A limit cycle for pressure oscillations in a premix burner
A.
Di Vita
1
, F. Baccino
2
, E. Cosatto
3
1
Ansaldo Energia, PDEISVTFC, Via N. Lorenzi 1 16152 Genova, Italy, Email:andrea.divita@aen.ansaldo.it
2
DICCA, Univ. Genova, Via Montallegro 1 16145 Genova,Italy, Email:bafiliba@gmail.com
3
Ansaldo Energia, PDEISVTFC, Via N. Lorenzi 1 16152 Genova, Italy, Email:ezio.cosatto@aen.ansaldo.it
The problem
Any satisfactorily description of humming (= dangerous,combustiondriven acoustic oscillation in gasturbine burners [1]) should include both acoustic and convective phenomena. In fact, the distance v
0
crossed by an elementof fluid (with speed v
0
10 m/s) during one humming period(
10 ms) is typically
the distance L between the flameand the inlet of the airfuel mixture (L
10 cm).Accordingly, no fully linear approach seems to besuccessful. Admittedly, hummingrelated pressure perturbation amplitude (p
1
10 mbar) is << the unperturbedworking pressure (p
0
10 bar), and the speed of a small fluidelement in an acoustic wave (v
1
m/s) is << the unperturbedsound speed (c
s0
500 m/s). Locally, however, v
1
may be
v
0
, e.g. near the vortices’ axis inside the burner, where v
0
vanishes. Moreover, available models often postulate theMach number M
v
0
/c
s0
to vanish, a scarcely justifiabletenet [2]. Furthermore, the combined effect of nonnormality and non–linearity [3][4] invalidates the familiar correspondence between humming occurrence and the signof the imaginary part of humming frequency as predicted bylinear theories. Finally, linear theories provide us with no prediction of humming amplitude; but such prediction isrequired in order
to assess humming impact. Accordingly, predictions of dangerous humming require nonlinear –evenif simplified– description. The aim of this work is to providesuch a simplified description. Firstly, we describe our model.Secondly, we compute the amplitude of pressure perturbation and discuss conditions for humming onset.Finally, a first application to Ansaldo burner is shown.
The model
An equation for acoustics
We assume humming is due to a feedback between perturbations of heat release at the flame and of stoichiometry at the inlet of the airfuel mixture. Wedescribe humming as a limit cycle [5][6]; its onset is a bifurcation between a steady state and the limit cycle [7].Before the flame, there is a mixture with total mass density
at temperature T
inlet
c
s02
, made of two perfect gases, ‘air’and ‘fuel’, with molar density n
air
and n
fuel
respectively, andwith air molar fraction
fuelair air
nnnz
. Any perturbation q*
1
in the density q* of power Q* exchanged between the flameand the fluid induces a pressure perturbation p
1
(with halfamplitude
p) which propagates in a time t
acoustic
up to theinlet, where it takes the value p
inlet 1
. (Here and below,
a
0
and
a
1
denote unperturbed value and perturbation of the genericquantity
a
respectively). The equation linking the sourceq*
1
and the signal p
1
is [8]:
*102012020
qt1c1 ptc1
vv
ss
(1)where
is the specific heat ratio before the flame. Whenapplied to a flame with area A
f
(
m
2
) and thickness d
0
(
mm), the first principle of thermodynamics provides us withQ* (= q*
A
f
d
0
):
tdAd1d pQQ
f 00c*
(2)where we take d
0
= const. as d
0
/v
0
<<
. The quantity Q
c
isthe heat release:
f f c
v
ρ
YAHQ
(3)Here H,
Y = m
fuel
p
fuel inlet
(RT
inlet
)
1
, R, m
fuel
, v
f
= v
fM
h,z
f
, v
fM
, and h = h(z
f
) are the amount of heat released by thecombustion of one unit mass of fuel, the fuel mass density,the perfect gas constant, the fuel molar mass, the flamevelocity (supposed to be known), the value of z at the flame,the value of v
f
for z = z
st
and a function of z
f
respectively,with 1
≤
h
≤
1, h(0) = h(1) = 0 and h = 1 only at z
f
= z
st
,where z
st
(= 10/11 for air/CH
4
) the stoichiometric value of z.Formally, (1) gives:
acoustic1*1inlet
ttQ
ζ
t p
(4)where
1f 10s0 N
Avc14
ζζ
and
N
is adimensionless, geometrydependent constant. Firstly, wediscuss the fixedflame case A
f
= const.; generalisation to thegeneric movingflame case dA
f
/dt
≠
0 follows.
