DOE / METC-96 / 1026
(DE96004366)
Distribution Category UC-111
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A Model for Premixed Combustion
Oscillations
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Technical Note
Michael C. Janus
George A. Richards
March 1996
U.S. Department of Energy
Office of Fossil Energy
Morgantown Energy Technology Center
P.O. Box 880
Morgantown, WV 26507-0880
(304) 285-4764
FAX (304) 285-4403/4469
http://www.metc.doe.gov/
1
Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States
Government. Neither the United States Government nor any agency thereof, nor any of their
employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product,
or process disclosed, or represents that its use would not infringe privately owned rights.
Reference herein to any specific commercial product, process, or service by trade name,
trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.
The views and opinions of authors expressed herein do not necessarily state or reflect those of
the United States Government or any agency thereof.
2
Contents
Page
Executive Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3
Model
3.1
3.2
3.3
3.4
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8
8
13
13
15
4
Comparisons to Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Open Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Inlet Air Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
16
20
5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
6
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Description . . . . . . . . . . . . . . . . . . . . . . . . .
Nozzle Region . . . . . . . . . . . . . . . . . . . . . .
Tailpipe Region . . . . . . . . . . . . . . . . . . . . .
Combustion Region . . . . . . . . . . . . . . . . . . .
Numeric Solution of the Governing Equations
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List of Figures
Figure
1
2
3
4
5
6
7
8
Page
Model Geometry for the Premixed Combustor . . . . . . . . . . . . . . . .
Details of the Swirl Vane Configuration . . . . . . . . . . . . . . . . . . . . .
Experimental Combustor Configuration . . . . . . . . . . . . . . . . . . . . .
Pressure Versus Time for a Combustor With No Cyclic Fuel Injection
(Experimental) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pressure Versus Time for a Combustor With No Cyclic Fuel Injection
(Numerical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pressure Versus Time for a Combustor With Cyclic Fuel Injection
(Experimental). Injector is Open When Signal is High. . . . . . . . . . .
Pressure Versus Time for a Combustor With Cyclic Fuel Injection
(Numerical). Injector is Open When Signal is High. . . . . . . . . . . . .
Time History of Pressure Signal for a Combustor With Cyclic Fuel
Injection (Numerical). Injector is Open When Signal is High. . . . . .
3
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10
10
17
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List of Figures
(Continued)
Page
9
10
11
Combustor RMS Pressure Versus Air Inlet Temperature
and Experimental) . . . . . . . . . . . . . . . . . . . . . . . . . .
Combustor RMS Pressure Versus Air Inlet Temperature
LPM Air Flow Rates (Numerical) . . . . . . . . . . . . . . .
Combustor RMS Pressure Versus Air Inlet Temperature
Equivalence Ratios (Numerical) . . . . . . . . . . . . . . . . .
(Numerical
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for Various
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for Various
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21
22
22
List of Tables
Table
1
Page
Conservation Equations for the Premixed Combustion Oscillation Model
(PCOM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
9
Executive Summary
Combustion oscillations are receiving renewed research interest due to the increasing
application of lean premix (LPM) combustion to gas turbines. A simple, nonlinear model for
premixed combustion is described in this report. The model was developed to help explain
specific experimental observations, and to provide guidance for developing active control
schemes based on nonlinear concepts. The model can be used to quickly examine instability
trends associated with changes in equivalence ratio, mass flow rate, geometry, ambient
conditions, and other relevant factors.
The model represents the relevant processes occurring in a fuel nozzle and combustor
that are analogous to current LPM turbine combustors. Conservation equations for the fuel
nozzle and combustor are developed from simple control volume analysis, providing a set of
ordinary differential equations that can be solved on a personal computer. Combustion is
modeled as a stirred reactor, with a bi-molecular reaction rate between fuel and air.
Although the focus of this report is the model development, a few comparisons to
experimental data are included. Model and experimental results are used to better understand
the effects of inlet air temperature and open loop control schemes. The model shows that
both phenomena are related to changes in transport time. A subsequent report will provide a
more comprehensive comparison of model results and experimental data.
