Modeling of Combustion Instability at Different
Damkohler Conditions
by
Sungbae Park
B.S., Mechanical Engineering
Seoul National University, 1996
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
BARKER
Master of Science in Mechanical Engineering
MASSAHUSETS
I~ilTTE
at the
OF TECHNOiLOG1Y
Massachusetts Institute of Technology
JUL 1 6 2001
May 2001
LIBRARIES
@ 2001 Massachusetts Institute of Technology. All rights reserved.
Signature of Author ..............
r....................
.......................
Department of Mechanical Engineering
February 1, 2001
................
Certified by .. .................................................
Anuradha M. Annaswamy
Principal Research Scientist and Lecturer
Thesis Supervisor
Certified by
Ahmed F. Ghoniem
Professor
Thesis Supervisor
Accepted by .....................
Ain A. Sonin
Chairman, Department Committee on Graduate Students
Modeling of Combustion Instability at Different Damkohler
Conditions
by
Sungbae Park
Submitted to the Department of Mechanical Engineering
on February 1, 2001, in Partial Fulfillment of the
Requirements for the Degree of
Master of Science in Mechanical Engineering
Abstract
Continuous combustion ranges from power generation, and heating to propulsion. One of
the characteristics of these processes is growing pressure oscillations which is referred to
as combustion instability. This instability becomes severe in a lean bum condition which
corresponds to reduced NO, emission-levels and hence a desired operating condition. To
avoid this instability, active control has been used over the past decade. However, a
systematic procedure to model the combustion system and design active control is not
currently available due to the complexity of process itself. Due to this inherent
complexity, we need to divide the modeling of the combustion system into several
categories based on the conditions. In this thesis, I apply three different approaches to
develop models at different Damkohler conditions. At low Damkohler and high Reynolds
number condition where the chemical reaction controls the heat release rate, a Well
Stirred Reactor model is used with a chemical reaction equation to represent the heat
release dynamics. At high Damkohler and low Reynolds number condition where the
wrinkled thin flame model applies, the nonlinear limit cycle phenomena is examined by
experiment and simulation. In the intermediate region of Damkohler and Reynolds
numbers, a System Identification method is used to develop a model. An LQG-LTR
controller is designed using the system identification model and implemented in a 30kW
combustor.
Thesis Supervisor: Anuradha M. Annaswamy
Title: Principal Research Scientist and Lecturer
Thesis Supervisor: Ahmed F. Ghoniem
Title: Professor
2
Acknowledgments
I would like to first thank Professor Anuradha Annaswamy and Professor Ghoniem
for their guidance through the past 1 and half years. Their efforts have been invaluable to
open my eyes and carry out my research.
I owe a great deal to Dr. Jean-Pierre Hathout for his generous help. He was the only
person who I can ask about anything. I am greatful to Shanmugan Murugappan with
whom I collaborated, for spending time to develop a system identification model and
implement controllers in a real combustor.
I owe a great deal to Dr. Young Joon Kim for his help to capture the flame motion
in the limit cycle region. I would like to thank my colleagues; Youssef Marzouk,
Daehyun Wee and Tongxun Yi.
I am very grateful to my parents, Hyunkyo Park and Boonle Kim for their love,
support and guidance. Finally, a warm and special thanks to my wife, Euene Kwon, for
her love and support during the past years.
This work has been sponsored by the National Science Foundation, contract no. ECS
9713415, and in part by the Office of Naval Research, contract no. N00014-99-1-0448.
3
Table of Contents
1. INTRODUCTION.......................................................................................................................9
2. HEAT RELEASE MODELING FOR KINETICALLY CONTROLLED
BU RN IN G ..................................................................................................................................
11
2.1 IN TROD U CTION ......................................................................................................................
11
2.2 ANALYTICAL MODELING OF THE WSR ..............................................................................
13
2.2.1
Governing Equations.......................................................................................
13
2.2.2
Linearized Heat Release Model ......................................................................
16
2.2.3
Physical Insight into the WSR Model..............................................................19
2.3 IMPACT OF OPERATING CONDITIONS ON THE WSR DYNAMICS .......................................
23
2.4 HEAT RELEASE DYNAMICS-ACOUSTICS COUPLING ..........................................................
28
2.5 EXPERIM ENTAL EVIDENCE .................................................................................................
30
2.6 THERMOACOUSTIC INSTABILITY SIMULATIONS.................................................................34
2.7 NUMERICAL CALCULATION OF THE WSR MODEL BASED ON MULTI STEP KINETICS...............38
3. NONLINEAR HEAT RELEASE DYNAMICS IN A WRINKLED THIN FLAME..........45
3.1 IN TR O D U CTION ......................................................................................................................
45
3.2 LINEAR HEAT RELEASE MODEL ........................................................................................
48
3.3 NONLINEAR HEAT RELEASE MODEL ..................................................................................
53
3.3.1
S and S ' .......................................................................................................
54
3.3.2
Time Stepping Algorithm and Spatial Discretization ..................
58
3.3.3
Simulation Results with a Fixed Boundary Condition.....................................59
3.3.4
Simulation Results with a Moving Boundary Boundary Condition ................
68
3.4 EXPERIMENTAL MEASUREMENTS .........................................................................
70
3 .4 .1
S E T -U P ................................................................................................................
70
3.4 .2
RE SU LT S .............................................................................................................
71
4. SYSTEM IDENTIFICATION BASED MODELING OF COMBUSTION
INSTABILITY FOR TURBULENT COMBUSTOR .......................................................
4 .1
IN TRO D U CTION ......................................................................................................................
4
76
76
4.2 EXPERIMENTAL SETUP ..........................................................................................................78
4.3
SYSTEM IDENTIFICATION OF A COMBUSTION SYSTEM ........................................................... 81
4.4 IMPLEMENTATION .................................................................................................................84
4.5 LQ G -LTR CONTROL ............................................................................................................86
4.6
CONTROLLER D ESIGN AND IMPLEMENTATION ......................................................................87
4.7 RESULTS ................................................................................................................................ 88
4.8 D ISCUSSION ...........................................................................................................................91
5. C O N CLU SIO N S ......................................................
........ oe. ............................... 95
R-EFERIEN CES ............................................................................................................................... 97
5
List of Figures
Figure 2-1 Definition of the blow-out Temperature T*. Data are for 0 = 0.8,
3
M/V =1040kg / m sandT* =
1800K . ......................................................................................
20
Figure 2-2 Plot showing conditions for in-phase relation between ;. and th,
corresponding to T > T** . Data are for 0=0.8,
and
IV=800kg /m 3s
Figure 2-3 Plot showing conditions for out-of-phase relation between n; and
corresponding to T* <T <T**. Data are for o =o.8,
an d
T**= 1882K .
V/V=1030kgm
T** =1882K . .........
',
3
s, T *=1800K
............................................................................................................
Figure 2-4 Plot showing the effect of # on
Data are for
Q,
IV =530kg /
21
3
s ...................................
22
23
Figure 2-5 The dependence of the cut-off and static gain on the mass flow rate at 0 = 0.8...............24
Figure 2-6 The dependence of the cut-off and static gain on the equivalence ratio at
S / V =530kg/m 3
.......
.............................
. . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . .. . .. .. . .. . . . .
25
Figure 2-7 Dependence of the phase of the heat release model on the mass flow rate at
different equivalence ratios ....................................................................................
26
Figure 2-8 Dependence of the gain of the heat release model on the mass flow rate at
different equivalence ratio ......................................................................................
26
Figure 2-9 Dependence of the p'- Q' phase on the mass flow rate for a quarter-wave
mode using the heat release model in Figure 2-7 and 2-8 ....................................
27
Figure 2-10 The Rayleigh Index for a quarter-wave mode using the heat release model in
Figure 2-8 and 2-9. ................................................................................................
27
Figure 2-11 The (p'- Q,) phase and gain for a quarter-wave mode at a fixed mass flow
rate (530kg/mA3 s), as a function of the equivalence ratio....................................29
Figure 2-12 Pressure amplitudes in a lean premixed combustor near the blow-out
con dition s [ 9 ]............................................................................................................3
Figure 2-13 Pressure amplitudes in a combustor at various equivalence ratios [ 3 ].................
1
33
Figure 2-14 Pressure amplitudes in a combustor at various flow times [ 3 ].............................33
Figure 2-15 Change of pressure amplitudes near the lean blow-out limit [ 10 ].........................34
Figure 2-16 The combustion feedback system with the WSR model.........................................36
Figure 2-17 Simulation of pressure oscillation in Case I.............................................................37
Figure 2-18 Simulation of pressure oscillation in Case II...........................................................37
6
Figure 2-19 Pressure oscillation map in LSU swirl stabilized rig [ 14 ].....................................38
Figure 2-20 Phase of WSR model in 4 step kinetic at fixed
....................................................
41
Figure 2-21 Gain of WSR model in 4 step kinetic at fixed
...................................................
42
Figure 2-22 Pole and zero map of WSR model in C3H8 4 step kinetics at $ = 0.7,
M=
732kg / m 3s and T = 600K ............................................................................
43
Figure 2-23 Magnified Pole zero map around acoustic frequency of WSR model in
C3 H 4 step kinetics at $ = 0.7, M = 732kg / m s and 7T = 600K ......................
44
Figure 3-1 Thermoacoustic instability feedback diagram...........................................................45
Figure 3-2 Initial Growth and limit cycle phenomena ................................................................
47
Figure 3-3 Definition of variables used to describe flame surface kinematics [ 16 ] ..................
49
Figure 3-4 Frequency response gain and phase of a premixed flame within the linear
regio n ..........................................................................................................................
Figure 3-5 Flam e shape in a Duct [ 20 ]......................................................................................
52
54
Figure 3-6 Approximated Flame shape using the 4th order polynomial function.......................55
Figure 3-7 Burning velocity profile using approximated flame profile and Chebychev
differentiation N =1000. ..........................................................................................
56
Figure 3-8 M ean curvature profile ................................................................................................
57
Figure 3-9 Overall flame shape change, u' = 0.3 -u max- sin(2953 -t) ,
K = 0.5 -u m ax/(h/ R), y = 0.5 ...............................................................................
60
Figure 3-10 Perturbation term change u' = 0.3 -u max- sin(2953 -t) ,
K = 0.5 -u m ax/(h / R), p = 0.5 .............................................................................
61
Figure 3-11 Overall flame shape change in polar coordiante, u' = 0.3 -u max- sin(2953 -t)
,K = 0.5 -u m ax/(h / R), p = 0.5 ..........................................................................
62
Figure 3-12 Perturbation term change in polar coordinate, u' = 0.3 -u max- sin(2953 -t) ,
/ = 0.5 -u m ax/(h / R ), p = 0.5 ...............................................................................
63
Figure 3-13 Phase between u' and q'.u' = 0.3 -u max. sin(2953 -t) ,
K = 0.5 -u m ax/(h/ R), p = 0.5 .............................................................................
64
Figure 3-14 Phase between u' and flame motion at the tip and
tail. u' = 0.3 -u max. sin(2953 -t) ,
7
K
= 0.5 -u max/(h / R), p =0.5 .......................
64
Figure 3-15 Phase change by the magnitude of u' ....................................................................
65
Figure 3-16 Gain change by the magnitude of u'......................................................................
66
Figure 3-17 Effect of
K
on the phase releationship.....................................................................67
Figure 3-18 Effect of p on the phase releationship ....................................................................
67
Figure 3-19 Phase between u'and q'. K = 0.1 -u max/(h / R), p = 0.1 ......................................
69
Figure 3-20 Perturbation term change in polar coordinate, u' = 0.3 -u max- sin(2953 -t) ,
K = 0.1 -u m ax/(h /R), p = 0.1 ...............................................................................
69
Figure 4-1 Schem atic of the combustor .......................................................................................
79
Figure 4-2 Baseline power spectra for # = 0.7 .............................................................................
80
Figure 4-3 Normalized p'rms and q'rms as functions of primary fuel equivalence ratio............81
Figure 4-4 Power spectra of the pressure signal with PRBS input at $ = 0.7 .............................
85
Figure 4-5 Power spectra of the system-identification model at $ = 0.7 ...................................
85
Figure 4-6 Q'rms spectra at the baseline, time- delay and LQG-LTR control at $= 0.7 ...........
89
Figure 4-7 p'rms spectra at the baseline, time- delay and LQG-LTR control at $= 0.7 ............
90
Figure 4-8 Normalized p' rms for different time delays at # = 0.7. ..............................................
91
Figure 4-9 Bode plot of LQG-LTR and the time-delay controller at 0 = 0.7 .............................
92
Figure 4-10 Open-loop transfer functions of the system (combustor*controller) ......................
93
8
1. Introduction
Continuous combustion processes are used in many applications ranging from power
generation and heating to propulsion.
