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Modern gas turbines suffer of the phenomenon of combustion instability, also known as “humming”. The main origin of the instability is considered to be related to the interaction between acoustic waves and fluctuations of the heat released by the

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A Novel FEM Method for Predicting ThermoacousticCombustion Instability
G. Campa
*1
, S.M. Camporeale
1
1
Politecnico di Bari
*
campa@imedado.poliba.it , DIMeG – Sez. Macchine ed Energetica – Politecnico di Bari – via Re David200, 70125 Bari, Italy
Abstract
Modern gas turbines suffer of the phenomenonof combustion instability, also known as“humming”. The main srcin of the instabilityis considered to be related to the interactionbetween acoustic waves and fluctuations of theheat released by the flame. Pressureoscillations may cause many damages of thegas turbine and loss of control of thecombustion process. This paper presents anovel numerical method in which thegoverning equations of the acoustic waves arecoupled with a flame heat release model andsolved in the frequency domain. The papershows that a complex eigenvalue problem isobtained that can be solved numerically byimplementing the governing equations inCOMSOL Multiphysics. This procedure allowsone to identify the frequencies at whichthermoacoustic instabilities are expected andthe growth rate of the pressure oscillations, atthe onset of instability, when the hypothesis of linear behaviour of the acoustic waves can beapplied. Some test cases and examples of applications are described in the paper.
Key Words:
combustion instability, gas turbine, acoustics,wave propagation, eigenfrequencies, growthrate.
1. Introduction
In modern gas turbines equipped with lowNOx emission combustion systems, thephenomenon of acoustically drivencombustion instability (known also as"humming") represents a severe risk for thesafe operation of the plant, since it causesintense vibrations that may damage the engine.The srcin of such phenomenon is notcompletely understood and, moreover, it is notclear which could be the best techniquessuitable for either decreasing the risk of instability or lowering its damaging effects [1-4]. This paper shows a method for predictingthe onset of acoustically combustioninstabilities in gas turbine combustors. Thebasic idea is that the governing equations of the acoustic waves can be coupled with aflame heat release model and solved in thefrequency domain. This procedure allows oneto identify the frequencies at whichthermoacoustic instabilities are expected andthe growth rate of the pressure oscillations, atthe onset of instability, when the hypothesis of linear behaviour of the acoustic waves can beapplied. A similar approach has been carriedout by Martin et al. [5], even if they used athree-dimensional finite element based in-house acoustic solver called AVSP.The present method can be applied virtually toany three dimensional geometry, provided thenecessary computational resources that are,anyway, much less than those required byComputational Fluid Dynamics (CFD)methods. Furthermore, in comparison with thelumped approach that characterize popularAcoustics Networks, the proposed methodallows one for much more flexibility indefining the geometry of the combustionchamber. The paper shows that different typesof heat release laws, for instance, heat releaseconcentrated in a flame sheet as well asdistributed in a larger domain, can be adopted.Moreover, experimentally or numericallydetermined flame transfer functions, giving theresponse of heat release to acoustic velocityfluctuations, can be incorporated in the model.To establish proof of concept, the method isvalidated against test cases taken fromliterature and moreover an application to anannular combustion chamber is proposed.