Fixed flame vs. moving flame
At the inlet, the fuel partial pressure p
fuelinlet
is constant inmost Ansaldo burners. This implies that p
1 inlet
perturbs air pressure, hence the value z
inlet
of z at the inlet. Then:
tz11 pt p
1inletinletfuel1inlet
(5)according to Dalton’s law. The fluid carries z
1
from the inletto the flame in a time t
convective
:
convectiveinletflame
ttztz
(6)We eliminate p
inlet 1
from (4) and (5), invoke (6), define thedimensionless quantity q
Q
*
/ p
fuel inlet
and obtain
tz11tq
1f 1
(7)where
= t
acoustic
+ t
convective
. We integrate both sides of (7)with the help of the boundary condition that no combustionoccurs (q = 0) when no air is present (i.e., z
f
= 0). We obtain:
1tz11tq
f
(8)If no fuel is present (z = 1) then p
fuel inlet
0 and q
. Weshift the time axis (t
t +
) and denote with
a
n
,
a
n+1
thequantities
a
(t) and
a
(t +
) respectively. Then, (8) gives:
,2,1n;
q111z
n1n
(9)Finally, if dA
f
/dt = 0 then (2), (3) and (4) give:
nn
zhq
(10)where
Mf 1inletfuelf
v pAYH
ζ
; (9) and (10) give:
n1n
zh111z
(11)Given
, (11) (‘return map’) links z
n
and z
n+1
. Hummingfreestate solves (11) and the ‘fixed point’ condition
n1n
zz
(12)We start with an userprovided value z
inlet
= 1–p
fuel inlet
/p
0
of zand investigate stability of the hummingfree state z
n+1
= z
n
=z
inlet
. Since 0
≤
z
n
≤
1, z
n+1
(1) = z
n+1
(0) = 0 and z
n+1
(z
n
,
) hasonly one maximum in z
n
, a necessary and sufficientcondition for stability at z
n
= z
inlet
reads [7]:
1z
ξ
,zz1n
inletn
(13)where the prime denotes derivation on z
n
at fixed
.According to (11), the L.H.S. of (13) increases withincreasing
. Then, a value
cr
of
exists such that theoperator ‘=’ appears in (13); the steady state is (un)stable for
≤
cr
(
>
cr
).As for the meaning of
, we observe that the definitions of
,M,
and
Y, together with the mass balance
02maxf f
vR
π
vA
in the unperturbed state of a burner withradius R
max
, give
01f
v
κ
v
and :
inlet321cr cr cr
zh
κ κ κ
11M;1M11M1
ξξ
(14)where
1
R
max2
A
f 1
,
2
(
–1)
m
fuel
H
(RT
inlet
)
1
, and
3
N
/(4
cr
). Accordingly,
is an increasing function of M.Moreover, in lean combustion z
n
is slightly less than 1 and
1n
z
< 0. Then, M
cr
is a strongly decreasing function of z
inlet
According to (13)(14), lack of humming requires M
≤
M
cr
.Violation of (13) triggers a limit cycle [7]. Here, a limitcycle is a stable oscillation bringing the system from a lower bound z
min
of z
n
to an upper bound z
max
, and vice versa (i.e.z
min
= z
1
= z
3
= …, z
max
= z
2
= z
4
= …). It correspondstherefore to a couple (z
min
, z
max
)(
) of stable fixed points of z
n+2
= z
n+2
(z
n
,
) = z
n+1
[z
n+1
(z
n
,
)]. We refer to the quantity
z
max
(
) – z
min
(
)
as to the ‘oscillation amplitude’ below.If dA
f
/dt
≠
0 then a term
O(d
0
) adds to the denominator of (11). Flame thinness ensures this is a small perturbation.Stability of limit cycles against such perturbations [7]ensures that the results obtained above still hold.In lean combustion, linear dependence of Q
c
on Y gives:Q
c1
/ n
fuel
= Q*
1
/ (n
air
+ n
fuel
) (15)Equation (3) gives:Q
c1
= (A
f0
v
f1
+ A
f1
v
f0
) H m
fuel
(RT
inlet
)
1
p
0
(1– z
inlet
)(16)Moreover, near
cr
we write:v
f 1
= (d v
f
/d z
n
)z
n1
v
f M
h’
z
max
– z
min
(17)Equations (2), (4), (14)(17) for M << 1 and
cr
give:
minmax0ncr *cr 0
zzzdhd12 p p
Ma
(18)where the subscript
0
denotes the derivatives computed at z
n
= z
inlet
,
=
cr
and
0f f 1*
vvlndAlnd1
. Inderiving (18) we have taken into account that if M << 1 then(14) and
01f
v
κ
v
imply
cr 320sMf
ξκ κ ξ
cv
.
