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A limit cycle for pressure oscillations in a premix burner - Conference AIA-DAGA 2013

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Any satisfactorily description of humming (= dangerous, combustion-driven acoustic oscillation in gas-turbine burners [1]) should include both acoustic and convective phenomena. In fact, the distance v0 ×  crossed by an element of fluid (with speed
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  A limit cycle for pressure oscillations in a premix burner A.   Di Vita 1 , F. Baccino 2 , E. Cosatto 3   1 Ansaldo Energia, PDE-ISV-TFC, 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, PDE-ISV-TFC, Via N. Lorenzi 1 16152 Genova, Italy, Email:ezio.cosatto@aen.ansaldo.it   The problem Any satisfactorily description of humming (= dangerous,combustion-driven acoustic oscillation in gas-turbine 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 air-fuel mixture (L  10 cm).Accordingly, no fully linear approach seems to besuccessful. Admittedly, humming-related 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 non-normality 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 non-linear –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 air-fuel 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 half-amplitude   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, geometry-dependent constant. Firstly, wediscuss the fixed-flame case A f  = const.; generalisation to thegeneric moving-flame case dA f  /dt ≠ 0 follows. Fixed flame vs. moving flame At the inlet, the fuel partial pressure p fuel-inlet 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 . Humming-freestate solves (11) and the ‘fixed point’ condition n1n zz   (12)We start with an user-provided value z inlet = 1–p fuel inlet /p 0 of zand investigate stability of the humming-free 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 half-amplitude   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 ’) z-min,   ≤ 1 and  (z n+2 ’) z-max,   ≤ 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 non-zeroroots 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 higher-order 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 humming-free states 3 = 0. We know that this is the case for   <  cr  . Self-consistency 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, humming-free 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 humming-free, 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 thehumming-free 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 M-MM-M11zz  (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 fuel-poor, 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 thermo-acoustic descriptionof the burner in the time domain with the help of COMSOLcommercial package. This description includes finite-Meffects. 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 References [1] Lieuwen, T. C., Yang, V. (Eds.): Combustion Instabilities In Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, And Modelling ,Progress In Astronautics And Aeronautics 210 , AIAA(2005)   [2] Nicoud, F., Wieczorek, K.: Intl. J. Spray and Combust.Dynamics 1 1 (2009), 67-112[3] Juniper, M. P.: J. Fluid Mech. 667 (2011) 272-308[4] Wieczorek, K., Sensiau, C., Polifke, W., Nicoud, F.:Phys. Fluids 23 (2011) 107103[5] Lieuwen, T.C.: J. Propulsion Power  18 1 (2002) 61-67[6] Pankiewicz, T, Sattelmayer, T.: J. Eng. Gas Turb. Power  125 (2003), 677-685[7] Feigenbaum, M.J.: Universal Behavior In Non-Linear Systems , Los Alamos Science 1 4-27 (1980), reprinted inCvitanovic, P. Universality And Chaos , Adam Hilger,Bristol (1984)[8] Dowling, A.:  Modeling And Control Of AcousticOscillations , Proc. GT2005, ASME TurboExpo June 6-92005, Reno, USA (2005)
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