5
1 Introduction
Combustion oscillations are receiving renewed research interest due to the increasing
application of lean premix combustion to gas turbines. Stationary gas turbines are now
commonly using premixed combustion to avoid the high levels of NOx emissions that are produced by earlier diffusion-style combustors. Beer (1995) and Lefebvre (1995) offer excellent
reviews of the status of LPM combustion technology. While the benefits to NOx emissions
are well established, experience has shown that LPM combustion is susceptible to oscillations.
Oscillating combustion should be eliminated during combustor development because associated pressure oscillations can severely damage engine hardware. As part of the Advanced
Turbine Systems Program (Alsup, Zeh, and Blazewicz 1995), the U.S. Department of Energy
is supporting the investigation of various solutions to this problem. Current studies include
work conducted at the Morgantown Energy Technology Center (METC) as well as several
university projects that are being supported through the South Carolina Energy Research and
Development Center (Fant and Golan 1995).
This report describes a simple, nonlinear model for premixed combustion, referred to
as PCOM (P remixed C ombustion O scillation M odel). The model represents the relevant processes occurring in a fuel nozzle and combustor that are analogous to current LPM turbine
combustors. Conservation equations for the fuel nozzle and combustor are developed from
simple control volume analysis, providing a set of ordinary differential equations that can be
solved on a personal computer. Combustion is modeled as a stirred reactor, with a
bi-molecular reaction rate between fuel and air. PCOM was developed to help explain specific experimental observations (Richards et al. 1995), and to provide guidance for the development of active control schemes based on nonlinear concepts. The focus of this report is the
model development, although a few comparisons to experimental data are included. A subsequent report will provide a more comprehensive comparison of model results and experimental data.
6
2 Background
Modeling of combustion oscillations dates back to the 1950s. Putnam and Dennis
(1953), and later Merk (1956), developed linear versions of the conservation laws for premixed burner flames from which they derived the criterion for oscillating solutions from the
well-known Rayleigh principle. The Rayleigh principle states that heat release and acoustic
fluctuations should be in-phase to drive oscillations, and out-of-phase to dampen oscillations.
During the 1960s, liquid rocket instabilities motivated similar linear analyses that described
the combustion response to pressure oscillations by a time lag, τ, and an interaction index, n.
The so-called τ-n analysis, as proposed by Crocco and Cheng (1956), has proven very useful
in analyzing rocket engine data. The method was more highly developed at a later time to
include nonlinear effects in the acoustic response; see the review by Culick (1994). Similar
time lag models have been used to describe oscillations in a variety of industrial burners
(Putnam 1971) and ramjets (Reardon 1989; Yu et al. 1991). The basic hypothesis behind the
τ-n model is that acoustic disturbances produce a change in the combustor heat release, but
delayed by some time, τ. The magnitude of the change (i.e., the gain) is controlled by the
parameter, n. Given the correct time delay, the heat release fluctuations can drive subsequent
pressure waves in accordance with the Rayleigh criterion. Assuming the gain is large enough,
an oscillation is established. The success of this approach depends on accurately predicting
the time lag. Prediction is complicated by factors such as uncertainties in mixing rates and
evaporation rates (in liquid-fueled combustion).
As an alternative to the time lag model, advances in computational fluid dynamics
(CFD) have made it possible to compute the time history of complicated reacting flow fields.
Recent papers based upon CFD analysis have described the oscillating behavior of ramjet
combustors (Menon and Jou 1991; Menon 1994) and pulse combustors (Benelli et al. 1993;
Najm and Ghoniem 1993). These computations follow the details of fluid motion and reaction and avoid the need to specify a combustion time lag. Inspection of the simulation results
can provide physical insight into the processes occurring. A drawback of CFD analysis is
that results are specific to the particular case modeled, and each case can take many hours of
computational time. General conclusions are therefore difficult to obtain.
As an alternative to linear analysis or detailed modeling, this report describes a simulation based on a time-dependent, nonlinear control volume analysis. A similar approach was
used by Richards et al. (1993) and Narayanaswami and Richards (1995) to successfully
describe experimental oscillations observed in several different styles of pulse combustors.
Daw et al. (1995) showed that this type of modeling can be particularly useful to understand
laboratory observations of nonlinear, chaotic behavior in oscillating combustion. Reardon
(1995) developed a similar analysis to the one presented here, the primary difference being
that Reardon models the combustion response with a specified time lag. In this report, the
combustion is modeled as a well-stirred reactor having finite kinetics. While a well-stirred
reactor is an obvious simplification of a premixed gas turbine combustor, it does represent a
valuable limiting case. Comparison to experimental data shows that much of the dynamic
behavior observed in the lab is also predicted by this model, including the effects of inlet air
temperature and some open loop control results.
