_{s} = 65536$ samples per second. A total of $n_{s} = 524288$ samples are recorded at each excitation frequency ranging from $60 – 400 Hz$ with a frequency step of $10 Hz$ .

Figure 2.

Atmospheric rich–quench–lean (RQL) test rig—top view setup for secondary zone; upper left: detailed view of mixing hole section and position of the reference plane $x_{u}$ and $x_{s}$ ; lower left: mixing hole section setup and view of variable mixing inserts.

Table 1.

Operating range and mixing hole section.

OP	$T_{p r e}$	${\dot{Q}}_{t h}$	${\dot{m}}_{f u e l}$	${\dot{m}}_{a i r, p z}$	${\dot{m}}_{a i r, s z}$	${\dot{m}}_{a i r, t o t}$	$ϕ_{p z}$	$ϕ_{t o t}$	$d_{I}$	$l_{i n s e r t}$	$J_{I}$	$y_{\max, I}$
$-$	$K$	$kW$	$g / s$	$g / s$	$g / s$	$g / s$	$-$	$-$	$mm$	$mm$	$-$	$mm$
$1$	$473.15$	$52.40$	$1.07$	$13.45$	$15.18$	$28.63$	$1.33$	$0.63$	$14.89$	$80.00$	$22.00$	$45.00$
$2$				$14.35$	$16.19$	$30.54$	$1.25$	$0.59$
$3$				$15.25$	$17.20$	$32.45$	$1.18$	$0.55$	1-row, 3 holes parallel opposed sides
$4$				$16.14$	$18.22$	$34.36$	$1.11$	$0.52$

Data processing

Operating modes and flame temperatures

As for the classical 2-port MMM technique the current setup also requires the characterization of the “cold” non-reacting flow first and then of the “hot” reacting flow case in order to separate the acoustic burner SM (BSM) and the flame dynamics. In the first “cold” operation mode without flame (BSM) the primary combustor is operated in the lean regime ( $ϕ_{pz} < 1$ ) at the same hot gas temperature $T_{u}^{BSM}$ as for the reacting case, and with the same air mass flow rates so only pure mixing takes place in the lean secondary zone. The three-source technique with forcing from upstream, side and downstream is used to determine the nine matrix coefficients ( $R_{n}$ , $T_{n m}$ ) of the BSM experimentally. This is represented by the left acoustic 3-port element in lower part of Figure 1 (small dashed blue line). In the “hot” case with flame in the secondary zone (BFSM), the rich mixture ( $ϕ_{pz} > 1$ ) from the primary zone forms an inverse diffusion flame in the shear layers of the secondary jets. Again the three-source technique is used to determine the “hot” BFSM. This BFSM includes the flame behavior and represents the whole MISO network system depicted in the lower part of Figure 1 (large dashed red line). Matching inlet conditions for both measurements are ensured by adjusting the fuel mass flow rate in the “cold” case so that $T_{u}^{BSM} = T_{u}^{BFSM}$ (measured at the inlet of the secondary zone).

The flame is considered as acoustically compact such that the RH relations between the two temperatures $T_{cold}$ and $T_{hot}$ can be invoked.²⁶ Both temperatures are measured using Type-N thermocouples downstream of the secondary zone (position is depicted in Figure 2). Measured in the “cold” (BSM) case, $T_{cold}$ represents the virtual mixture temperature of the primary and secondary flows before the heat addition from the inverse diffusion flame. $T_{hot}$ is measured in the BFSM case with flame and is the flame temperature. A detailed discussion on the operating modes and the definition of the RH flame temperatures is given by Renner et al.¹⁰ With the predetermined “cold” matrix coefficients, the temperatures $T_{cold}$ , $T_{hot}$ and the second “hot” measurement the MISO network system can be solved for the two FTFs.

