Sensitivity analysis of combustion instability Position Lecture

Transcription

1 Department of Engineering Sensitivity analysis of combustion instability Position Lecture Matthew Juniper, Hans Yu, Nicholas Jamieson, José Aguilar

2 If more heat is injected at times of high pressure and less heat at times of low pressure, then heat is converted into wor pressure 2 Z q in wor out: Z v v 1 p 23 dv 1 3 wor in: Z v1 p 1 dv q out v 1 Z v v volume

3 A simple thermoacoustic model contains the acoustic momentum and energy equations, with heat release applied at a point in space Diagram of the Rije tube air flow hot wire or flame x m Non-dimensional governing equations acoustic momentum acoustic @x dilatation due to @x ( 1)q D(x x f )

4 In the simplest possible model, the heat release is linearly proportional to the velocity at a measurement point, with a time delay. Diagram of the Rije tube air flow hot wire or flame x m Non-dimensional governing equations acoustic momentum acoustic @x dilatation due to @x ( 1)q D(x x f ) n-tau heat release model q(t) nu(x m,t ) delay, and x is the

5 The coefficients of the heat release model are found from models, experiments, or numerical simulations. The behaviour is very sensitive to errors in these coefficients. Diagram of the Rije tube air flow hot wire of flame x m Non-dimensional governing equations acoustic momentum acoustic energy acoustics n-tau heat @x dilatation due to @x ( 1)q D(x x f ) q(t) nu(x m,t ) delay, and x is the n and tau are found from: models experiments numerical simulations

6 The thermoacoustic mechanism is simple and can be easily modelled

7 To explain why this thermoacoustic model is so sensitive, we reduce it to a single equation for the eigenvalue, s, by projecting onto the natural acoustic modes. Diagram of the Rije tube air flow hot wire or flame x m Non-dimensional governing equations; acoustic momentum acoustic energy acoustics n-tau heat @x dilatation due to @x ( 1)q D(x x f ) q(t) nu(x m,t ) delay, and x is the single mode Galerin model project onto an acoustic mode, 1 N u(x, t) u (t) cos( x) p(x, t) p (t)sin( x) perform a Laplace transform u (t) U e st p (t) P e st s 2 + n e s +( ) 2

8 Even for small heat release, n, we see that small changes in time u(x, t) can cause u (t)t) cos( x) (1) u(x, uunstable (t) cos( x) delay, the most mode to switch. u(x, t) tau, u (t) cos( x) (1) p(x, t) st u(1) (t) U est Laplace transform growth rate sr /n (si π)/n (si π)/n frequency sr /n xm (1) st st p(x, t) p(x, p (t)t)sin( x) (2) p (t) P e p (t) P e p (t) sin( x) (2) p(x, t) up(t) (t)cos( x) sin( x) (2) (x, t) (t) cos( x) (2)L s22 + n e s (x, t) (1) st s + ( )22 (x, u (t) cos( x) (1) st (t) t) stu e (3) L s + n e u (t) U e (3) + ( ) U e (3) (x, t) (t)sin( x) sin( x) (3) heat release, n (x, t) P (t) sin( x) (2) st pp(t) ss (x, t) (2) Single mode Galerin small stmodel; analysis in the limit of (t) (4) st e p (t) P e (4) Pst est (4) st U e (4) 2 e (3)Galerin travelling wave model model with N 4 n UUs (3) n e+22( ) (5) s most unstable (5) ( ) s + n e + ( ) (5) st st 2 st ss P e (5)2 3 P e (4) 1 Pe (4) s s + s (6) 1 ssmall s1release (6) s s + s1 (6) 2 s1 +heat 22s impose 2 ( ) (6) s1 3 ( ) (5) (7) ) n! n (5) n -.2! n (7) n! n (7)Real(s1 ) -.2 s s + s (7) Real(s 1 s s + s (6) s 4 6 (6) ) ss + s 12 1 s ± i (8) ± i (8) s (8) τ ± i τ ss1 n! n (8) s n! n (7) 1 ns! n (7) s s /2s (9) )e (n at/2s find solutions ε(n and ε s )e (9) 1 s1 (n /2s )e (9) s ± i (9) s) ± i ± i (8) (1) Real(s sreal(s) (8) /2 ) (n /2 ) sin( ) 1) al(s (n sin( ) (1) Real(s 1) 1 s Real(s ) (n /2 ) sin( ) (1) 1 sss (n /2sns)e )e (1) /2s n s1 (n (n /2s )e (9) sss111 (9) n s s1 -.2 e 2s e (11) (11) -.2 s e (11) sin( al(s) (n /2 ) ) (11) 1 2s l(s ) (n /2 ) sin( ) (1) al(s ) (n /2 ) sin( ) (1) 11 2s n n τ nnn sin( Real(s) ) n (12) τ time delay al(s ) ) (12) sss sin( s e (12) 1 ss11 2 ee2 1 ) (11) (11) Real(s sin( ) (12) 2s extreme sensitivity to time delay, tau 2s 2s 2 (13) n (13) to geometric parameters strong sensitivity nn sin( ) al(s) (13) (13) l(s11)) 2 sin( (12) al(s sin( )) (12) 2 2 (14) (13) (13) 1