Equation (18) is a link of humming halfamplitude
p andthe oscillation amplitude. Below, we are going to computethe latter as a function of
, i.e. of M.
Oscillation amplitude
Analytical properties of the return map
In the following we investigate (11) at
cr
, where
,zz1n
inletn
z
= – 1 +
(
) with
(
)
<< 1,
(
=
cr
) = 0,define s
z
n
– z
inlet
, and take
s
<< z
inlet
. When applied to
z
min
(
) and z
max
(
), (13) gives
(z
n+2
’)
zmin,
≤
1 and
(z
n+2
’)
zmax,
≤
1. Chain rule ensures that both z
inlet
, z
min
(
)and z
max
(
) solve the relationships:G = 0 ; G’
≤
0 ; G’’
≥
0 (19)where the function G(z
n
,
)
z
n+2
(z
n
,
) – z
n
is unambiguouslyknown once the return map is known. The oscillationamplitude is just the difference between stable nonzeroroots of G. We develop G(z
n
) = G(s) in a power series in s;the coefficients will depend on
. Since G(s = 0,
) = 0, G(s,
) has the form G(s,
) = s
H(s,
). In order to compute H,we recall that we are interested in
z
max
(
)
– z
min
(
)
. Then,we may take z
inlet
– z
min
(
) = z
max
(
)
– z
inlet
, i.e. if s
1
(
) is aroot of H then s
2
(
)
– s
1
(
) too is a root of H. Accordingly,we may write H(s,
) = H(–s,
); hence G(s,
) = –G(–s,
) isa power series which contains only odd powers of s. Finally,if oscillation is triggered (
=
cr
) at z
n
= z
inlet
(i.e. at s = 0),then at
=
cr
the value s = 0 satisfies the 3 conditions (19)with the operator ‘=’, i.e. it is an inflexion point of G. Then,G depends on 3 distinct parameters at least, say
a
(
),
b
(
)and
c
(
). We neglect higherorder terms near s = 0:G(s) =
a
(
)
s +
b
(
)
s
3
+
c
(
)
s
5
For simplicity, we drop the dependence on
below. Let uscompute
a
,
b
and
c
. The derivative of G at s = 0 is
a
andvanishes at the bifurcation
=
cr
. We invoke the chain ruleagain and again and obtain:
055033cr 021n
sG1201;sG61;
ξξξ
dzd
cba
where we neglect both d
b
/ds and d
c
/ds for simplicity. In particular,
a
(
>
cr
) > 0. Both
a
,
b
and
c
are known once thereturn map is given. Roots of G(s) are:
cabcbssscbss
4;2;0;2
254321
Depending on which real roots are stable,
z
max
– z
min
isequal either to s
1
–s
2
= 2s
1
or to s
4
–s
5
= 2s
4
or to s
3
= 0. In allcases, the only real root for
< 0 is the hummingfree states
3
= 0. We know that this is the case for
<
cr
. Selfconsistency requires therefore that
is an increasingfunction of
, hence of
a
. This is only possible for
c
< 0, sothat
=
b
2
+ 4 
c

a
(
). The sign of
b
affects the dependenceof
z
max
– z
min
on
–i.e., of
p on M.
b
< 0 (no hysteresis) vs.
b
> 0 (hysteresis)
If b < 0 then both s
1
and s
2
are never real. Both s
4
and s
5
arereal (and
≠
s
3
= 0) for
>
cr
. Root s
3
= 0 corresponds to asteady, hummingfree state z
n
= z
inlet
; it is stable for
<
cr
.Roots s
4
, s
5
correspond to a limit cycle bounded by z
max
=z
inlet
+ s
4
and z
min
= z
inlet
+ s
5
= z
inlet
– s
4
; they are stable for
>
cr
. Stability swap (bifurcation) occurs between s
3
and s
4
,s
5
as
crosses
cr
. Figure 1 is a bifurcation map, i.e. itdisplays all real roots as functions of
. Clearly, the value
cr
of
where humming is triggered as we raise
from below isthe same value where humming is suppressed as we lower
from above: no hysteresis is observed.
