thermoacoustic modeling of a gas …guethe/pub/pap/asme_2009_schuermans_final.pdfthermoacoustic...

Proceedings of ASME Turbo Expo 2009: Power for Land, Sea and AirGT2009

June 8-12, 2009, Orlando, USA

GT2009-59605

THERMOACOUSTIC MODELING OF A GAS TURBINE USING TRANSFERFUNCTIONS MEASURED AT FULL ENGINE PRESSURE

Bruno Schuermans,Felix Guethe, Douglas Pennel

AlstomCH-5405 Baden

Switzerland

Daniel Guyot,Christian Oliver PaschereitTechnische Universitat Berlin

Germany

ABSTRACTThermoacoustic transfer functions have been measured

of a full-scale gas turbine burner operating at full enginepressure. Excitation of the high-pressure test facility wasdone using a siren that modulated part of the combustionairflow. Pulsation probes have been used to record theacoustic response of the system to this excitation. In ad-dition, the flame’s luminescence response was measured bymultiple photomultiplier tubes and a light spectrometer.

Three techniques to obtain the thermoacoustic trans-fer function are proposed and employed: two combinedacoustical-optical technique and a purely acoustic tech-nique. The first acoustical-optical technique uses one sin-gle optical signal capturing the chemiluminescence inten-sity of the flame as a measure for the heat release in theflame. It only works, if heat release fluctuations in the flamehave only one contribution, e.g. equivalence ratio or massflow fluctuations. The second acoustic-optical acoustic-optical technique makes use of the different response of theflame’s luminescence at different optical wavelengths bandsto acoustic excitation. It also works, if the heat releasefluctuations have two contributions, e.g. equivalence ratioand mass flow fluctuation. For the purely acoustic tech-nique, a new method was developed in order to obtain theflame transfer function, burner transfer function and flamesource term from only three pressure transducer signals.The purely acoustic method could be validated by the re-sults obtained from the acoustic-optical techniques.

The acoustic and acoustic-optical methods have been

compared and a discussion on the benefits and limita-tions of the methods is given. The measured transferfunctions have been implemented into a non-linear, three-dimensional, time domain network model of a gas turbinewith an annular combustion chamber. The predicted pulsa-tion behavior shows a good agreement with pulsation mea-surements on a field gas turbine.

NOMENCLATUREA f flame areaH f chemical enthalpyHe Helmholtz number, He = ωL/(2πc)I chemiluminesence intensityIBB spectral black body radianceQ heat release rateS f flame speedSxy estimate of the cross power spectral densityT temperatureTBB black body temperaturec speed of soundclight speed of lighthP Planck constantkB Boltzmann constantm mass flux through flame surfacep pressureu velocityy fuel mass fractionγ ratio of specific heats

1 Copyright c© 2009 by Alstom

λ optical wavelengthρ densityω angular frequencyx steady component of xx′(t) unsteady component of x, x′(t)≡ x(t)− xx(ω) Fourier coefficient of x′(t)

INTRODUCTIONThermoacoustic analysis is an integrated part in Al-

stoms gas turbine technology and product development pro-cess. Alstom’s approach to thermoacoustic analysis is touse measured flame transfer functions and source terms ina non-linear, 3-dimensional, acoustic network model. Theacoustic wave propagation through combustion chamberand plenum are represented in the model via a modal ex-pansion. The required acoustic modes are obtained from afinite element calculation of the detailed geometry. Thiscombined experimental and numerical approach to ther-moacoustic modeling is discussed in detail in [1] and [2].

A crucial aspect of this modeling approach is to ob-tain a correct representation of the interaction between theheat release and the acoustic field. Alstom’s approach is tomeasure flame transfer functions in a single burner test fa-cility and to fit models to this transfer function data. Thisapproach has proven to give very good results in predictingstability behavior and pulsation spectra of multi-burner gasturbine combustion systems [3, 4].

The influence of pressure on transfer functions is partic-ularly important for fuel flexibility (high hydrocarbon andhydrogen content, liquid fuels). In order to correctly pre-dict thermoacoustic behaviors for cases where such fuels areused, transfer functions measured at elevated pressure areneeded.

This paper is about the measurement of flame transferfunctions of a full scale swirl-stabilized EV-type burner atfull engine pressure and about using these transfer functionsto predict stability and pulsation spectra of a gas turbinewith annular combustion chamber.

Due to the limited access in an industrial high pressuretest facility, new techniques had to be developed to obtaintransfer functions from only three pulsation probes and op-tical sensors. Two different techniques have been used tomeasure the flame transfer function: the first technique usespulsation probes and multiple chemiluminescence sensors,the second only uses pulsation probes. Using chemilumines-cence signals to obtain transfer function is not very straightforward, because fluctuations of the fuel to air ratio may bepresent in this type of burner and hence the chemilumines-cence intensity is not necessarily proportional to the heatrelease. To overcome this problem a technique has been

used that uses chemiluminescence signals of different wave-lengths bands. This method has been discussed in detailin [5] and was validated at atmospheric pressure conditions.

At high-pressure the correct measurement of the flamechemiluminescence is more challenging than at atmosphericconditions, however. This is due to constrains regarding op-tical access to the flame, an overlapping of the wavelengthsbands of flame luminescence and the combustor walls’ heatradiation, and the generally lower probability of the forma-tion of the chemical species associated to the flame chemi-luminescence under high pressures. Therefore, the error ofthe acoustical-optical technique was expected to be largerthan under atmospheric conditions.

For this reason a second new technique has been devel-oped to obtain transfer functions from only three pulsationprobes. A comparison of the acoustical and the combinedacoustical-optical method for obtaining the transfer func-tions and source terms is given in this paper, together withan error analysis.

