(PSP) 23-1 Modeling and Control of Combustion Instability using Fuel Injection Jean-Pierre Hathout...

(PSP) 23-1

Modeling and Control of Combustion Instability using Fuel Injection

Jean-Pierre Hathout*, Anuradha Annaswamy, and Ahmed Ghoniem

Department of Mechanical Engineering

MIT

NATO AVT SymposiumMay 8-11, 2000

* Dr. Hathout joined the Robert Bosch Corporation Research and Technology Center in Pittsburgh, PA, since July 2000. Email: jean-pierre.hathout@rtc.bosch.com

(PSP) 23-2

Continuous Combustion Processesand Thermoacoustic Instability

Power Generation• Boilers• Burners • Gas turbines

Propulsion• Commercial: Environmentally friendly • Military: high power • Rockets• Shuttle main engine

Combustion instability inthe form of Screech can be seen in the heat-release signature

F-22 Raptor

(courtesy of UTC)

(PSP) 23-3Overview• Model

– 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

(PSP) 23-4

Modeling

Acoustics

HeatRelease

p

uCoupling

mechanism

q

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

Longitudinal modes Bulk mode

Heat-release kinematics Mixture inhmogeneity

(UTRC, 100kW)

(PSP) 23-5

Heat Release Model: Flame KinematicsKinematics:

12

rS

rvu

t u

drr

KqR

0

2

1

Linearized PDE Model:

t

i

t

i

fiifii

dtdutu

ddttdtutududQ

00

43210

)()( ,)()(

))()(())()((

uf SR /

r

• For small , and conical flames reduces to:

fpfqf u

uuAQQ

Flame surface

u

uS: Propagation delay

f

(PSP) 23-6

Acoustics: Longitudinal Modes in an Organ-pipe 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:

n

iipp1

ODE Model:fiiii qb 2

t

q

x

pc

t

p f

12

22

2

2

p,u

fq

Flame

MIT combustor

(PSP) 23-7Organ-pipe Combustor (MIT): Coupling caused by Velocity Fluctuations

n

iipp1

fiiii qb 2

iif

fffffff

cu

ugqbq

~

~Acoustics

HeatRelease

p

u

q

b

c

Summary of the model predictions: (2-modes)

MIT Model prediction: 115.8 620.3Experimental (Lang et al.’87): 113 630

Growthrate(1/s)

Frequency (Hz)

c: fn of velocity mode shapeb: fn of pressure mode shape

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

bb

cc

Three-quarter-wave modeQuarter-wave mode

bc>0: system becomes unstable

(PSP) 23-8Acoustics Model: Dump Combustor with a Large

Bulk• 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):

(1) )1(1 22

feeii QmcmcVdt

pd

(2) ),(j

jjj

L

ptL

x

pA

dt

md

(assume ducts open to atmosphere; pressure distribution is negligible)

dt

Qd

Vp

dt

pd

dt

pd f

)1(

)2( 22

2

VL

Ac

VL

Ac

e

ee

i

ii22

Where the effective Helmholtz frequency is

Flame surface

Reactants inlet

Productsoutlet

V

eL

iL

im iA

eA

em

(PSP) 23-9

UTRC Combustor: Coupling caused by Inhomogeneity Dynamics

• Acoustic velocity perturbation in cavity is small, negligible effect onarea 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?

fQ

uuuu

/1/1

i

ss u

L

Delay:

scomb

combsn

iui

inf LS

AtpQ

),(

iu

sL

Fuel

Air

fQ

(PSP) 23-10

ac

s

0 1 2 3

Unstable bands

UTRC Combustor: Combustion Instability due to Inhomogeneity Dynamics

• United Technologies combustor:• Instability due to

•Model prediction:

0)( 22

2

ptpdt

pds

0.62 UTRC instability

0 0.005 0.01 0.015 0.02 0.025 0.03-1000

-500

0

500

1000

1500

Pre

ssur

e (P

a)

Time (s)...4,2,0for 2

2

2

1

n

nn

ac

s

Unstable:

/2ac102

when

112

e

o

T

T

•Stability bands identified in experiments (Putnam 1971, Richards 1995, Zinn 1998)

(PSP) 23-11

Summary of Instability Models

General Model:

dt

dddtudud

dt

d

dt

d ft

t f

321

22

2

)(2

)( sf t

ac

fs

or

0 1 2 3Unstable bands

When ’ fluctuations are dominant

0)(2 22

2

2

stddt

d

dt

d

When u’ fluctuations are dominant

0)()()2( 222

12

2

ftdddt

dd

dt

d

Time-delay instabilityPhase-lag instability

(PSP) 23-12Model Predictions: ’ oscillations

(Lieuwen and Zinn et al., 1998)(Richards and Yip, 1997), “--”

• Experiments:

- Mongia et. al,1997- Richards and Yip, 1997- Lieuwen and Zinn et al., 1998

(Cohen et al., 1998)ac

s

0 1 2 3Unstable bands

• UTRC (Cohen et al., 1998)

...4,2,0for 2

2

2

1

n

nn

ac

s

Unstable:

/2ac

0.62 UTRC instability

,102

when and,112

e

o

T

T

Heat release

Bulk Mode

Feed system impedance

’

Time-delay

(Similar dynamics also in rockets, Crocco 1960,Tsien 1962)

(PSP) 23-13

Heat release

Longitudinal mode

Impedance

u’

MIT Model prediction: 115.8 620.3Experimental (Poinsot 1989 ): 113 630

Growthrate(1/s)

Frequency (Hz)