One of the characteristics of these processes
growing pressure oscillations that transition to a sustained limit cycle. These oscillations
occur due to the coupling between acoustics and heat release dynamics. In acoustics, the
heat release oscillation supplies volume expansion and this expansion acts as a driving
force of the pressure oscillation inside the combustor. This pressure oscillation generates
velocity, temperature, and equivalence ratio perturbations, and these perturbations again
induce unsteady heat release through heat release dynamics. If this feedback is positive,
the combustion system becomes unstable, and the resulting dynamics is referred to as
combustion instability.
Combustion instability occurs especially in a lean bum condition where the
efficiency is high and emission is low, or in a rich bum condition where the thermal
output is high. These imply that the operating condition affects the feedback mechanism
and switches the positive feedback to negative feedback or vice versa. If one understands
this feedback mechanism, it may be possible to stabilize the combustor in the regions
where combustion instability originally occurs by switching the positive feedback to
negative feedback. This stabilization will widen the attainable operating conditions, and
aid to achieve the goals, e.g., low emission and high thermal output combustors.
However, the underlying mechanism is so complex and the mechanism changes as
the geometry and operation condition change and hence it is very difficult to develop a
single model that describes the dynamics in the whole region. Instead, one needs to
divide the operating condition into several sub-categories and develop models that can
represent distinct characteristics in those operating conditions.
Several investigations
have been attempted to develop models at different combustion conditions. At high
Damkohler and low Reynolds number condition, a wrinkled thin flame model is used to
represent heat release dynamics [ 16 ]. In [ 16 ], the heat release oscillation is due to the
9
change of the flame area, and the flame area changes by the velocity oscillations. The
flame area change by the velocity perturbation is represented using G equation. At other
regimes where the Damkohler is low and the Reynolds number is high, the chemical
reaction controls the heat release rate. In this case, a Well Stirred Reactor model has
been used to represent the heat release dynamics [ 3 ]-[ 6 ]. In other regions, where
important, physics based modeling of the
flame vortex and other mechanisms are
combustion system has not been undertaken rigorously.
In this thesis, the modeling of the combustion instability is addressed in three
different combustion condition. 1) Low Damkohler and high Reynolds number, 2) High
Damkohler and low Reynolds number, and 3) Intermediate Damkohler number
conditions. In low Damkohler number condition, the heat release dynamics is modeled
as a Well stirred reactor and linearized heat release model is developed using chemical
reaction equation and conservation equations. At high Damkohler number condition
where the thin flame model is applicable, nonlinear characteristics of the flame that lead
to limit cycle is investigated as an extension of the modeling in [ 16 ]. At intermediate
Damkohler number condition, the System Identification approach is used to develop a
model of the combustion system and the model is used to develop a controller.
10
2. Heat Release Modeling for Kinetically Controlled
Burning
In this chapter, we present a heat release dynamics model which utilizes a well-stirred
reactor (WSR) model and one-step kinetics to describe the unsteady combustion process.
The model incorporates linearized mass and energy equations to describe the response of
the reactor to external perturbations, and is cast in the form of a first order filter. The
model is able to predict the phase between the mass flow rate oscillations and the
resulting heat release fluctuations, as function of the operating conditions, e.g., the mean
equivalence ratio and mean mass flow rate. The model predicts a sudden shift in phase in
the region between the maximum reaction rate and the blow-out limit. We show that this
phase change may trigger combustion instability. We use this novel model to predict
combustion instability conditions in high swirl combustion, and demonstrate that these
predictions agree qualitatively with experimental studies.
2.1 Introduction
Combustion in high performance engines utilizes strong swirl, recirculation and
interacting jets to enhance the mixing rate of the fuel, air and products, and hence
maximize the burning rate. The ideal limit for these systems is often modeled as a wellstirred reactor [ 1 ], in which the mixing rate is faster than the fuel conversion rate, and
products exit the reactor at their interior uniform state. The operation of a well-stirred
reactor is governed by a characteristic residence time, res, which is the nominal time the
reactants spend inside the reactor;
re
res =
11
t
where pi is the density of the reactant, V is the reactor volume, and
rate at the inlet.
rh
is the mass flow
Stable operation is achieved when the residence time is larger than the
characteristic chemical time, otherwise blow-out should be expected.
Combustion dynamics, resulting from coupled heat release-pressure oscillations, has
been suspected to occur when oscillations in the mass-flow rate, equivalence ratio, inlet
temperature and pressure, etc., occur at the same time-scale. However, the mechanisms
that support the positive coupling between the heat release dynamics and acoustic
perturbations have not yet been investigated or modeled thoroughly. The condition under
which a combustion system becomes unstable has been expressed in terms of the
Rayleigh criterion [2]:
IL
L
- -e'Adx =
Ct f
)
i7
Pcpc
0
,2
where e'=
(2.1)
p'q'Adx AL (E'A) - C >0
;
2
2p~
- r2
2 + Tv
2
and E'= P'
are the acoustic energy density and acoustic energy
flux, respectively, p'is the perturbation in the density of the unburned mixture, A is the
cross-sectional area of the combustor, 4 is the perturbation in the rate of energy
dissipation, x and
t
are the distance and time, respectively, and AL signifies the difference
over the combustor length L. The conclusion drawn from this mathematical condition is
that a combustion system becomes unstable when the heat release increases at a moment
of a rise in pressure, i.e., Z(q'-p')s 900. The Rayleigh criterion also shows that acoustic
energy depends on the dissipation in the system, and hence the gain in the (p'-q')
relationship also plays an important role in determining the characteristics of instability.
Combustion instability has been modeled using a well-stirred reactor and one-step
kinetics by Richards et al. [ 3 ], Janus and Richards [ 4 ], Lieuwen et al. [ 5 ], and
Lieuwen and Zinn [ 6 ]. Richards et al. [ 3 ] investigated the effect of heat loss, flow rate
12
and friction in a tailpipe of a pulse combustor. The governing flow equations were
reduced to a set of ODEs assuming a well-mixed combustion zone and choked inlet flow.
The authors showed that the simulation results of the ODEs agree qualitatively with the
experimental data.
A similar approach was used by Janus and Richards [ 4 ] for a
premixed combustor. In that study, the authors showed that the model could predict the
effect of the inlet temperature and open loop control by comparing the simulations with
experimental results. Lieuwen et al. [ 5 ] investigated the impact of the equivalence ratio
oscillation on the heat release. Given a perturbation in the equivalence ratio, as the mean
equivalence ratio is decreased, they show that a well-stirred reactor model yields an
increase in the magnitude of the corresponding heat release perturbations. In [ 6 ], the
same model was coupled with acoustics and a convective time delay for the equivalence
ratio perturbation, and instability was predicted over a range equivalence ratio of 0.6-1.
In this chapter, we investigate the linear response of a WSR model to the mass flow
rate, or residence time oscillations, using one-step kinetics. We show that as the mean
equivalence ratio or the mean residence time approach the blow-out limit, the operating
point may transition from stability to instability due to a sudden phase change between
pressure and heat release oscillations.
2.2 Analytical Modeling of the WSR
2.2.1 Governing Equations
The governing equations of a well-stirred reactor are obtained using the conservation
laws and a set of reaction-rate equations. The conservation equations of the mass, energy
and species in the WSR are given by:
13
Mass Conservation: dM = 6i
dt
Energy Conservation:
dE
(2.2)
,
-
= 6ih, - thh +Q,,
(2.3)
dM,
Species Conservation: dMk =
di
where M
,
E
,
IYk,
-
(2.4)
Yk - Wk,
and Mk are a total mass, energy and mass of species k inside the
combustor, respectively,
,. is the heat release rate due to the chemical reaction, Wk is a
consumption rate of species k , rh is the mass flow rate, h is the enthalpy, Y is the mass
fraction, and subscript i refer to the inlet condition. We assume that the condition at the
exit are the same as inside, consistent with the assumption that mixing is much faster than
the chemical reaction. Equation 4 can be written for all species; e.g., CnH,,, 02, C0 2 ,
H 2 0,
etc. In case of one-step kinetics, one differential equation is sufficient and the mass
fractions of other species are related by stoichiometry. Equation (2.2) and (2.3) can be
simplified as follow:
pVcP
dT'
dp
dt
dt
TVd-=
rhicP(T - T) + Q,
(2.5)
where p is the density of the mixture, V is the volume, c, is the specific heat, T is
temperature, and p is the pressure. In deriving Equation (2.5), we assume that the c,,
c, and V are constants. The V "" term can be expressed as a function of T using the inlet
dt
and exit conditions, and the ideal gas law. Assuming that the pressure oscillations are
weak, the pressure energy term is negligible, and equation (2.5) reduces to
pVcd
= rnc,(T - T)+,..
(2.6)
Using Equation (2.2), equation (2.4) reduces to
14
(2.
(2.7)
pV dYk = th (Yk -Yk)-W.
dt
The source terms,
,., and Wk for the fuel, can be represented as function of Y and
T using a one-step kinetics mechanism [ 7 ] as follow:
W
Af
ff npf(pOY 0 2
-T
)n02 exp(-T-) and
Qr=Ah,
(2.8)
Wf
where Af is the frequency factor, Ah, is the enthalpy of reaction (measured per unit mass
of fuel), and Ta = Ea / R where Ea is activation energy and R is the gas constant.
At a fixed #, Y2 and Yf are related by the stoichiometric mass ratio V/s as follows
1
Y02 =
1
Yf +(Y
V/s
2
(2.9)
-- 1Y)
V'S
Near stoichiometric conditions, Y ~: 0
, and far from the stoichiometry, i.e., in a fuel
2
lean mixture, Y02 ;zconst. In a fuel lean mixture, pnl can taken as a constant around the
equilibrium point because the strongest dependence of the reaction rate on temperature
comes from exponential term. Therefore, Equation (2.8) can be simplified to
.Af
Ah,.VP"Y" exp(
T
(2.10)
)
where Y = Yf and n=nf +n0 2 at near stoichiometric condition, and n
condition.
15
=
nf under fuel-lean
2.2.2 Linearized Heat Release Model
While the dynamics of a well-stirred reactor can be investigated by integrating these
nonlinear ODEs directly, a linearized model makes it possible to examine its properties,
such as the blow-out limit, and the gain and phase relations between the heat release rate
and mass flow perturbations. A linear heat release model can be obtained from Equations
(2.6), (2.7) and (2.10) assuming small perturbations around a steady state. In deriving the
linear heat release model, it is assumed that the air and the fuel lines are choked.
Therefore, equivalence ratio oscillations are absent, while the mass flow rate and
temperature oscillations in the combustion zone are the forcing terms of the heat release
model. The dependent variables T, Y, p and mihare represented using steady-state and
perturbation terms, e.g. Y = F + Y'. The linearized reaction rate equation (2.10) is:
-T
5nI-
Q' =At AhV[n"'Y" exp(-
-
-n"1~X(Ta
-,n-nT
-a-)
T
T
±pY
exp(
T
_ )T'
T
.
(2.11)
Moreover, p' is expressed in terms of T' (assuming constant pressure and molecular
weight),
T',
(2.12)
p'=-P-.
TO
Using Equation (2.11) and (2.12), one linearizes Equations (2.6) and (2.7) as
dT'P~Vc-v
=~c(-T')
t
PC
+
'c,(Tj - T) + A'f AhVx
rV
-TpI T'
[~n~n
[-n-n"Y" exp(a T
dY' = N(-Y')+ r'(Y -Y)
dt
T
T
-T
+nYn-]Y'exp(a)+ -T
- A' V x
-nP"Y" exp(
-T
16
T
-T
-f~f
T
exp( -a
"
T
2
T
)T'],
a )+ nyY "Y'exp(
(2.13)
T
a)+
Since the ratio (
/(-)=
Tm
Ma (7 -1);
for low Mach number flows, we neglected T;' in
equation 13. Using Laplace transform of Equations (2.11), (2.13), and (2.14), we obtain
the following linear heat release rate model:
Q =J(s)m'
(2.15)
s+a
where
((T
T=
Tj)
(T - T)
(Y-Y)
T
Y
a
(2.16)
and
3= A Ahji
-T
Y exp( T
(-
T
T)
(T - T) T(Yi
Ta +n
-Y)
]
(2.17)
Note that i(s) is a first order filter.
The cut-off frequency a and the static gain p of the linear model are functions of the
mean residence time, the equivalence ratio, and the inlet temperature.
At a fixed
equivalence ratio, if the residence time is much larger than the chemical reaction time,
almost all the fuel is burnt, i.e., Y~ 0.
In this case, a and 6 are much larger than the
acoustic frequency (due to the Y term in the denominator in equation (2.16) and (2.17)),
and the heat release responses instantaneously to the acoustic perturbations.
