Sep 24, 1999 - Secondary peaks are created while using these controllers due to ...... The combustor as well as the controller respond dynamically over a ...
The Origin of Secondary Peaks with Active Control of Thermoacoustic Instability M. Fleifil, J.P. Hathout, A.M. Annaswamy and A.F. Ghoniem September 24, 1999
Abstract This paper deals with the generation of new peaks in the pressure spectrum in controlled combustors in experimental investigations of active control of thermoacoustic instabilities. Typically, the reported experiments have demonstrated that the dominant thermoacoustic instability can be suppressed, but secondary peaks at different frequencies which were not excited in the uncontrolled combustor appear. We develop a physically-based model of an actively controlled premixed laminar combustor which takes into account (a) laminar flame kinematics, (b) linear acoustic dynamics with coupling between the acoustic modes, and (c) actuation using side-mounted and end-mounted loudspeakers. Using this model, we analyze some of the experimental controllers proposed in the literature and explain the origin of secondary peaks observed in these studies. Secondary peaks are created while using these controllers due to resonant coupling between various mechanisms in the combustor that are distinct from those responsible for thermoacoustic resonance and mechanisms in the controller other than those that enable suppression of thermoacoustic instability. Other than at acoustic frequencies, premixed laminar combustors respond at lower frequencies due to flame dynamics, and at higher frequencies due to antiresonance. The experimental controllers are usually implemented using analog electronic circuitry whose components are designed so as to provide the functionalities of a phase-shifter, a filter, and an amplifier. Since analog filters tend to provide a phase compensation over a wide range of frequencies and not just at the isolated (unstable) frequency, they can initiate resonances by coupling with various mechanisms present in the combustor. Low frequency secondary peaks are typically due to coupling between the flame dynamics and the filter components in the controller, while high frequency peaks are due to either the interaction between various components of the active controller themselves, or the interaction between the controller components and antiresonance. Such phenomena clearly underscore the need for an active control design which accounts for the combustor dynamics over a range of frequencies with the goal of obtaining the desired performance over this entire range.
Submitted to Combustion Science and Technology: 1
Nomenclature
A Af As a0 bf ,gf ,!f e b,ce c D Do dp E e G(s),(Gi(s)) ∆ qr
ka kao ki kci,kpi L nb nf p pe peh,pedirect ,peindirect qf ,(qfh ,qfi ) i
Su s T T (s) t u
i
Combustor cross-sectional area ua,ya Flame surface area Loudspeaker cross-sectional area ug ,yg Acoustic constant Flame parameters uf Acoustic vector parameter uef Speed of sound V c Diameter of the flame-holder v c Duct height W (s) Diameter of perforation in Wf (s) the flame-holder Wqf (s),Wi(s) Energy in the mode w Specific internal energy x Controller (and its components) xf transfer functions xs Heat of reaction per unit mass xa of the mixture ∆x Speaker calibration constant y Speaker location effect zf Wave number r Approximate controller and
plant gains Length of the combustor duct i0 Orthogonal vector Number of holes in the flame-holder Pressure Nondimensional unsteady pressure f Nondimensional pressure components ! Heat release rate per unit area i (and its components) i Laminar burning velocity Time-derivative operator H() Period () Closed-loop transfer function (x) Time (x)0 Velocity
2
Input signal to combustor and controller (Appendix A) Output signal from combustor and controller (Appendix A) Axial velocity at xf Effective velocity at xf Displaced volume by the flame Diaphragm velocity Open-loop transfer function Flame transfer function “Antisound” transfer functions Duct width Axial coordinate Flame location Sensor location Actuator location Width of speaker Diaphragm displacement Flame zero Ratio of As to A Specific heat ratio Flame base diameter parameter Phase shift Density Flame parameter Dummy variable Flame time constant Frequency ith basis function Amplitude of i Damping ratio Heaviside function Dirac delta function Mean of the variable x Perturbation of the variable x
1
Introduction
Thermally driven acoustic instabilities occur commonly in premixed systems designed to reduce NOx formation by operating at lean conditions. While passive approaches such as changing the flame anchoring point, the burning mechanism, the acoustic boundary conditions, installing baffles and acoustic dampers, etc., have been sought to counter the instability, a desire to operate over a wide range of conditions without running the risk of self-destruction, and maintaining their NOx emission levels within desirable limits, has led to exploring active control as possible strategies for achieving the desired performance. Active control solutions that have been suggested in the literature can be broadly classified into two categories: (i) controllers designed on the basis of an experimental strategy, (ii) controllers designed on the basis of a theoretical model derived from first principles. Examples of the first category include (Lang et al. , 1987; Bloxsidge et al. , 1987; Poinsot et al. , 1989; Langhorne et al. , 1990; Gulati and Mani, 1992; McManus et al. , 1990; Billoud et al. , 1992), wherein the dominant pressure mode has been shown to be suppressed successfully. Typically, the implementation of these controllers utilizes an electronic circuit made up of analog components including a filter and a phase-shifter. The control strategy consists of tuning the parameters of this circuit by trial and error so as to reduce the amplitude of oscillations corresponding to the unstable frequency. However, in many of these studies, it has been found that such a control procedure introduces other secondary pressure peaks in the closed-loop system. These peaks occur at various frequencies that do not correspond to any of the natural modes of the open loop indicating that it is a result of the specific control design employed in the problem. Examples of the second category include (Bloxsidge et al. , 1987; Billoud et al. , 1992; Fung et al. , 1991; Fung and Yang, 1992; Yang et al. , 1992; Tierno and Doyle, 1992). Here, the fundamental laws that govern the behavior of the thermal acoustics in the combustor are employed to obtain a theoretical description of the underlying dynamics. Conservation equations and constitutive relations make it possible to analyze the behavior of the combustor, quantify its properties in relation to various system parameters such as its geometry, operating points (equivalence ratio, flow rate, etc), and design (for flame stabilization, cooling, etc.), and predict its behavior as these system parameters are changed. Our focus in this paper is on the first category. In particular, we are interested in explaining the origin of secondary peaks encountered when experimental controllers are utilized. For this purpose, we apply a finite-dimensional model based on one-dimensional inviscid flow and laminar 3