7
3 Model Description
Flow properties and species concentrations in the nozzle, combustion, and tailpipe
regions are determined using a control volume formulation of the conservation equations.
The development is based upon the integral form of the conservation laws as presented in
most texts (Moody 1990), and also in Richards et al. (1993). PCOM assumptions include
ideal gas behavior, variable specific heats, and uniform conditions within each region. The
assumption of uniform conditions within each region reduces the volume and surface integrals
to algebraic expressions, resulting in a set of first-order differential equations. The eight
conservation equations to be derived in this section are consolidated in Table 1.
The model geometry for the premixed combustor is shown in Figure 1. The subscript
u is used to denote conditions at the upstream end of the nozzle, and the subscript n refers to
the nozzle. A mixture of fuel and air, mn,u , enter the nozzle at a specified equivalence ratio
and temperature, Tu. The premixed flow passes through the swirl vanes and past a so-called
bypass fuel port, b, where additional fuel can be injected at a rate, mf,b(t) . Bypass fuel is a
term denoting fuel that is not included in the fuel/air mixture, mn,u , entering the nozzle.
Bypass fuel injection may incorporate both a steady and a fluctuating component if desired.
The injected fuel mixes with the nozzle mass flow in a specified volume, V m, and continues
to the end of the nozzle where it enters the combustion region. The subscript b refers to the
bypass fuel flow or to conditions downstream of the mixing volume. The combustion region
is treated as a perfectly stirred reactor with length Lc and diameter Dc. The pressure and
temperature are calculated in this region along with the oxygen and fuel mass fractions (P, T,
Yo, Yf). The flow exits the combustion region into the tailpipe region, which has the same
diameter as the combustor.
3.1
Nozzle Region
The upstream boundary of the nozzle can be set to one of two distinct conditions.
One condition specifies the temperature, equivalence ratio, and pressure, and allows the mass
flow rate to fluctuate. The other condition sets the temperature, equivalence ratio, and mass
flow rate, and allows the inlet pressure to fluctuate. This latter condition is equivalent to a
choked fuel/air inlet, and will be modeled here for comparison to the experiment described
later. Relevant aspects of the nozzle flow to be discussed include the fluid motion in the
nozzle, the effect of the swirl vanes, and the effect of bypass fuel injection.
Fluid motion in the nozzle can be described as unsteady plug flow for most cases of
interest. The longest nozzle tested to date in our experimental investigation is 0.2 m. At the
limiting case of room temperature, the time required for an acoustic wave to travel this
distance is approximately 0.58 ms. In comparison, a 500-Hz oscillation has a period of 2 ms,
which therefore results in a transit time that is approximately 30 percent of the cycle time.
Typical geometries and flow conditions result in transit times that are less than 10 percent of
the cycle time. Plug flow is thus accepted as a reasonable approximation. Note that elevated
8
Table 1. Conservation Equations for the Prem ixed
Com bustion Oscillation Model (PCOM)
Nozzle Region Equations
dū n
dt
dYf,b
(P̄ u
P̄)
1
ū Y
τm n f,u
dt
2
RT A
RT A
ū n
Ln
Ln
2
1
τm
Yf,b
Ln
1
Dn
mf,b
ρA A n
1
1
(5)
f
Yf,b
(9)
RT A
Tailpipe Region Equations
dū
dt
RT A
P 1
L
ρ
3
f Lc u
2 D c u
(10)
Combustion Region Equations
dP
dt
γ
T oi
T oe
τi
τe
(γ 1)
τc
dρ
dt
dY f
dt
dY ox
dt
1
τi
1
( Yf,i Y f )
τiρ̄
1
( Yox,i Y ox )
τiρ̄
P
ρT
9
(γ 1)
Tw
τHT
1
τe
T
(12)
(17)
RTA 1 1
∆H f ρ̄ τc
(18)
RTA 1 1
S
∆H f ρ̄ τc r
(19)
(20)
u
b
e
i
Combustion
Region
mf,b (t)
DN
Swirler
TU = TA
PU
P,T
Yf,u
YOX,U
mn,u
Mixing
Volume
Vm
Dc
Lb
Ln
Lc
L
M96001763C
Figure 1. Model Geometry for the Premixed Combustor
inlet temperatures common to gas turbines will reduce the transit time even further via the
increased speed of sound.