As the
residence time decreases, the unburned fuel Y_ increases, so the values of a and p
decrease. Moreover, the change of the residence time affects the equilibrium temperature
T. As the residence time decreases, the equilibrium temperature T decreases, while a
17
and 8 change from positive to negative values because of the - (
-
T term. When
a becomes negative, the heat release model itself becomes unstable since a perturbation
grows exponentially as e-'. The system is critically stable when a =0. As we will see
in the next section, this corresponds to blow-out. The value of T which leads to a =0 in
equation (2.16) is defined as T* which is the blow-out temperature;
T* satisfies the
following equation:
1+n (T*
-Ti)
*
T
(Yi - F
_(T* -Ti)
2
-
(T )
(2.18)
0
Y
Equation (2.15) shows that when 8 changes sign, it introduces 1800 phase change
between th' and 0 . If the heat release dynamics is coupled with acoustics, this phase
change may trigger a thermoacoustic instability as an out-of-phase relationship between
(p', q') becomes in-phase.
That is, at 85=0, the system can transition from stability to
instability. This thermoacoustic driven instability is different from the instability of the
flame dynamics itself, which is defined by the sign of a in the above paragraph. The
critical value of 7 which corresponds to /5=0 in equation (2.17) is called as T**, and is
determined from
n (T**i)
T**
T,+n
(T** )2
'
(2.19)
Y
As will be shown in the next section, /5=0 corresponds to burning at the maximum
heat release rate. Equations (2.18) and (2.19) are similar expect for the extra "1" in
equation (2.18).
Based on this, one expects / to become negative before a as the
residence time decreases.
Therefore, just before blow out (a
=
0 ), the heat release
experiences a phase change. That is, the onset of thermoacoustic instability may occur
before blow-out.
18
The change of the equivalence ratio at a fixed residence time also changes the
equilibrium temperature T, thereby affecting a and 6. One can expect that a and fl
become negative as the equivalence ratio decreases due to the drop of the equilibrium
temperature F. Therefore, the linearized model shows that by decreasing the residence
time or the equivalence ratio, one expects phase change or blow-out to occur.
2.2.3 Physical Insight into the WSR Model
The heat release dynamics model presented in Section 2.2.2 has two parameters a and
8.
To gain insight into the meaning of a and 8, we examine the critical steady state
response of the WSR. We define
Qf = mhic, (T - T)
as the energy added to the flow across the reactor, and draw Q, and of as fi changes, as
shown in Figure 2-11. The equilibrium or steady-state temperature is determined by the
intersection of two curves. As known in the well-stirred reactor theory , three solutions
exist; hot and cold stable solutions and an unstable hot solution. As the slope of the Qf curve increases due to an increase in mass flux (or decrease of the residence time), the
two hot solutions collapse onto one. There is no hot solution for higher values of mass
flux. Therefore, the equilibrium point in Figure 2-1 where Qf -line becomes tangent to
the Q, -curve is a critically stable point, and it can be calculated by solving the following
equations:
All the Figures in Chapter 2 for one step kinetics are calculated for C2 H 6 , for which Af = 4.24. 10',
nf = 0.1, no 2 =1.65, Ta =15098K and T, = 600K .
19
and der _ dQf
QrdT
(2.20)
dT
The solution of these equations is given by equation (2.18), indicating that a =0 captures
the static blowout limit.
2
x 106
--
ur
Oro....
1.5
-4.
U)
cE
1
h .t
unsOf
h.
stabl
X o
~
hot
unstabe
2
soluolutio\
0.5
solution
0
0
5 00
2
p
1500
1000
2000
2500
Temp(K)
Figure 2-1 Definition of the blow-out Temperature T*. Data are for 4 = 0.8,
m/
V =1040kg Im 3 s and T*
=1800K
Another critical point exists in P, -curve. It occurs when
, reaches a maximum, as
shown in Figure 2-2. The condition corresponding to maximum heat release rate,
dQr = 0,
dT
(2.21)
is shown in Figure 2-2. The equation defining this temperature is exactly the same as
T** obtained from equation (2.19).
The Figure shows that for T > T** (1,, ' ) are in-
phase; however, for T* < T < T**, their phase changes by 1800.
20
This is confirmed in
Figure 2-3, where the equilibrium solution corresponds to T* < T < T**.
the mass flow rate, P, increases
For T > T**, as
, also increases, i.e., (th, P') are in-phase.
On the
other hand, for T < T**, as the mass flow rate, rhi, increases, P, decreases, indicating an
out-of-phase relation.
2
x 106
.
.....
Or
1.5k
Conditions corresponding
cE
to maximum heat release rate ~.'..*Sal
0
-.equilibrium
temperature
dl?
1
Polio4
0
.0 0
0.5V
0
50 0
1000
1500
Temp(K)
2000
25 00
Figure 2-2 Plot showing conditions for in-phase relation between ;. and -h,
corresponding to T > T**. Data are for 0 =0.8, M/IV=800kg/m3s and T** =1882K .
21
x 10
6
Of
---
Stable
equilibrium
temperature
1.6
co
cv,
E
T
*0
1.5Blow-out
0.
temperature
Maximum heat release
rateI
170.
1900
1800
Temp(K)
Figure 2-3 Plot showing conditions for out-of-phase relation between n; and th,
corresponding to T* < T < T**. Data are for 0 =0.8,
T
M/V =1030kg/m
3
s , T*=1800K
and
=1882K.
Therefore, the phase between (Ph,Q) changes by 1800 as the point of the maximum
heat release rate is crossed. In Figure 2-2 and 2-3, the equilibrium condition shifts due to
the change of the residence time (mass flow rate), which also leads to a phase change.
Changing the equivalence ratio also can introduce phase change, as shown in Figure 2-4.
As the equivalence ratio decreases, or curve moves down causing the equilibrium point
to cross the maximum heat release point. We conclude that a phase change of 1800 occurs
either by decreasing the residence time, or equivalence ratio, in the regions between the
maximum heat release point and the blow-out point.
In summary, the heat release dynamics is modeled as a first-order filter with a transfer
function J(s) given by (2.15). It is worth noting that even with such a simple form, the
heat release model is capable of capturing blow-out, and the transition across the
maximum heat release rate point. The first-order filter is able to characterize both of
22
these characteristics utilizing two degrees of freedom in J(s) expressed in terms of the
two parameters a and fi.
X 10,
18
1614
=0.6
.----- e=0.7
-
%'0.
4 =0.8
12C,,
cE
108
-
6440
2 -
01
500
asing4 2
11Decr
1000
1500
2000
25( 0
Temp (K)
Figure 2-4 Plot showing the effect of # on ,. Data are for
I V=530kg /m 3 s.
2.3 Impact of Operating Conditions on the WSR
Dynamics
The heat release model, J(s) , describes the linearized dynamics around a fixed
operating condition. The operating condition is determined by 0, J , and T,. While the
structure of the heat release model does not change as the operating condition changes,
its parameters, the gain fl and the cut-off frequency a, depend on the #,
23
i and T7
through equations (2.16) and (2.17). We now show how these quantities change with #
and '
for '; =600K .
Figure 2-5 depicts the impact of m on a and 8 at a fixed
equivalence ratio. For values of M' / V is less than 700 kg /m 3 s, the cutoff frequency a is
about 3khz. In this region, the heat release model J(s) responds to the acoustic
perturbation instantaneously when the frequency of the latter is of order of hundred Hz.
As M
increases, 8 becomes negative beyond the maximum heat release rate point.
/V
Around this area, a is close to the acoustic frequency. For a narrow range of rn /V , the
; and iW changes by 1800 .
phase between
Figure 2-6 shows the effect of the
equivalence ratio on a and 6 at a fixed mass flow rate. As the equivalence ratio
decreases, a and 6 decrease. In a narrow range of equivalence ratio from 0.7 to 0.705, '8
is negative which introduces 1800 phase change between (rh', Q;). In both cases, the 180*
phase change for 8 <0 and a >0 may trigger a thermoacoustic instability near the blowout limit, as shown next.
4
x 10
7
x 10
4
3
3
C,
2
4
-
-
----
--
-*---------
---
--
,2
Blov -out
1
0z
0
Reaction
Rate
.41
600
.
*
11
700
800
900
1000 1100 1200
mass flux(kg/m 3 s)
Figure 2-5 The dependence of the cut-off and static gain on the mass flow rate at 0 = 0.8.
24
20
x 106
- -- - - -
15
C/)
10
nnnn
_&2
- - -
--
15000
.
--
- -- - - - --
-- - - - - -
- ---
-- -
10000
C,)
(t
Blovy -out
c,)
.*
5000
5
- ----
-- - -
--
- ..-
CO
Maximurr
reaction
rate
-1
0.7
0.71
--
0.73
0.72
0.74
-------------.-
~I
I 5000
II
0.75
equivalence ratio
Figure 2-6 The dependence of the cut-off and static gain on the equivalence ratio at
3
1V =530kg/rm
The characteristics of the heat release model are exhibited from Figures 2-7 to 2-10 at
a given acoustic frequency, e.g., 200 Hz. The phase between rh' and 0 changes from 00
at low Mi to small negative values as we approach
**(the point of the maximum heat
release) as shown in Figure 2-7. A 1800 increase in phase is experienced at T**. For iMi
corresponding to T* <T <T**, the phase decreases to 900.
The sudden phase jump at the
maximum heat release point corresponds to the sign change of 8, while the continuous
phase change is due to the decrease of a. Figure 2-8 shows the dependence of the gain
on M . Note the sharp increase for T <T**.
25
-4=0.6
-----.
=0.7
--- =0.8
150
rJ)
-a)
CO,
'i
L
100-
50
Blow-out
i-Blow-out
")
(
(6r) miax--_W
0
(Q)
r max :
M'M'M'M
BI ow-out
-.-
-~
-i
r max
--
-501-
103
102
mass flux(kg/m 3 s)
Figure 2-7 Dependence of the phase of the heat release model on the mass flow rate at
different equivalence ratios
6000
_
-
5000
=0.6
Blow-out
4=0.7
=0.8
----
4000
-E 3000
.b
a 2000
05
I
I
S
I
U
S
S
.- Blow-out
C:
- . -
.
. . . .. . .
. .. . .
..
1000
ft
*
.
--.
Blow-out
-l
9'
0
102
103
flux(kg/m 3
mass
s)
Figure 2-8 Dependence of the gain of the heat release model on the mass flow rate at
different equivalence ratio
26
50
-4=0.6
------ e=0.7
v
ii
- =0.8
c,)
Blow-out
0k
-a
0
BI ow-out
I
--
'L-Blow-out
C/)
-c
-50 k
rmax :r
r max
CO
9
max
-... .
-100
-.
-150
103
102
mass flux(kg/m 3 s)
Figure 2-9 Dependence of the p'- ' phase on the mass flow rate for a quarter-wave
mode using the heat release model in Figure 2-7 and 2-8
I
I
1
51
C
-c
n
- =0.7
=0.8
0
*
0
*
.
0
*
0
-
ri
- -----
0
3
00
*
0
0
I
U0
13
0
*
I
00000000000q.o.
N
3
I
0
E
-4
0
z
-1
400
100
700
1000
mass flux(kg/m 3 s)
Figure 2-10 The Rayleigh Index for a quarter-wave mode using the heat release model in
Figure 2-8 and 2-9.
27
--
2.4 Heat Release Dynamics-Acoustics Coupling
The possibility that a phase change of 1800 at T** may trigger a thermoacoustic
instability is now demonstrated.
To model the latter, we must determine the p' -v
relationship. The momentum equation shows that the phase between p' and v' (inlet
velocity perturbation) is 900. In open-closed boundary conditions, the first two acoustic
modes correspond to a quarter-wave and a three-quarter-wave.
For a quarter-wave
mode, p' leads v' over the entire combustor, i.e. Zp'-v'=-900.
For a three-quarter-
wave, p' leads v' on either sides of the left and right nodes,
Zp'-v'=-900, while
v'leads p' between the two nodes, Zp'- V=900. Moreover,
' can be represented as
(2.22)
m' = piv'A
where pi is the density and A is the cross sectional area of the combustor. Using the heat
release model, the phase Zp' - V and the relation in equation (2.22), the phase between
p' and Q' can be determined. Figure 2-9 shows p'- Q' phase as a function of M for
three different equivalence ratios, assuming a quarter-wave mode for the p' - v relation.
It shows that p' and o' are in-phase between the point of maximum heat release rate and
the blow-out limit. As discussed in section 2.1, thermoacoustic instabilities occur when
p' and o' are in-phase.
Moreover, as shown in Figure 2-8, as the mass flow rate
increases, the gain decreases first reaching a minimum at T**, and then increases again.
Both effects indicate that one should expect strong pressure oscillation near the blow-out
limit.
Given the magnitude and the phase relation as shown in Figure 2-8 and 2-9, it is
possible to compute the Rayleigh Index, IR, which is defined as
inR
= fp'q'dtdV ,
28
where q' = , / V .