2. Equations
The problem is solved in the frequency domainusing the eigenfrequency “Pressure Acoustics”application mode. Since in gas turbinecombustion chamber the flow velocity isgenerally far below the sound velocity, theflow velocity is negligible, except that in someareas, such as conduits of the burners. Theseareas, in terms of propagation of pressure
waves, can be treated as separate elements thatcan be modelled by means of specific transferfunction matrices, obtained experimentally ornumerically through CFD or aeroacousticscodes. Therefore, the flow velocity isconsidered negligible in comparison with thesound velocity, within the computationaldomain. Moreover, the effects of viscouslosses and heat conduction will be neglected,and the fluid considered an ideal gas, thatmeans that the ratio of the specific heats issupposed constant. Under such hypotheses, inpresence of heat fluctuations, theinhomogeneous wave equation can beobtained:
t qc pt pc
∂′∂−=
′∇⋅∇−
∂′∂
2222
111
γ ρ ρ
(1)where q' is fluctuation of the heat input perunit volume, overbar denotes a time averagemean value and the prime a perturbation. Theterm at the RHS of Eq.(1) shows that the rateof non-stationary heat release creates amonopole source of acoustic pressuredisturbance. In the eigenvalue analysis as wellas in a frequency response analysis, pressurewave can be splitted into a function of position)(ˆ
x
p
multiplied by a complex exponential thatis a function of time
)exp()(
ˆ'
t i p p
ω
x
=
, (2)where is complex variable, comprising areal part that gives the frequency of oscillations )2 /()Re(
π ω
=
f
, while imaginarypart gives the growth rate at which theamplitude of oscillations increases per cycle)Im(
ω
−=
g
. If the growth rate is positive,fluctuations over time will grow exponentiallywith time. Within the harmonic analysis, heatrelease fluctuation
q'
and acoustic velocity
u’
are also functions of time of the type)exp(ˆ'
t iqq
ω
=
and)exp()(
ˆ'
t iuu
ω
x
=
. Then,using Eq.(1) and Eq.(2), the acoustic pressurewaves are governed by the following equation
qc p pc
ˆ1ˆ1ˆ
222
λ γ ρ ρ λ
−−=
∇⋅∇−
(3)where
i
ω
=
λ
−
and c is the velocity of thesound. Eq.(3) shows a quadratic eigenvalueproblem that can be solved by means of aniterative linearization procedure [8].The finite element method allows us to modelthe heat release fluctuation
q'
as a variable of space, so that it is possible to describe the formof the flame in a very flexible manner. In thiswork, however, the flame is modelled as astraight flame sheet placed at the exit of theburner, in order to accomplish the aim of comparing the numerical results with those of the cited cases available in literature.The boundary conditions are consideredbasically as three types: sound hard (wall),sound soft and normal acceleration. The solidwalls are considered as sound hard, while theplenum inlet and the combustion chamberoutlet walls changes with the different testcases analyzed.
3. Tests on Linear Combustion Chamber
The preliminary application tests are carriedout on a duct, with uniform cross-sectionalarea, mean temperature, constant density andno mean flow. Such a scheme is the sameexamined by Dowling and Stow in [6] and waschosen as a benchmark. The duct is connectedto a large plenum at the inlet and is provided of a restriction at the exit (Figure 1). In this one-dimensional case, the planar wave hypothesiscan be adopted and the only abscissa x alongthe duct is used to describe the variation of acoustic pressure and velocity in the duct. Aplanar flame sheet is supposed to be located atthe abscissa
x=b
. Under the no flowhypothesis, the following boundary conditionsare assumed
0)(',0)0('
==
lu p
. (4)
Figure 1 – Simplified scheme of flame location in astraight duct with uniform cross section.
Dowling and Stow offer an example where theheat release fluctuation is supposed to beconcentrated in a single plane placed at
x=b
(Figure 1) and related to the velocityfluctuation of the oncoming flow with a timedelay
τ
)()('),('
b xt Qt xq
−=
δ
(5)
)(')]1 /([)('
12
τ γ ρ β
−−−=
t uct Q
(6)where
Q
’(
t
) is the rate of heat input per unitarea of the cross section of the duct andsubscript 1 denotes conditions just upstream of this region of heat input, that is),(')('
1
t but u
−
=
. The Eq.(5) relates thefluctuation of heat input rate per unit volume,
q’(x,t),
to the fluctuation of the heat input rateper unit area of the cross section,
Q
’(
t
),through the Dirac’s delta)(
b x
−
δ
. Thenondimensional number
β
gives a measure of the intensity of the heat fluctuations while
τ
can be estimated as the convection time fromfuel injection to its combustion.In the FEM eigenvalue analysis, the heatrelease fluctuations are supposed to occur in avery thin volume with thickness
s
and theDirac’s delta)(
b x
−
δ
that appears in Eq.(9)can be approximated as:
+>+≤<−
−≤≅−
2 / 02 / 2 / / 1
2 / 0)(
sb xsb xsbs
sb xb x
δ
. (7)Using Eq.(2), setting
i
ω
=
λ
−
and simplifying,Eq.(3) becomes
( )
( )( ) ( )
[ ]
λτ λ βδ ρ λ ρ
−−−−=∇∇−
−
exp)(ˆ)(ˆ1ˆ11
22
bub x p p
c
(8)that is the governing equation to be solved inthe internal domain by the FEM method.
