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**Modeling and Control of Combustion Instability using Fuel Injection**

Jean-Pierre Hathout*, Anuradha Annaswamy, and Ahmed Ghoniem Department of Mechanical Engineering MIT NATO AVT Symposium May 8-11, 2000 * Dr. Hathout joined the Robert Bosch Corporation Research and Technology Center in Pittsburgh, PA, since July

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**Continuous Combustion Processes and Thermoacoustic Instability**

Power Generation Boilers Burners Gas turbines Propulsion Commercial: Environmentally friendly Military: high power Rockets Shuttle main engine Combustion instability in the form of Screech can be seen in the heat-release signature F-22 Raptor (courtesy of UTC)

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**Overview Model Fuel-injector dynamics Model Control**

Heat release Acoustics Coupling dynamics; combustion instability due to Area fluctuations due to velocity fluctuations Mixture inhomogeneity Fuel-injector dynamics Model Proportional actuation Two-Position actuation Control No delay control: LQG-LTR Time-delay control “Posi-cast” control Impact of injector dynamics Bandwidth and authority Nonlinearities

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**Modeling 1. Organ-pipe combustor (MIT, 1kW) 2. Dump combustor**

(UTRC, 100kW) Bulk mode Longitudinal modes Acoustics Heat Release Coupling mechanism Mixture inhmogeneity Heat-release kinematics

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**Heat Release Model: Flame Kinematics**

Linearized PDE Model: Flame surface : Propagation delay For small , and conical flames reduces to: r

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**Acoustics: Longitudinal Modes in an Organ-pipe Combustor**

MIT combustor Assumptions: 1-D flow, Inviscid Perfect gas, Linear model (perturbations around a constant mean) No velocity and heat release. Using Conservation Equations: PDE Model: p,u Flame ODE Model:

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**Organ-pipe Combustor (MIT): Coupling caused by Velocity Fluctuations**

Acoustics Heat Release b c c: fn of velocity mode shape b: fn of pressure mode shape 0.2 0.4 0.6 0.8 1 -1 -0.5 0.5 b c Three-quarter-wave mode Quarter-wave mode bc>0: system becomes unstable Summary of the model predictions: (2-modes) MIT Model prediction: Experimental (Lang et al.’87): Growth rate(1/s) Frequency (Hz)

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**Acoustics Model: Dump Combustor with a Large Bulk**

Flame surface Reactants inlet Products outlet Assumptions: 1-D flow, Incompressible in the ducts, Volume of cavity>>Volume of ducts, Inviscid Perfect gas, Linear model (perturbations around a constant mean) Mass and energy conservation in the cavity: Mass and momentum conservation in the jth duct: Substitute (2) in (1): (assume ducts open to atmosphere; pressure distribution is negligible) Where the effective Helmholtz frequency is

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**UTRC Combustor: Coupling caused by Inhomogeneity Dynamics**

Acoustic velocity perturbation in cavity is small, negligible effect on area perturbation. Only perturbations in the equivalence ratio are important Instantaneous at fuel nozzle due to perturbations in the air flow rate: Recall: effect of on is static, but effect of on is delayed! Can a delay trigger the instability? Fuel Air Delay:

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**UTRC Combustor: Combustion Instability due to Inhomogeneity Dynamics**

United Technologies combustor: Instability due to Model prediction: Pressure (Pa) Time (s) Unstable: when 0.62 UTRC instability 1 2 3 Unstable bands Stability bands identified in experiments (Putnam 1971, Richards 1995, Zinn 1998)

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**Summary of Instability Models**

General Model: When u’ fluctuations are dominant Time-delay instability Phase-lag instability When f’ fluctuations are dominant 1 2 3 Unstable bands

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**Model Predictions: f’ oscillations**

(Cohen et al., 1998) 1 2 3 Unstable bands UTRC (Cohen et al., 1998) Unstable: 0.62 UTRC instability when Heat release Bulk Mode Feed system impedance f’ Time-delay Experiments: - Mongia et. al,1997 - Richards and Yip, 1997 - Lieuwen and Zinn et al., 1998 (Lieuwen and Zinn et al., 1998) (Richards and Yip, 1997), “--” (Similar dynamics also in rockets, Crocco 1960, Tsien 1962)

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**Model Predictions: u’ oscillations**

Longitudinal mode Phase-lag instability: - MIT combustor - Poinsot et. al, 1989 - Gulati and Mani, 1992 - Sivasegaram and Whitelaw, 1992 - Seume et. al, (Siemens), 1997 Time-delay instability: - Santavicca et. al, 1998 - Richards, 1999 u’ Heat release Impedance MIT Model prediction: Experimental (Poinsot ): Growth rate(1/s) Frequency (Hz) Two-modes simulation Frequency Gain Phase Agrees with Experiments by Bloxsidge et al., 1987

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**Overview Model Fuel-injector dynamics Model Conrol**