Heat release - chemiluminescence

In Renner et al.,¹⁰ it was shown that in this specific case of the inverse diffusion flame the ${OH}^{*}$ -radical CL is a good representative of the fluctuating heat release. The signal from the PMT $I (t)$ can be related to the heat release $Q$ as $I \propto Q$ . The combination with the MMM to determine the velocity fluctuations at the reference position and the PMT allowed an alternative route and led to a second formulation of the FTF. Since both FTFs showed a good agreement, ${OH}^{*}$ CL measurements are also evaluated in this work. As for the pure acoustic approach it is assumed that the total intensity fluctuation at the PMT is a superposition of two independent contributions. Each of them are a function of the corresponding velocity fluctuation at the reference position upstream and in the mixing inserts and can be evaluated as follows

\frac{I_{O H^{*}}^{'}}{{\bar{I}}_{O H^{*}}} = {FTF}_{u, PMT} \frac{u_{u}^{'}}{{\bar{u}}_{u}} + {FTF}_{s, PMT} \frac{u_{s}^{'}}{{\bar{u}}_{s}}

(5)

Both FTFs found with the

{OH}^{*}

CL method are compared with the ones measured with the pure acoustic MISO approach (subscript

(\cdot)_{. . ., M}

) in the results chapter.

Optimization problem

The presented ${BFSM}_{MISO}$ contains the reflection- and transmission coefficients from the “cold” measurement, temperatures before ( $T_{c o l d}$ ) and after ( $T_{h o t}$ ) the flame and the FTFs. Using a “hot” measurement the system forms an optimization problem and a regression algorithm is applied to determine the desired complex numbered ${FTF}_{u, M}$ and ${FTF}_{s, M}$ . Genetic algorithms (GAs) in particular have proven to be robust in finding the optimum solutions for various engineering optimization problems.²⁷ GAs do not rely on gradients and have superior parameter search patterns. The GA is applied to the resulting MISO optimization problem ( ${BFSM}_{MISO}$ ) as well as the CL-PMT measurements and equation (5) to determine ${FTF}_{u, PMT}$ and ${FTF}_{s, PMT}$ .

Results

Table 1 summarizes the four operation points which are presented below. With respect to the heat release in the secondary zone OP1 has the highest and OP4 the lowest value. To keep the thermal power constant the fuel rate was fixed. The primary and secondary air flow rates are increased in order to reduce the primary zone equivalence ratio $ϕ_{p z}$ while maintaining the momentum flux ratio $J$ in the secondary zone. Thereby the JIC mixing field characteristics of the secondary zone are preserved.

Verification

To qualify the MISO FTF data analysis proposed above, the results were compared with results presented previously by Renner et al.¹⁰ There, the acoustic response of the mixing ports is suppressed ( $u_{s}^{'} = 0 \to R_{s i d e} \approx 1$ ) using choked orifices upstream of the mixing jet nozzles. As a result, the response to primary zone velocity fluctuations $u_{u}^{'}$ could be isolated. Furthermore, this allowed to determine the FTF more confidently using the well established 2-port RH technique. The suppressed setup is shown in Figure 3. In the following, this procedure is only applied to gain the verification data and the resulting FTF is marked with the subscript $(\cdot)_{u, 2 p o r t}$ .

Figure 3.

Mixing hole section setup and view of mixing inserts and variable boundary; lower right: reflection coefficient determined at simplified test section in mixing holes resulting in $u_{s}^{'} = 0$ .

Figure 4 shows the resulting FTFs for the most reactive operation point OP1. In the upper graph, the verification FTF amplitude of ${FTF}_{u, 2 p o r t}$ (black crosses) is compared with the data obtained from the MISO method ${FTF}_{u, M}$ (light red dots). The middle graph shows the comparison of the FTF phase and on the lower graph the relative velocity fluctuations upstream and in the mixing ports obtained in the 3-port measurements for upstream forcing are shown. Three frequency ranges , , and can be discussed. In range , from $60 – 120 Hz$ , good agreement between ${FTF}_{u, 2 p o r t}$ and ${FTF}_{u, M}$ is observed. While the amplitudes still compare well in range the phase of $∠ {F T F}_{u, M}$ jumps to just below $- π / 4$ , whereas $∠ {F T F}_{u, 2 p o r t}$ continues the same slope. The plot of the relative velocity fluctuation amplitudes in Figure 4 for the upstream $u_{u}^{'} / {\bar{u}}_{u}$ (red dots) side mixing ports $u_{s}^{'} / {\bar{u}}_{s}$ (blue squares) indicates that the side branch goes through a damped resonance frequency and reaches rather high amplitudes up to $40 %$ . The effect of non-linear forcing resulting in a saturation limit for the FTF is often discussed by Külsheimer and Büchner,²⁸ Balachandran et al.,²⁹ Armitage et al.,³⁰ Eckstein et al.,⁵ Kim et al.³¹ Nevertheless, the role of amplitude dependence of the phase between the velocity and heat-release fluctuations is less clear from the shown experimental data and not fully understood here and in the literature. One possible explanation could be that the shift to the non-linear regime strongly affects the mixing and combustion process of the inverse diffusion flame. With the remaining low fuel to be burned in the secondary zone the flame is sensitive to changes in the momentum flux- and velocity ratio $J$ and $R$ . Both are known to be key parameters in the design of JIC. With increasing $J$ and over-penetration of the jets the mixing characteristics changed for the worse. Consequently, the flame shape changes and shifting its flame center of gravity $x_{C O G}$ upstream towards the reference plane $x_{u}$ . Similar to the change in flames length caused by a variation of the operational point, the time delay $τ = φ / 2 π f = x_{C O G} / u$ decreases thus shifting the phase. For more information on the time delay modeling in this specific case the interested reader is referred to the Appendix by Renner et al.¹⁰ At any rate the substantial variation in oscillation amplitude through this region makes the evaluation of the FTF difficult. The last region shows amplitudes of almost zero for both FTFs. The velocity fluctuations are back in the linear regime but the flame seems unaffected by these. Due to the very low amplitudes the associated phase values are arbitrary in this frequency range and the data becomes questionable in this region.