Both methodologies have been used to obtain transferfunctions and predict engine pulsation spectra for a widerange of operating conditions (varying flame temperature,burner exit velocity, pressure, air temperature, fuel typeand fuel staging ratio) and hardware configurations. Due tolimited length of this paper only a small part of the resultswill be discussed here.

EXPERIMENTAL SET-UPHigh-Pressure Combustion Test Facility

The process of Alstom burner development and im-provement includes combustion tests under high-pressureconditions. The facility is shown in Fig. 1. It allows quick,cost-effective and therefore extensive testing of single AL-STOM machine burners. The test-rig consists of a plenumchamber upstream of the burner, two tubular pressure ves-sels and the rectangular chamber liner. The hot exhaustgases are quenched before the pressure reduction throttleand then discharged to the chimney. During the combus-tion tests the flow through the throttle was choked to mini-mize acoustic perturbations traveling in upstream directionthrough the throttle. The combustor liner is convectivelycooled to prevent contamination of actual burner emissionsby introducing additional film cooling air into the com-bustor and to avoid the introduction of unrelated acousticdamping effects.

The test were performed on an Alstom EV-type testburner. Such a burner has the unique property of flame sta-bilization in free space near the burner outlet utilizing thesudden breakdown of a swirling flow, called vortex break-down. Gaseous or liquid fuels are injected into the com-bustion air by means of fuel distribution tubes comprising


preheater

mass flow controller

fiber optic probe for flamechemiluminescence detection

light spectrometer

PMT 1: UV filterPMT 2: CW 307nmPMT 3: CW 431nmPMT 4: CW 450nmPMT 5: CW 515nm

air supply

splitter

exhaust

preheater

siren actuator

burner

pulsation probes

Figure 1. Experimental set-up: High-pressure test facility equipped withsiren, pulsation probes and optical access via an fiber optic probe.

rows of small holes perpendicular to the inlet ports of theswirler. Complete mixing of fuel and air is obtained shortlyafter injection.

Acoustic ForcingAt high-pressure conditions loudspeakers provide nei-

ther the required robustness nor sufficient acoustic forc-ing amplitudes for good transfer function measurements.Therefore, a siren actuator has been designed (Fig. 2). Thesiren features an insulated air vessel to which preheated airis supplied. The pressurized air exits the air vessel throughan orifice into the test-rig’s plenum chamber. The orifice isperiodically blocked by a rotating disc. This disc has eightholes. Hence, the orifice is opened and closed eight timesduring one revolution of the disc. The disc is driven by a6.2 kW motor. The disc’s rotational speed is measured bya trigger placed onto the motor axle, which generates oneshort pulse signal per disc revolution. The free cross sec-tion of the orifice was designed in a way that resulted in analmost sinusoidal modulation of the siren air flow with onlyminor response at higher harmonics.

Acoustic excitation can thus be forced onto the com-bustion test-rig by feeding a fraction of the combustion airvia the siren into the plenum chamber. The forcing ampli-tude can be adjusted by changing the siren air mass flow.The excitation frequency is controlled by defining the sirendisc speed and can be varied between 0 and 400 Hz.

To perform transfer function measurements in a givenfrequency range the test-rig was generally consecutively ex-cited at discrete frequencies (i.e., mono-frequent siren exci-tation). In most cases frequency steps of 5 to 10 Hz werechosen. Alternatively, the siren can also sweep through a

support

air supply

siren actuator

insulation

plenum

motor

air vessel

Figure 2. The siren actuator.

given frequency range during one recording. The generatedacoustic forcing amplitude was sufficiently high to limit therecording time per frequency step 10 to 20 seconds, whilestill achieving a good signal to noise ratio.In fact, the maxi-mum acoustic forcing was not limited by the siren actuator,but rather by the mechanical integrity of the test facility.

Measurement Set-upPressure fluctuations in the test-rig are measured using

three pulsation probes. One probe is situated in the plenumchamber and two in the combustion chamber. Their posi-tions are indicated in Fig. 1.

The light emitted by the flame is collected by a water-cooled fiber optic probe equipped with a lens system sit-uated in the downstream half of the combustion chamber.The optical detection is accomplished by a non imaging sys-tem of quartz lenses focusing the light on one thick fiber of15 m length. From this thick fiber the light is transmit-ted into 7 different fibers by an optical splitter. This setupassures that each fiber obtains the same light and avoidsangle dependency. The lens has been optimized in orderto minimize the flame position dependence of the intensity.This is important in order to avoid that flame movementis interpreted as intensity fluctuation. The opening angleof the system was chosen such that the entire heat releasezone was covered. The geometrical extent and movementof the heat release zone are known from previous flame vi-sualization studies.

The light is then distributed to a system of filters andphotomultiplier tubes (PMTs) as indicated in Fig. 1. Thisyields five separate signals for five different optical wave-length regions. The chemiluminescence signals are recordedsimultaneously with the pulsation probe signals on the samedata acquisition system.

The optical setup incorporates five different band pass-


filters of about 10 nm width at central wavelengths (CW)of 307 nm, 431 nm, 450 nm, 515 nm and one UV transpar-ent broad band filter (DUGX11, Schott). The transmissioncurves are shown in Fig. 3 together with the transmission ofthe fiber optic probe1. As an example, Fig. 4 presents thelight spectrum of the premixed natural gas flame of an EV-type burner at atmospheric conditions. Using this opticalset-up the different channels can be attributed to the oc-currence of chemiluminescence of mainly the species (OH∗,CH∗, C∗2) as indicated in [6, 7]. To monitor the broad emis-sion (attributed to the CO + O → CO∗2 chemiluminescence)which contributes as background in the spectrum [8] a filterwith CW 450 nm is used.

