 Open Access
 Total Downloads : 56
 Authors : Myunggon Yoon
 Paper ID : IJERTV6IS120059
 Volume & Issue : Volume 06, Issue 12 (December 2017)
 Published (First Online): 12122017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A ThermoAcoustic Model of a Gas Turbine Combustor Driven by a LoudSpeaker
MyungGon Yoon
Department of Precision Mechanical Eng. GangneungWonju National University Wonju 26403, Republic of Korea
AbstractWe develop a thermoacoustic model of a gas turbine combustor where acoustic perturbations are generated by an external loud speaker. Our analysis shows how the electromechanical dynamics of a speaker effects on the overall thermoacoustic dynamics of a combustor. A numerical example is included to illustrate our results.
KeywordsOnedimensional acoustic model, Acoustic Transfer Matrix

INTRODUCTION
Gas turbine combustor are prone to combustion in stability where the dynamics of combustion and that of thermoacoustic field have a positive feedback (synergy effect) with each other to result in a large magnitude of
speaker since the speaker serves as a new boundary condition, which leaves us an important question ; how the speaker dynamics will modify the thermoacoustics of combustor ?. This is a key motivation of our development presented in this paper.
Our thermoacoustic model for a combustor also al lows us to indirectly and experimentally validate the soundness of our acoustic model in the process of obtaining a flame transfer function.

ACOUSTIC MODEL

Speaker Model
A speaker can be seen as a RL (resistorinductance) circuit whose dynamics can be described as
pressure and velocity perturbations [1].
For a theoretical investigation of a possible occurrence
di
V = Ri + L + k
dt
d
b dt
(1)
of combustion instability it is essential to develop pre cise dynamic models of both combustor thermoacoustic and combustion process. The thermoacoustic part is commonly approximated with onedimensional acoustics and this approximation allows an analytical model with relatively ease.
In contrast, however, the dynamic properties of a com bustion process is so complicated and highly nonlinear that in many cases experimental approach are preferred. For an experimental identification of the combustion
where t denotes time, R and L are electrical resistance and inductance, i(t) is current, V (t) is a driving voltage, kb is the coefficient of the counterelectromotive force and denotes the position of speaker diaphragm (cone) which is assumed to have a simple onedimensional motion.
The mechanical force generated by the speaker coil is f = kfi for a constant kf and it drives a mechanical vibration system
f = kfi = mÂ¨ + b + k + Aepl(x, t)x=0
dynamics, a loud speaker is widely employed to gener ated a usercontrollable acoustic perturbations and thus
(t) = ul(x, t)
x=0
(2)
to obtain a frequency response of combustion flame in terms of heat rate perturbation with respect to a velocity perturbation, i.e., the socalled flame transfer function.
The use of a loud speaker however effects the thermo acoustic property of a combustor system without a
where x denotes the onedimensional coordinate of an acoustic element attached to the speaker such that x = 0 correspond to = 0, Ae denotes the effective area of the speaker cone, constants m, b, k are mechanical pa rameters of the speaker. In addition pl(x, t) and ul(x, t)
are the perturbations of pressure and velocity. See an illustration in Fig. 1.
Using Laplace transformation we rewrite (1)(2) as
'
Â± := x1/(c u) where c denotes the sound speed.
Â±
1 + R2 1 + R2
p (x1, t)
'(x1, t)
= 0. (8)
The boundary condition at x1 specified with a reflec tion coefficient R, the ratio of incident and reflected waves at x = x1, can be written as
(
\
V (s) = (Ls + R)I(s) + kbu (0, t)
kfI(s) =
ms +
k
+ b
u'(0, s) + Aep'(0, s)
(3)
l r ' l
s
In addition, a combination of (4) and (6) gives
and an elimination of the term I(s) in those two equal
ities gives
u
p
0
'(x1, t)
u
p
G G l (Px1)1 rp'(x1, t)l = G G V. (9)
' (0, s)
V (s) = 1/Gp 1/Gul rp'(0, s)l , := cu' (4)
kf
From this equation and (8), one can find the following relation between '(s) and V (s) ;
2
p'(x1, s) 1 1 + H(s)es
e
Gp(s) := (Ls + R)A
(5)
V (s) = 2 Â· 1 H(s)R es Gu(s)Gp(s) (10)
ckf s
G (s) :=
'(x1, s) 1 1 H(s)es
u
kbkfs + (Ls + R)(ms2 + bs + k)
V (s) = 2 Â· 1 H(s)R es Gu(s)Gp(s) (11)
2
and , u denote the mean density, velocity and c denotes the sound speed.