flame kinematics. This model captures the behavior of the stable and unstable acoustic modes as well as their coupling over a wide range of frequencies which encompasses those corresponding to acoustics, flame characteristics, and controller components. This feedback model also includes actuation effects from a loudspeaker on the conservation equations. We study end-mounted as well as side-mounted loudspeakers both of which have been used in the experimental investigations. Since the use of feedback control can shift natural frequencies of a system and more importantly, is capable of producing resonant frequencies over a wide range, the study of the combustor behavior over a range not limited to acoustics is essential. In our previous work (Annaswamy et al. , 1997b), a feedback model was constructed and used to analyze the effect of acoustic coupling in the presence of active control. In this paper, we explain, using the same model, the behavior of controllers used in experimental investigations. The latter is synthesized using, for the most part, analog filter components. These components are chosen so as to provide a requisite phase at the thermoacoustically unstable frequency. Since the primary mechanism responsible for the instability is the phase lag between the heat release rate and the pressure, such a design procedure is successful in suppressing the pressure oscillations. However, analog components, by and large, tend to have a gradual phase change over a wide range of frequencies and therefore change the overall gain and phase variations in the combustor. As such, these components can induce new mechanisms which can couple with other dynamics in the combustor to produce resonance at previously stable modes. Precisely how this occurs is what is delineated in this manuscript. In Section 2, we present the physical laws that will be used to derive the finite dimensional model. Effects of an external actuator are included by using a deformable control volume analysis of an acoustic actuator. Using Galerkin expansion and linearization, a finite dimensional model that relates the dynamic effects of a loudspeaker action on a pressure output measured by a microphone is derived. The characteristics of this model over frequencies comparable to the flame characteristic frequency as well as acoustic frequencies are analyzed. In Section 3, we discuss two examples of experimental controllers. Through numerical studies, we show that our model predicts the secondary resonances observed in the laboratory experiments. We then provide an explanation for these resonances and the possible coupling mechanisms that induce such a behavior. In our discussions, we have assumed that the effects of mean flow and mean heat addition are negligible. While indeed they are important in the design of practical combustors, they are not central to the points that we wish to make in this paper. While more practical actuators employed in large scale combustors are typically air/fuel flow modulators, we have chosen to use loudspeakers since they have been used in the specific experimental studies that we have chosen to validate our model. We believe that a similar study can be carried out for different actuation mechanisms, and is the subject of an on-going investigation. 4
Premixed air + fuel flameholder
loudspeaker
,, ? @@ x
-Products xf
a
xs
microphone Figure 1: Schematic of the combustor with an end-mounted loudspeaker.
2
The Feedback Model with a Loudspeaker as Actuator
A commonly used actuator to control pressure oscillations in combustion systems is a loudspeaker. The ease of its implementation as an actuator is one of the predominant reasons for its ubiquity as an actuator in many of the investigations of active control of thermally driven acoustic instability. In this section we examine the basic role of the loudspeaker by modeling its effect as a source in an acoustic field. The starting point is the conservation equations applied on a deformable control volume. The dynamic effect of the loudspeaker, which is either flush-mounted on the combustor wall or end-mounted, on the acoustic field is shown to be a source of flow, momentum, and energy.
2.1
Contributions of a Loudspeaker to the Acoustic Dynamics
As in (Annaswamy et al. , 1997b), the flow is one dimensional with negligible transport processes,
combustion occurs within a flame zone localized at x = xf , gases in the reactant and product side are perfect, and unsteady components of the flow variables are small variations about the mean flow. Whether a loudspeaker is end-mounted (see Figure 1), or flush-mounted on the wall of a duct, (see Figure 2), its diaphragm motion can be modeled as a piston. In the former case, the motion is parallel to the flow, while in the latter, it is in the direction normal to the flow. We assume
that the width of the loudspeaker, ∆x, is relatively small compared to the smallest wavelength of the relevant acoustic modes, and that the diaphragm motion, in the side-mounted case, is small
compared to the diameter of the combustor (see Figure 3). The contributions of the diaphragm motion can be represented in the mass, momentum, and energy equations as follows:
@ + @u + u @ @t @x @x
@ (u) + u @ u2 @t @x
+
=
@p @x 5
vcAs ; w∆x(Do , y) =
vcAsu ; w∆x(Do , y)
(1)
(2)
Premixed air + fuel flameholder
? xa
,, @@ @ ,
-Products xf
xs microphone
loudspeaker Figure 2: Schematic of the combustor with side-mounted loudspeaker.
,,
Piston (loudspeaker)
@@@@ @@ @I@@ @@@ @, , As @@ @@@w @@@@@ ,@ @@ @@@@R @@@ @ ? y @@ 6 @@ @@ - ∆x x
6
Do
? @I@ @@Duct
Figure 3: Schematic diagram of the combustor with a side-mounted loudspeaker to incorporate the contribution of the loudspeaker modeled as a movable piston of cross-sectional area As to the conservation equations in 1-D flow dynamics.
6
"
@e + p @u @e + u @t @x @x
#
vcAs p + vc2 ; w∆x(Do , y) 2
=
(3)
where vc , y , and As are the speaker’s diaphragm velocity, diaplacement, and cross-sectional area,
respectively. , p, u, and e denote the density, pressure, velocity and specific internal energy of the fluid, respectively. The duct has a width w and a height Do . Our assumptions regarding the loudspeaker imply that ∆x, y=Do and vc2 =(p=) are small. In the presence of a heat source due to a flame at x = xf , neglecting the effect of mean heat addition and mean flow, the conservation equations of the perturbations can be obtained from as follows (Annaswamy et al. , 1997b):
@ 2 p0 , c2 @ 2 p0 @t2 @x2
! =
, 1)
(
!
@qf0 @t (x , xf )
+
c (x , x );
p r @v a @t
(4)
@p0 + p @u0 = ( , 1)q0 (x , x ) + p v (x , x ); (5) f r c a f @t @x where r = As =A, A is the area of cross-section of the combustor, is the specific heat ratio, xa is the speaker’s location, c¯ is the mean speed of sound, qf0 is the unsteady heat release rate per unit area, and () is the Dirac delta function. If the loudspeaker is end-mounted, the same analysis can be applied, setting xa = 0, and r 1. In Equations (4) and (5), we assume that the heat release source is concentrated at the flame
holder, xf . We note also that the loudspeaker, at xa , is an acoustic source. The boundary effects, which can be considered as additional virtual sources, are introduced implicitly by imposing a spatial distribution of the pressure since its specific shape depends on the boundary conditions.