9 The thermoacoustic mechanism is simple and can be easily modelled But thermoacoustic models are very sensitive to parameters

10 For the F1 engine of the Saturn V, thermoacoustic instability was eliminated with baffles on the injector plate. But finding this design required 2 full scale tests. ~ 4"

11 Despair: Practical thermo-acoustic systems are extremely sensitive to geometric parameters and to the heat release time delay, tau Some of these parameters, e.g. tau, are difficult to model or simulate accurately.

12 Despair: Practical thermo-acoustic systems are extremely sensitive to geometric parameters and to the heat release time delay, tau Some of these parameters, e.g. tau, are difficult to model or simulate accurately. Hope: Usually (due to damping) only a few eigenvalues are unstable Many parameters can be changed (including flame parameters) To find stable operating points, linear stability analysis is sufficient Experience shows that most systems can be stabilized

13 We model a thermoacoustic system as an acoustic networ and exploit the features described earlier to predict how to stabilize it: Thermoacoustic model (flame + acoustics)

14 A linear analysis is sufficient. Thermoacoustic model (flame + acoustics) Linear modes of the model Growth rate Frequency

15 A linear analysis is sufficient. Only a few modes are unstable Thermoacoustic model (flame + acoustics) Linear modes of the model Growth rate unstable stable Frequency

16 A linear analysis is sufficient. Only a few modes are unstable. Many parameters can be changed. Thermoacoustic model (flame + acoustics) Linear modes of the model Growth rate unstable stable Frequency

17 The 'brute force' approach would be to change each parameter in turn and examine its influence on the eigenvalues. Thermoacoustic model (flame + acoustics) Linear modes of the model Growth rate unstable stable Frequency

18 But with a single adjoint calculation, one obtains the sensitivity of a single eigenvalue to every parameter. Thermoacoustic model (flame + acoustics) Linear modes of the model Growth rate unstable stable Frequency

19 Conceivably, one could use this gradient information to stabilize all eigenvalues at a reasonable cost Thermoacoustic model (flame + acoustics) Linear modes of the model Growth rate unstable stable Frequency

20 We have applied adjoint-based sensitivity analysis in thermoacoustics to stabilize a model of a practical burner.

21 We want to use the model in an optimization algorithm. If the growth rate and frequency are sensitive to the model parameters, their gradients will be severely prone to error.

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23 The thermoacoustic mechanism is simple and can be easily modelled But thermoacoustic models are very sensitive to parameters This sensitivity can be exploited to stabilize a model

24 We want to use the model in an optimization algorithm. If the growth rate and frequency are sensitive to the model parameters, their gradients will be severely prone to error. Model growth/decay rate frequency d(growth rate) / d(parameter) d(frequency) / d(parameter) feed into optimization algorithm

25 The thermoacoustic mechanism is simple and can be easily modelled But thermoacoustic models are very sensitive to parameters This sensitivity can be exploited to stabilize a model But models need to be accurate and the sensitivity to parameters introduces large systematic error

26 Ideally, therefore, we would measure growth/decay rates, frequencies, and their sensitivity to experimental parameters. Model growth/decay rate frequency d(growth rate) / d(parameter) d(frequency) / d(parameter) feed into optimization algorithm Update model Perform experiment growth/decay rate frequency d(growth rate) / d(parameter) d(frequency) / d(parameter) Experiment

27 We carefully characterize a laboratory-scale thermoacoustic system heat release q at mesh side view of heater Jamieson, N. P., Rigas, G., and Juniper, M. P.