The nozzle swirl vanes are treated as an ideal cascade, producing no change in
entropy. The swirl vane configuration is depicted in Figure 2. The vanes are positioned at
45° from the nozzle axis, and flow through the vanes is assumed incompressible due to the
low Mach numbers expected in the nozzle. The axial velocity across the vanes is constant
due to mass conservation. It can be shown from a momentum balance that the pressure drop
Figure 2. Details of the Swirl Vane Configuration
10
across the vanes is equivalent to the dynamic pressure of the upstream flow, as shown in
equation (1).
1/2 ρ23 u22 .
P2 P3
(1)
The overall momentum balance for the nozzle is obtained by next applying momentum
conservation independently to the nozzle section upstream (section 1-2) and downstream
(section 3-4) of the swirl vanes. Derivation of these momentum balances assumes a quasisteady friction force (F12), adiabatic walls, spatially uniform density, and unsteady plug flow
(u1 = u2 = u3 = u4 = un). The spatially uniform density assumption implies that changes in the
combustor pressure affect the pressure force terms in the momentum balance, but do not
affect the density calculated from the conservation of mass. The momentum balance for
nozzle section 1-2 is as follows:
du n
P1
P2
dt
L12 ρ12
F12
A n L12 ρ12
.
(2)
A similar expression is developed for the nozzle section downstream of the swirl vanes
(section 3-4) assuming mass addition from the bypass port is neglected. The mass flow of
injected fuel is typically less than 20 percent of the total fuel flow, which is less than
6 percent of the total air flow. Thus, the momentum balance neglects 1.2 percent of the nozzle mass flow. This assumption may be invalid for cases where the bypass fuel is injected at
high percentages. The momentum balance for section 3-4 is as follows:
du n
P3
P4
dt
L34 ρ34
F34
A n L34 ρ34
.
(3)
If the velocity in the nozzle is assumed not to reverse, the pressure at the exit of the
nozzle will equal the combustion region pressure ( P 4 = P). This therefore limits the analysis
to positive nozzle velocities only. Detailed CFD modeling and experimental testing show this
approximation to be valid for most cases where the inlet flows are choked at the upstream
station u. Extreme pressure oscillations at low flow rates may invalidate the assumption.
Since the nozzle length is much larger than the swirler length, it is assumed that
L = L12 + L34. Equations (1), (2), and (3) are thus combined to yield the overall momentum
balance for the nozzle. The friction forces F12 and F34 are written in terms of the Fanning
friction coefficient f; see the nomenclature. The resulting equation is nondimensionalized,
with the exception of time, by defining the reference velocity as
ur
RT A ,
11
(4)
and normalizing the nozzle pressure, temperature, and density by their ambient values. The
resulting nozzle momentum balance is as follows:
dū n
dt
(P̄ u
P̄)
2
RT A
RT A
ū n
Ln
Ln
2
1
Ln
Dn
f .
(5)
Although the effect of bypass fuel injection is neglected in the momentum balance, the
fuel mass fraction must be tracked downstream of the bypass port. The assumption of
positive velocity negates the need to calculate the fuel mass fraction upstream of the bypass
port. A simple, one-dimensional finite difference grid is established for the post-bypass
section of the nozzle. Fuel mass fraction is calculated at each grid point using a simplified
version of the method of characteristics in which only the position of the pathlines are
tracked. Mixing between bypass fuel and the premix flow occurs in the volume Vm, which
has length Lm. A characteristic mixing time, τm, is defined for the volume as
τm
Vm
Lm
An Ur
.