Positive values of IR lead to strong pressure oscillation, whereas
negative IR indicates a stable system. Figure 2-10 shows the Rayleigh Index normalized
by its maximum value at the same conditions shown in Figure 2-8 and 2-9. The Rayleigh
Index experiences a sharp increase between the point of maximum heat release and blowout as the mass flow rate increases. The maximum Rayleigh Index is achieved at the
blow-out point.
Figure 2-11 shows the impact of the equivalence ratio on (p',
') gain and phase
relations. Near blow-out, p' and d' becomes in-phase while their gain increases sharply.
Note the narrow range of 0 within which conditions support an instability.
2500
100
Phase
----- --
50
- ---
-
-----
-
Bow-ait
CD
c)
C,
0
-
-- - --
---- -------
--
- - -- -
_0
heat rdae rae
I
20
-- --
1500OE
.9CL
.100
----
-A-0
0.7
-
0
0.72
0.71
0.73
ecM\dence ratio
Figure 2-11 The (p'-Q,) phase and gain for a quarter-wave mode at a fixed mass flow
rate (530kg/mA3 s), as a function of the equivalence ratio.
29
2.5 Experimental Evidence
There exists ample experimental evidence that as the equivalence ratio is decreased at
a fixed mass flow, or the mass flow rate is increased as a fixed equivalence ratio, the
system develops self-sustained oscillation. Soon after these oscillations are observed,
blow-out is often encountered. In this section, we review some of these results and use
the theory developed here to explain some of concomitant observations.
In an experiment conducted to examine the response of a lean premixed, swirl
stabilized combustor, it was observed that the system remained stable until rather low
values of 4, where thermoacoustic instabilities seem to become strong [ 8 ]. Soon after
the onset of the instability, and within a small decrease in 0, combustion blows out in a
way that is qualitatively similar to the prediction in Figure 2-11.
Results of a lean premixed combustor in which a flame was stabilized behind a
rearward-facing step [ 9 ] exhibited the dependence of the pressure amplitude on the
equivalence ratio shown in Figure 2-12.
As the equivalence ratio decreased, the
amplitude of a 48 Hz mode increased, while that of a 124 Hz mode decreased within the
same range. According to the system configuration in [ 9 ], the 48Hz mode corresponded
to a quarter-wave mode, while the 124Hz mode corresponded to a three-quarter-wave
mode.
The theory presented in this paper predicts this mode selective behavior. Since
the flame was located in the middle of the combustor, p' leads
mode while
v'
v'
in the quarter-wave
leads p' in the three-quarter-wave mode, as mentioned above. Therefore,
if p' and 0 are in-phase in one mode, they are out-of-phase in the other, i.e., the
pressure amplitudes should respond to change in 0 in opposite ways, as explained in the
previous section. We should mention that this agreement is only qualitative since the
heat release dynamics in the experiment may be governed by flame surface motion.
However, since the chemical time scale governs the heat release rate near the lean blowout limit, the combustion dynamics can be approximated by a well-stirred reactor in that
30
region. Note that the pressure amplitudes increase sharply prior to blow-out, as captured
by the WSR model.
.
0.07
0.08
IX
.......
*...w..*124
005
.......... .
0.03
..... ... .....
........
.....
0.02
...........
001
-N\
0,7
0.75
_
----------
08
.
40
..
..........
0AS
0.0
:095
Equavntnce Ratk
Figure 2-12 Pressure amplitudes in a lean premixed combustor near the blow-out
conditions [ 9 ]
The experimental results of Richards et al. [ 3 ] also agree with the prediction of the
WSR dynamics. In that study, the combustor used to investigate the effect of the heat
loss, flow rate and friction was composed of a choked inlet, well-mixed combustion zone
and a tail pipe. Because the inlet was choked, equivalence ratio fluctuations were absent.
As shown in Figure 2-13, the pressure amplitude increased as the equivalence ratio was
decreased at a fixed residence time (39ms).
Figure 2-14 shows the impact of the
residence time at a fixed equivalence ratio. As the residence time was decreased (by
increasing the mass flow rate), the pressure amplitude increased. The dependence of the
stability of the system on the equivalence ratio and the residence time qualitatively match
the predictions based on the WSR heat release dynamics model. Figure 2-13 and 2-14
also show that the mode changes to a lower frequency as the pressure amplitudes grow.
This may be due to our prior observation that different phase relations for Zp'-mh' should
31
be considered for different modes, and that the phase strongly depends on # and Mi
through the model parameters a and 8J.
Another sets of experimental result [ 10 ] where a three-nozzle sector combustor was
used with full-scale engine hardware to examine the characteristics of an annular
combustors showed sharp rise of pressure oscillation within the narrow range of
equivalence ratios between 0.41 and 0.42 as shown in Figure 2-15. This is similar to the
simulation result as shown in Figure 11.
In summary, these experimental studies support the characteristics of the heat release
dynamics model:
1)
As the equivalence ratio decreases or the mass flow rate increases, the system
becomes unstable. The transition seems abrupt.
2) The instability is due to a sudden phase change near the lean blow-out limit.
While gain increases there as well, it cannot explain mode switching.
3) The combustion instability region is narrow (Ao~ 0.1), and it exists just before
the lean blow-out.
32
-- ------- --
SUG? ...
50k
OAL
a.
.50
I
so
C
.5~I
VVVV
601
1A A h AA AkA
A A
V 'jyV V VVVyVVVV
AA
ci Dhk AAA AWA
:1
0
20
40 60
0140T100
00 106
1600M
T :0
.06
*, 0.72
905K
T
-145 4z
us5 Ht
110 H2
A
A
Figure 2-13 Pressure amplitudes in a combustor at various equivalence ratios [ 3 ]
too ---f
A:
---iVV'JVVVVV'J
VVjvyjvpjx
--
r40
A ~4
I
n I ----4-
460
so
p'~
.~~
U
VA
VVVVVUUV U
0
.0
V VV
20
lj Uj'-.
'TVVVTIIU
4060W10
120 140 10018020
rv~aims)
,,,0.64
133 HZ
--0.64
T 1 ' 4lrns
133 Hi
4" 0.64
Tt39 cm
in! HI
4'b 0.64
'f,33 w4
24 Hz
Figure 2-14 Pressure amplitudes in a combustor at various flow times [ 3 ]
33
te
V.
I.$
1.5
1.4
_rwfm
_M
142
11
IA
O's
OA
4
0,42
0.43
Eqdvalnc0 Rao
0.44
0.45
0.48
Figure 2-15 Change of pressure amplitudes near the lean blow-out limit [ 10]
2.6 Thermoacoustic Instability Simulations
The model presented in Section 2.2 can be used to predict combustion instability once
an acoustic model is derived. Using a Galerkin approximation [11]
-[
12 ], we express
the unsteady pressure p' as:
n
p'(x,t)= P
y/j x77;(W ,
where V/i (x) and ;i (t) are modal shape and amplitude. Assuming that one acoustic mode
is dominant, and that the heat release is localized at
x= xf
, the amplitude this mode can
be shown to be governed by (see Ref. [ 13 ]):
d 2q)
dt2
2
=E-I1(xf)ayr
/-"7
(2.23)
dt 3
34
where w is the acoustic frequency, a,=
and E=
(x)2dx.
Using the configuration
of the LSU-swirl stabilized combustor [ 14 ], in which co = 1257rad / s for a quarter mode,
L =0.6m ,
Xf
=0.03m , A=0.0196m2 , y=
1
.4,
pi =0.6kg/r 3 , k=2.618 and p =1atm , the
following acoustic model is obtained
r=F(s)'=
2
(2.24)
0.0133s
s 2 +1.579 x10 6
The feedback relationship between Q' and p' can be obtained as follows:
dependence of 0' on t'
can be expressed using equation 2-15.
The
Moreover the
relationship between i' and p' can be expressed using the momentum equation and
equation 2-22 (see Ref. [13])
rh'=piAv'= piA I d V k-2 drq
Y dx X
dt
(2.25)
where p, is the density, A is the cross sectional area of the combustor, and k is the wave
number. Using the data of the LSU combustor, we get
th'= -2.512 x10_4 dq
dt
(2.26)
The parameters a and 6 in the heat release model are evaluated for two different
operating conditions. In both cases, T; = 600K and
= 0.6, while for
Case 1. mT /V =100kg /M 3 s,
J(s)- 6.6x 106
s +5594
(2.27)
,
and to
35
Case II.
J(s) =
i / V = 230kg / m's,
-3.475x105
.+75
s+746.5
(2.28)
Using equations (2.24), (2.26), and heat release models, we develop the combustion
feedback system shown in Figure 2-16. For the given data, the maximum reaction is at
T** =1605K , while the blow-out is at T* = 1555K.
The equilibrium temperature is 1815K
in Case I and 1588K in Case II. Note that the equilibrium temperature is T** < T in Case
I, while T* < T < T**in Case II. Figure 9 shows that in Case I Zp'-Q,' =-100 0, and in
Case II Zp'-0' =0 . Therefore, one can expect stable operation in Case I and pressure
oscillation in Case II based on the Rayleigh Criterion.
This is supported by the
simulation as shown in Figure 2-17 and 2-18.
Acoustics
0.0133.s
52 +1.579 x10
WSR model
J(s).512
x10
Figure 2-16 The combustion feedback system with the WSR model
36
x 10-
32 i 0ill
iiii 1..
1
0.
CO
1-)
0
U,
-1
-2
-3-
0
1
0.5
2
1.5
time(s)
Figure 2-17 Simulation of pressure oscillation in Case I
4
x 10~
3,F
2k
C20
If!
~
r'I~
II
)
I
~
-2
0
0.1
0.3
0.2
0.4
time(s)
Figure 2-18 Simulation of pressure oscillation in Case II
37
0.5
As shown in Figure 2-19, the same trend is observed in LSU experiment.
As
mi,
increases or # decreases, the magnitudes of the pressure oscillations increase.
--
1.5
CIO
0~
.. -..
C2L
..
0
0
200
2
60
mass flux (scfm)
80
3
Figure 2-19 Pressure oscillation map in LSU swirl stabilized rig [14]
2.7 Numerical calculation of the WSR model based on
multi step kinetics
The WSR model is depend on the reaction mechanism. If one uses a more detailed
reaction mechanism, the WSR model will be more accurate. In this section, 4 step
38
mechanism [ 15 ] is used to get a WSR model. For C3H8, the following 4 reaction step
is used.
-+3/2C2H 4 +H 2
C 3H 8
C2 H 4 -> 2CO+ 2H
CO+1/20
2 --+
2
CO 2
H 2 +1/202 -+ H 2 0
In this case, the order of WSR model can be up to 5 because of 4 reaction rate equations
and 1 energy equation. To get a linearized WSR model, numerical differentiation is used.
The detailed calculation method is described as follow:
x=[YCH
YC2H 4H
02
2
CO 2
H 2O
(2.29)
T
(2.30)
Q,. = h (x,M-)
(2.31)
Equations (2.30) and (2.31) can be acquired by the 4 step kinetics mechanism. To
get a linearized equation in s equilibrium point, Jacobian is used as follows:
a
OYC3 H8
C2H4
Of8
Of8
(2.32)
018
C H
2
rh
OT
4
aYC^H
IrOyC3HS
ah
x'
08
+
Oh
_._
m.
(2.33)
39
The transfer function J(s) can be calculated as follow:
J(s) = [C(sI - A)-' B + D]
(2.34)
where
Of,
afY
aC 3 H8
1YC
C2H,
aT
,C=
B a8
aC3H8
and
af
ay
8
C 2 H4
Of8
af8
OT
_
h
aYC3H
Oh
- T-
-Ei
L
ah
D= .
(2.35)
As one can see in Figure 2-20 and Figure 2-21, the phase change is gradual
instead of nonlinear 180 degree change at T**. One can expect the structure of the
numerator is changed. The WSR model acquired by 4 step kinetics has 5 poles and 5
zeros as shown in Figure 2-23, and the frequencies of 4 zeros and 4 poles are over
10kHz, so the effect on the system is negligible. Therefore the WSR model can be
represented one pole and one zero model. Figure 2-24 shows those pole and zero
whose frequencies are comparable to acoustic frequency.
The difference in heat release model is that the zero is included. It means the
180 degree phase change is dependent on the frequencies. One pole and one zero
moves from negative real axis to positive real axis as the mass flux is increased or
equivalence ratio is decreased as the pole following the zero.
40
phi=0.6
phi=0.7
phi=0.8
50
40-
0
--....-.-
1. ..200
400
80
600
mass flux(kg n:s)
1000
Figure 2-20 Phase of WSR model in 4 step kinetic at fixed
41
1400
1200
#
2400
phiO".6
phiO0.7
phi=O.8
-
0 --2
2000
4N
N8N N2
4
81:800,
E
1800
1400
A:
1200.