2.2
A Finite Dimensional Feedback Model
To facilitate our analysis, we separate the spatial and temporal changes in the dependent variables using the Galerkin method (Myint-u and Debnath, 1987) commonly applied in this context (Zinn and Lores, 1972; Culick, 1976):
p0(x; t)
n X
=
p
=
sin(ki x + i0 );
i=1
i (x)i (t);
(6)
where the basis functions i (x) are
i (x)
(7)
ki and i0 are determined by the boundary conditions, and correspond to the spatial mode shapes, and ki are the wave numbers. 7
2.2.1 Acoustic Dynamics Using the Galerkin expansion, Eqs. (4) and (5), after some manipulations, can be written for a single actuator as
p
n X i=1
i (x)¨i , c
2
u0f (t)
p
=
n X i=1
i (x)i (t) n X
1
k
2
i=1
=
@q0 x , xa ) ;
, 1) @tf (x , xf ) + p r @vc (@t
(
: i (t) ddxi (xf ) + a0 q0f (t) + r vcH(xf , xa );
(8)
(9)
where is a parameter that reflects the contribution of the velocity behind and ahead of the flame
to u0f (Annaswamy et al. , 1997b). The loudspeaker and the heat release act as sources at xa and xf , respectively, to the acoustic pressure and velocity as shown in Eqs. (8) and (9). The effect of
boundary conditions is implicitly introduced the mode shapes i (x). 2.2.2 Heat Release Dynamics
The unsteady heat release from a premixed flame arises due to the sensitivity of the burning rate to flow perturbations. These flow perturbations, which correspond to the acoustic field of the host resonator (the combustor), affect the rate of heat generation through changing the reaction rate and/or the area of the flame. In most cases, the latter is the more dominant effect. For a premixed flame stabilized in a low velocity region, such as behind a perforated disc, the flame can be represented by a thin sheet moving with a constant burning velocity Su in a normal direction to its surface relative to the reactants. Since the burning velocity is constant, the heat release rate is
directly proportional to the flame surface area. For a flame stabilized on a perforated disc with nf holes, the dynamic relation between the unsteady heat release rate qf0 and the unsteady velocity u0f is derived as (Fleifil et al. , 1996)
where
: q0f
4 Su
=
,!f qf0 , gf u0f ;
!
(10) !
2 d p !f = d and gf = D nf ∆qr ; p ∆qr is the heat release rate per unit mass of the mixture, D is the diameter of the flameholder, and is a factor greater than unity included to represent the effect of the increase in the flame base diameter with respect to the perforation diameter dp . In this paper, we set = 1:2. We refer
the reader to (Annaswamy et al. , 1997a) where sensitivity studies of the model proposed here to
changes in and are carried out. To assess their roles in the dynamics, we performed parametric studies and chose their values so as to conform with the experimental results. Increasing , as well 8
0
Magnitude
10
−1
10
−2
10
−2
10
−1
0
1
−1
0
1
10
10 10 Normalized Frequency (Frequency/ωf)
2
10
Phase (deg.)
0
−50
−100 −2 10
10
10 10 Normalized Frequency (Frequency/ωf)
2
10
Figure 4: The gain and phase characteristics (Bode plot) of the flame dynamics. Frequency/!f is the normalized frequency with respect to the flame bandwidth. as , increases the growth rate slightly but does not affect the frequency of the instability. Thus,we chose = 0:5, which assumes that the flame is affected equally by the velocity upstream and
dowstream, and = 1:2 which reflects an approximate value for the flaring of the flame base that we observed in our experimental set-up (Annaswamy et al. , 1997c).
The parameter !f is the flame characteristic frequency (or bandwidth), and gf is the amplification
between u0f and qf0 (or gain). Also, f = 1=!f is the flame time constant, which can be interpreted as the time taken by the flame to displace a volume Vc 0 if the initial consumption rate is held constant i.e., f = Vc 0 =Su Af 0 , where Af 0 is the unsteady flame surface area. f is essentially the time lag between a velocity perturbation and the flame response to it. Figure 4, which shows the flame
frequency response, indicates that for frequencies ! > !f , the phase approaches ,90 and the gain gets attenuated at the rate of -20 db/decade. It is worth noting that (1) the resonant frequencies imposed by the combustor dimensions and the speed of sound are, in most cases, higher than !f
for typical laminar premixed flame, (2) at these resonant frequencies, where the thermoacoustic instability occurs, the flame shows a phase shift of ,90 with very little change as the frequency varies, and (3) although the flame gain is attenuated significantly at the resonant frequency, it is still of sufficient amplitude to sustain the instability (Annaswamy et al. , 1997b). 9
It should be mentionned that the flame model described above (and in detail in (Fleifil et al. , 1996)) is applicable for weak flow perturbations which may result in the growth of small waves along the flame front without folding or convolution. These conditions prevail downstream of a perforated plate if the flow is laminar and at low Reynolds number. In this case, the velocity perturbations affecting the flow are predominantly induced by acoustic perturbations, and not flow perturbations. On the other hand, the same approach can be extended to cases of more complex mean flow and flow perturbations using the appropriate modifications (Dowling, 1996; Peracchio and Proscia, 1998; McManus and Magill, 1997).