A Simple Duct Model
duct
Ae
u
m
(t)
p(x, t)
u(x, t)
speaker
k
R2
b
x = 0 x = x1
Fig. 1: Duct driven by a speaker
r l = P r l (6)
The wave propagation over the interval [0, x1] is given
p'(x1, t) x1 p'(0, t)
'(x1, t) 0 '(0, t)
1 re +s + e s e +s e sl
0
+
where = + + and H(s) is given in (15). The transfer function (16) will be used later.
Note that the pole of the above transfer function are composed of two parts ; poles of the electromechanical transfer functions GuGp in (5) and acoustic poles which are the roots of the next equation
1 H(s)R2es = 0. (12)
Let us suppose the speaker is removed and the left reflection coefficient of the duct at x = 0 is R1. In this case it is easy to show that the corresponding acoustic poles are given as roots of
1 R1R2es = 0. (13)
A comparison of this fact and (12) suggests that the duct with a speaker in Fig. 1 can be seen as a simple duct along with a frequencydependent reflection coefficient R1 = H(s).
Note that if
b Â» cAe, (14)
then we have H(s) 1.
Px1 =
2
e
s e
s e
+s + e s
, (7)
Lms3 + [(b c Ae) L + Rm] s2 + [Lk + (b c Ae) R + kbkf] s + Rk
H(s) := Lms3 + [(b + c A ) L + Rm] s2 + [Lk + (b + c A ) R + k k
(15)
] s + Rk
e e b f
e e b f
2kf s K(s) := Lms3 + [(b + cA )L + Rm] s2 + [kL + (b + cA )R + k k ] s + Rk
(16)

Thermoacoustic Combustor Model
Making use of the previous speaker model, here we develop a thermoacoustic model for a combustor as
The mass, energy and momentum relations across an area expansion and a flame give the following relation
M
(1 2)
1
illustrated in Fig. 2.
p
= 1 1M1
1
l p
x
combustor
speaker nozzle u2
1
+ 2M1
1
1
c1
x+
q (t) (23)
u1
f 1\
u1(x1, t) R
where q(t) denotes the combustion heat source in the
q(t)
2 unit of [Joule/m2] and two constant 1, 2 depend on
particular acoustic model of the area expansion at x1 [2], [3]. In this paper, following [2], we have
x = 0 x = x1
Fig. 2: Acoustic Model of Combustor
x = x2
1 :=
2( 1) + 1 2
2
, 2 :=
(24)
Acoustic waves propagate from x1+, just fter an area expansion which is the same as the location of a thin flame, to the outlet x3 as
where , denotes the ratio of sound speed and area before and after the area expansion or flame.
By multiplying the matrix in the left hand side of (22) to (23), one can obtain
x1
p = Px2 p
x2
(17)
0 =
P
x1+
Px2 := 1
2
2
2
2
, (18)
e + + e e + e l
x1
2
e + e
e + + e
1 M l
1 + R2 1 + R2
x2 x1
x1
2
2
2
2
2
(12) M
1
1
1
p
1
x1
2Â± := (x2 x1)/(c2 Â± u2) (19)
where c
denotes the sound speed after a flame.
+
P
1 + R2 1 + R2
x2 x1
c1
1
q (t). (25)
2M1 1 l
2
In addition, the right boundary condition at x2 can be written as
c1
This equation and (22) constitute two simultaneous
1 + R2 1 + R2
p
x2
= 0 (20)
equations between the waves (p , )x1 and two sources
V (s) and q (s).
In particular, regarding {V, q } as two driving inputs
A combination of (20) and (17) gives
2
2
x1
1 + R 1 + R Px2 p = 0. (21)
x1+
and u (x1) = (x1)/1c as a velocity output, the corresponding MISO (multiinput singleoutput) transfer
function representations can be written as
x1+
By replacing x1 in (9) with x1, just before an area expansion or flame location, we have
u (x1, s) = Gs(s)V (s) + Gf(s)q(s) (26)
u
p
0
u
p
G G (Px1)1 p = G G V (22)
x1
where the speaker transfer function Gs(s) and the heat
transfer function Gq(s) are given
x1
where {c, u, , , } in the previous section should be
G (s) := Ns(s) , G (s) := Nq(s) (27)
rewritten as {c1, u1, 1, 1, 1}.
s D(s) q
D(s)
= 1.38
= 6
u1 = 60
c1 = 345
1 = 1.177
2 = 0.0327
L1 = 0.6
L2 = 1.5
L = 0.02
R = 4
m = 0.01
b = 0.6
k = 500
Ae = A1
kb = 1
kf = 0.5
= 10
A1 = 0.005
A2 = 0.05
R2 = 1
where
Ns(s) := K(s)1(1 2R2e2s)e1s (28)
N (s) := 1 (1 H(s)e1s)
f c2 3
(1 + R e2s) (29)
TABLE I: Parameters for the example
4 2
D(s) := 1 6R2e2s
+ 7H(s)(1 5R2e2s)e1s, (30)
This resonance frequency clearly effects on the Bolde plots of the speaker transfer function Gs(s) in Fig. 6 and
i := + + (i = 1, 2), K(s) in (16) and
G (s) in Fig. 5.
i i
+ (2 1)M1
f
The second peak near 157 (Hz) in Fig. 6 and Fig. 5
1
1 := + 1 + (2
2 := + (2 1)M
(2 1)M1
1
)M1
seems to be an acoustic resonance frequency of the nozzle part (0, x1). In fact, the sudden area expansion and the flame at x1 result in an acoustic impedance jump and thus acoustic waves in either nozzle or combustor
3
:= 1 2M1
4
+ 1 + (2 1 )M1 := 1 + 2M1
1 2M1
are largely reflected at x1 [4] (p. 151). This gives rise to a standing wave inside the nozzle whose resonance frequency is given
Â· Â· Â· (34)2
(31)
5
:=
+ 1 (2 1
)M1
1 + 2k
fq := , k = 0, 1,
1
1 + (2 + 1 )M1
1 (2 + 1 )M1
and the smallest frequency fr 140 (Hz) with k = 0 is
1
6 := + 1 + (2 )M
1
close to the observed peak frequency.
1 + (2 + 1 )M1
7 := + 1 + (2 )M 0
Magnitude(dB)
1
1
It is a common situation where Â» 1 and M1 0. In this case we have i = 0 for all i = 1, Â· Â· Â· , 7 except 3 1/ and thus obtain simple representations ;
N (s) = K(s)(1 R e2s)e1s,
2
4
6
0 50 100 150 200 250 300 350
s 2
N (s) = 1 (1 H(s)e1s)(1 + R e2s), (32)
Freq (Hz)
400
f c2
2
Phase (deg)
300
D(s) = (1 + H(s)e1s)(1 R2e2s).