.0
200:
400
II
800
600
3
mass flux(kg/m s)
00
1200
Figure 2-21 Gain of WSR model in 4 step kinetic at fixed
10
#
In this case, the maximum reaction point is when the zero of the heat release model
becomes zero value. The blow out point is when the pole of the heat release model
becomes zero value (same as one step kinetic model).
As the zero moves from high
frequency to low frequency as the equivalence ratio or residence time are decreased,
the zero affects the high acoustic frequency mode first to change the phase, and the
low acoustic frequency mode gets phase change after the high frequency gets phase
change. It implies that the heat release model itself will react to the acoustic modes
differently. However, the physical explanation of structure change of the heat release
model is not available, and it needs to be addressed.
42
Pole-zero map
x
0
4.
E
0
_61
-.
16
............ ............
...... ......
......
.......
-----. --...
...
---...............
-14
*12
-8
4
-i(
..................... ...
......
-6
Real Axis
-2
0
Il)
Figure 2-22 Pole and zero map of WSR model in C3 HA 4 step kinetics at
m
= 732kg/m 3 s and 71 =600K
43
#i = 0.7,
Pole-zero map
...... .....
I ..
.......
. ....
....
I
E
-8**
- 1 0)
-2000
------- - -----
ThOO
-1000
-500
.......-
.4-
0
............. ; .................L .................................... 1:
504
1000
150"
200')
AniY'ds
Figure 2-23 Magnified Pole zero map around acoustic frequency of WSR model in
= 732kg / m 3s and T =600K
C 3H 4 step kinetics at t =0.7,
44
3. Nonlinear Heat Release Dynamics in a Wrinkled
Thin Flame
3.1 Introduction
Thermoacoustic instability occurs due to the coupling between acoustics and heat
release dynamics as can be seen in Figure 3-1. The former is the traditional generation
and sustenance of acoustic oscillations via expansion generated by heat addition, while
the latter is the generation of oscillation in the combustion process, or heat release rate,
through oscillations in the pressure or velocity field in its vicinity.
velocity perturbation
Acoustics
Release
Heat Release Perturbation |Heat
Figure 3-1 Thermoacoustic instability feedback diagram
Due to the complexity of combustion, it has been most challenging to capture these
oscillations in models that can be used to analyze thermoacoustic instability and design
control algorithms. Modeling of thermoacoustic instability has been carried out in many
ways in different conditions. One example is at a high Damkohler number, where the
flame thickness is assumed to be infinitely small, and laminar flow rates, a linear heat
45
release model is acquired by area variation of a flame [ 16 ].
However, the linear model is limited in its scope. As shown in Figure 3-2 , which is
measured in the 1Kw combustor at MIT, the pressure oscillation reaches a limit cycle
instead of unlimited growth of the pressure oscillation. Therefore, to explain the limit
cycle behavior of the pressure oscillation, it is important to examine the characteristic of
the nonlinear heat release model and determine the mechanisms that generate a limit
cycle. In this chapter, we examine the nonlinear heat release model and try to understand
the limit cycle phenomena.
46
1I0
100
-50-
S
0.04
0.06
008
01
012
014
016
018
02
0.2
0.04
08
0.08
01
0.12
0.14
016
0.18
0.2
0.02
0.04
0.06
0.08
0.1
%mes (S)
0.12
0.14
0.16
0.18
0.2
.8
002-
0 0g
0.01
-
0es
=002-I
.03 .....
0
Figure 3-2 Initial Growth and limit cycle phenomena
47
3.2 Linear Heat Release Model
In this section, we review the linear heat release dynamics model at high Damkohler
condition [ 16 ], and discuss its limitations.
Assuming that the flame is axisymmetric, the flame surface is described by the
following nonlinear wave equation [ 18 ],[ 19]
= ---
- S
(3.1)
+
where - is the flame surface z-coordinate, (u,v) are the velocity components in the axial
z and radial r directions, respectively, t is time, and Su is the laminar burning velocity.
We also assume that the flame surface is very weakly convoluted, i.e. that 4'(r,t) is a
single valued function. The heat release rate is given by the intergal over the flame area:
R
Q = 21p
,rSphr
(3.2)
+1dr
0
where R is the radius of the flame base, p is the reactants mixture density and Ahr is the
enthalpy of reaction per unit mass of mixture.
48
I
VF
V
U
z Reactams
ProducLs
F<o
I
-IO
Flame surface
Figure 3-3 Definition of variables used to describe flame surface kinematics [ 16 ]
To examine the growth of small perturbations along the flame front, Eqs. (3.1) and
(3.2) are linearized by considering the effect of small perturbations (u ,V ), around the
mean (u, v), on the deviation of the flame surface
4
from the mean
After some
manipulations, we find that the flame surface and heat release rate perturbations are
governed by:
aU
L
andy =KfrSu
0
dr
j
Su1(0
(3.3)
Q
(3.4)
/
-
where K = 27fpAhr and q7(()= - /
dr (dr}
2
+ 1. The average flame location is governed
by:
49
-2
U
-
+1 = 0
-S
dIr
U (d)
(3.5)
For the case in which (1) v<<u (a boundary layer approximation) and (2)
d{;
-r > 1 , r
dr
-+ 1, and the equations governing the average flame shape, perurbations in
the flame shape and heat release rate reduce to;
--
2
-1
_
dr
S
(3.6)
,
-
d
(3.7)
9;= ai -SU 0, 1
and
Q
=
KcfrSu
0
(3.8)
-- dr
O
This situation is consistent with the condition that:
(3.9)
u/Sn > 1;
which can be used to further reduce Eq. (3.6) to
(3.10)
r= +u
dr
~s;4
Solutions of Eq.(3.7), for constant laminar burning velocity, can be obtained for the case
in which
50
U'=
&Um.sin(Ot)
(3.11)
where un is the mean average velocity, and e is the amplitude of the perturbation. In this
case, taking {'(R) = 0, we get
4'
R
(3.12)
cos [cot - G 1-;)]- cos ct}
=-
R _
where the flow Strouhal number is F = wR/ urn and the flame Strouhal number is
G = wR / S,. Substituting in Eq.(3.8), we find that the heat release perturbation is,
UG 2
-Rc2(
1 - cos G)sin ot - Gyl -
sinG>]
sj
G
(3.13)
As shown in Figure 3-4, the results in Eq. (3.13) can be approximated accurately in the
frequency domain using the simple expression,
+G(3.lz
H(jG)~
)
with Go ~ 2. In the time domain, Eq. (3.14) takes the form;
dQ'
dt
+W
fQ
=
(3.15)
guf
where co = G S, / R and gf =
d2 cjpAhr.
51
0
-
0
-*--
388
-10
-20 -
-30
.
-
-80
15.55
-19.44
-
-
-
-
-
-
-23.33
inodel
-
-
-27.22
- -
-
-
-
a
-90
101
-
Flame model
-
.-
-
-
Appr xrha
-70
-
-
- -'-
-60 -
-11.68
- - --
-40
-50 .
-7.77
-.-....
-- -
A
-31.11
35
100
10
G
Figure 3-4 Frequency response gain and phase of a premixed flame within the linear
region
The heat release dynamics model described by Eq. (3.15) shows that the phase
between the flame response and the applied velocity perturbation is, as expected from a
linear model, independent of the velocity amplitude. This leads to a growing instability
in which the velocity amplitude increases indefinitely when the heat release model is
coupled with acoustics. The growth of Iu' I will eventually violate the conditions of the
linear analysis and will require looking at the nonlinear response of the flame surface to
velocity perturbation. In the following we investigate conditions under which this may
occur, and its consequence on the phase relationship.
52
3.3 Nonlinear Heat Release Model
The nonlinear wave equation as in Eq (3.5) has several nonlinear characteristics. These
nonlinear characteristics may lead to the limit cycle. The nonlinear characteristics that
need to be considered are as follows:
1)
The burning velocity, S., is function of a flame temperature. The temperature of
the flame is determined by equivalence ratio and a heat loss. The heat loss can be
changed if the distance between the flame holder and flame changes. Also, the
flame temperature can rise or drop locally if the curvature of the flame changes.
2) If the perturbation is large enough, the flame can be detached from the flame
holder.
3)
-
dr
> 1 condition cannot be applicable in the large amplitude velocity change
region because larger velocity oscillation leads to strong perturbation of the flame
area, therby at a certain time the flame can be more flatter.
If we assume that v
=u(r)-Su(T)
term in Eq. (3.5) is negligible, it becomes
+1.(3.16)
where T is temperature of the flame. Su (T) can be computed using the heat transfer
equation. However, this can make the problem complicated. Instead, one can get SU by
the mean flame profile g using Eq. (3.6), and the perturbation term, s',
approximated by the perturbation of the flame location
53
;'.
can be
3.3.1
- and s,
To compute 9, one needs to know the mean flame profile, ;7.
Figure 3-5 shows
general flame shape. Using the flame shape as shown below, one can expect the two
boundary conditions at r=R, which is ;(R) =0 and dR =0.
dr
Also the flame profile
should be symmetric. On can approximately represent g(r) using polynomials and the
boundary conditions.
4.5
-
4V
3.5 L
,/1
3-
~
/
2.5
/
12
a) 1.5-
/
1-
0.57
0-
-0.5
0
r/R
0.5
1
Figure 3-5 Flame shape in a Duct [ 20 ]
Using two boundary conditions, the lowest order polynomial can be acquired in the form
of
54
g(r)= h (
)4 -2(
+
r)2
-
(3.17)
where h is the height of the flame, r is the normalized distance from the center, and q
R
is the flame height. Figure 3-6 shows the flame shape acquired by Eq (3.17). Then SU is
computed using Eq (3.6). Figure 3-7 shows the mean burning velocity profile. It should
be noted that the burning velocity rapidly increases near the center. This may due to the
curvature of the flame shape. The concave shape will increase the temperature of the
reactant, thereby increasing the flame temperature and the burning velocity. The gradual
temperature increase around r/R=0.5 may due to the distance between the flame holder
and the flame.
interpolated data
4
3.5
3
2.5
k
2
CO
0)
N
1.5
/
0.5k
/
/
/
0
-1
-0.5
0
OR
0.5
1
Figure 3-6 Approximated Flame shape using the 4th order polynomial function
55
Su
1.61.4
1.2 k
1
0. 8
0.6
/
0.4
/
/
K
K
0.2
-1
-0.5
0
r/R
1
0.5
Figure 3-7 Burning velocity profile using approximated flame profile and Chebychev
differentiation N= 1000.
Now the perturbation burning velocity, S' , needs to be computed. The burning
velocity is function of temperature of the flame as mentioned above, and the temperature
is function of distance between flame and the flame holder, and curvature of the flame.
On can approximately represent the these two effects by linear combination of the two
parameters as follows:
2 /3 2
SU =
(g-
where
K
ar ar
)2/3
a
ar
T
/(1+(
a4
ar
2)2/3
(3.18)
)
represent sensitivity of Su' on the distance between the flame holder and the
flame, and p represent the sensitivity on the curvature. Both
K
and p need to be
calculated by solving a heat transfer equation. However, let's assume both values to
simplify the problem. S' value have significant meaning on stability of the equation.
The
K
value is related on the transverse movement of the flame, and it tries to move the
56
flame back to the original equilibrium location. If
thereby reduces q to
.
>0, and q > 7, S; becomes positive,
K
If q < , S; becomes negative, thereby increases q to
.
The
curvature effect on the SU is to converge the wrinkled flame to the original curvature.
Figure 3-8 shows the mean curvature profile. One can expect that this curvature effect
plays a role at the tip and the tail especially.
20
I
I
I
I
I
I
I
I
1510-
/
5
ci)
/
/
7/
0
C,
/
-5
7
/
-
/
/
/
-10
-
-15/
/
/
/
-90
0
0.1
0.2
0.3
0.4
0.5
r/R
0.6
0.7
Figure 3-8 Mean curvature profile
57
0.8
0.9
1
3.3.2 Time Stepping Algorithm and Spatial
Discretization
Time stepping algorithm is important because large time stepping can produce
numerical instability and unsophisticated time stepping algorithms also requires small
time step value, which requires longer simulation time. In this problem, also the
ambiguity of the governing equation, e.g. S,
makes the problem more complicated. In
the simulation, we cannot distinguish whether the divergence is from large time stepping
or wrong selection of S,. Therefore, it is appropriate way to use unconditionally stable
time-stepping algorithms. In this simulation, I used Implicit method using Jacobian
matrix [ 21]. The procedure is described as follow:
Aq(x)=
ni
(x)-
(x)
(I- At J)[Ag]=[Atf. + (At)2 f
where J is Jacobian matrix, and f = L. Note that q and f are arrays. For At, it needs to
dt
be small enough to capture the detailed flame motion.
As shown in Figure 3-7, the S, profile increases rapidly near the r-0. It means that
the S, value has large amount of high frequency content. This sharp peak can limit the
accuracy of the spectral method, and spectral method may give the same accuracy with
increased computation time. Therefore, in this study, centered difference Finite
Difference Method is used.