2.2.3 The Combustor Model The complete combustor dynamics with n modes can be summarized using the following 2n + 1 differential equations (Annaswamy et al. , 1997b):
¨i + !i i : q0f +bf qf0 2
and ue 0f where kao
=
1 if xa
= = =
bi q: 0f +ebci i;
p
e
e =
n X i=1
ceci i;
(11)
!f gf ue0f ; Z n X : e ci i +kaoka r id
(12) (13)
i=1
< xf and 0 otherwise, i is the current into the loudspeaker assuming that v: = k i; (14) a
bf = !f (1,a0gf ) is the effective flame characteristic frequency, pe denotes the normalized pressure p0=p¯, ebi = aE i(xf ), cei = k1 i ddxi (xf ), ebci = E ka r i(xa ), ceci = i(xs), ao = ( , 1)=( p¯), R and E = 0 L i i T dx. :0 Equation (11) implies that the measured pressure output pe is due to two sources q f and i. It :0 should however be noted that q f is not an external source and is in fact affected by the loudspeaker’s 0
2
action. This is because the loudspeaker action affects the flame independent of its effect on the acoustics. As seen from Eqs. (12) and (13), the total heat release is not only due to the coupling with acoustics but also the effect of diaphragm motion. That is, the additional velocity component due to the loudspeaker at the flame generates additional heat release (see Eq. (13)). In particular, we can write:
qf0 where
qf0 h
=
Wf
"
X
i
#
cei : i ;
=
qf0 h + qf0 i ; qf0 i 10
=
Z
Wf kaoka r i d ;
(15)
and
Wf = s!+f gbf
f
is the transfer function representing the flame dynamics. qf0 h can be viewed as the homogeneous component of the heat release, which would be present without the loudspeaker. This, in turn, implies that the total pressure pe can be represented as
pe
=
peh + pedirecti + peindirecti
(16)
where Z X c X c eci e eci e c b s b b s c i i e e p = s2 + !i2 qfh ; pdirecti = s2 + !i2 i; pindirecti = s2 + !i2 Wf kaoka r id :(17) peh, pedirecti , and peindirecti are the components of the pressure due to the homogeneous part of the heat eh
X ec ei i
release, the direct power added from the diaphragm motion, and the indirect power added from the
diaphragm motion, respectively. pedirecti is the contribution of the loudspeaker action on the acoustic pressure, whereas peindirecti is the effect of the change in the heat release due to the additional velocity component due to the loudspeaker action. The difference between the two lies primarily in the frequency range over which the phase lag is significant. From Eq. (17), it can be seen that both peh
and pedirecti are responsible for the phase lag between i and pe primarily in the acoustic range. This is not the case with the component peindirecti . The latter generates a significant phase lag between i and
pe at frequencies comparable to the effective flame characteristic frequency, bf which, as mentioned above, is governed by the burning speed Su and the perforation diameter of the flameholder. Due to the fact that Su 300Hz, the unstable acoustic mode dominates the behavior 1
1
of the combustor as can be seen in Figure 5. When n = 2, the transfer function W (s) can be factored as
2 2 W (s) = K (s + !e )(s2 , 2 ! ss ++!!2 z)(s2 + 2 ! s + !2 ) : (21) f 1 a 2 a a a e f ) is once again due to the flame dynamics. The complex factors (s2 ,21 !a s+!a2 ) The factor (s+! and (s2 + 22!a s + !a2 ) are due to the two acoustic modes. The factor (s2 + !z2) represents the 1
1
2
2
1
2
1
2
antiresonance effect, discussed in detail in (Annaswamy et al. , 1997b). For certain actuatorsensor locations, the linear coupling between the acoustic modes contributes significantly to the antiresonance damping. In this case, however, the antiresonance damping is zero since the direct effect is zero and hence, all damping mechanisms, both directly and indirectly through linear coupling, are not excited. As such, the effect of linear coupling between acoustic modes is negligible in the end-mounted case. 13
1
10
0
Magnitude
10
−1
10
−2
10
−3
10
0
1
10
2
10
ω (Hz)
10
3
10
Phase (deg.)
300 200 100 0 −100 0 10
1
2
10
ω (Hz)
10
3
10
Figure 6: The gain and phase characteristics of an uncontrolled combustor with an end-mounted loudspeaker (open-open combustor). !a1 : unstable frequency at 357 Hz; !a2 : stable frequency at 714 Hz. The overall gain and phase characteristics of W (s) are depicted in Figure 6, taking into account the unstable fundamental mode at 357 Hz and an additional stable second harmonic at 714 Hz. Similar to the single mode case, we divide the frequency behavior of the combustor into two parts; : for ! < 200 Hz, the phase between v c and pe is due to the flame dynamics, while for ! > 200 Hz, the phase is due to the acoustic dynamics. The two spikes in the magnitude plot are due to the two acoustic modes. The spike at 700 Hz in the phase plot is due to the combined action of the antiresonant frequency at 687 Hz and the stable mode at 714 Hz, since the former adds a phase : of ,180 , and the latter 180 at their corresponding characteristic frequencies, between v c and pe. Between 400 Hz and 600 Hz, which is the range between the unstable acoustic frequency and the stable frequency, the phase is almost a constant at 90 . That is, we can approximate the combustor behavior as
W (s) kp s
for !
2
2 [400Hz; 600 Hz]
(22)
where kp2 is the gain. This approximation can be explained as follows. At the unstable frequency,
qf0 and pe are in phase, but at slightly larger frequencies, the acoustic pressure lags the heat release
rate by 90 . This phase however varies as one approaches the stable frequency. The acceleration of
the loudspeaker diaphragm causes a change in heat release rate which lags by 180 as seen by Eq. (15). As a result, the phase between the diaphragm acceleration and pe is ,270 which is the same 14
:
as 90 . Hence, W (s) in (22) represents the relation between v c and pe for ! 2 [400 Hz, 600 Hz]. The behavior of the combustor as represented by Eqs. (20) and (22) will be shown in Section 3 to be responsible for secondary peaks when an experimental controller similar to the one in (Gulati and Mani, 1992) is used to stabilize the combustor.
2.2.5 With a side-mounted loudspeaker If the loudspeaker is side-mounted (see Figure 2), considering only a single mode and when the : loudspeaker is placed upstream of the flame, the transfer function between v c and pe is given by h
W (s)
i
s bf )ebccec + kagf !f rebcec : (s2 + ! 2 )(s + bf ) , gf !f e bces2 ( +
=
(23)
For the same combustor as before, it can be shown that the plant transfer function can be factored as
W (s)
=
K (s + zf ) e f )(s2 + 2!a s + !a2 ) (s + !
where !a corresponds to the frequency of the dominant acoustic mode, is its damping ratio which e f bf . The transfer function has a zero at ,zf which is negative if the mode is unstable, and !
:
determines the the phase-lag between v c and pe at an intermediate range of frequencies between the flame characteristic frequency and the acoustic frequency. The physical mechanism responsible for the zero can be described as follows. As mentioned in Section 2.2.3, there are two effects for the added power, direct (pedirecti ) and indirect (peindirecti ), which arise due to the motion of the loudspeaker diaphragm. Due to the impact of the flame dynamics on peindirecti , there is an additional phase lag ef , at low frequencies. Hence, at these frequencies which are comparable to the flame bandwidth ! the contribution peindirecti to the sensed pressure is significant, while that from pedirecti is negligible.
As the frequency increases, the latter effect increases and near the acoustic frequencies, only the direct effect is important. As a result, in the frequency range that lies between the flame bandwidth
:
and the acoustic frequency, the phase lag between v c and pe gradually diminishes. This combined effect on the phase lag over various frequencies can be represented by a zero at ,zf together with
e f with zf larger than ! e f , since the effect of a zero is to provide a phase lead over the the pole at ,! range [0:1zf ; 10zf ].