300300300300 400400400400 500500500500 600600600600 7007007007000000

50505050

100100100100

wavelength (nm)wavelength (nm)wavelength (nm)wavelength (nm)

tran

sm

issio

n (

%)

tran

sm

issio

n (

%)

tran

sm

issio

n (

%)

tran

sm

issio

n (

%)

UV bandUV bandUV bandUV bandCW 307 nmCW 307 nmCW 307 nmCW 307 nmCW 431 nmCW 431 nmCW 431 nmCW 431 nmCW 450 nmCW 450 nmCW 450 nmCW 450 nmCW 515 nmCW 515 nmCW 515 nmCW 515 nmoptical probeoptical probeoptical probeoptical probe

Figure 3. Light transmission of the optical filters an the fiber optic probe.

OHOHOHOH****

CHCHCHCH****

C2C2C2C2****

CO2CO2CO2CO2****

Figure 4. Light spectrum of a premixed flame at atmospheric conditions.

All channels are mainly detecting emission from differ-ent species probing a different part of the flame chemistry.

1Note that for the fiber optic probe only the relative transmissionis presented. The maximum of the relative transmission is interpreted

as 100% transmission.

Each of these detection windows scale differently with theoperating parameters burner velocity and flame tempera-ture. With the calibration of at least three averaged steadyoperating points this ratio of the channels allows the inter-pretation as instantaneous measurements of these operatingparameters. This reading can be interpreted as fluctuationsof the governing heat release parameters (adiabatic flametemperature and mixture mass flow) as described in [9,10].A further correlation of the signals to individual molecularemissions utilizing the spectra can be undertaken and wouldimprove the quality of the interpretation. However, becauseit is not necessary for the approach described in this workthis has been omitted here.

The fiber optic probe in the combustion chamber doesnot only collect the light emitted by the flame, but also theheat radiation of the combustor walls. Therefore, an OceanOptics QE65000 high-sensitivity spectrometer measured thespectrum of the light captured by the fiber optic probe inthe range of 200 to 850 nm to allow for correction for thecaptured heat radiation in the data post-processing.

TRANSFER FUNCTION MEASUREMENT USINGACOUSTIC SIGNALS

The combustion system can conceptually be dividedinto six subsystems: the siren, the plenum, the burner, theflame, a noise source and the combustion chamber. A causalnetwork representation of this system is given in figure 5. Inthis diagram the measured signals are represented by plainsymbols. The unsteady pressure signals recorded by thepulsation sensors in the plenum and combustion chamberare represented by the symbols p1, p2 and p3. The lightintensity of the different chemical species recorded by thephoto multipliers are represented by the symbols I1 to I5.The siren actuator provided a reference signal, which gavea pulse for each revolution of the siren. This signal is rep-resented by r. The italic symbols in the diagram representrelevant signals that can not be measured directly. Thevelocity fluctuation imposed by the siren is represented byusiren. The flame is considered as a compact interface be-tween the cold and hot products, represented by underscoresc and h respectively. Because of the low Mach number, con-tinuity of pressure is assumed across the flame sheet (pc =ph). The velocity fluctuation just before and after the flamelocation are denoted as uc and uh. The relation between ucand uh is given by the flame transfer function H. The flamewill also act as a source of sound independent of uc. Thephysical nature of this source is believed to be mainly dueto turbulent unsteady heat release. Nevertheless, it can berepresented by an equivalent velocity source term us. Thesum of the velocity fluctuations right after the flame andthe source term is represented as ut .


Plenum Burner

Flame

Combustion

Chamber

SourceSiren

p1 p2 p3

I1:5

uc uh utusiren

pc= ph

us

r

Figure 5. Thermoacoustic network representation of the test facility.

The aim of the present system identification is to obtainthe flame transfer function H and the source term us fromthe measured quantities. It is evident from figure 5 thatthe system has two input signals: the naturally occurringsource term us and the externally impose excitation by thesiren: usiren. Thus all measured quantities are a superpo-sition of the response to the siren and to the source term.For system identification it is crucial to separate these twocontributions of the signal. This can be done via a cor-relation analysis with the excitation signal. However, theexcitation is not measured. This is why an artificial exci-tation signal ε(t) is constructed from the reference signal rprovided by the siren. This is done by constructing a se-ries of sine waves such that exactly eight periods (the sirendisc has eight holes) fit between every two consecutive trig-ger pulses. The response of the microphone signal i to thesiren excitation will be denoted as pi,ε. An estimate of theFourier transform of pi,ε is given by:

pi,ε =Sε pi√

Sεε

, (1)

where Sxy is the estimate of the cross power spectral density,which is the Fourier transform of the cross correlation.

Using the two microphone method [11,12], the velocityfluctuations uh can be calculated from the pressure signalsp2 and p3:

uh =−iρ

cos(k2) p2− cos(k3) p3

sin(k2− k3), (2)

where kn ≡ ωxnc , and xn denotes the axial position of the nth

pressure transducer.The acoustic velocity at the exit of the burner is as-

sumed to be a function of the unsteady pressure dropover the burner and the burner transfer function: uc =B(ω)(p1,ε − p2,ε). From a test case without combustion

(F = 1, thus uc = uh) the burner transfer function (B) isobtained as:

B(ω) =p1,ε− p2,ε

uh(3)

The flame transfer function H = uhuc

is obtained from atest case with combustion from:

H(ω) =uh,ε

uc,ε=

uh,ε

B(p1,ε− p2,ε), (4)

where uh,ε is obtained from p2 and p3 using the two micro-phone method, and B was already obtained from the testcase without combustion.