Figure 4.

Upper: absolute value of $| {F T F}_{u, M} |$ (light red dots) from network MISO approach compared to $| {F T F}_{u, 2 p o r t} |$ (black crosses) gained with a distinctive experimental approach as by Renner et al.¹⁰; middle: corresponding phase of the two FTFs $∠ (\cdot)$ ; lower: normalized velocity fluctuations $u_{r e f}^{'} / {\bar{u}}_{r e f}$ in the upstream (red dots) and side (blue squares) reference plane during upstream forcing with the standard deviation (shaded surface) for the operating range.

Figure 5 shows the resulting side FTFs for the most reactive operation point OP1. In the upper and middle plot of Figure 5, the amplitude and phase of ${F T F}_{s, M}$ are displayed in light blue square markers. Starting at $f = 60 Hz$ the amplitude rises to its global maximum at $f = 130 Hz$ with $| F T F_{s, M} | = 0.95$ . From here, the FTF falls linearly towards a local minimum at $f = 260 Hz$ . Towards $310 Hz$ the amplitude rises up again to a local maximum of $| F T F_{s, M} | = 0.40$ . For higher frequencies, the $| F T F_{s, M} |$ shows a rather random behavior. The phase of $∠ F T F_{s, M}$ , shown in the middle plot of Figure 5, starts with a value of zero at $f = 60 Hz$ and shows a constant decreasing linear trend until $310 Hz$ . From there on the phase starts to scatter and shows the same random behavior as seen for the amplitude. As for the upstream FTF a second evaluation was performed to verify the $F T F_{s, M}$ from the acoustic model. Since the upstream velocity fluctuations could not be suppressed with a stiff boundary an alternative approach was used. During a side forcing cycle, regions with very low $u_{u}^{'} / {\bar{u}}_{u}$ could be identified. In those regions equation (5) reduces to a formulation, where the global intensity fluctuation is only a function of $u_{s}^{'} / {\bar{u}}_{s}$ , coupled via an FTF denoted by ${F T F}_{s, P M T}^{*}$ and displayed with the gray diamonds. The ${F T F}_{s, P M T}^{*}$ is calculated based on a side forcing and compared to $F T F_{s, M}$ in Figure 5. Again, three ranges , and , can be discussed. In range and , $u_{u}^{'} / {\bar{u}}_{u}$ still shows a significant amplification of about $4 % - 5 %$ shown in the lower plot of Figure 5. In this region, the amplitude and the phase of both FTFs show a large deviation since the flame reacts to both fluctuations and ${F T F}_{s, P M T}^{*}$ is only referred to the side. In between, in range very low upstream velocity fluctuations are observed. Here the flame only reacts to velocity fluctuations from the side. In this region, the amplitude and phase of both FTFs show a very good agreement with smaller deviations at $f = 260 Hz and 300 Hz$ . These results confirm the findings based on the pure acoustic MISO approach.

Figure 5.