Frequency

Gai

nP

hase

Agrees with Experiments by Bloxsidge et al., 1987

Model Predictions: u’ oscillations

Two-modes simulation

• 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

(PSP) 23-14Overview• Model

– 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

(PSP) 23-15Fuel-Injector DynamicsProportional Injection

BliFFkxdt

dxb

dt

xdm

dt

dxBlVV

dt

diLiRE

mm

,

,

2

2

• Electro-magnetic and mechanical components dynamics:

• Fluid dynamics

pLp

pv

dt

vdifluid

cfluid

2 ,2

xkAAvvAm of ,

- Fuel inlet choked:-

0vfuels) gaseous(for /1, ,1 3mkgpppmL coi

1/))(1()(

)(222

s

k

RslBkbsmss

k

sE

sm

m

v

e

vf

kRlBbm /)/( 22

armature Magneticcoil

spring

poppet

fm

x

op

cp

E

RLe /

fluid

(PSP) 23-16Fuel-injector DynamicsTwo-position (on-off) injection

• Dynamics: Same as proportional + effect of physical stops (saturation) + Dead-zone

E(s)

Driver gain

+ 1s

m1

on

offDead-zone

vk)(sm f

kRlBbonm /)/(| 22• Hysteresis On:

Off: kboffm /|

(PSP) 23-17

model

experiment

100 Hz, 50% duty cycle

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

model

experiment

100-Hz sweep

model

experiment

50 Hz, 50% duty cycle

model

experiment

50-Hz sweep

(PSP) 23-18Overview• Model

– 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

(PSP) 23-19

Model:• 2 Acoustics modes, and flame dynamics• Fuel Injector: - Proportional

- 200 Hz bandwidth - 1st order dynamics

• 5th order modelController: LQG/LTR (5th order)

Using Pulsed-fuel Injection (on flame)

LQG/LTR

0 10 20 30 40 50 60 70 80 90 100-0.4

-0.2

0

0.2

0.4

0 10 20 30 40 50 60 70 80 90 100-100

-50

0

50

100

Control on (’)

Equ

ival

ence

rat

io

’P

ress

ure

p’ ,(

Pa)

(PSP) 23-20

Using Pulsed-fuel Injection (on flame)

0 50 100 150 200 250 300

0.7

0.72

0.74

0.76

0.78

0.8

0 50 100 150 200 250 300-100

-50

0

50

100

Equ

ival

ence

rat

io

Pre

ssur

e p

’ (P

a)

Time (ms.)

Time (ms.)

Control on (’)

Model:• 2 Acoustics modes, and flame dynamics• Fuel Injector: - Two-position (on-off)

- 200 Hz bandwidth - 1st order dynamics

• 5th order modelController: LQG/LTR (5th order)

LQG/LTR

(PSP) 23-21

Time-delay Control (injection at main fuel supply)

• Idea: cancel the perturbations in the main fuel causing the instability, stability depends on naturaldamping in the combustor.

• Choose control:

)()(2 22

2

scs tdt

dptp

dt

pd

dt

pd

)( dttpK ccc

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-200

-100

0

100

200

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-0.2

-0.1

0

0.1

0.2

Pres

sure

(Pa

)C

ontr

ol in

put, c

*

Time (s)

c

Secondary fuel

Primary fuel

c

UTRC combustor

cExperimental results (UTRC, Cohen et al.’98): “c”

Stable and unstable zones, model predictions

unstablestable

stable

(PSP) 23-22

Cancel!Stabilize!

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

Controller structure:

sie

sce )(

)(

sR

sZK

p

pp

)(

)(

sh

so)(

)(

sh

sc

)(

)()( 21

sR

esnsn

p

si

p’’

sppopcp

spp

i

i

esZKsshsnsRsshsnsR

eshsZKsM

)]()()()()()([)())()((

)()()(

21

Closed-loop:

)(

)()(

sR

esZKsM

m

spp

i

•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:

(PSP) 23-23

Simulation with Time-delay Compensator Control

MIT combustor model: i ~50ac. (mean velocity <<)

0 200 400 600 800 1000 1200-100

-50

0

50

100

0 200 400 600 800 1000 1200-0.5

0

0.5Time (msec)Control on

Con

trol

inpu

t, c

*Pr

essu

re (

Pa)

Time (msec)

(PSP) 23-24Overview• Model

– 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

(PSP) 23-25

Actuator Limitations (Sec. Injector)

0 50 100 150 200 250 300

0.7

0.72

0.74

0.76

0.78

0.8

0 50 100 150 200 250 300-100

-50

0

50

100

p’

(P

a)

Time (ms.)

Time (ms.)

Control on (’)

0 50 100 150 2000.7

0.8

0.9

1

0 50 100 150 200-100

0

100

Time (ms.)

p’ (

Pa)

Higher authority, sec. Fuel flow rate

Faster settling time

0 50 100 150 2000.7

0.8

0.9

1

0 50 100 150 200-100

0

100

Time (ms.)

p’ (

Pa)

Lower bandwidth

Unsuccessful control

Results similar to observations in Yu (1997)

Impact of Nonlinearities in the Actuator

Heat release

Acoustics

f(.)nonlinearity

u’

Control

• 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.

Controlled (stable) limit cycle

Unstable limit cycle

Asymptotic stability

pres

sure

% secondary fuel

G

Actuator dynamics

Combustor dynamics

Open-loopStable limit-cycle

Unstablelimit-cycle

p

p

(PSP)23-26

(PSP) 23-27

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

(PSP) 23-28

Current Work

• Open-loop subharmonic control using fuel injection

Richards et al., 1999

-1 -0.5 0 0.5 1-4

-2

0

2

4

Time(sec)

Nor

mal

ized

pre

ssur

e,

75.02 7.0av

65.01

Prasanth,Annaswamy, Hathout and Ghoniem, 2000

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

for further details

• Extend models to turbulent combustion

• System ID Models