The response of a pressure signal to the combustionsource term us is obtained from a separate test case wherethere was combustion, but where the siren is switched off.The noise driven response is then obtained from the crosspower spectral density between pressure signal i and thereference microphone 1.

pi,s =Sp1 pi√Sp1 p1

. (5)

The source term is then obtained as:

us = ut,s−H uc,s = ut,s−H B(p1,s− p2,s), (6)

where H and B where obtained from the previous test cases,and ut,s is obtained by applying the two microphone methodto p2,s and p3,s. Thus, the burner transfer function, theflame transfer function and the source term are calculatedfrom three pressure signals using three test cases (no com-bustion with forcing, combustion without forcing and com-bustion with forcing).

TRANSFER FUNCTION MEASUREMENT USING OPTI-CAL SIGNALS

The heat release flux of the flame is given by the fol-lowing expression:

Q = m f H f = myH f (7)

where m f and m are the instantaneous values of the massfluxes of fuel and mixture entering the reaction zone,y the


mass fraction of fuel (y = m f /m), the chemical enthalpy ofthe fuel is represented as H f .

The mass flux of the mixture through the flame frontfulfills the following equality: m =

∫ρSdA = ρS f A f . Please

note that in the following, all relevant quantities are con-sidered as averages of the quantity over the flame surface(hence assuming homogeneity of the time-averaged quanti-ties and linearity of the perturbations).

Flame chemiluminescence intensity has frequently beenreported to depend linearly on the mass flux and exponen-tially on the fuel mass fraction of the mixture entering theflame (see e.g. Higgins et al. [9]). Therefore, the recordedchemiluminescence intensity for the nth optical signal is as-sumed to have the following dependency of mass flux andfuel mass fraction:

In = kn myαn , (8)

where kn and αn are constants for the nth optical signal. It isassumed here that the Mach number is sufficiently small, see[5] for a more elaborate discussion on this topic. The presentwork deals with acoustic phenomena. That is why onlysmall perturbation of the flow conditions are considered,which justifies a linearization of Eqns. 7 and 8 :

Q′

Q=

m′

m+

y′

y(9)

I′nIn

=m′

m+ αn

y′

y(10)

These relations show that in case of αn = 1 or y′ = 0 thechemiluminescence is proportional to the heat release. How-ever, as pointed out in [5], typically αn 6= 1 so, the chemilu-minescence intensity is not proportional to the heat releaseif equivalence ratio fluctuations are present.

The method used here overcomes this problem by us-ing chemiluminescence of at least two species as input andcalculates the heat release from these signals via an inverseoperation. Thus, measurement of the intensity of N specieswould result in N relations like equation 10, which is writtenin matrix form as:

I′1I1...

I′NIN

=

1 α1...

...1 αN

[ m′my′y

]= Cα

[m′my′y

], (11)

with

Cα.≡

1 α1...

...1 αN

. (12)

If αi 6= α j, then the system can be inverted in leastsquares sense, in order to calculated m′

m and y′y from mea-

sured I′nIn

.The flame transfer function can be obtained if the

acoustic velocity fluctuation in front of the flame is mea-sured. In case of a negligible effect of the influence of pres-sure on the heat release fluctuation, the transfer function isthen obtained as:

F(ω) =Qu1

u1

Q= (

mm

+yy

)u1

u1. (13)

The optical method proposed here has the advantagethat it does not only provide quantitative heat releasefluctuations, but it also quantifies the underlying physi-cal mechanisms that cause the heat release fluctuations: itshows what part of the heat release is caused by equivalenceratio fluctuations and what part by flame front dynamics.

The relation by the transfer function F and H is foundby making use of the Rankine-Hugoniot relations for lowMach number flows:

uh = uc +(Th

Tc−1)uc

QQ

= uc (1 +(Th

Tc−1)F) (14)

H = 1 +(Th

Tc−1)F (15)

Note that in case of the negligible equivalence ratio fluc-tuations (y′ = 0), the flame chemiluminescence is propor-tional to the heat release and Eqn. 13 can by simplified to

F(ω) =Qu1

u1

Q= (

In

In)

u1

u1, (16)

where In denotes one of the recorded chemiluminescence sig-nals.

Optical signal correction for wall radiationThe fiber optic probe in the combustion chamber does

not only collect the light emitted by the flame, but also theheat radiation of the combustor wall.


An Ocean Optics QE65000 high-sensitivity spectrome-ter measured the spectrum of the light captured by the fiberoptic probe in the range of 200 to 850 nm. As an example, atypical light spectrum recorded at high pressure is shown inFig. 6. Note that the light spectrum has already been cor-rected for the spectrometer sensitivity and the transmissionof the optical probe.

As evident from the figure, the recorded spectra fea-tures increasingly high light emission above approximately500 nm. This increase is believed to be due to heat radiationof the combustor walls within the field of view of the opticalprobe. By fitting a black body radiation spectrum to therecorded spectrum an average combustor wall temperaturecan be estimated. For this fit the black body radiation isassumed to follow Planck’s law:

IBB(λ,TBB) =2hPc2

light

λ5

(e

hPclightλkBTBB −1

)−1

, (17)

where IBB is the spectral black body radiation, λ the opti-cal wavelength, TBB the black body temperature, clight thespeed of light, hP the Planck constant, and kB the Boltz-mann constant.

The estimated combustor wall temperatures are withinthe range of temperatures measured by several thermocou-ples distributed along the liner walls, thus giving confidenceto the interpretation of the recorded spectra.

300 350 400 450 5000

20

40

60

80

100

wavelength (nm)

light

inte

nsit

y (a

rb.)

recorded light spectrumblack body fit (T

BB = 1185 K)

estimated flame luminescence

Figure 6. Optical spectrum measured inside the combustion chamberand fit of the black body radiation.