Upper: absolute value of $| {F T F}_{s, M} |$ (light blue squares) from network MISO approach compared to $| {F T F}_{s, P M T}^{*} |$ (gray diamonds) gained from a PMT ${OH}^{*}$ CL measurement with forcing from the side; middle: corresponding phase of the two FTFs $∠ (\cdot)$ ; lower: normalized velocity fluctuations $u_{r e f}^{'} / {\bar{u}}_{r e f}$ in the upstream (red dots) and side (blue squares) reference plane during side forcing with the standard deviation (shaded surface) for the operating range.

As shown in both Figures 4 and 5, the FTFs gained with the MISO approach fit the two verification’s in the ranges , , and very well. Next equation (5) is evaluated. Based on the “hot” measurements and the three-source technique $F T F_{u, P M T}$ and $F T F_{s, P M T}$ were calculated with a regression approach and shown in Figure 6. On the left hand side in Figure 6, the results for the upstream FTFs are compared. Over the entire frequency range good agreement between $F T F_{u, M}$ and $F T F_{u, P M T}$ is observed with minor deviations at very low frequencies. The phase of the PMT measurements show a similar behavior to the MISO FTFs. Recalling the three ranges from Figure 4 one can see that in range , $F T F_{u, P M T}$ has the same monotonic trend as $F T F_{u, M}$ . In range , the PMT approach is also sensitive to resonance and non-linear behavior of the side velocity fluctuations and shows the same phase-jump from $- π$ towards $- π / 4$ . In range , the PMT phase shows some scattering but remains the linear decreasing characteristics towards higher frequencies. The $F T F_{u, P M T}$ seems more robust than the pure acoustic MISO model resulting from a better signal-to-noise ratio of the sensitive PMT signal and the calculated velocity fluctuations. At very low amplitudes, phase values remain questionable.

Figure 6.

Left: ${F T F}_{u, M}$ (light red dots) from network MISO approach compared to ${F T F}_{u, P M T}$ (dark red triangles) extracted from PMT measurements and in the background as a reference ${F T F}_{u, 2 p o r t}$ (black crosses) gained with the distinctive experimental approach see Renner et al.¹⁰; right: ${F T F}_{s, M}$ (light blue squares) from network MISO approach compared to ${F T F}_{s, P M T}$ (dark blue triangles) extracted from PMT measurements and in the background as a reference ${F T F}_{s, P M T}^{*}$ (gray diamonds) gained from a PMT ${OH}^{*}$ CL measurement with forcing from the side.

On the right hand side in Figure 6, the results for the side FTFs are compared. Hereby the $F T F_{s, P M T}$ is displayed with dark blue triangles. For the amplitude, the $| F T F_{s, P M T} |$ shows a very good agreement with the $| F T F_{s, M} |$ . Minor deviations are seen in the area of the side resonance frequency around $f = 150 Hz$ and at $260 Hz$ . The same applies for the phase, here the $∠ F T F_{s, P M T}$ shows the same linear falling slope and thus the same time delay as the $∠ F T F_{s, M}$ . Again remembering the ranges shown in Figure 5 the advantages of the three-source PMT approach is now clearly visible. In range , the amplitude and phase of $F T F_{s, M}^{*}$ based only on a side forcing overshoots the $F T F_{s, M}$ . In contrary, $F T F_{s, P M T}$ matches almost perfectly. In range with the very low upstream fluctuations all FTFs show a similar good agreement. For the last range , both FTFs based on the PMT show the same trend and the MISO FTF is prone to scattering in this frequency area.