We numerically verify this behavior when a side-mounted loudspeaker is utilized in Figure 7, which illustrates the gain and phase characteristics of a combustor with a single mode at 542 Hz, e f = 10 Hz, and zf = 207 Hz. Unlike the end-mounted closed upstream, open downstream, with !
case, the frequency response can be divided into three distinct ranges, ! < 50 Hz, ! 2[50 Hz, 200 Hz], and ! > 200 Hz. As in the end-mounted case, the frequency response for ! > 200 Hz is due to 15
−1
Magnitude
10
−2
10
−3
10
0
1
Phase (deg.)
10
2
10
ω (Hz)
10
ω (Hz)
10
3
10
100
0
−100 0 10
1
2
10
3
10
Figure 7: The gain and phase characteristics of an uncontrolled combustor, with side-mounted loudspeaker (closed-open combustor), with one dominant mode at 542 Hz.
the acoustic dynamics. However, over low frequencies, there is first a gradual decrease in the phase (over 0-50Hz) to ,90 and then an increase (over 50-200 Hz) to 0 . This is due to the combined presence of the direct and indirect power contributions from the diaphragm motion on the pressure. Over 0-50 Hz, the frequencies are comparable to the flame characteristic frequency. Hence, over
this range, the contributions from pedirecti are very small and hence the frequency behavior is similar to that in the end-mounted case shown in Figure 5. As the frequency approaches the acoustic range,
pedirecti is of comparable magnitude to peindirecti and hence, the phase lag from the flame dynamics is not that significant which in turn contributes to the rise in the phase. It should be noted that leaving out the contribution of the stable acoustic mode which occurs at a lower frequency than the unstable mode in a combustor with closed-open boundary conditions may lead to inaccurate conclusions (We have however included the above example to illustrate the effect of pedirecti vs. peindirecti on the frequency response). Including this mode, i.e., with n transfer function W (s) can be factored as
=
2, the
2 + !z2 ) W (s) = K (s + !e )(s2 (+s 2+ zf!)(ss ++ !22 z)(!sz s2 , (24) : 2a !a s + !a2 ) f a a a as before, where !a and !a are the acoustic frequencies. If a and a are positive, the component associated with the frequency !a grows whereas that corresponding to !a decays. The factor s + zf is, as mentioned in the single-mode case, due to the combined effects of pedirecti and peindirecti . 1
1
1
2
2
1
1
1
2
2
2
2
16
0
10
−1
Magnitude
10
−2
10
−3
10
−4
10
0
10
1
2
10
ω (Hz)
10
ω (Hz)
10
w1
w2
3
10
Phase (deg.)
0 −50 −100 −150 −200 0 10
1
2
10
3
10
Figure 8: The gain and phase characteristics of an uncontrolled combustor, with a side-mounted loudspeaker (closed-open combustor). !a1 : stable frequency at 173 Hz; !a2 : unstable frequency at 542 Hz. The zero polynomial s2 + 2z !z s + !z2 represents the antiresonance effect, discussed in detail in (Annaswamy et al. , 1997b). The linear coupling between the acoustic modes contributes significantly to the antiresonance damping since the direct effect of the active control action affects the acoustic pressure not only through the unstable mode but also through the stable mode. The
damping ratio z as well as the relative location of the antiresonance frequency !z with respect to the acoustic frequencies, depend on the actuator-sensor locations. When z < 0, we have a nonminimum phase system which, especially in the presence of an unstable mode and when !z is
close to the unstable frequency, poses a stern challenge for designing a robust active controller that can suppress the pressure oscillations. Figure 8 illustrates the gain-phase characteristic of this case. The modes are at 173 Hz and 542 Hz, with the former being stable and the latter being unstable. The actuator is located at 0.12 m and the sensor at 0.25 m from upstream end. The remaining parameters are the same as in the single mode case. We discuss this case by focusing on two ranges, ! 250 Hz.
For ! 250 Hz, the unstable mode yields a positive phase change of 180 at 542 Hz. Since we chose the actuator location at 0.12 m and sensor location at 0.25 m, the antiresonance frequency occurred at 708 Hz with a negative damping ratio z = ,0:05. Hence, there is a phase change of
,180 at this frequency. In fact, if we focus on ! > 600 Hz, we can argue that the only contributor
to the combustor dynamics is antiresonance and hence, the transfer function can be approximated as
W (s) kp (s2 , 2z !z s + !z2)
for !
4
> 600 Hz
(26)
where kp3 is the gain. As will be seen in Section 3, the frequency characteristics in (25) and (26) are responsible for the generation of secondary peaks if one uses experimental controllers similar to those suggested in (Bloxsidge et al. , 1987).
3
An Explanation of Secondary Peaks
As mentioned in Section 1, a plethora of results have been reported in the literature pertaining to a satisfactory suppression of pressure instabilities using active control but some of the investigations report secondary peaks exhibited in the pressure power spectrum while using active control. The frequencies at which these peaks occur do not correspond to the combustor’s natural modes. This indicates that the closed-loop system exhibits a new “instability” which was not present in the uncontrolled combustor. In this section, we provide an explanation for this phenomenon. Our discussions will focus on the specific experimental control configurations used in (Gulati and Mani, 1992) for the end-mounted case, and in (Lang et al. , 1987; Poinsot et al. , 1989) for the side-mounted case using similar active controllers. We have chosen these works to carry out the analysis of secondary peaks since our model is more directly applicable to the premixed laminar combustor used in these experiments. Our investigations will show that (i) the unstable mode can be successfully suppressed, (ii) the stable mode can remain stabilized, and (iii) secondary peaks at frequencies both lower than and between the acoustic frequencies can be generated. Our comments are applicable for the results reported in (Bloxsidge et al. , 1987; Heckl, 1986; Gutmark et al. , 1993) as well. In order to analyze the experimental controllers, we carry out “numerical experiments” of active control where the combustor model is placed in a feedback loop together with a controller model and the resulting closed-loop system is numerically simulated. The parameters of the controller model 18
Premixed air + fuel flameholder
? -
-Products
,, @@ @ , Control
Figure 9: Active control of the combustor with side-mounted loudspeaker. are chosen by trial and error so that the numerically simulated controller duplicates the functionality of the experimental controller under consideration as closely as possible. A typical experimental strategy consists of measuring the microphone output, modulating it through a controller, and using its output to actuate the loudspeaker (shown schematically in Figure 9 for the side-mounted loudspeaker). The structure of these active controllers typically consists of a filter block, a phase shifter, and an amplifier cascaded together. Analytically, the experimental controller can be represented by the transfer function
G(s)
=
KAGa(s)Gb(s);
(27)
where KA corresponds to the amplifier gain, Ga (s) is the transfer function of a phase-shifter and
Gb(s) is a Butterworth filter.