Figure 1
: Bifurcation map for b < 0 and
cr
= 1. Continuous(dotted) lines correspond to stable (unstable) roots.
If
b
> 0 then three cases are possible:I)
< 0: the only real root is s
3
= 0 (nohumming), and is stable.II)
0 <
<
b
2
: all roots are real. Roots s
1
, s
2
areunstable. Both s
3
, s
4
(limit cycle with amplitude2s
4
)
and s
5
(no humming) are stable.III)
b
2
<
: the only real roots are s
3
, s
4
and s
5
.Root s
3
is unstable; s
4
and s
5
are stable. Limitcycle is the only stable configuration.Figure 2 is the bifurcation map. Starting from a hummingfree, steady state at
<
cr
, we may raise
up to the bifurcation at
=
cr
. Correspondingly, we follow thehorizontal orange line s = s
3
= 0 up to
=
cr
(= 1 in Figure2). Then, the orange line becomes dotted, i.e. our steadystate loses stability. The only stable state available above
cr
is the couple of stable roots s
4
and s
5
, which the systemoscillates between in a limit cycle with amplitude 2s
4
. Whencoming back from right to left, the system lies in the stablehumming state as far as
remains
≥
1
(
= 0); further decrease of
makes the system to drop back to thehummingfree state. Since
1
<
cr
, hysteresis occurs.
Figure 2
: Bifurcation map for b > 0 and
cr
= 1. Continuous(dotted) lines correspond to stable (unstable) roots.
Figure 3 displays
p (
z
max
– z
min
) vs. M/M
cr
(anincreasing function of
); points A and B correspond to M
cr
M(
=
cr
) and to M
1
M(
(
= 0) ) (<M
cr
) respectively.
Figure 3
: Hysteresis (see text).
Stable values of
p are found following the green (violet)arrows as we raise (decrease) M/M
cr
. Green and violet pathsdo not coincide: hysteresis occurs. Near bifurcation
–
cr
M – M
cr
; as a function of M, the oscillation amplitude is:
1cr cr minmax
MMMM11zz
(20)(the sign + corresponds to hysteresis). Together, (18) and(20) give
p. According to (14), the lower z
inlet
, the lower M
cr
. According to (3), once z
inlet
is given the larger Q
c
T
inlet
/p
0
the larger M. According to (18) and (20),humming is more likely to occur in fuelpoor, hot mixture atlow working pressure in a burner with larger heat release.Finally, preliminary results suggest that hysteresis occurs(i.e.
b
> 0) at very low content of fuel.
A first application
In order to check validity of the model in a real burner,Ansaldo has been developing a thermoacoustic descriptionof the burner in the time domain with the help of COMSOLcommercial package. This description includes finiteMeffects. The source of the acoustic oscillation is described inequations (2) and (3). Stoichiometry at the flame depends onstoichiometry at the inlet according to equation (6). Thelinearized equation (1) is replaced by a system of linearized balance equations of mass, momentum and energy; CFD provides us with the unperturbed fields of temperature,
v
0
etc. Figure 4 displays an output, with the pressure field in anannular chamber at a given time.
Figure 4
: p
1
(
x
) at fixed time.
To date, preliminary results only are available. All the same,the spontaneous onset of an acoustic oscillation as Mexceeds a threshold is clearly visible in Figure 5, whichdisplays the transient behavior of p
1
as a function of time.Figure 6 shows sensitivity of
p against tiny variations of z
inlet
for M > M
c
.
Figure 5
: p
1
(t) at fixed position and different values of M.
Figure 6
: p
1
(t) at the inlet vs. time with M = 1.1· Mc. Green: z
inlet
=0.9471, blue: z
inlet
= 0.94705, red: z
inlet
= 0.9470001
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Combustion Instabilities In Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, And Modelling
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210
, AIAA(2005)
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1
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667
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Universality And Chaos
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Modeling And Control Of AcousticOscillations
, Proc. GT2005, ASME TurboExpo June 692005, Reno, USA (2005)