By subtracting the contribution of the combustor wallradiation from the recorded spectra an estimation of the

flame luminescence spectra is obtained, as indicated inFig. 6. Four main observation can be made:

1) The estimated flame spectrum features a broad back-ground, which is believed to correspond to CO∗2.2) The estimated flame spectrum also feature the CH∗

peak at 431 nm, although it is less pronounced com-pared to the atmospheric flame. (Note that generally apressure increase results in weaker chemiluminescenceemission as the excited radicals are more likely to dis-sipate their energy in collisions with other molecules oratoms.)3) No C∗2 peak can be observed.4) At 309 nm the high-pressure light spectra do not fea-ture the expected peak corresponding to OH∗ formationin the flame front, but instead a distinct dip in the spec-tral intensity. This effect is attributed to the presents ofOH radicals in the exhaust between flame and fiber op-tic probe. While the excited OH∗ radicals in the flamefront emit light at 309 nm, the unexcited OH radicals inthe exhaust absorb light at exactly this wavelength.

From the filter transmission and the light spectrometerresults for each burner operating condition the offset in thephotomultiplier signals due to wall radiation is estimatedand accounted for in the further evaluation.

NETWORK MODELINGThe in-house acoustic network modeling tool Ta3 has

been used to predict stability characteristics and pulsationamplitudes of a gas turbine combustion chamber. In orderto obtain dynamic models of such systems, a hybrid ap-proach is used: numerical, experimental and analytical tech-niques are combined to describe the system. The system ismodeled as a modular network, where the input–output re-lation of the modules can be based on analytic models, ex-perimental data or numerical analysis. The modules arerepresented as state-space realizations. A modal expan-sion technique is used to obtain a state-space representationof the acoustic propagation through complex 3-dimensionalgeometries. The modal expansion can be based on an an-alytic model (for relatively simple volumes), or on a finiteelement analysis (for geometries of any complexity). Theflame transfer functions and source terms are incorporatedby fitting state space models to the measured data. Themethod is not restricted to symmetries of any kind: configu-rations with geometrically or operationally different burnersare simulated. The state-space network approach allows ei-ther for time domain simulations , including non-linearities.Alternatively an eigenvalue analysis can be performed whichis very straightforward due to the state-space formulation.Frequency spectra can either be obtained directly from a


u1p2

Sysnoise

u1p2

SysnoiseFlame Source

Modal Analysisclick to plot system poles

Globals Spectral Analysis

pi

pj

ui

uj

p1

u1

p4p2

u2

u3

p3 u4

Figure 7. Causal network interconnection of the elements of the com-bustion chamber

frequency domain analysis or by applying a Fourier analysisof the time traces in a post-processing step. The method isvery computational efficient: all results shown in this workhave been calculated in only a few seconds of computationtime.

This modeling approach relies on the assumption thatthe transfer function of a burner in single burner and multiburner configurations are similar. This assumption has beenvalidated experimentally on down-scaled but very realisticgeometry in [13].

For the analysis shown in this work, the elements ofthe gas turbine combustion chamber are interconnected ina Matlab/Simulink block diagram a shown in figure 7. Themodeling approaches for the different blocks are detailedbelow.

Modeling 3-D wave propagationA key aspect of this modeling approach is that the wave

propagation through a volume of any complexity is repre-sented by a transfer matrix between m velocity inputs andn pressure outputs:

pn

um= ρc2 Am

ψk(xn)ψk(xm)Λ2

k (s2 + ω2k)

(18)

where ψk(xn) is the value of the kth mode at location xn onthe boundary and Λ2

k =∫

ψ2k(x)dx is a normalization factor

for the modes. This equation can be expressed in state-space form as:

∂

∂ t

[~η~η

]=[

0 I−Ω2 0

][~η~η

]+[

0ΨT A

]~u (19)

~pρc

= c [0,Ψ][~η~η

], (20)

where Ψ is a N×K matrix which K columns contain ψk(xn)Λk

,the diagonal matrix A contains the area corresponding tothe mth velocity inputs. The K×K diagonal matrix Ω con-tains the eigenfrequencies of the solid walled geometry, I and0 are the K×K unit matrix and zero matrix respectively.The vectors ~η, ~p and ~u, contain the modal amplitudes, theinput velocities um and the output pressures pn. In the limitcase of K approaching infinity, the system of equations 19 isan exact solution of the wave equation. However the solu-tion converges rapidly, so in practice only a limited numberof modes K is required in the modal expansion.

Note that although the modal expansion is based on aset of orthonormal modes of the solid walled geometry, thisrepresentation remains exact even in the case when (com-plex, frequency dependent) boundary conditions are appliedto Equ. 19 or if the geometry represented by Equ. 19 takespart in a more complex network interconnection. Clearlyin such a case, the modes of the final, interconnected sys-tems will generally not be orthogonal anymore. Howeverin this methodology no assumption of orthogonality of theresulting modes has been made.

Time domain transfer function and source modelsThe flame model has been incorporated in the model by

fitting a state space model to the measured transfer functiondata. The fitting routine minimizes the magnitude squareddifference between the measured frequency response dataand the frequency response of the state space system underthe constraint that the fitted system must be stable. Thesource term is modeled in a similar way: a state space sys-tem is fitted to the measured source term. This state spacemodel is then used to filter a normally distributed randomsequence (white noise). The phase of the source term isundefined. However, in order to obtain the filter transferfunction, a phase needs to be generated. This was donehere by using the so-called complex cepstrum algorithm.

Burner transfer functionThe burner geometry was incorporated in the FEM

analysis of the plenum geometry and was hence included inthe plenum state space representation. However, this rep-resentation does not take into account the damping in theburner and the length correction at the exit of the burner.These have been modeled separately by using a so-calledL–Zeta model. The L-zeta models relates the acoustic ve-locity at an area discontinuity to the pressure drop over thediscontinuity:

u1 =p1− p2

iωLρ + u1 ρζ. (21)


The value of L has been obtained by a fit to the finite ele-ment model results of the entire combustion system (includ-ing plenum, burner and combustion chamber), the value of ζ

has been obtained from the measured mean flow conditions:ζ = p1−p2

12 ρu2 .