The filter is usually either low-pass or high-pass, depending on the location of the unstable frequency in the acoustic range, which in turn depends on the boundary conditions. For example, in (Gulati and Mani, 1992), where a combustor is open on both ends with a flame held at its mid-span, the one half-wave mode is unstable. Thus, a Butterworth low-pass filter with an appropriate bandwidth is needed to filter out all but the unstable frequency. A second-order low-pass Butterworth filter with a corner frequency !b has the form
Gb(s)
=
!b2 s2 + 2b!bs + !b 2 :
(28)
On the other hand, for the same setup but with the upstream end closed, the three-quarter mode is unstable. Hence, a Butterworth high-pass filter is more appropriate (see (Bloxsidge et al. , 1987)) since it filters frequencies lower than the unstable frequency. For the simplest high-pass filter, the transfer function takes the form
Gh(s) = s +s h 19
(29)
which filters out frequencies lower than h. On the other hand, the phase shifter provides an appropriate phase at the unstable frequency by judicious design of the filter parameters. It is realized by an all-pass filter in (Gulati and Mani, 1992) which is of the form
Ga (s)
=
s , a 2 ; s+a
(30)
when the filter is centered at a frequency a. It can also be realized by a phase-locked loop circuit (as in (Gutmark et al. , 1993)), or by a phase-lead compensator whose transfer function is of the form
Gp(s)
=
s2 + 2N !N s + !N 2 : s2 + 2D !D s + !D 2
(31)
This compensator adds a positive phase or a negative phase between !N and !D depending on whether !N < !D or !N > !D , whose magnitude is determined by the difference between !N and
!D as well as the damping ratios N and D . For such a second-order compensator, the maximum phase addition is 180 . Finally an amplifier of gain KA is incorporated to provide an additional freedom to stabilize the unstable mode.
3.1
Example I
A bench top premixed combustor rig similar to (Gulati and Mani, 1992) is simulated to verify the performance using the experimental active control methods proposed in Sections 3. The combustion chamber is chosen to have an effective length of 0.49 m, with the flame anchored at 0.24 m from the upstream end. The burner diameter is 0.04 m, and the flame holder has 80 holes, each with 1.5
mm diameter. The combustor is treated as being acoustically open both upstream and downstream of the flow so as to be similar to the conditions of (Gulati and Mani, 1992). The loudspeaker was
end-mounted and the sensor was placed at 0.075 m from the upstream end as in (Gulati and Mani, 1992). Assuming perfect gas, =1.4, and atmospheric conditions, p¯ = 1atm and T = 350 K ,
c = 350m=s and = 1:15kg=m3. Using these conditions, the acoustic parameters a0, k1, !a , k2, and !a , and the mode shapes 1 (x) and 2 (x) are calculated. The flame parameters gf and !f are computed assuming that the heat of reaction is ∆qr = 2:563 106 J=kg ( the heat of reaction of the propane/air mixture at an equivalence ratio of 0.8), Su = 0:4m=s, = 0:5, and = 1:2. The 1
2
resulting transfer function is
W (s)
=
98 103
103) (s + 59)(s2 + 28s + 20147 103 )(s2 , 29s + 5037 103 ) s
( 2 + 18606
(32)
The power spectrum for the uncontrolled combustor, as illustrated in Figure 10, shows two frequen-
cies !a1 = 357 Hz and !a2 = 714 Hz which are close to the experimental results in (Gulati and Mani, 1992) where frequencies are 380 Hz and 760 Hz, respectively. Also, the transfer function 20
4
10
"−−"Uncontrolled Combustor
Power spectral density (Pa^2/Hz)
"−"Controlled Combustor, Controller A
3
10
2
10
1
10
0
10
0
100
200
300
400
500 ω (Hz)
600
700
800
900
1000
Figure 10: Power spectrum for the uncontrolled system “- -”, and the controlled system using Controller A “–”. (32) shows that the first thermoacoustic frequency is unstable with a growth rate of 14.3 s,1 while the second frequency is stable. Similar to (Gulati and Mani, 1992), we consider two experimental controllers, Controller A and Controller B, where the former has a structure similar to (27), while the latter has a transfer function
G(s)
=
KAGa (s)Gb(s)Gp(s)
(33)
where the additional Gp (s) is a phase lead compensator of the form of (31). The purpose of the phase-lead compensator is to provide a larger stability margin for the closed-loop system. Note that in (Gulati and Mani, 1992), the authors have included in Controller B, a notched filter to attenuate a spike that occurred in the open-loop response close to 600 Hz, which may be due to external noise. Since such a spike does not occur in our simulation of the open-loop model (Figure 6), we excluded the notched filter in our control design.
3.1.1 Controller A In (Gulati and Mani, 1992), it was shown that a control strategy consisting of a Butterworth filter and an all-pass filter was successful in suppressing the unstable mode of the bench-top combustor. We performed our first “numerical experiment” with the combustor modeled by (32), by placing an active controller with a transfer function G(s) given by Eq. (27) in a feedback loop. The Butterworth parameter !b
=
286 Hz, the all-pass parameter a = 485 Hz, b 21
=
0:7; and KA
=
9.
1
Magnitude
10
0
10
−1
10
0
1
10
2
10
3
10
10
Frequency (Hz)
Phase (deg.)
0
−200
−400
−600 0 10
1
2
10
3
10
10
Frequency (Hz)
Figure 11: Gain and phase characteristics of Controller A in Example I.
The frequency response of the resulting transfer function is shown in Figure 11.
The resulting
pressure response and its power spectrum are shown in Figures 12 and 10, respectively. These results are similar to those of (Gulati and Mani, 1992), with Figure 10 being similar to Figure 3 in (Gulati and Mani, 1992). The reduced peak near !a1 = 357 Hz is the result of the stabilization action of the controller which is achieved by choosing a near !a1 (see Figure 11). As mentioned earlier, the Butterworth filter was chosen so as to filter out frequencies higher than !a1 and hence
!b was chosen near !a
1
as well.
