A Thermo-Acoustic Model of a Gas Turbine Combustor Driven by a Loud-Speaker

A Thermo-Acoustic Model of a Gas Turbine Combustor Driven by a Loud-Speaker

Myung-Gon Yoon

Department of Precision Mechanical Eng. Gangneung-Wonju National University Wonju 26403, Republic of Korea

AbstractWe develop a thermo-acoustic model of a gas turbine combustor where acoustic perturbations are generated by an external loud speaker. Our analysis shows how the electro-mechanical dynamics of a speaker effects on the overall thermo-acoustic dynamics of a combustor. A numerical example is included to illustrate our results.

KeywordsOne-dimensional acoustic model, Acoustic Transfer Matrix

1. INTRODUCTION

   Gas turbine combustor are prone to combustion in- stability where the dynamics of combustion and that of thermo-acoustic 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 thermo-acoustics of combustor ?. This is a key motivation of our development presented in this paper.

   Our thermo-acoustic 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.

2. ACOUSTIC MODEL

   1. Speaker Model

      A speaker can be seen as a RL (resistor-inductance) 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 thermo-acoustic and combustion process. The thermo-acoustic part is commonly approximated with one-dimensional 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 counter-electromotive force and denotes the position of speaker diaphragm (cone) which is assumed to have a simple one-dimensional 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 = m¨ + b + k + Aepl(x, t)|x=0

      dynamics, a loud speaker is widely employed to gener- ated a user-controllable 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 so-called 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 one-dimensional 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

      '

      ± := x1/(c u) where c denotes the sound speed.

      ±

      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 · 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 · 1 H(s)R es Gu(s)Gp(s) (11)

      2

      and , u denote the mean density, velocity and c denotes the sound speed.

   2. 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 electro-mechanical 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 frequency-dependent reflection coefficient R1 = H(s).

      Note that if

      b » 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)

   3. Thermo-acoustic Combustor Model

      Making use of the previous speaker model, here we develop a thermo-acoustic 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

      2± := (x2 x1)/(c2 ± 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 (multi-input single-output) 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

      · · · (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 » 1 and M1 0. In this case we have i = 0 for all i = 1, · · · , 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).

   4. 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. 3-4 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)

3. CONCLUSION

We developed a thermo-acoustic 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

1. 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.

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

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

4. 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 thermo-acoustic 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 real-world combustor systems.