28 We carefully characterize a laboratory-scale thermoacoustic system heat transfer coefficients, h pressure loss coefficients, heat release q at mesh side view of heater Jamieson, N. P., Rigas, G., and Juniper, M. P.

29 We carefully characterize a laboratory-scale thermoacoustic system automated iris to change boundary condition top view of iris heat transfer coefficients, h Stepper motor 6 mm pressure loss coefficients, heat release q at mesh side view of heater Jamieson, N. P., Rigas, G., and Juniper, M. P.

30 We carefully characterize a laboratory-scale thermoacoustic system automated iris to change boundary condition top view of iris heat transfer coefficients, h Stepper motor thermocouples 6 mm pressure loss coefficients, heat release q at mesh side view of heater Jamieson, N. P., Rigas, G., and Juniper, M. P.

31 We use nonlinear regression to find the values of heat transfer coefficients and pressure loss coefficients that best fit the experimental data Exit temperature (K) Entrance speed of air (m/s) Jamieson, N. P., Rigas, G., and Juniper, M. P.

32 Having characterized the base state, we measure growth/decay rates and frequencies of thermoacoustic oscillations in order to infer n and tau n n q heat release at mesh q(t) nu(x m,t ) delay, and x is the two microphones loudspeaer for active control or pinging.2 d wire U inlet Jamieson, N. P., Rigas, G., and Juniper, M. P.

33 We use a linear fit to obtain the linear growth rates between pre-determined thresholds. Typically (depending on the growth/decay rate) there are around 5 cycles in the linear regime. Growth rate Decay rate Threshold Threshold ~ 5 cycles active control off log(amplitude) (Pa) -5-1 Decay rate s!1 Fitting acoustic ping Upper limit Lower limit Time (s) Jamieson, N. P., Rigas, G., and Juniper, M. P.

34 With nonlinear regression on many thousand measurements, we infer models for n and tau. n 13.8q Frequencies as function of heater power d wire U inlet Growth / decay rates as fn of heater power Jamieson, N. P., Rigas, G., and Juniper, M. P.

35 The thermoacoustic mechanism is simple and can be easily modelled But thermoacoustic models are very sensitive to parameters This sensitivity can be exploited to stabilize a model But models need to be accurate and the sensitivity to parameters introduces large systematic error One solution is to infer parameters from experiments

36 We then use these to predict the behaviour of a new experiment hot wire mesh on automated traverse heat release at mesh, q two microphones loudspeaer for active control or pinging Jamieson, N. P., Rigas, G., and Juniper, M. P.

37 We operate in the stable regime and obtain a decay rate every 6 seconds. We obtain 15 datapoints at each operating point and step to a new operating point automatically. Measured decay rates as the secondary heater power, P2, changes -1-2 Decay rate P 2 P Shift in decay rate -3-2 <r(s!1 ) -4-5 /<r(s!1 ) Experiment number Experiment number Jamieson, N. P., Rigas, G., and Juniper, M. P.

38 This provides growth/decay rates, frequencies, and their sensitivities with respect to the model parameters. Shift in decay rate and frequency as the secondary heater power changes 5 Shift in decay rate 4 Shift in frequency /<r (s!1 ) P 2 (W) /<i (s!1 ) P 2 (W)

39 We characterize the model with a single heater, add a secondary heater, and compare predicted and measured growth rate sensitivities to the secondary heater power. This forces us to improve our model d(growth rate) / d(secondary heater power) Experiments Model 216 Model.1.1 /<r/p2 (W!1 s!1 ) /<rp2 (s!1 W!1 ) x c L x c L Jamieson, N. P., Rigas, G., and Juniper, M. P.

40 The thermoacoustic mechanism is simple and can be easily modelled But thermoacoustic models are very sensitive to parameters This sensitivity can be exploited to stabilize a model But models need to be accurate and the sensitivity to parameters introduces large systematic error One solution is to infer parameters from experiments This leads to accurate models, which can predict stabilization strategies