In (Gulati and Mani, 1992), it was also observed that a secondary frequency of 240 Hz was excited. In our “numerical experiment”, we observed a similar secondary peak, though at the lower frequency of 135 Hz. We now explain the origin of this peak. While the purpose of adding a controller was to add a phase at !a1 so as to suppress the instability, the filters in Controller A add a nonzero phase over a wide range starting from a value as low as 10 Hz, as shown in Figure 11. Focusing on ! !
while the phase of G2 (s) between the pressure oscillation and the diaphragm acceleration (i.e. the controller contribution) is about 90 at !b0 . Clearly !b0 is a function of the control parameters !b and
a.
The specific resonant value is somewhat smaller than !b since it represents a combined effect of both the Butterworth and the all-pass filter action. While the resonance occurs due to the phase addition from the flame, the flame characteristic frequency does not directly affect !b0 . 3.1.2 Controller B We now discuss Controller B, which is the second class of controllers tested in (Gulati and Mani,
1992). This has a transfer function similar to Eq. (33) where KA = 3, and Ga (s) and Gb (s) are similar to our choices in Controller A. The parameters of the phase-lead compensator Gp (s) are:
N
0:07, !N = 334 Hz, D = 0:04, and !D = 557 Hz, and are similar to those used in (Gulati and Mani, 1992). Figure 14 shows the power spectrum of the closed-loop response with controller B, it exhibits two new peaks. These frequencies can be observed also in the closed-loop =
system Bode plot in Figure 15: a low frequency at 135 Hz, and a higher frequency at 560 Hz. The peak at 135 Hz is smaller than the secondary peak caused by Controller A, but a new peak at 560
24
2
10
0
Magnitude
10
−2
10
−4
10
1
2
10
3
10 ω (Hz)
10
Phase (deg.)
0
−200
−400
−600 1
2
10
3
10 ω (Hz)
10
Figure 15: Gain and phase characteristics (Bode plot) for the closed-loop system with Controller B, for Example I.
Hz has appeared between the fundamental frequency and the second harmonic. These peaks are in agreement with the results of (Gulati and Mani, 1992) wherein two additional frequencies, 240 Hz and 550 Hz, were observed (see Figure 6 in (Gulati and Mani, 1992)). Similar to Controller A, the lower secondary peak at 135 Hz occurs due to the interaction of the controller with the flame. This is evident from Figure 16 which shows that Controllers A and B provide an identical phase responses for all frequencies less than 200 Hz (This is because, the contribution of a phase-lead compensator, which is the additional component in controller B, lies around the unstable frequency over 400-600 Hz.). On the other hand, the secondary peak at 560 Hz arises due to the action of the controller components alone. In particular, it forms because of the coupling between the all-pass filter and the phase-lead compensator. This can be argued as follows: Focusing on the range 400-600 Hz, : as shown in Section 2.2.4, the relation between v c and pe can be approximated by s (see Eq. (22)). Over the same frequency range, Controller B can be approximated by a fourth-order filter ,G2 (s) of the form
G2 (s)
550Hz, !s0 1
=
, s(s + !0 )(s2 +kc2 ! s + !2 ) s s s s 2
1
1
1
1
with !s1 = =700 Hz, s1 = 0:05, and kc2 is the approximate controller gain for this case. The resulting behavior of the closed-loop system is therefore equivalent to the feedback loop
25
0
Phase (deg.)
−50 −100
"−" Controller A
−150
"−." Controller B
−200 −250 −300 −350 −400 −450 0 10
1
2
10
10
3
10
Frequency (Hz)
Figure 16: The phase characteristics of Controllers A “–”and B “-.”.
in Figure 19(b) where
W (s)
G(s) = G2(s) : which leads to resonance at a frequency very close to !s (see appendix A). At this frequency, v c and yc are in phase because the combustor contribution over 400-600 Hz is 90 , while the phase contribution of the controller is -90 (at !s , the phase of ,G2 (s) is -270 which implies that the phase of G2 (s) is ,90 ). An equivalent, simplified, feedback loop is illustrated in Figure 19(b), =
kp s 2
1
1
which illustrates that instability occurs essentially due to one component of the controller inducing others into resonance. As in the previous case, the exact location of the resonance frequency, !s1 , is a function of a, !N , and !D , which are the filter parameters.
3.2
Example II
In this example, the chamber was chosen to be acoustically closed upstream and acoustically open downstream (see Figure 2), while possessing the same parameters as in Section 3.1. The loudspeaker was side-mounted and placed at 0.12 m while the sensor was placed at 0.25 m from upstream. This configuration is similar to that in (Lang et al. , 1987) and the boundary conditions described in (Bloxsidge et al. , 1987), though the latter discusses the implementation of an experimental controller on a turbulent combustor. For the chosen parameters, using the procedure outlined in
26
Section 2.2, the transfer function W (s) can be shown to be
W (s)
=
21 103
s , 404s + 19777 103) : (s + 61)(s2 + 866s + 1177 103 )(s2 , 271s + 11610 103 ) s
( + 1300)( 2
(34)
In what follows, we will report the results of a “numerical experiment”, designed in a manner similar to Section 3.1, following the control designs adopted in (Lang et al. , 1987; Bloxsidge et al. , 1987). The experimental control designs consist of an amplifier KC , a high-pass filter Gh (s) similar to Eq. (29) that identifies the unstable frequency, and a phase shifter Gp (s) similar to Eq. (31). We therefore choose
G(s)
KC Gh(s)Gp(s):
=
(35)
KC = -200 (Note that a positive feedback was necessary to stabilize the combustor in this case). Gh (s) is cornered at h = 239 Hz and ensures that the stable mode is not excited by filtering it out. Gp (s) is chosen with N = 0:19, !N = 239 Hz, D = 0:14; and !D = 605 Hz. Since !N < !a < !D , this ensures that the unstable mode is stabilized. with
2
3.2.1 Secondary Peaks with the Experimental Controller The experimental controller in Eq.(35) indeed stabilizes the unstable mode, but introduces secondary peaks at 50 Hz and 637 Hz, as can be seen from the power spectrum and the Bode plot of the closed-loop system shown in Figsures 17 and 18, respectively (Note that the unstable frequency at 542 Hz, gets attenuated and shifted to 400 Hz with control). A similar observation has been made in Figure 2 in (Bloxsidge et al. , 1987) where secondary peaks were evident at 66 Hz, 600 Hz and 700 Hz. Other results that have reported similar behavior include (Poinsot et al. , 1989; Billoud et al. , 1992). The low frequency peak at 50 Hz is the result of the high-pass filtering action of the controller on the combustor. As in the previous example, we provide an explanation using a feedbackloop configuration. Over the range 0-250Hz, as shown in Section 2.2.5, the combustor can be approximated by
W (s)
=
kp (s + zf ) (s + !f )(s2 + 2a !a s + !a2 ) 3
1
1
Similarly, the compensator in the controller has a negligible phase over the same range and hence,
G3 (s)
=
kc s s + h0 3
(36)
where h0 = 330 Hz and kc3 is the gain for this approximate controller. The resulting feedback loop has W (s) given by (25) and G(s) = G3 (s). A simplified loop is illustrated in Figure 19(c) where
W (s)
=
s ! s
( +
kp
f )( 2 +
3
2a !a1 s + !a21 ) 27
;
G(s)
=
kc s 3
"−−"Uncontrolled Combustor "−"Controlled Combustor
6
Power spectral density (Pa^2/Hz)
10
5
10
4
10
3
10
2
10
1
10
0
100
200
300
400
500 ω (Hz)
600
700
800
900
1000
Figure 17: Power spectrum of controlled combustor with experimental controller “–” compared to the uncontrolled response “- -”, for Example II.