(6)
RT A
Fuel injection through the bypass port is considered choked and is therefore unaffected
by downstream pressure fluctuations. Assuming the fuel is mixed uniformly in the volume,
the conservation of mass and conservation of species respectively yield equations (7) and (8)
below. To derive (7), note that the plug flow, uniform density assumption implies mn,u is the
same at station u and just upstream of the mixing volume.
mn,u
mf,b
mn,b
dρ
dt
Vm
(7)
and
m n Yf,u
mf,b
mn,b Yf,b
Yf,b V m
dρb
dt
d
Y .
dt f,b
ρb V m
(8)
Combining equations (6), (7), and (8) yields the fuel conservation equation for the
nozzle downstream of the bypass port:
dYf,b
dt
1
ū Y
τm n f,u
Yf,b
1
τm
12
mf,b
ρA A n RT A
1
Yf,b .
(9)
3.2
Tailpipe Region
Flow in the tailpipe region is also modeled as unsteady plug flow. The tailpipe region
is the same diameter as the combustor, but with length L. Note that tailpipe flow is not
restricted to positive velocities, and may therefore reverse into the combustion region. The
momentum balance is derived similarly to that for the inlet region, and results in the
following:
dū
dt
3.3
RT A
(P 1)
L
ρ
(10)
3
f L u
2 D c u
Combustion Region
Properties within the combustion region are calculated using the conservation of mass
and momentum, species conservation for fuel and oxygen, and the ideal gas law. Combustion
is modelled as a bi-molecular reaction. Fuel and oxygen react at a rate determined by a onestep Arrhenius kinetic mechanism. Fuel properties used in the model are those of methane.
A conservation of energy equation is developed for the region by accounting for
energy entering with the inlet mass flow and exit mass flow, denoted m i and m e . These
flows enter and/or exit with stagnation temperatures, Toi and Toe, respectively. The heat
release per unit volume is Q . Heat loss occurs by convection through a convection coefficient, h, to the combustion zone walls with surface area As and specified temperature Tw.
Noting that ideal gas law relations imply P (γ 1)ρe , and employing the assumption that
pressure is constant throughout the combustion region volume Vc, the energy balance for the
region is
Vc
dP
γ 1 dt
miCpToi
m eCpT oe
QV c
hA s T
Tw .
(11)
The ratio of specific heats, γ, varies with temperature and represents an average for
the properties of air and combustion products. Equation (11) is normalized by dividing by the
ambient pressure (PA) and combustion region volume (Vc). Temperatures are normalized by
the ambient temperature TA. Other variables are normalized as indicated in the nomenclature.
Denoting the normalized values by an overbar, modest algebraic manipulation of equation (11) results in the following:
dP
dt
γ
T oi
T oe
τi
τe
1
τc
13
(γ 1)
Tw
τHT
T .
(12)
The various characteristic times appearing in equation (12) are defined in equations (13) through (16) below:
τi
τe
ρA V c
mi
ρA V c
me
PA
τc
Q (γ 1)
τHT
inlet flow time ,
(13)
exit flow time ,
(14)
combustion time , and
(15)
heat transfer time .
(16)
ρA R V c
hA s
Notice that the reciprocals of these time scales enter into equation (12) and represent
rates for the various processes.
The approach to developing the remaining conservation laws for the combustion region
is similar to that used for the energy equation. Full details of a similar control volume
analysis for energy, mass, and species conservation can be found in Richards et al. (1994).
The resulting equations are a balance of a storage term versus input, exit, and generation
terms. The balance equations are as follows:
Overall Combustor Mass Conservation:
dρ
dt
1
τi
1
.
τe
(17)
Fuel Species Conservation in the Combustor:
dY f
dt
1
( Yf,i Y f )
τiρ̄
RT A 1 1
.
∆H f ρ̄ τc
(18)
RT A 1 1
S .
∆H f ρ̄ τc r
(19)
Oxygen Species Conservation in the Combustor:
dY ox
dt
1
( Yox,i Y ox )
τiρ̄
14
In addition to the above conservation equations, the normalized form of the ideal gas
law is needed to complete the combustion zone description:
ρT .
P
(20)
Note that the above set of equations depends on characteristic times that are not
constants, but will vary throughout the combustion cycle. The inlet and exit times have
instantaneous values determined from conservation laws in the inlet and tailpipe. The combustion time is determined from an Arrhenius rate law for bi-molecular reaction of fuel and
oxygen in the combustion region. The form of this rate law and the specific rate constants
are taken from Kretschmer and Odgers (1972). After considerable algebraic manipulation of
the rate law in Kretschmer and Odgers (1972), the combustion time is
1
τc
K(γ 1)
PA
PSTD
0.5
T STD
T
1.5
A
∆H f
R
2
P T
1.5
Y ox Y f exp
T act / T .