Finally, the low frequency limits of both FTFs are discussed. For the low frequency limit $f \to 0$ of the upstream FTFs, the amplitude can be extrapolated towards $| F T F_{u} | \approx 1$ and the phase becomes $∠ F T F_{u} \approx 0$ , comparable to characteristic for non-premixed flames with stiff fuel injector as found by Bade et al.,⁶ Kaufmann et al.¹⁷ Here the amplitude is indicating a quasi-stationary thermal power modulation for the lower frequencies. The variance of the absolute values result from the temperature measurements used in the MMM and the MISO approach for $T_{c o l d}$ and $T_{h o t}$ . The temperatures are not corrected for radiation effects, thus the resulting amplitude values of the model $F T F_{u, M}$ should be little lower. This would shift $F T F_{u, M}$ more towards $F T F_{u, P M T}$ . The linear falling phase slope speaks for a constant convective time delay between the velocity fluctuations in the reference plane $x_{u}$ and the heat release of the inverse diffusion flame. This is also covered by the time delay model presented in the Appendix of the authors previous research. For the side FTFs on the right of Figure 6, the amplitude can be extrapolated towards $| F T F_{s} | \approx 0$ for $f \to 0$ . The side FTFs represent the flames reaction to $u_{s}^{'}$ at the position of air injection into the chamber. Without significant modulation of the air flow the entrainment of the inverse diffusion flame is not affected and thus there is no modulation of the heat release and the flame characteristics do not change. In case of an amplitude low frequency limit of zero the phase is arbitrary. Nevertheless, it can be stated that both phase slopes of the side FTFs have the same trend for low frequencies.

Influence of burnout heat release rate on flame dynamics

In Figures 7 and 8, the FTFs representing the flame response to upstream and side velocity fluctuations are displayed for the full operating range from Table 1. Above the legend in each figure the solutions from the pure acoustic MISO approach are shown. The FTFs based on the PMT ${OH}^{*}$ CL approach are shown below the legend in each figure. Hereby the most reactive operating points are represented by dark colored dot and triangle markers in red for ${F T F}_{u}$ and blue for ${F T F}_{s}$ . The least reactive points are colored in bright red for ${F T F}_{u}$ and light blue markers for ${F T F}_{s}$ .

Figure 7.

FTFs upstream for the entire operating range ordered from dark red color for the most reactive OP1 to light red color for the least reactive OP4; upper: ${F T F}_{u, M}$ (dot markers) from network MISO approach; lower: ${F T F}_{u, P M T}$ (triangle markers) extracted from PMT measurements.

Figure 8.

FTFs upstream for the entire operating range ordered from dark blue color for the most reactive OP1 to bright pink yellow for the least reactive OP4; upper: ${F T F}_{s, M}$ (dot markers) from network MISO approach; lower: ${F T F}_{s, P M T}$ (triangle markers) extracted from PMT measurements.

In Figure 7, all ${F T F}_{u, M}$ amplitudes show the same trend, starting from their global maximum value at $f = 60 Hz$ , falling towards a local minimum at around $f = 120 Hz$ , increasing in the regions of high resonance in the sides towards a local maximum at $f = 140 - 150 Hz$ and than falling to values close to zero for higher frequencies. Also the phase data shows the same behavior for all OPs. Here OP4 shows the highest deviation and spread. All FTFs in Figure 7 follow the low frequency limit of $| F T F_{u} | \approx 1$ and the phase becomes $∠ F T F_{u} \approx 0$ .

In Figure 8, ${F T F}_{s, M}$ show an increasing amplitude from OP1 to OP4 and the phase changes with a steep phase for OP1 to the flattest for OP4 indicating the change in flame length towards shorter flames for less reactive OPs. The latter also applies for the phase of ${F T F}_{s, P M T}$ . The amplitudes of ${F T F}_{s, P M T}$ show a different behavior than the MISO ones. The $| {F T F}_{s, P M T} |$ for each individual operation point deviates about $- 10 %$ from the most (OP1) to the least (OP4) reactive point. All PMT FTFs for the side show a local minimum in amplitude around the resonance frequency of the side ducts at $f = 140 - 150 Hz$ . Rising back to a local maximum at $f = 170 Hz$ and decreasing from here to almost constant values around $| F T F_{s, P M T} | = 0.5$ for $f = 220 - 400 Hz$ . The phase of the side FTFs shows a constant falling behavior for all OPs. The low frequency limits corresponds to the expectations of $| F T F_{s} | \approx 0$ for $f \to 0$ with an arbitrary phase value.