OPTICAL VS. ACOUSTICAL TRANSFER FUNCTIONRESULTS

For two burner configurations (A and B) the transferfunction H was measured. Figures 8 and 9 present a com-parison of this transfer function obtained from the acousti-cal approach (Eqn. 14) and the combined acoustical / op-tical approach (Eqn. 4). The purely acoustical results arelabeled acoustical. For the combined approach results ob-tained from multiple chemiluminescence signals accordingto Eqn. 13 are labeled optical, while results obtained fromone single chemiluminescence signal according to Eqn. 16are labeled with the center wavelength of corresponding fil-ter (e.g. CW309nm).

For the multi-signal optical method the matrix Cα,which contains the coefficients αi and links the intensityfluctuations to mass flow and fuel mass fraction fluctuation(see Eq. 11), was determined in a calibration measurement.This calibration was permormed for each burner configura-tion individually.

Note, that although the two optical methods provideinteresting additional insights into to combustion process,their main purpose in this work was to validate the purelyacoustic transfer function results. For this validation it hasto be considered that the reliability of the two acoustic-optical methods depends on the cause of heat release fluc-tuations in the flame, i.e., equivalence ratio fluctuationsand/or mass flow fluctuations. In case only one of thesetwo contributions to heat release fluctuations is present, theacoustic-optical method based on only one optical signal isexpected to work well according to Eqn. 16.

The multi-signal optical technique on the other handalso works, if the heat release fluctuations have two contri-butions, e.g. equivalence ratio and mass flow fluctuations.However, this method relies on the difference in the responseof the flame chemiluminescence at different wavelength re-gions to these fluctuations. These differences in response arecommonly very small. Therefore, even small measurementerrors in the intensity signals can have a significant impacton m′

m and y′y obtained from Eq. 11, and hence the transfer

function H. Additionally, errors in the determination of Cα

will also effect the accuracy of H. Especially large errors inH have to be expected, if Cα is badly conditioned. This isthe case, if the coefficients αi are very similar.

Figure 8 shows the transfer function H for burner con-

figuration A. The optical results based on multiple chemi-luminescence signals are in good agreement with the acous-tical results in terms of absolute values and phase. The re-sults obtained from only one chemiluminescence signal showan increasing difference in absolute value for St > 0.3 anddo not capture the falling slope of the phase obtained fromthe other two methods. This deviation validates the ex-pectation that equivalence ratio fluctuations are present inthe flame of this burner, and that therefore heat releaseoscillations can not be captures accurately from only onechemiluminescence signal. However, from the results of themulti-signal optical method it was still found that the maindriver of heat release fluctuations are flame dynamics ratherthen equivalence ratio fluctuations.

For burner configuration B the results are presentedin Fig. 9. The figure shows a good agreement between theacoustical and the one-signal optical method, which indi-cates a negligible impact of equivalence ratio fluctuationson the heat release of the flame. The multi-signal opti-cal method captures the main trends in absolute value, butwith some deviations in amplitude, and features significantphase deviations for St < 0.25.

0.10.10.10.1 0.20.20.20.2 0.30.30.30.30000

0.50.50.50.5

1111

abs(

H)

abs(

H)

abs(

H)

abs(

H)

St (-)St (-)St (-)St (-)

UV bandUV bandUV bandUV bandCW 307 nmCW 307 nmCW 307 nmCW 307 nmCW 431 nmCW 431 nmCW 431 nmCW 431 nmCW 450 nmCW 450 nmCW 450 nmCW 450 nmopticalopticalopticalopticalacousticalacousticalacousticalacoustical

0.10.10.10.1 0.20.20.20.2 0.30.30.30.3

-2-2-2-2

0000

2222

St (-)St (-)St (-)St (-)

ph

s(

H)

ph

s(

H)

ph

s(

H)

ph

s(

H)

Figure 8. Flame transfer function of burner configuration A.

In order to identify the reason for these deviations anerror analysis has been performed. In this error analysis itwas found that for burner configuration B the condition ofCα was significantly worse compared to burner configura-tion A due to very similar coefficients αi, which will causelarger errors in H for burner configration B. To visualize thesensitivity of the transfer function H to measurement errors5% relative noise was artificially added to the Fourier coef-


0.10.10.10.1 0.20.20.20.2 0.30.30.30.3 0.40.40.40.40000

0.50.50.50.5

1111

abs(

H)

abs(

H)

abs(

H)

abs(

H)

St (-)St (-)St (-)St (-)

UV bandUV bandUV bandUV bandCW 307 nmCW 307 nmCW 307 nmCW 307 nmCW 431 nmCW 431 nmCW 431 nmCW 431 nmCW 450 nmCW 450 nmCW 450 nmCW 450 nmopticalopticalopticalopticalacousticalacousticalacousticalacoustical

0.10.10.10.1 0.20.20.20.2 0.30.30.30.3 0.40.40.40.4

-2-2-2-2

0000

2222

St (-)St (-)St (-)St (-)

ph

s(

H)

ph

s(

H)

ph

s(

H)

ph

s(

H)

Figure 9. Flame transfer function of burner configuration B.

ficients II (obtained from the measured chemiluminescence

intensity time traces) and the determined αi, respectively.Then, the transfer function H was determined based on thecontaminated values. This procedure was repeated 1000times to obtain statistically meaningful results.

Figures 10 and 11 shows the obtained results for burnerconfiguration A and B, respectively. The original resultsobtained from the measured data are indicated by blackcrosses. The results obtained with 5% artificial relativenoise in the intensity signals are plotted in blue, the resultsobtained with 5% artificial relative noise in the coefficientsαi are plotted in red. The red and blue solid lines indicatethe median absolute value and phase based on the 1000repetitions, while the dotted lines represent the upper andlower quartil as a measure of the relative error.