(21)
Values for the various constants are included in the nomenclature. PSTD and TSTD
represent standard conditions of 101 kPa and 300 K, respectively. The ambient conditions PA
and TA represent conditions of the air supplied to the combustor, which therefore correspond
to conditions at the compressor discharge in a gas turbine.
3.4
Numeric Solution of the Governing Equations
Solution of these equations is obtained by applying the Euler predictor-corrector
algorithm. The solution method is straightforward and involves direct marching from one
time to the next. The simulation is started by simply specifying a high initial temperature.
Depending on the geometry and operating conditions, combustion will continue as a steady
flame, blow out, or oscillate. It is emphasized that this model will predict any of these
responses depending on the conditions specified. Periodic heat release is not imposed.
The solution speed of PCOM is relatively fast. Given the same combustion geometry
as described here and a specified time step, we have shown that a detailed CFD model
running on an SGI Indigo2 machine with a R400 MIPS processor could take more than
30 days to simulate a single second of combustion. In comparison, PCOM running on a
personal computer with a 100 MHz Pentium processor can simulate a second of combustion
in less than 15 minutes.
15
4 Comparisons to Experimental Data
The simple model described here is valuable in understanding the origin of combustion
instabilities and the various means to control them. One can quickly examine instability
trends associated with changes in equivalence ratio, mass flow rate, geometry, ambient conditions, and other relevant factors. PCOM obviously has somewhat limited use as a precise
design tool due to its simplicity. Experimental evidence has shown that multiple, interdependent mechanisms affect the stability performance of a combustor, and not all of these complicated mechanisms are captured by the model. As an example, experimental results have
shown that a pilot flame can have significant effects upon the stability of a combustor. The
behavior of a pilot flame cannot be modeled in a stirred reactor. Given these limitations,
PCOM has nonetheless proven valuable to our understanding of LPM combustion instability.
Although a more comprehensive comparison of model results and experimental data will be
included in a subsequent report, this report briefly examines some open loop control results
and the effects of inlet air temperature.
4.1
Open Loop Control
Combustion oscillation control using low frequency, cyclic fuel injection was successfully tested on the experimental combustor shown in Figure 3. A complete description of the
experiment and details of the combustor configuration can be found in Richards et al. (1995).
A time history of a pressure signal from the combustor with no cyclic fuel injection is shown
in Figure 4. A 4.5 kPa RMS pressure signal is established at 300 Hz given premix fuel and
air flow rates of 0.4 g/s and 9.9 g/s, respectively. Figure 5 shows model results using a
similar geometry and operating conditions. A 4.0 kPa RMS pressure signal is established at
approximately 280 Hz.
Active control of the oscillation using low frequency injection was attempted by
adding 0.067 g/s of fuel through the pulse injector. The fuel injector was operated with a
frequency of 50 Hz and a pulse width of 7 ms. The resulting pressure oscillation of the combustor along with the injector status is shown in Figure 6. A step in the injector signal signifies the injector is open. The low frequency injection is shown to decrease the pressure oscillation by a factor of 0.3 from the uncontrolled case shown in Figure 4. Figure 7 shows model
results using a low frequency injection through the bypass port. Pressure oscillations are
shown to decrease by a comparable percentage.
Figure 8 shows a time history of the pressure signal and the injector status from the
time at which the injected fuel is first introduced into the nozzle until the oscillations are
significantly suppressed. The model demonstrates that under these specified conditions, the
limit cycle can be broken by periodically altering the equivalence ratio. The low frequency
fuel injection is able to break the limit cycle by shifting combustion back and forth between
stable and unstable conditions. This was demonstrated numerically by observing the oscillating behavior at (steady) equivalence ratios representing the instantaneous conditions
corresponding to the injector open or closed states.
