(psp) 23-1 modeling and control of combustion instability using fuel injection jean-pierre hathout...
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(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: [email protected]
(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