A Numerical Example
We computed various transfer function appeared in our developments with speaker and combustor parameters summarised in Table I.
Two Bode plots of the transfer function H(s) and
K(s) shown in Fig. 34 shows minimal and maximal peaks, respectively, which seems to come from the mechanical speaker resonance frequency
fs := 2
= 35.59 (Hz). (33)
m
1 I k
200
100
0
0 50 100 150 200 250 300 350
Freq (Hz)
Fig. 3: Bode plot of H(s)


CONCLUSION
We developed a thermoacoustic model of a gas tur
bine combustor under the condition that the nozzle inlet
Magnitude(dB)
0
50
100
150
100
Phase (deg)
0
100
200
0 50 100 150 200 250 300 350
Freq (Hz)
0 50 100 150 200 250 300 350
Freq (Hz)
Fig. 4: Bode plot of K(s)
100
Magnitude(dB)
150
200
250
300
0
Phase (deg)
100
200
300
0 50 100 150 200 250 300 350
Freq (Hz)
0 50 100 150 200 250 300 350
Freq (Hz)
Fig. 6: Bode plot of Gq(s)
Magnitude(dB)
0
50
100
150
0 50 10 150 200 250 300 350
Freq (Hz)
200
Phase (deg)
0
200
REFERENCES

T. Lieuwen and V. Yang, Combustion instabilities in gas turbine engines: operational experience, fundamental mechanisms and modeling, ser. Progress in astronautics and aeronautics. Amer ican Institute of Aeronautics and Astronautics, 2005.

M. L. Munjal, Acoustics of Ducts and Mufflers, 2nd ed. Wiley, 2014.

J. Li, D. Yang, C. Luzzato, and A. S. Morgans, Open source combustion instability low order simulator (osciloslong), Im perial College, Tech. Rep., 2017.

L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders,
Fundamentals of Acoustics, 4th ed. John Wiley & Sons, 2000.
400
0 50 100 150 200 250 300 350
Freq (Hz)
Fig. 5: Bode plot of Gs(s)
is acoustically driven by a loud speaker. Our model was given as a transfer function representation in which an electrical voltage for speaker and a heat perturbation are two inputs and the velocity perturbation at the flame location is a single output. A numerical case study sug gested a possibility that the thermoacoustic properties of the combustor can be effected by the resonance dynamics of the speaker. It is left as a further work to validate our model with realworld combustor systems.