16
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Figure 3. Experim ental Com bustor Configuration
17
Pressure (kPa)
110
105
100
95
90
0
10
20
30
40
50
Time (ms)
Figure 4. Pressure Versus Time for a Combustor W ith No Cyclic Fuel Injection
(Experimental)
110
108
106
Pressure (kPa)
104
102
100
98
96
94
92
90
0
10
20
30
40
50
Time (ms)
Figure 5. Pressure Versus Time for a Combustor W ith No Cyclic Fuel Injection
(Numerical)
18
11 0
10 8
10 6
Pressure (kPa)
10 4
10 2
P ressu re
Injecto r
10 0
98
96
94
92
90
0
10
20
30
40
50
T im e (m s)
Figure 6. Pressure Versus Time for a Combustor W ith Cyclic Fuel Injection
(Experim ental). Injector is Open W hen Signal is High.
110
108
Pressure (kPa)
106
104
102
Pressure
Injector
100
98
96
94
92
90
0
10
20
30
40
50
Tim e (ms)
Figure 7. Pressure Versus Time for a Combustor W ith Cyclic Fuel Injection
(Num erical). Injector is Open W hen Signal is High.
19
110
108
Pressure (kPa)
106
104
Pressure
Injector
102
100
98
96
94
0
80
160
241
322
Time (ms)
Figure 8. Tim e History of Pressure Signal for a Combustor W ith Cyclic Fuel Injection
(Num erical). Injector is Open W hen Signal is High.
Experiment and model both show that a relatively small amount of fuel injected appropriately
can break a limit cycle and thus stabilize combustion oscillations.
4.2
Inlet Air Temperature
Initial difficulties in repeating experimental data from day to day led to the identification of inlet air temperature as a relevant parameter in combustion instability research.
Although differences in inlet air temperature were severely affecting oscillation levels, the
root cause was not immediately apparent. Figure 9 shows experimental and numerical data
from a combustor similar to the one shown in Figure 3 operating at an equivalence ratio of
0.75 and an air flow rate of 19.3 g/s. The only parameter altered in Figure 9 is the temperature of the fuel and air mixture entering the nozzle. Experimental data shows that a significant increase in the pressure oscillation occurs at approximately 283 K. Numerical data
shows a more severe, yet similar transition at approximately 293 K.
It should be noted that the simplicity of the model in conjunction with the multiple,
interdependent instability mechanisms alluded to previously make exact prediction of an
20
6
RMS Pressure (kPa)
5
4
Model
Experiment
3
2
1
0
270
275
280
285
290
295
300
305
310
315
320
Inlet Temperature (Degrees K)
Figure 9. Com bustor RMS Pressure Versus Air Inlet Temperature
(Num erical and Experimental)
experimental temperature transition very difficult. Nonetheless, a better understanding of the
root cause of the transition is provided by the model. The model demonstrates that changes
in air temperature will alter reaction rates, and even nozzle velocities to a lesser extent. As
the inlet air temperature increases, the reaction rate will increase and the nozzle velocity will
also slightly increase. Both of these responses to an increasing inlet temperature result in a
decrease in the transport time. The transport time is defined as the total amount of time it
takes for a particle of fuel to advect down the nozzle to the flame front, effectively mix, and
subsequently combust. Transport time has been shown to be directly related to the occurrence
of instability (Richards and Yip 1995). The change in transport time could possibly move the
combustor from a stable to an unstable region, or vice-versa. Figures 10 and 11 show the
effect of inlet air temperature as air flow rate and equivalence ratio are respectively changed.
A change in inlet air temperature has a varied effect on each of the cases modeled, as would
be expected. Figure 10 shows that flow at 29.4 g/s is highly stable and that modest changes
in temperature will not have an effect. On the other hand, flow at 9.1 g/s is highly unstable,
but inlet air temperature changes can still not alter the transport time significantly enough to
move the flow out of the unstable region. As alluded to previously, flow at 19.3 g/s
experiences a transition at approximately 293 K and 353 K.