Figure 2 – Variation with
β
of the normalized frequency for the first mode of the duct with heat release fluctuation concentrated in a flame sheet.Time delay
τ
=0 and b=l /10. Symbols represent theresults from the acoustic code. Line is obtained fromthe analytic solution.Figure 3 – Variation with
β
of the growth rate for the first mode of the duct with heat release fluctuation concentrated in a flame sheet. Caseb=l/10. Symbols represent the results from theacoustic code. Lines are obtained from the analyticsolution.
Figure 2 and Figure 3 show the comparison of the results obtained from the acoustic code andthe results obtained from the application of theplanar wave theory. It appears a very goodagreement for the frequency as well as for thegrowth rate. The numerical results obtainedhere appear to be much better than thoseobtained from the one-term Galerkinapproximation given in [6]. The dependence of the resonant frequency on
β
is shown in Figure2. As
β
increases, the intensity of the heatrelease in the combustion chamber and theacoustic pressure increase. Consequently, thereis a decrease of the resonant frequency in ameasure to comply with inlet and outletboundary conditions. The influence of
β
and
τ
on the growth rate is shown in Figure 3.Growth rate increases if the rate of heat inputhas a component in phase with pressureperturbation. The differences between theresults obtained from the FEM analysis are soclose to those obtained analytically thatgraphically any difference can be observed,confirming the success of this test.In the previous examples, only a theoreticalgeometry has been considered, suitable forbenchmark tests, since the analytical solutionsexist for such tests. In [6] Dowling and Stowexamine a more realistic quasi-one-dimensional geometry, composed by threecylindrical ducts: diffuser, premixer andcombustion chamber. In this case, a low Machnumber flow was considered in [6], while herethe flow is neglected according with Eq.(1). Itis assumed, following the approach adopted in[6], the blockage at the premixer inlet so that itacts approximately like a hard end (
u
’=0). Anopen end is instead assumed at the combustorexit (
p’
=0).
Figure 4 - Simplified scheme of flame location and boundary conditions, for benchmark tests onstraight duct with variation of section.
The flame sheet is supposed to be placed at theexit of the premixer, just at the inlet of thecombustion chamber. The heat releasefluctuation, given by Eq.(6), is related to thefluctuations of the flow velocity at the fuelinlet point through a time delay
τ
. The flamemodel used by Dowling and Stow in [6] was
( )
ωτ
immk QQ
ii
−−=
expˆˆ
(9)where
m
i
is the air mass flow at the fuelinjection point (assumed to be located at theinlet of the premixer).
k
is a nondimensionalnumber, used for varying the intensity of theunsteady heat release. The time average of heatrelease rate per unit area of the combustionchamber is
( )
.comb.chambpremix12
A AT T cuQ
pii
−=
ρ
(10)where
c
p
is specific heat at constant pressureand
1
T
and
2
T
are the temperatures in thepremixer and in the combustion chamber,respectively. From these equations, anexpression of
Q
ˆcan be obtained. Then theheat input per unit area can be written as afunction of time )exp(ˆ'
t iQQ
ω
=
. So that, byusing Eq.(5), it is straightforward to obtain theexpression of the coefficient
β
to be used toquantify the fluctuations of heat release inEq.(8) where, coherently with the flame modelassumed here, the velocity fluctuation at thepremixer inlet,
i
u
ˆhas been considered insteadof the fluctuation )(ˆ
−
bu
just upstream of theflame.The grid used for this case, composed of 11382 elements uniformly distributed, isrepresented in Figure 5:
Figure 5 – Mesh visualization of the quasi-one-dimensional combustion systemFigure 6 – Resonant mode of a simple combustor.Crosses represent the results from the acoustic code.Circles represent the results from [6]
The results obtained from the calculations of the eigenvalues of the system with heat releasefluctuations with
k
=1 are given in Figure 6,where the values of growth rate values areplotted against the frequency values. In thisfigure such results are compared with thecorresponding values given in [6]. It appearsthat the code is able to identify the all themodes and their frequencies, carefully. Alsothe growth rate is well estimated taking intoaccount that calculations in [6] are carried outunder the hypothesis of planar waves andtaking into account the damping effects due tothe mean flow. In many cases heat releasefluctuations have the effect to create instabilityas the growth rate has a positive value. Theanalysis of the pressure patterns of each mode,show that the lowest frequency of 29 Hzcorresponds to a resonance of the wholesystem, 105 [Hz] is the frequency of the firstmode of the plenum, that behaves like anacoustic tube with a closed end at the inlet andan approximately closed end at the otherboundary due to the variations of the crosssection at the conjunction with the premixer.Further eigenfrequencies appear at 199, 290,387 and 502 Hz that are the harmonics of themode at 105 Hz. A good correspondence with
the results shown in [6] is obtained. In fact allthe important frequencies are valuated. Thegrowth rate are not equal because mean flow isneglected, following a conservative analysis.In this way growth rate are greater in absolutevalue than those obtained in [6] and tightenedto instability.