Robustness

For both pure acoustic MISO FTFs a stronger scattering towards less reactive OPs in almost the entire frequency range is observed in Figures 7 and 8. This is now discussed on the basis of OP4. After the fuel rich combustion in the primary zone, only $10 %$ of additional heat release in the lean secondary zone remains for OP4. This leads to a small temperature increase from $T_{cold, OP 4} = 856 K \to T_{hot, OP 4} = 951 K$ . This makes the FTF more vulnerable to measurement inaccuracies especially in the case of very lean $ϕ_{t o t}$ operational points. Additionally, the small temperature increase results in a very small differences between the underlying BSM and BFSM measurements and both matrices get too similar. The deviation between both matrices is mathematically determined by the Frobenuis norm,³² normalized with the overall maximum deviation and is shown in Figure 9. Larger values indicate a strong variance between BSM and BFSM indicating a larger impact of the flame heat release. Lower values indicate that the matrices are almost identical. For the least reactive OP4 the variance is traveling around $1 %$ for frequencies above $200 Hz$ . With this small difference between “cold” and “hot” measurement the GA solution of the optimization problem becomes difficult and the results become questionable. Nevertheless, the PMT is not prone to those small changes since it relies only on the “hot” measurement. This is why the PMT method shows a robust behavior also for less reactive OPs as seen for ${F T F}_{u, P M T}$ and ${F T F}_{s, P M T}$ . The amplitude and phase data for less reactive OPs with low temperature increase must be interpreted cautiously.

Figure 9.

Normalized Frobenuis norm $| A_{O P_{i}} |_{F, n o r m}$ for each frequency between the individual BSM and BFSM for the operating range. BSM: burner scattering matrix; BFSM: burner flame scattering matrix.

Conclusions and outlook

In the presented article, the lean secondary zone of a separated RQL combustion chamber was investigated in terms of flame dynamics. For the first time FTFs depending on acoustic velocity fluctuations coming from the primary zone and from the mixing ports have been measured. With a new pure acoustic MISO approach the lean combustion chamber was treated as a network system by combining an acoustic 3-port and a 2-port flame element. The former was described by a $3 \times 3$ SM with inputs of the two incoming streams from upstream and the mixing ports. Application of the RH jump conditions to the latter 2-port flame element and substitution of the flame heat release fluctuations ${\dot{Q}}^{'}$ with a formulation based on two linear superimposed FTFs depending on the two velocity fluctuations from upstream $u_{u}^{'}$ and the side $u_{s}^{'}$ leads to a $3 \times 3$ system matrix describing the secondary zone. Using two independent measurements - “cold” and “hot”—the two FTFs could be determined. The resulting acoustic $F T F_{u}$ and $F T F_{s}$ were compared to FTFs based on a hybrid approach with a PMT using the ${OH}^{*}$ CL as a measure for the heat release. The following conclusions could be drawn:

The FTFs depending on upstream and side velocity fluctuations show a very good agreement with the verification data gained with distinctive experiments on the test rig. This supports the linear superposition approach of the two FTFs in the acoustic MISO model.

In the investigated operating range, the flame of the lean secondary zone reacts to acoustic velocity fluctuations from upstream ( $F T F_{u}$ ) with heat release fluctuations in the low frequency range of $f = 60 - 180 Hz$ and shows a clear low pass behavior.

As a reaction to the velocity fluctuations from the side, $F T F_{s}$ shows a wider, more broadband behavior than $F T F_{u}$ , with regions of higher amplitudes from $f = 60 – 200 Hz$ . From $f = 200 – 400 Hz$ almost constant amplitudes around $| F T F_{s} | = 0.5$ were observed.

In regions from $f = 60 – 180 Hz$ both FTFs affect the heat release fluctuations of the inverse diffusion flame. At higher frequencies only the fluctuations from the side contribute to the heat release fluctuations.

The FTFs measured with the pure acoustic MISO model and the ones from the hybrid PMT match in terms of their amplitude and phase values very well. High scattering for leaner OPs were related to the low change between the underlying “cold” and “hot” measurements. Again, ${OH}^{*}$ CL seems to work good as a measure of the heat release fluctuations for the inverse diffusion flame presented in this research.

In the secondary zone, the remaining

10 % - 25 %

of heat is released considering fuel rich combustion in the first stage. In relation to the primary zone the FTF amplitudes presented can be weighted by the heat release rate allocation. Consequently lowering the FTF amplitudes, thus the secondary zone has a significantly minor contribution to the combined system dynamics.

Footnotes

Acknowledgements

The authors would like to thank Dominik König for his support with the experiments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by the Bundesministerium für Wirtschaft und Technologie (BMWi) as per resolution of the German Federal Parliament under grant number 20E1715, which is gratefully acknowledged.

ORCID iDs

Julian Renner

Moritz Merk

Thomas Sattelmayer

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