21
7
RMS Pressure (kPa)
6
5
Air = 9.1 g/s
Air = 19.3 g/s
Air = 29.4 g/s
4
3
2
1
0
270
280
290
300
310
320
330
340
350
360
370
Inlet T emperature (Degrees K)
Figure 10. Com bustor RMS Pressure Versus Air Inlet Temperature
for Various LPM Air Flow Rates (Numerical)
10
9
RMS Pressure (kPa)
8
7
6
Equiv. Ratio = 0.6
Equiv. Ratio = 0.75
5
Equiv. Ratio = 0.9
4
3
2
1
0
270
280
290
300
310
320
330
340
350
360
370
Inlet T emperature (Degrees K)
Figure 11. Com bustor RMS Pressure Versus Air Inlet Temperature
for Various Equivalence Ratios (Num erical)
22
5 Conclusion
As an alternative to linear analysis or detailed modeling, a simple, nonlinear model for
premixed combustion has been developed to help explain experimental observations and to
provide guidance for the development of active control schemes. The Premixed Combustion
Oscillation Model (PCOM) represents the relevant processes occurring in a fuel nozzle and
combustor that are analogous to current LPM turbine combustors. Conservation equations for
the fuel nozzle and combustor are developed from control volume analysis, providing a set of
ordinary differential equations that can be solved on a personal computer.
The model can be used to quickly examine instability trends associated with changes
in equivalence ratio, mass flow rate, geometry, ambient conditions, and other relevant parameters. PCOM will have somewhat limited use as a design tool due to its simplicity. Given its
limitations, the model has nonetheless proven valuable to our understanding of LPM combustion instability. Comparison to experimental data shows that much of the dynamic behavior
observed in the lab is also predicted by PCOM, including the effects of inlet air temperature
and some open loop control results. A subsequent report will provide a more comprehensive
comparison of model results and experimental data.
23
6 Nomenclature
Note: The text uses an overbar to denote the normalized counterpart of variables
listed here. The normalizing factor is listed with the variables below.
As
surface area of the combustion zone
Cp
constant pressure specific heat
Cv
constant volume specific heat
f
Fanning friction coefficient, 0.03
F12
friction force in nozzle section 1-2, F12 = πDNL12 f ρ12un3/(8|un|)
F34
friction force in nozzle section 3-4, F34 = πDNL34 f ρ34un3/(8|un|)
h
heat transfer coefficient for the combustion zone walls, 120 W/m2/K
K
kinetic coefficient, Equation (19), 1.576 x 109 s-1
L
length of the tailpipe region
Lb
nozzle length post bypass
Lc
length of the combustion region
Ln
total nozzle length
Lm
length of the bypass mixing volume
L12
nozzle length from inlet to swirl vanes
L23
length of the swirl vanes
L34
nozzle length from swirl vanes to exit
mf,b
mass flow rate of fuel added at the bypass port
mi
mass flow rate through the combustion zone inlet station i
me
mass flow rate through the combustion zone exit station e
P
pressure in the combustion chamber; normalized with the ambient pressure PA
24
PA
the ambient pressure, upstream of the combustor
P STD
standard pressure, 101 kPa, used as a reference in the reaction rate term
Pu
pressure at the upstream end of the inlet pipe (station u); normalized by PA
Q
heat release per unit volume
R
specific ideal gas constant, 287
SR
the stoichiometric ratio for the mass of oxygen consumed per mass of fuel
burned
TA
the ambient temperature, upstream of the combustor
T act
the dimensionless activation temperature, 50
T oi
combustion zone inlet stagnation temperature (station i); normalized with TA
T oe
combustion zone exit stagnation temperature (station e); normalized with TA
T STD
standard temperature, 300 K, used as a reference for reaction rate, equation (19)
Tw
combustion zone wall temperature, 1200 K; normalized with TA
u
velocity in the tailpipe region; normalized with ur
un
velocity in the nozzle; normalized with ur
ur
reference velocity;
Vc
combustion region volume
Vm
bypass port mixing volume
Yf
fuel mass fraction in combustion zone
Y f,b
fuel mass fraction at the bypass port, downstream of the mixing volume
Y f,i
fuel mass fraction, combustion zone inlet, station i
Yo
oxygen mass fraction in combustion zone
Y o,i
oxygen mass fraction, combustion zone inlet, station i
J
K
kg
R TA
25
ρ
combustion region density; normalized by PA
ρA
ambient density
ρb
nozzle density at the bypass port
ρ1 2
nozzle density in section 1-2
ρ2 3
nozzle density in section 2-3
ρ3 4
nozzle density in section 3-4
γ
temperature-dependent ratio of specific heats
τc
combustion time
τe
exit flow time
τHT
heat transfer time
τi
inlet flow time
τm
mixing time
∆H f
fuel heat of combustion per unit mass 5 x 107
26
J
kg
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