58
3.3.3 Simulation
with
Results
a
Boundary
Fixed
Condition 2
First, assuming the flame is attached to the flame holder ( g(R) = 0) , the
nonlinearities of the equation (3.16) is examined by simulation. Figure 3-9 and Figure
3-10
show
the
overall
flame
motion
and
perturbation
onlly
term
when
u'= 0.3 -um sin(2953 -t) , K = 0.5 -u max/(h /R) and p = 0. 5 . Figure 3-11 and Figure
3-12 show the same values in the polar coordinate. As shown in Figure 3-11, the height
of
the
u'=0.3 u,
flame
sin(1 800)
becomes
.
The
maximum
velocity
between
u' = 0.3 -u,
perturbation
is
and
sin(90')
maximum
when
U'= 0.3 urn sin(90'). Therefore, the heat release perturbation lags velocity perturbation
between 0' to 90'. Figure 3-13 shows the exact phase between u' and q', which is 36'
in this case. It is interesting to note that the flame move differently at the tip and the tail
as can be seen in Figure 3-12. The tip lags velocity perturbation around 90', however
the flame motion in the middle moves more in-phase with the velocity perturbation,
thereby generating nipple shape at u' = 0.3- urn -sin(180*).
Figure 3-14 clearly shows
that the flame moves differently at the tip and the tail.
2 Parameters
used in the simulations are the same as the MIT
59
lkw rig, and are described in Section 3.4.
61
0
2
x 10
0.6
40.
0.2
time(s)
8
r/R
0
Figure 3-9 Overall flame shape change, u'= 0.3 -u max- sin(2953 -t) ,
K =0.5
-u max/(h/ R), p =0.5
60
0.
0.5-
00.60.
N
00
-0.5
2
4
r/R
6
x 10-
time(s)
Figure 3-10 Perturbation term change u' = 0.3 -u max. sin(2953 t)
K = 0.5 -u max/(h/ R), p = 0.5
61
0
u', 90 deg
u',0 deg
<ID
N
0
1 0 0
1
--
1
u', 270 deg
u', 180 deg
5 -
1
0
-1-1
1
1
Figure 3-11 Overall flame shape change in polar coordiante, u' = 0.3 -u max. sin(2953 -t)
, K =0.5 -u max/(h / R), p =0.5
62
u', 90 deg
u',0 deg
0.2 - --..
0.2
-0.2 --
-0.2
-
N
0
1
0
I
u, 270 deg
u', 180 deg
0.2
-
--
-0.2
-0.1
0
-1
-1
1
-
Figure 3-12 Perturbation term change in polar coordinate, u' = 0.3 -u max sin(2953 -t)
K
=0.5 -u max(h / R), p =0.5
63
.-
- -- ---
1,j
0.5#-
-
1.,.
---
--
--
- ----- - --
------
I
-
3
45
6
7
x 10 -
time(s)
1L
Figure 3-13 Phase between u' and
i0.5.
.
-
q'
- u max/(h
u0.3
u max
-
sin(2953 t)
R),
-1.5
-
p =0.5
--
-~
ies
x
tail
1.54I
bewe
-~-
0
~f-
-0.5
Fiur3-14~ Phs
I
I
I
xA
'4n4lm*mto
tth
i
n
tail. u' =0.3 - umax- sin(2953 t) , K = 0.5 - umax/(h /R), ui =0.5
64
1y0
tip
Figure 3-15 and Figure 3-16 show that in the fixed boundary condition, the nonlinear
flame model can not represent the limit cycle behavior. To reprsent the limit cycle, the
simulation model should show the phase change in Figure 3-15 or saturation in Figure
3-16. However, the phase change is less than 20, and the saturation is not shown in
Figure 3-16. It implies that the nonlinearities such as burning velocity perturbation and
1+
)2
term cannot generate limit cycle.
40.2
I
I
I
I
0.35
0.4
40
39.8
39.6
-
N
N
N
N
N
N
-
N N
N
NN
39.4
D 39.2
N
.
39
N
N
N
N
38.8
38.6
38.4
38.2
0.05
0.1
0.15
0.2
0.3
0.25
u/umax ratio
0.45
Figure 3-15 Phase change by the magnitude of u'
65
0.5
2.5
0
'D20,
CO,
E
1.5
.
Ca,
.0
a.
0 .5 0
0
.01
.5
02
0.15
0.2
02
.
y.
.5
04
04
.
0.35
0.4
0.45
0.5
0
0.05
0.1
0.25
0.3
u/umax ratio
Figure 3-16 Gain change by the magnitude of u'
Figure 3-17 and Figure 3-18 show the impact of K and p . Increasing
K
affects the
flame move more fast to the imposed perturbation, so it decreases the phase difference
between u' and q'. However ,the effect of p on the phase is negligible.
66
75
70
V
65
60
C:
CU
55
C
CL
50
I
45
40A
35
0
0.1
0.2
0.3
0.4
kappa
0.5
0.6
0.7
0.8
Figure 3-17 Effect of K on the phase releationship
38.85 -/
38.8
7
38.75,\
~0
C
38.7
\
38.65
38.6
Ca
.C
38.55-
7
7/7
---- 4'
38.538.45
0. 1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
mu
Figure 3-18 Effect of p on the phase releationship
67
1
3.3.4 Simulation Results with a Moving Boundary
Boundary Condition
In the previous section, it is observed that the gain or the phase of the heat release
model do not change by the magnitude change of the imposed velocity perturbation. It
implies that the heat release model cannot explain the limit cycle phenomena in the fixed
boundary condition.
In this section, it is assumed that the flame is lifted when the
velocity perturbation is over a certain value3 , and the boundary condition is changed to
a -(R) = 0. We know that the flame moves differently at different
positions as shown in
at
Figure 3-14.
The changed boundary condition will increased the amplitude at the
boundary, and it may lead to the phase change. Figure 3-19 shows the change of the
phase when the magnitude of u' changes. The magnitude of the phase change is
increased to order of 4 compared to the fixed boundary condition 4 . Also, we can expect
the different value of
certain the
K
K
depending on the flame location may increase the difference. It is
value will be larger at the tail than that at the center because the heat loss at
the tail can be drastically changed by the lifted flame. The lifted flame loses the heat to
the air, whereas the anchored flame loses the energy to the solid plate whose heat transfer
coefficient is much higher than that of air. It requires spatial variation of
requires the solution of the heat transfer equation.
3 In this paper, it is assumed the flame is lifted when U' > 0. 1 -Ur
4 In the experiment, it is observed that the phase changes about 300.
68
Kc,
which in turn
78
77
76 V
75
0)
a 74
m
IL
73-
72
71
70
0.35
0.3
0.25
0.2
0.15
0.1
0.4
uprime/max(u)
Figure 3-19 Phase between u'and q'. K = 0.1-u max/(h / R), p = 0.1
u', 90 deg
u',0 deg
0.5N
-0
-1
-1
1
-0.
1
u', 270 deg
u', 180 deg
Ar
5
0.5N
-0.5 ----
-
0
---
0--
-.
1
Figure 3-20 Perturbation term change in polar coordinate, u'= 0.3 -u max- sin(2953 -t)
K
=0.1-umax/(h/R),u=O.1
69
3.4 EXPERIMENTAL MEASUREMENTS
3.4.1 SET-UP
The combustor rig is illustrated in Figure 3-21. It consists of an air supply through a
low-noise blower, a settling chamber, a rotameter for adjusting and measuring the air
flow rate, a fuel (propane) supply through a pressure regulator and another rotameter. The
cold section is a 5 cm diameter, 26 cm long tube. At the downstream end of section, the
flame is stabilized on a perforated disc with 80 holes, 1.5 mm in diameter each,
concentrated in a 2 cm diameter circular area. The flame is contained in the hot section
which consists of a 22cm long pyrex tube to allow visual access to the burning zone.
Several ports exist in the cold section for mounting sensors. Measurements on the test rig
are recorded using a Keithley MetraByte DAS- 1801 AO data acquisition board with a
maximum sampling frequency of 300 KIz. A software package, ExcelLINX, is used for
data processing. The board is hosted in a Pentium PC. The combustor is equipped with
several sensors including a Kystler pressure transducer, a TSI hot-film anemometer, and a
Hamamatsu photodiode to measure the dynamic pressure, velocity and heat release
(through light intensity). The first two are measured from a port which is 1 cm upstream
the perforated plate (in the cold section), the latter is pointed at the flame from outside the
pyrex glass at a distance of 0.5 cm. This allowed it to measure an integrated value of the
light projected from the side-view area of the burning zone.
70
Combustion products out
t
Pyrex tube
Photodiode
Flame
Propan C
Air blower
Pressur gauge Rotameter
Pressure
\Hot-filxn
anemometer
transducer
I
Reactants
Figure 3-21 Schematic of MIT 1kw rig.
3.4.2 RESULTS
The combustor exhibited an instability at 490 Hz with steady-state pressure
amplitudes of 0.1% of the atmospheric mean and velocity amplitudes of 30% of a mean
velocity of 0.16 m/s in the cold section. For this condition, the equivalence ratio, 0, was
0.7. The acoustic boundary conditions are closed upstream end and open in the
downstream end, and the unstable frequency corresponded to a three-quarter-wave mode.
Our interest was to understand the dynamics that lead to limit cycles through changes
in the heat release in response to growing pressure and velocity fluctuations. To realize
an experiment that captures the changes in the dynamics of heat release/acoustics when
the system is in transition from small perturbations (linear growth) to sustained
71
oscillations (nonlinear limit cycles), the following procedure is implemented: (i) We set
#0.68, which
corresponds to a stable operating for this combustor. (ii) The fuel flow rate
is suddenly increased to 0=0.7 which corresponds to an unstable operating condition. At
this instant, we start recording measurements for the pressure, p',the velocity, u', and the
heat release, q', which increase through a transient until it settles to a limit cycle as
shown in Fig. 3-2. The data is then filtered around the unstable frequency to remove any
possible low frequency noise (e.g., from the blower or other electrical devices
surrounding the rig).
The experimental results agrees with simulation results in Section 3.3.4. The data
obtained from the experiment show that the limit cycle phenomena is from the phase
change between u' and q' as shown in Figure 3-22.
Figure 3-22 shows the phase
between u' and q' in the initial grow and limit cycle range. At the initial growth range the
phase between u' and q' is around -60* and it changes to around -200.
Using p' and
u' relation from the momentum equation, one can get the phase relation between p' and
q'as shown in Figure 3-23. In Figure 3-23, the phase between p'and q'changes from
-300 to -700 .
As explained in Section 2.1, the Rayleigh criterion implies that the
system is more unstable in the initial growth range. Even thought, p' and q' are slightly
in-phase in the limit cycle region, the pressure amplitude does not increase. It may due to
the dissipation energy that needs to be compensated by the energy source. Figure 3-24
shows the gain change between of u' and
q' oscillation as the magnitude of u'
oscillation changes. Saturation, which can also explain the limit cycle phenomena [ 22] [
23] is not observed in this experiment as shown in Figure 3-24.
72
Phase difference between q' and u'
0
-10 ------------
- ----------
-
- -- ---- ----- -----. --.
-20 - - -- - --
-30 --
-
-
-40~~~11
--+ -4
-40
-80
--
- -... -
......
11.....
.. .......
--------
o
0-6
- - ---- --------
0
----------
-70
-70
- ---- ------ - --
- - -------
+
50
-GO
---------- - -------
+--------+
o
--------..
. . . .. . . ---- -- - - - -------. ...--+ ---
-------------------
---
005
0.1
0.15
0.2
---- ---
-------
---
0,25
0.3
0,35
0.4
ime (S)
Figure 3-22 Phase change between u' and q' in the initial growth and limit cycle
region
73
----
Phase difference between p' and q'
0
;4
-10
++
420
---- 4 -. . .+.
-30
-- - -- - - - - - - - -
C
00
P-40
-o
------- - + -- --------------- ----------- ---- - -------
0
-s0
++
++
- ------- + ---
-60
---- ------------------------------- -----------
+
+0+
0
0.1
0.2
04
0.3
0.4
-
---
-
0.5
0o
0.6
0.7
IuImm/s)
Figure 3-23 Phase change between p' and q' in the initial growth and limit cycle
region
74
0.025
---------------
0.02
#***
**
C
I0.015
*
-
-
M: 0.01
0006
I-
010
0'1
0.2
04
0.3
0.5
0.6
0.7
0.8
lu'l
Figure 3-24 Gain change between u' and q' in the initial growth and limit cycle
region.