As evident from Figs. 10 and 11 the impact of artificialnoise on the transfer function H is much stronger in case ofburner configuration B, indicating that the results obtainedfor this burner configuration are likely to be less reliablethen for burner configuration A.

NETWORK MODELING RESULTSThe detailed geometry of an annular gas turbine

plenum, burner and combustion chamber was modeled in afinite element package to obtain the mode shapes and eigen-frequencies required for modal expansion. The measuredtransfer functions and source terms were then incorporatedinto this network in order to calculate eigenvalues and pul-sation spectra of the engine. The investigated burner con-figuration allows for local enrichment of the fuel-air-mixtureby fuel staging. Increasing the fuel stage ratio commonlyhas a stabilizing effect on combustion pulsations. As a typ-

0.1 0.2 0.3

-2

0

2

St (-)

phs(

H)

0.1 0.2 0.3 0.4

012

abs(

H)

data as measurement5% noise on optical signals5% noise on coeff. α

0.1 0.2 0.30

0.3

0.6

abs(

H)

0.1 0.2 0.3

-2

0

2

St (-)

phs(

H)

Figure 10. Error sensitivity analysis for flame transfer function H ofburner configuration A.

0.1 0.2 0.3

-2

0

2

St (-)

phs(

H)

0.1 0.2 0.3 0.4

0

1

2

abs(

H)

Figure 11. Error sensitivity analysis for flame transfer function H ofburner configuration B. (Legend identical to Fig. 10.)

ical example, the influence of the fuel staging ratio on theflame transfer function, source term, eigenvalues and spec-tra will be shown. A comparison with spectra measured inthe engine is given.

The acoustic transfer functions used for this analysisare shown in Fig. 12. The normalize fuel staging ratio var-ied from 0 to 2.0, where the nominal value was 1. The effect


0.10.10.10.1 0.150.150.150.15 0.20.20.20.2 0.250.250.250.25 0.30.30.30.3 0.350.350.350.350000

0.50.50.50.5

1111

abs(

H)

abs(

H)

abs(

H)

abs(

H)

St (-)St (-)St (-)St (-)

0.10.10.10.1 0.150.150.150.15 0.20.20.20.2 0.250.250.250.25 0.30.30.30.3 0.350.350.350.35

-2-2-2-2

0000

2222

St (-)St (-)St (-)St (-)

ph

s(

H)

ph

s(

H)

ph

s(

H)

ph

s(

H)

staging 0staging 0staging 0staging 0staging 1.0staging 1.0staging 1.0staging 1.0staging 2.0staging 2.0staging 2.0staging 2.0

Figure 12. Measured transfer functions for different values of the nor-malized fuel staging ratio.

0.10.10.10.1 0.150.150.150.15 0.20.20.20.2 0.250.250.250.25 0.30.30.30.3 0.350.350.350.350000

0.20.20.20.2

0.40.40.40.4

0.60.60.60.6

0.80.80.80.8

1111

St (-)St (-)St (-)St (-)

abs(

sou

rce)

abs(

sou

rce)

abs(

sou

rce)

abs(

sou

rce)


Figure 13. Measured source terms for different values of the normalizedfuel staging ratio.

of increasing fuel staging show –as expected– that the abso-lute value of the transfer function is significantly reduced.The slope of the phase decreases for increasing fuel staging,which is also expected because staging will cause the flameto stabilize closer to the burner exit.

The dependence on the source term on fuel staging ratiois less distinct, as can be seen in Fig. 13.

The eigen analysis of the system clearly shows the stabi-lizing effect of the fuel staging on the stability of the system.In Fig. 14 the real part of the eigenvalue is plotted versusthe imaginary part. Positive real parts of the eigenvaluesindicate instability. The dominant eigenvalues close to nor-malized frequency of 0.35 are clearly stabilized by increasingthe fuel staging.

A simulation in the time domain has been performedto obtain time traces of pulsation in the combustion cham-

0.10.10.10.1 0.20.20.20.2 0.30.30.30.3 0.40.40.40.4 0.50.50.50.5 0.60.60.60.6 0.70.70.70.7 0.80.80.80.8 0.90.90.90.9

-25-25-25-25

-20-20-20-20

-15-15-15-15

-10-10-10-10

-5-5-5-5

0000

5555

x 10x 10x 10x 10-3-3-3-3

frequency (-)frequency (-)frequency (-)frequency (-)

grow

th r

ate

(-)

grow

th r

ate

(-)

grow

th r

ate

(-)

grow

th r

ate

(-)


Figure 14. Calculated eigenvalues of the annular engine configuration.

0.10.10.10.1 0.20.20.20.2 0.30.30.30.3 0.40.40.40.4 0.50.50.50.5 0.60.60.60.6 0.70.70.70.7 0.80.80.80.8 0.90.90.90.90000

0.20.20.20.2

0.40.40.40.4

0.60.60.60.6

0.80.80.80.8

1111


pu

lsat

ion

am

pli

tude

(-)

pu

lsat

ion

am

pli

tude

(-)

pu

lsat

ion

am

pli

tude

(-)

pu

lsat

ion

am

pli

tude

(-)


Figure 15. Calculated pulsation spectra of the annular engine configura-tion.

ber. The pulsation spectra shown in Fig. 15. These resultsreflect the general behavior already seen in the eigen anal-ysis: the dominant pulsation peak is reduced by increasedfuel staging.

In Fig. 16 the measured pulsation spectra of the pul-sations in the actual engine are shown. The qualitativeagreement is very good: all dominant pulsation modes arecorrectly predicted. The predicted reduction of the mainpulsation peak with increasing fuel staging is clearly ob-served in the measurements as well.