Heat release Acoustics Coupling dynamics; combustion instability due to Area fluctuations due to velocity fluctuations Mixture inhomogeneity Fuel-injector dynamics Model Proportional actuation Two-Position actuation Conrol No delay control: LQG-LTR Time-delay control “Posi-cast” control Impact of injector dynamics Bandwidth and authority Nonlinearities

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**Fuel-Injector Dynamics Proportional Injection**

Magnetic coil armature Electro-magnetic and mechanical components dynamics: x E spring poppet Fluid dynamics - Fuel inlet choked: -

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**Fuel-injector Dynamics Two-position (on-off) injection**

Dynamics: Same as proportional + effect of physical stops (saturation) + Dead-zone S E(s) Driver gain + 1 s on off Dead-zone Hysteresis On: Off:

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**Two-position (on-off) injection: Velocity Response**

model experiment 100 Hz, 50% duty cycle model experiment 50 Hz, 50% duty cycle model experiment 50-Hz sweep model experiment 100-Hz sweep

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**Overview Model Fuel-injector dynamics Model Conrol**

Heat release Acoustics Coupling dynamics; combustion instability due to Area fluctuations due to velocity fluctuations Mixture inhomogeneity Fuel-injector dynamics Model Proportional actuation Two-Position actuation Conrol No delay control: LQG-LTR Time-delay control “Posi-cast” control Impact of injector dynamics Bandwidth and authority Nonlinearities

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**Using Pulsed-fuel Injection (on flame)**

Model: 2 Acoustics modes, and flame dynamics Fuel Injector: - Proportional - 200 Hz bandwidth - 1st order dynamics 5th order model Controller: LQG/LTR (5th order) f LQG/LTR Control on (f’) Equivalence ratio f’ Pressure p’ ,(Pa)

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**Using Pulsed-fuel Injection (on flame)**

Model: 2 Acoustics modes, and flame dynamics Fuel Injector: - Two-position (on-off) - 200 Hz bandwidth - 1st order dynamics 5th order model Controller: LQG/LTR (5th order) f LQG/LTR Equivalence ratio f Pressure p’ (Pa) Time (ms.) Control on (f’)

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**Time-delay Control (injection at main fuel supply)**

Secondary fuel Primary fuel UTRC combustor Idea: cancel the perturbations in the main fuel causing the instability, stability depends on natural damping in the combustor. Choose control: Experimental results (UTRC, Cohen et al.’98): “c” Stable and unstable zones, model predictions unstable stable Pressure (Pa) Control input, fc* Time (s)

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**Pole-Placement Control for a Combustor with a Delayed Control Input**

Controller structure: S p’ f’ Closed-loop: Stabilize! Cancel! Stable synthesis (Manitius & Olbrot’79, Ichikawa’85) Robust (Niculescu & Annaswamy, ACC’99) Amenable to adaptation with uncertainties (Niculescu & Annaswamy, ACC’99) Validation in turbulent combustors (Evesque, Annaswamy & Dowling, NATO Symposium’00) Properties:

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**Simulation with Time-delay Compensator Control**

MIT combustor model: ti ~50tac. (mean velocity <<) Pressure (Pa) Time (msec) Control on Control input, fc* Time (msec)

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**Overview Model Fuel-injector dynamics Model Conrol**

Heat release Acoustics Coupling dynamics; combustion instability due to Area fluctuations due to velocity fluctuations Mixture inhomogeneity Fuel-injector dynamics Model Proportional actuation Two-Position actuation Conrol No delay control: LQG-LTR Time-delay control “Posi-cast” control Impact of injector dynamics Bandwidth and authority Nonlinearities

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**Actuator Limitations (Sec. Injector)**

f p’ (Pa) Time (ms.) Control on (f’) Results similar to observations in Yu (1997) Time (ms.) p’ (Pa) f Higher authority, sec. Fuel flow rate Faster settling time Time (ms.) p’ (Pa) f Lower bandwidth Unsuccessful control

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**Impact of Nonlinearities in the Actuator**

(PSP)23-26 Impact of Nonlinearities in the Actuator Acoustics Combustor dynamics Controlled (stable) limit cycle Heat release f(.) nonlinearity u’ pressure Unstable limit cycle G Control Asymptotic stability Actuator dynamics % secondary fuel Saturated/on-off injectors: limited control authority Stability (asymptotic, or stable limit-cycle) depends on control authority Stable solutions depend on Initial conditions, define an unstable limit-cycle In agreement with K. Yu 1997. Unstable limit-cycle Open-loop Stable limit-cycle

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**Summary Reduced-order models for combustion instability**

Heat release Acoustics Coupling dynamics; combustion instability due to Area fluctuations due to velocity fluctuations Mixture inhomogeneity Model-based control Optimal Accommodates large time-delays Injection dynamics Bandwidth and authority limiations Nonlinearities

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**Visit us at http://centaur.mit.edu/rgd**

Current Work Open-loop subharmonic control using fuel injection Time(sec) Normalized pressure, h Prasanth,Annaswamy, Hathout and Ghoniem, 2000 Richards et al., 1999 Extend models to turbulent combustion System ID Models Visit us at for further details

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