75
4. System Identification Based Modeling of combustion
instability for Turbulent Combustor
An optimal controller for a 30 kW swirl stabilized spray combustor using a systemidentification (SI) based model is developed. The combustor consisted of a dual-feed
nozzle whose primary fuel stream was utilized to sustain combustion, and the secondary
stream was used for active control. An LQG-LTR (Linear Quadratic Gaussian-Loop
Transfer Recovery) controller was designed using the SI based model to determine the
active control input, which was in turn used to modulate the secondary fuel stream. Using
this controller, the thermoacoustic oscillations, which occurred under lean operating
conditions, were reduced to the background noise level. A simpler time-delay controller
was also implemented for comparison purposes. The results showed that the LQG-LTR
controller yielded an additional pressure reduction of 14 db compared to the time-delay
controller. This improvement can be attributed to the added degrees of freedom of the
LQG-LTR controller that allow an optimal shaping of the gain and phase of the
controlled combustor over a range of frequencies surrounding the unstable mode. This
leads to the observed further reduction of the pressure amplitude at the unstable
frequency while avoiding generation of secondary peaks.
4.1 Introduction
For intermediate Damkohler number condition where flame vortex plays a role, the
WSR model and the thin flame model presented in Chapter 2 and 3 are no more valid.
Due to the spatially changing heat release characteristics, it is more difficult to get a
reduced order model, which captures the dynamics of the heat release dynamics in a
simple form. Because of this difficulty, a simple time delay controller has been used [ 24
]-[ 28 ], which can be applied without a model of the combustor. However, this time
76
delay controller often generates secondary peaks [ 29 ], and the performance is limited by
the restricted degree of freedom of the controller parameters [ 30 ] . An alternate
approach is to develop model-based active control designs using System Identification
methods (e.g., [ 31 ]-[ 41) to derive the model. The system identification method can be
viewed as a black-box approach where data from the system is used to fit a particular
system model structure, the choice of which is dependent on the main system
characteristics that need to be captured. One of the most important features of the
pressure/heat-flux sensor signal during unstable combustion is the presence of nonlinear
limit cycle oscillations. Following an initial growth in the pressure or heat-flux response,
a limit cycle is established due to the effect of system nonlinearities. In [ 31 ]-[ 34 ],
nonlinear model structures are employed to derive the SI model. In [ 31 ], the authors use
a nonlinear feedback model where the forward loop contains the linear acoustics, and the
feedback loop includes a convective time delay and a nonlinear heat release model. The
parameters of these blocks are then identified separately using appropriate input-output
data sets. In [ 32 ]-[ 34 ], a nonlinear model of the form
77+a 7+bq
=
(
+ g( (t))
is used, where a, b, and f correspond to the self-sustained oscillations in the combustor
and g((t)) denotes the effect of an exogenous random noise
. In [ 34 ],f is chosen to be
a polynomial function, and data from an experimental rig is used to identify the
parameters a and b and the polynomial coefficients. Burgs method [ 35 ] and a leastsquares method [ 36 ] are used to carry out the parameter identification in [ 33 ] and [ 34]
respectively.
An alternate approach can be used to model combustion oscillations. Even though
the combustion response is nonlinear, in an experimental run, one seldom captures the
signal growth within the linear range and transition phase due to its brevity. It is the
periodic pressure/heatflux signal, which is the more persistent feature and the one that is
experimentally recorded. If it is the periodic oscillations that need to be modeled, one
can choose a linear model structure to capture the pressure characteristics. The approach
77
in as well as in this Chapter belongs to this category, where the SI model is linear. The
implication of such an approach is that in a neighborhood of the limit-cycle oscillations,
the SI model can accurately predict the combustor response and therefore can be used to
design a controller that reduces the amplitude of these oscillations. In [ 36 ]-[ 39 ] as well
as in this Chapter, a linear dynamic input-output model structure is chosen as the SI
model, whereas in [ 40 ], a Fourier-series expansion is used to represent the pressure
response. Once the model structure has been selected, several identification methods can
be used to determine the model parameters. In [ 39 ], and in this Chapter, since the
parameters appear as linear coefficients of a differential equation, least squares methods
are employed to estimate the parameters [ 41 ]. In [ 40 ], a nonlinear observer is used to
identify all of the parameters in the Fourier series expansion.
The distinction between [ 36 ]-[ 39 ] and this Chapter is in the process of the
validation of the SI model. In [ 36 ] and [ 37 ], a laminar combustor is used as an
experimental test bed for model validation whereas in [ 38 ], simulation studies were
carried out using a solid-rocket. In this Chapter, as well as in [ 39 ], a turbulent combustor
is the experimental platform for validating SI model-based controllers.
4.2 Experimental Setup
The experiments were performed in a swirl-stabilized combustor operating at 30 kW
heat release. A schematic diagram of the nozzle and the combustion chamber are shown
in Figure 4-1. Air stream with a swirl number equal to 0.8 was used to atomize the fuel.
The air stream entered the combustion chamber at standard temperature, 298 K, and
pressure, 1.01x105 Pa. Ethanol was used as a liquid fuel. It was pressurized to an absolute
pressure of 8.27xl 05 Pa in a fuel tank using high-pressure inert nitrogen, metered, and
supplied to a dual feed nozzle through a tube mounted in the center of the air chamber.
The primary fuel flow rate was kept constant at 2.02 gm/sec and the average secondary
78
fuel flow rate was set to 0.26 gm/sec under all operating conditions. The secondary fuel
stream could be modulated using an automotive fuel injector driven by a signal processor
over a bandwidth of 0 to 400 Hz. The airflow rate was varied between 0.014 and 0.035
m 3/sec. The combustion shell was 0.6 m in length and 0.14 m in diameter. A high
sensitivity, water-cooled pressure transducer was mounted at a normalized axial distance
z/D=1.45, where z is measured from the nozzle base, as shown in Figure 4-1, to measure
pressure oscillation. Light emissions recorded at the 430nm CH wavelength using a
photodiode was taken as a measure of the heat flux fluctuations from the flame. These
signals were then processed in real time using a digital signal processor (DS 1103,
DSPACE, 333 MHz Motorola power PC) to be used in active control.
Combustion
Chamber
P ressure
Sensor
Photodiode
Sensor
J.A
Dual Feed Nozzle
C.
Figure 4-1 Schematic of the combustor
79
In order to investigate the combustor dynamics, pressure and photodiode measurements
were taken at different equivalence ratios. The entire combustor operating envelope was
mapped out as a function of the equivalence ratio, whose value was based on the main
fuel stream. A single peak at 205 Hz was observed over the entire operating range. Figure
4-2 shows a typical pressure spectrum at the unstable condition. The frequency of the
largest amplitude oscillation corresponded to the quarter wave mode of the combustor [
24 ]. The amplitude of this peak varied depending on the Equivalence ratio. Both
and
pressure
heatflux fluctuations were normalized by the corresponding maximum
rms fluctuations and was used as measure of the instability. These are shown in Figure 43. The recorded rms fluctuation of p' varied from 0.2 to 2.7 millibar over an equivalence
ratio of 0.6 to 1.5.
The figure illustrates that both the pressure and heatflux oscillations
are high near the lean blow out limit. An equivalence ratio of 0.7, which corresponds to
peak instability where p'rms=2.7millibar was chosen for the closed loop control study.
1.4-
E
1.2
-c 0.8E
CLO.6-
E 0.4cL 0.2
0
100
300
200
Frequency (Hz)
400
Figure 4-2 Baseline power spectra for $ = 0.7
80
500
1 .00 S0.80
-U
S
-0.60
*0.40
o 0.20
z
0.00
0.5
.6
.7
0.8 0.9
1
1.1
1.2 1.3
1.4
1.5
1.6
1.7
Equivalence Ratio +
Figure 4-3 Normalized p'rms and q'rms as functions of primary fuel equivalence ratio
4.3 System Identification of a Combustion System
System-identification modeling consists of using the input-output data and a
black-box approach to derive the model structure and parameters. A typical system
identification procedure includes (i) model-structure selection; (ii) determination of the
'best' model in the structure as guided by the data; and (iii) selection of an appropriate
excitation signal that includes a wide range of frequencies in order to accurately estimate
the model parameters. As mentioned earlier, a typical pressure response in a combustor
consists of an initial period where the signal consists of diverging oscillations that are
followed by sustained oscillations. We focus on the latter part of the pressure response
and choose a linear input-output dynamic model to describe these oscillations. This
81
model combines the acoustics, heat release, fuel injector and solid state relay into a single
lumped transfer function, which is directly used to design the controller.
With the model-structure selected as a linear dynamic model, we then proceed to part
(ii) of the SI procedure, which consists of finding the most accurate linear model given
the combustor input-output data. In the current system, the input for system identification
is a voltage to the fuel injector, and the output is the pressure signal. The general form of
a linear discrete input-output model is given by
n,
y(t)
=
nk +nb
aiy(t -iAt)+
n,
Zbiu(t -iAt)+Lcie(t
Ii=n
-iAt)
(4.1)
i=O
where u(t) is the voltage to the injector, y(t) is the pressure signal, e(t) is white noise,
At is the sampling time, na, nb , nc and nk represent the number of poles, zeros, order of
noise and delay in the combustor respectively, and ai, b and ci are the model parameters.
We employ a two-level iteration in order to determine these quantities. The first level of
iteration is in the parameter space 9, where
0 =[a,, a2....a
, I b 2....
(4.2)
b,c1, c2..cnc ]
for a given dimension D = [n a Inbn,nk],
and the second is in the dimension space. At
each iteration, the parameters are adjusted so that a suitable error that reflects the model
accuracy is minimized. The details of the two-level iteration are summarized below.
Since the model structure described in equation (4.2) can be used to capture the
periodic nature of the pressure response, our starting point is a model whose output is a
weighted sum of the past inputs, outputs and the noise. We first select a certain value for
D. Denoting y(t 10) as the model output, we choose a model as
82
y(t
10)
(4.3)
T
= O (P(t)
where qp(t) is a regression vector that is a combination of the past inputs, outputs and
noise and is given by
S(t)=[-y(t
At),., y(t -n. At), u(t -nk At),..., u(t -(nk +nb -1)At), e(t - At)...e(t - n At)]T
-
(4.4)
The goal is to find the optimal value of 0 so that y(t 10) predicts the pressure y as
accurately as possible. To achieve this, we construct the error, e(t,9), defined as
A
(4.5)
e(t, 9) = y(t) - Y(t 10)
and a normalized value of the error V(9) given by
N
(y(t,0))2
N
V(O)=
-('1
N
(y(t))2
(4.6)
)
where N is the total number of samples. The SI model is then obtained by minimizing
V(9) over 9. That is,
*D
(4.7)
= arg min(V(0))
and the resulting V is denoted as
VD
(4.8)
=V(9D*)
83
It should be noted here that in order to carry out the minimization in (4.5), sufficient
number of frequencies must be present in the input u so that accurate parameter
identification can be carried out. This corresponds to part (iii) of the system-identification
We note that the minimum error
procedure.
VD
also varies with D. Hence having
determined 9* and VD for a particular dimension D, in the second-level of iteration, we
evaluate O* andVD for different D = [nb ,
n
kn]
to identify the dimension that gives
the best SI model. That is, we compute
(4.9)
V =Min(V )
D
where the best SI model is that which yields V
4.4 Implementation
In order to derive a SI model, an operating condition which corresponded to an
equivalence ratio of 0.7 and p'rms=2.7millibar was chosen. A PRBS (Pseudo Random
Binary Sequence) signal, low pass filtered at 400 Hz, was chosen to drive the fuel
injector so that sufficient number of frequencies are present in the input. The resulting
pressure response was recorded using a pressure transducer, and the corresponding power
spectrum is shown in Figure 4-4. The figure clearly shows a dominant mode at 205 Hz,
the same mode captured in the unforced case. There are two other distinct peaks around
60 Hz and 10 Hz. The 60 Hz mode is due to the inherent electric noise, while the 10 Hz
mode is associated to the injector dynamics. The latter was confirmed by velocity
measurements recorded at the exit of the injector for an input white noise.
84
1 .2 r
I
I
I
*
I
I1
I
a
a
I
0.81------.I
0.6
0..4.
0.2
I
I
a-Ia
I
I
I
*
I
I
I
I
P
1
I
I
----
I
a
a............
a
I
a
*
a
a
a
a
I
a
a----------------I
a
a
a
a
a
a
a
-- aa
a
I
I
150
200
I.
I.
I
I
I
I
*
I
a
a
-
300
250
a.
I
*
p
I
i
I
I
a
a
a
100,
50
0
I
I.