CONCLUSIONThree different methods have been used to measure

source terms, flame and burner transfer functions in a full-scale high pressure test facility. A purely acoustical methodhas been compared with two optical methods. In the sim-plest optical method, the relative heat release fluctuationis assumed to be equal to the relative chemiluminescenceintensity fluctuation. This is correct for cases where con-tribution of equivalence ratio fluctuations can assumed to


0.10.10.10.1 0.20.20.20.2 0.30.30.30.3 0.40.40.40.4 0.50.50.50.5 0.60.60.60.6 0.70.70.70.7 0.80.80.80.8 0.90.90.90.90000

0.20.20.20.2

0.40.40.40.4

0.60.60.60.6

0.80.80.80.8

1111


pu

lsat

ion

am

pli

tude

(-)

pu

lsat

ion

am

pli

tude

(-)

pu

lsat

ion

am

pli

tude

(-)

pu

lsat

ion

am

pli

tude

(-)

staging 0.5staging 0.5staging 0.5staging 0.5staging 1.0staging 1.0staging 1.0staging 1.0staging 1.5staging 1.5staging 1.5staging 1.5staging 2.0staging 2.0staging 2.0staging 2.0

Figure 16. Measured pulsation spectra of the engine.

be negligible, which has been demonstrated experimentally.In case the equivalence ratio fluctuations do have an influ-ence, only meaningful results can be obtained by using themethod that uses multiple chemical species. This could bedemonstrated by comparing the two methods with transferfunctions that were obtained from acoustic data only. Themeasured transfer functions and source terms have been in-corporated in an acoustic network model of a gas turbine.As a typical example the stabilizing influence of fuel stagingon pulsation behavior is shown. The model with measuredtransfer functions proved to accurately reproduce the mea-sured pulsation spectra. Being able to measure transferfunctions at high pressure conditions on full scale burnersopens up the possibility to predict thermoacoustic behav-ior of gas turbines using fuels whose combustion propertiesdepend on pressure such a natural gases with higher hydro-carbon content, syngases and liquid fuels.

ACKNOWLEDGMENTThis work has been conducted in the framework of AG

Turbo / COOREFF-T with support from the German Fed-eral Ministry of Economics and Technology.

REFERENCES[1] Schuermans, B. “Modeling and control of

thermoacoustic instabilities”. PhD Thesis nr.2800 (2003), EPFL Lausanne, Switzerland.http://library.epfl.ch/theses/?nr=2800.

[2] Schuermans, B., Bellucci, V., and Paschereit, C., 2003.“Thermoacoustic modeling and control of multi burnercombustion systems”. ASME 2003-GT-38688, Proc.ASME Turbo Expo 2003, Atlanta, June 16-19.

[3] Bellucci, V., Schuermans, B., Nowak, D., Flohr, P.,and Paschereit, O., 2003. “Thermoacoustic model-ing of a gas turbine combustor equipped with acoustic

dampers”. ASME GT 2004-53977, Proc. ASME TurboExpo 2004, Vienna, June 14-17.

[4] Bellucci, V., Nowak, D., Geng, W., and Steinbach, C.,2007. “Thermoacoustic modeling and control of multiburner combustion systems”. ASME GT2007-27329,Proc. ASME Turbo Expo 2007, Montreal, June 14-17.

[5] Schuermans, B., Guethe, F., and Mohr, W., 2008.“Transfer function measurements for technically pre-mixed flames using a novel optical method”. ASMEGT2008-51500, Proc. ASME Turbo Expo 2008, Berlin,June 9-13.

[6] Hardalupas, Y., and Orain, M., 2004. “Local mea-surements of the time-dependent heat release rateand equivalence ratio using chemiluminescent emissionfrom a flame”. Combustion and Flame 139, (2004)188207.

[7] Kojima, J., Ikeda, Y., and Nakajima, T., 2000. “De-tailed distributions of oh*, ch* and c2* chemilumines-cence in the reaction zone of laminar methane/air pre-mixed flames”. 36th AIAA/ASME/SAE/ASEE JointPropulsion Conference and Exhibit (2000) AIAA-3394.

[8] Nori, V., and Seitzman, J., 2007. “Detailed distribu-tions of oh*, ch* and c2* chemiluminescence in the re-action zone of laminar methane/air premixed flames”.AIAA-2007-0466 at the 45th Aerospace Sciences Meet-ing, Reno, NV, Jan 8-11, 2007.

[9] Higgins, B., McQuay, M., Lacas, F., Rolon, J., Dara-biha, N., and Candel, S., 2001. “Systematic mea-surements of oh chemiluminescence for fuel-lean high-pressure, premixed, laminar flames”. FUEL 80, 2001,PP 67-74.

[10] Higgins, B., McQuay, M., Lacas, F., and Candel, S.,2001. “An experimantal study of pressure and strainrate on ch chemiluminescence on premixed fuel-leanmethans /air flames”. FUEL 80, 2001, PP 1583-1591.

[11] Cremer, L., 1971. “The treatment of fans as blackboxes”. Journal of Sound and Vibration, 16(1), 1-15.

[12] Schuermans, B., Bellucci, B., Guethe, F., and Meili,F., 2003. “A detailed analysis of thermoacoustic in-teraction mechanisms in a turbulent premixed flame”.ASME GT 2004-53831, Proc. ASME Turbo Expo 2004,Vienna, June 14-17.

[13] Fanaca, D., Alemela, P. R., Ettner, F., Hirsch, C., Sat-telmayer, T., and Schuermans, B., 2008. “Determina-tion and comparison of the dynamic characteristics ofa perfectly premixed flame in both single and annularcombustion chambers”. ASME GT2008-50781, Proc.ASME Turbo Expo 2008, Berlin, June 9-13.


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