4. Tests on Annular Combustor
The examination of an annular combustor hasbeen carried out taking as reference the simplegeometry examined by Pankiewitz andSattelmayer in [7]. The combustor, shown inFigure 7, is characterized by a diffusionchamber ring (plenum) and an annularcombustion chamber linked by 12 "swirler"burners. The hot gases leave the combustionchamber through 12 choked nozzles, whichallow the achievement of high pressure insidethe chamber. The conditions are set accordingto a case of an experimental combustor thatwas operated with natural gas, the air enteringthe plenum being preheated to
T
1
= 770K andthe total thermal power being
P
= 1020kW. InFigure 7 the grid composed of 66519 elementsuniformly distributed is represented.
Figure 7 – Computational grid of the annular combustion chamber.
For inlet and outlet boundary conditions thesame approach used in [7] is adopted in orderto take into account the effects of the flowvelocity on acoustic impedance of suchboundaries. The flow both enters and exits thecomputational domain from and to a largeplenum, where pressure perturbations can beneglected. Defining the inward normalacceleration at the boundary as
t a
n
∂∂•−=
''
un
, (11)taking into account the equation0'1'
=∇+∂∂
pt
ρ
u
and considering that
( )
t i p= p
'
ω
expˆ, the expression of normalacceleration can be written as
p
ρ
+uK K
ρ
i
ω
=a
nn
ˆ11ˆ
⋅−
(12)where, like other fluctuating quantities,
n
a
ˆisobtained from the relation )exp(ˆ'
t iaa
nn
ω
=
.Assuming a suitable value for the constant
( )
∞
−=
p puK
n
2 /, Eq.(12) gives theexpression of normal acceleration that is usedin COMSOL to set the boundary conditions atinlet and outlet of the combustion system [8].The value
K =
0 represents the condition of choked or closed end;
K =
∞
represents thecondition of open end. As in [7], the values of K = 0.1 at inlet and K = 0.2 at outlet, areassumed in the calculations.The eigenvalue analysis of the system has beencarried out in order to obtain the characteristicmodes of the system. At the beginning thesystem is considered without heat release. Theeigenfrequencies have been normalized withthe frequency
000
/
Lu= f
where
u
0
is equal tothe sound speed in the plenum and the length
L
0
is equal to the mean diameter of thechamber. The obtained results for the first fourmodes are given in Table 1 and compared withthe results given in [7]. The modes are denotedwith the nomenclature
(l,m,n)
, where
l
,
m
and
n
are, respectively, the orders of the pure axial,circumferential and radial modes that appear inthe eigenmode obtained from simulation. In adistribution of pressure within the wholecombustion assembly, for the first foureigenmodes, is presented: pressuredistributions appear to be very similar to thoseproposed in [7]. The first mode is a pure axialmode, the second one is a circumferentialmode that involves only the plenum where thelargest pressure oscillations can be seen. Thethird mode is a coupled axial-circumferentialmode where the largest oscillations occur inthe combustion chamber. The fourth mode is acircumferential mode of the second order thatinvolves only the plenum. The main differencebetween the results obtained here and thosegiven in [7] appears to be related to the valueof the eigenvalue corresponding to the axialmode. This is most probably due to somegeometrical differences in the axial length of

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