I
I
a
I
0
i
I
I
S
I
~
'*
i
I
I
I
I
a
-
a
I
I
a
I
I
P
I
I
I
I
*
a
I
*.
a
I
I
I
a
I
I
---------------------. 4
*
I
-----
I
a
a
I
a
*I
I
*
I
I
I
I
I
-.------
I
a
S~.-~
I
I
I
I
a
I
I
*
*
a
*
I
I
I
I
*
I
I
I
I
I
I
a--a
a
a
I
I
I
I
I
p
350,
4001
w
Figure 4-4 Power spectra of the pressure signal with PRBS input at q0 = 0.7
1.2
1
--
-
- -
-
-
- -
- -
-
IL
--
-
-
-
0.8
- - - - - -
x
--
-
-
-
I
-
-
-
-
- -
-
-
-
-
- - - - - - - - - - - - - - - - - - - - -
c15
~c0 E 0. 6
- - -
--
- -
-
-I
-
-
-
I
0.4
0.2
0
0
50
100
150
200
250
300
350
-7
Figure 4-5 Power spectra of the system-i denti fi cation model at 0 = 0.7
85
400
The velocity measurements also indicated that the fuel injector has another mode at 300
Hz. Since the goal of the SI modeling was to represent the combustion dynamics, the
fuel injector dynamics was ignored by choosing a band pass filter with a lower and an
upper cut-off frequency of 100 and 300 Hz, respectively.
The filtered pressure signal
was chosen as the output y that had to be modeled. The SI model was then chosen based
on the discussion in the previous section. It was found that the optimal model
corresponded to D
=
[3,1,1,0] , OD* =[-2.44, 2.32, -0.82; 4.6x10 5 ], V(9)
=
12.5% . The
structure of D indicates that a third order model was sufficient to predict the combustor
dynamics. This is also corroborated in Figure 5, which shows the power spectrum of the
SI model predicting the peak at 205 Hz.
4.5 LQG-LTR Control
For a high order unstable system, a classical time-delay controller ( see ref. [ 42 ] and
[43 ] ) is inadequate to stabilize the system because it lacks requisite degree of freedom
in gain and phase. One way to overcome this deficiency is to use the LQG-LTR method [
44 ]. This method provides sufficient performance and robustness over a wide range of
frequencies [ 42 ]. An LQG-LTR controller has the form:
u =-[K(sI - A - BK - HC)
(4.10)
H]y
where the matrices AB and C are obtained from the combustor state-space model, and
the estimator gain, H, and the state feedback gain, K, are to be designed. The feedback
gain, K, is determined using the performance index J given by
86
(4.11)
(YTQY+uTRu)dt
J=
0
(4.12)
Q= I, R = pI
where p is a scaling factor that determines the trade-off between fast transients and the
magnitude of the control input. H can be found in a similar way as K by posing the
problem as the design of a Kalman filter where one introduces input noise with a variance
I and output noise with a variance uI where p represents the model uncertainty. H and
K can then be found by fine tuning p and p using the MATLAB control system
toolbox.
4.6 Controller Design and Implementation
The discrete time combustor model obtained previously is cast in continuous-time
using Tustin's method [ 45 ]. The resulting expression is
2
TF
= - 7.010*10-6 (s - 4000)(s + 4000)
(s2 + 2qw.s + w, 2 )(s + 364.7)
where g=0.0185 and w,=1287 radian/sec. Using this model, an LQG-LTR controller
was designed using MATLAB. The control parameters p and p, were varied to obtain
the maximum attenuation in pressure oscillation. The controller has the form:
0.4963S 2 -1004s-2.53*10'
TFLQG-LTR = s3 +732.
S +1.9*10 6 S + 7.6 *10
with p =l and p
=10-
6
.
87
8
In order to perform real-time control a super scalar microprocessor Motorola power
PC 604e running at 333 MHz and a slave DSP TMS320F240 were used. The latter has 16
input channels and 8 output channels with A/D's at 16 bit and D/A at 14 bit with the
latter having a +-10V range and a 20 MHz clock rate. Code generation, compiling and
downloading was done with SIMULINK and DSPACE real time interface. A sampling
time of 0. 1msec was chosen to implement the control algorithms. The output of the D/A
board was then fed to a solid-state relay to run the automotive fuel injector on the
secondary stream.
4.7 Results
An operating condition corresponding to an equivalence ratio of 0.7 and
p 'rn=2.7mbar was chosen to implement the active controllers. The LQG-LTR controller
resulted in pressure and heat release responses whose spectra are shown in Figure 4-6 and
Figure 4-7. These figures also show the power spectra of the uncontrolled (baseline)
system. The performance of the LQG-LTR controller is also compared with the more
commonly used time-delay controller [ 24 ]. The latter consisted of a filter-time delayamplifier combination, where the filter attenuated frequencies outside the band [150, 350
Hz]. The time delay, i-,p, was varied between 0 and 4.8msec, and the amplifier gain was
fixed at 100. The gain was chosen so as to reduce the pressure to the levels shown in
figure. The choice of the time delay was on an empirical basis. The impact of the time
delay on the pressure amplitude is shown in Figure 4-8. As can be seen in Figure 4-8, a
maximum pressure reduction(defined as p'ns/ p'rms,baseline) of 60 % was obtained at r,=
4.26msec. In contrast, the maximum pressure reduction with the LQG-LTR was 80 %. In
addition, a frequency-domain figure of merit was computed as
88
A
R,. = A mLQG-LTR
P m Phasedelay
A
A
where P =
A
Max P(w) and P denotes the power spectrum of the pressure response.
wE[100,300]
It was found that R,. = 0.22. A similar computation corresponding to heat flux response
yielded Rq, =0.52.
w
.F)
(U
-o
(I)
1.4
-
1.2
-
- -
1-
Baseline
LQG-LTR
Time-dela
I
a
I
0.8-
d
U)
'3
1111
I
0.60.4-
I
L
0.2
I
I ~L~Li
Cs
I
All..
CA
a
~
0
0
100
200
300
Frequency (Hz)
400
500
Figure 4-6 Q'rms spectra at the baseline, time- delay and LQG-LTR control at
89
1.4-
Baseline
Time-delay
-... ..LQG-LTR
-
a 1.2
53
10.8-
E
0.6-
E 0.40.2
0.
0
916.
.
100
AL4."'
300
200
Frequency (Hz)
400
500
Figure 4-7 p'rms spectra at the baseline, time- delay and LQG-LTR control at
90
1.2
.
_
__............~-.
~
.
--
.
-
0.8
0.6
0.4
00.2
-
0
0
0.61
1.22 1.83 2.44 3.05 3.66 4.27 4.88
Time delay (Tos)
Figure 4-8 Normalized p' rms for different time delays at 0 = 0.7.
4.8 Discussion
The results in the previous section show the improvement achieved when using an
LQG-LTR controller, compared to the time-delay controller. In this section, we discuss
possible reasons for this improvement. As will be shown, the time-delay controller adds a
fixed gain and time delay to the pressure signal, whereas the LQG-LTR controller
optimizes the profile of the gain and phase to achieve the desired goal.
By construction, the gain of the time-delay controller is a constant over all
frequencies. To increase the effectiveness of the controller at the unstable frequency, this
gain must be large. At frequencies where the phase of the forward-loop transfer function
of the system together with controller is close to 0' , a large gain can excite the
corresponding frequency. Thus, the gain must be kept reasonably low to avoid exciting
secondary modes. This limits the effectiveness of the controller. On the other hand, the
91
gain of the LQG-LTR controller reaches a maximum around the unstable frequency. This
allows the controller to suppress the dominant oscillation effectively. At the same time,
the gain of the LQG-LTR drops rapidly on either sides of unstable frequency. Since
secondary peaks are generated at points where the gain of the open-loop transfer function
of the system (controller+combustor) is greater than 1 millibar/volt and the phase is near
O' (positive feedback), and since the LQG-LTR controller has a small gain at all values
away from the unstable frequency, the controller prevents the excitation of secondary
modes.
2
------ ----
I
LQG-LTR
-
Time-delay
- - -- - -
-.
1.5
- ---
---
-
C
0.5
150
0
160
-1-
170
180
200
Hz
190
210
220
230
9
LOG-LTR
Time-delay
-.
-180
--
-
------ -
-------
1
-540
150
160
250
240
170
180
190
I
I
200
Hz
210
220
230
Figure 4-9 Bode plot of LQG-LTR and the time-delay controller at b
92
250
240
=
0.7
6
-
--
4-------------------
.F
0
150
170
160
8 -LQG-LTR*combustor
.ime-delay*combustor
-----
180
200
Hz
190
210
220
230
240
4P
180
...
m...
-360
150
250
160
170
180
200
Hz
190
210
LQG-LTR*combustor
Time-delay*combustor
220
230
240
250
Figure 4-10 Open-loop transfer functions of the system (combustor*controller)
The time-delay controller has a single parameter, which is the value of the time
delay, that can be adjusted to affect the slope of the phase, as shown in Figure 4-9. Even
though the added time delay corresponds to a correct phase at the primary mode, it may
give the wrong phase at other frequencies. Figure 4-10 shows the forward-loop transfer
function of the controller together with the combustor. The resulting closed-loop system
can generate a secondary peak with the time-delay controller because the phase crosses
0* line at co = 185Hz. At this frequency, any perturbations present can be amplified if
the gain is larger than 1 millibar/volt. If the gain at this frequency is reduced to be less
than one, the gain plot of the phase-shift*combustor transfer function in Figure 4-10
indicates that the gain at the unstable frequency is also reduced to a value less than 4.8.
This value, however, may be too small for the time-delay controller to be effective
enough to result in pressure suppression. This limitation is not present in the LQG-LTR
93
controller, since as shown in Figure 4-10, the corresponding phase does not cross 00 at
any frequency.
In summary, two properties of the LQG-LTR controller contribute towards not
exciting any secondary peaks. These include: the rapid roll-off of the gain around the
unstable frequency, and the small change of the phase away from the unstable frequency
so as to avoid cross-over of the 00 line, both of which are not present in the time-delay
controller.
Both of these properties are due to the fact that the LQG-LTR controller
allows many degrees of freedom in its gain and phase by virtue of the fact that it has
several parameters (- twice the order of the controller). This is in contrast to the timedelay controller which has only two parameters, the gain and the delay.
94
5. Conclusions
The focus of thesis is modeling of combustion instability in three different regions to
understand the underlying mechanism and control the system to achieve the goals. In
each chapter, different approaches are used to develop models and understand diverse
characteristics. We summarize the main results in each chapter as follows:
In Chapter 2, we obtain a linearized heat release dynamics model based on the
assumptions used in a well-stirred reactor, and express the heat release oscillation as a
function of the mass flow rate. The heat release dynamics model has the form of a first
order filter, having a pole and a static gain. The model captures static blow-out as the
pole becomes unstable, and shows that the phase between mass flow rate and the heat
release oscillations changes by 1800 at the point of the maximum heat release,
corresponding to the change of the sign of the gain. The phase and gain between mass
flow oscillation and heat release perturbation depend on the mean residence time and
equivalence ratio. Phase change occurs soon before blow-out. For certain cases, while it
depends on the nature of the acoustic mode and the location of the heat release zone, the
phase between (P',0,) changes from close to -90* before the maximum reaction point to
close to + 900, following a transition across this point, to around 00 at blow-out as the
residence time or the equivalence ratio is decreased. Based on the Rayleigh Criterion, the
combustor may become unstable due to the positive coupling between the heat release
dynamics and acoustics at the maximum power, or at lean bum condition close to lean
blow-out. Experimental studies ([ 3 ], [ 9 ], [ 10 ] and [ 14]) show similar characteristics.
In Chapter 3, nonlinear heat release model is investigated to understand the limit
cycle phenomena in thermoacoustic instability. Using Implicit method and FDM, the
nonlinear PDE is solved. Three different nonlinearities are examined. 1) S change 2)
Changed boundary condition(Lifted flame) 3)
1+(
ar
)2
term. It is assumed that S,
changes by the change of the transverse location change and curvature change, and the
95
parameters
K
and p are introduced to represent these effects. It is observed that changed
boundary condition may introduce phase change. The magnitude of the phase change
may be larger when the spatially varying
K
values is used.
It requires to solve the
nonlinear flame equation with the heat transfer equation simultaneously. Experimental
results show that the phase change is the cause of the limit cycle.
In Chapter 4, a system-identification method was used to develop a model for a swirl
stabilized spray combustor operating at 30 KW. An LQG-LTR controller designed using
the SI model reduced the pressure and photodiode oscillations to the background noise
level. A simpler time-delay controller was also implemented for comparison purposes
and it was observed that the LQG-LTR controller provided 12-14 db higher reduction
over the former. Analysis using SI based model showed that the LQG-LTR controller
allows many more degrees of freedom than the time delay controller, as a result of which,
the LQG-LTR controller effectively suppresses the pressure oscillations by carefully
tailoring the gain and phase over the entire spectrum.
However, these approaches should be extended further.
For example, in
chemically controlled combustion, the valid range of the model and the appropriate
control algorithm should be investigated. For system identification approach, nonlinear
model should be investigated to improve the model prediction, and justification of the
linear model also should be made clear.
For the limit cycle phenomena, more
sophisticated measurement is indispensable to unveil the dynamics at the onset of
instability and the main mechanism of the nonlinearity should also be addressed.
96
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