2
10
1
Magnitude
10
0
10
−1
10
1
10
2
10 ω (Hz)
3
10
Phase (deg.)
0
−200
−400
−600 1
10
2
10 ω (Hz)
3
10
Figure 18: Gain and phase characteristics (Bode plot) of Closed-loop with experimental controller, for Example II.
28
:
The feedback forces v c and yc to be in phase at a frequency between !f and !a1 , which leads to resonance. This is because, at frequencies slightly higher than !f , the acoustic pressure lags the
acceleration by ,90 due to the indirect power component, which is compensated by the high-pass filtering action of the controller which adds a phase of +90 . It should be noted that, in this
example, there is a wide separation between the natural frequency of the component that instigates the resonance, !a1 at 173 Hz, and the secondary peak at 50 Hz. This is in contrast to the case in Example I where the separation is between 123 Hz and 135 Hz. This is because the controller has a phase contribution of 90 over the range 0-250 Hz while for the combustor, the phase shift can e f and !a1 . This means that this phase lag comes partly from be -90 only at a frequency between ! the flame dynamics and partly from the acoustic dynamics. A similar frequency shift occurs at the unstable frequency, from 542 Hz to 400 Hz. Both
frequency shifts, at 173 Hz and at 542 Hz, are due to the specific choice of the controller as in G3 (s) and the positive feedback action. The phenomenon is similar to what occurs in active control of a mass-spring-damper system when a position-force is used as the sensor-actuator pair. In this case, active control increases the stiffness of the system and hence increases the natural frequency if negative feedback is applied. On the other hand, when positive feedback is used, it tends to makes the system more compliant and hence decreases the frequency. The equivalent feedback gain is higher in this case than in Example I due to the fact that the flame dynamics has a higher gain at 50 Hz compared to 135 Hz. To explain the secondary peak at 637 Hz, we focus on frequencies greater than 400 Hz. We showed in Section 2.2.5 that at this frequency, the flame dynamics as well both the acoustic modes : contribute little to the phase between v c and pe, and the combustor dynamics is primarily governed by the antiresonance mechanism. As a result, the combustor transfer function can be approximated by
W (s)
=
kp (s2 , 2z !z s + !z2) 4
The phase contribution of the antiresonance is similar to an oscillator with a natural frequency !z and a damping ratio z , and hence,
6
W (s) 6 s2 + 2 1! s + !2 : z z z
As far as the controller is concerned, the high pass filter has no phase lag at this frequency. The combined phase-lead and phase-lag effects can be represented by the transfer function
G4(s)
=
, s2 + 2 k!c s + !2 s s s 4
2
2
(37)
2
where !s2 =610 Hz and s2 = 0:07. Note that the minus sign accounts for the contribution from the phase-lead component and !s2 !D . The resulting feedback loop is therefore of the form 29
as when two oscillators are connected in negative feedback and therefore force each other into resonance. Thus, the secondary peak in this case is due to the interaction of the antiresonance behavior with the phase-shifter in the controller. While the phase-shifter was adequate at the unstable frequency, it added a significant phase at higher frequencies; when this coincided with the antiresonant frequency, it induced a resonant behavior since near this frequency, the combustor goes through a large phase change. In particular, at !s2 , G4 (s) has a phase of 70 while W (s) has a phase of ,70 and hence the secondary peak.
4
Discussion and Concluding Remarks
In the previous sections, we considered two examples in which the same combustor had different upstream boundary conditions: either open as in (Gulati and Mani, 1992), or closed as in (Bloxsidge et al. , 1987). In the first case, the first acoustic mode was unstable while in the second, the second mode was unstable. This in turn implied that in order to identify the unstable frequency, the filtering component in the active control circuit had to be low pass in the first example and high pass in the second example. The experimental investigations in (Gulati and Mani, 1992) and (Bloxsidge et al. , 1987) showed that secondary peaks were created at 240 Hz and 550 Hz in the first, and 66 Hz, 600 Hz, and 700 Hz in the second. Using the feedback models developed in Section 2 for the combustor, and models of the experimental controllers used in these investigations, we predicted that secondary peaks would arise at 135 Hz and 560 Hz in Example I, and at 50 Hz and 637 Hz in Example II. We also discussed in Sections 3.1 and 3.2 the possible coupling mechanisms which are responsible for producing these peaks. Figure 19 summarizes the feedback loops involved in these coupling mechanisms. In each case, resonance occurs due to a different form of coupling between components of the combustor and the controller. In the combustor, at low frequencies, the flame dynamics is a contributor to resonance (see Figures 19(a),(c)). If acoustic modes occur at the lower end of the frequency spectrum, they contribute to the resonance as well (see Figure 19(c)). Coupling between the acoustic modes and in particular antiresonance is yet another mechanism that can be responsible for resonance (see Figure 19(d)). These figures illustrate that aspects of the controlled combustor other than the unstable acoustic mode contribute to secondary resonance. In Figure 19(a), a Butterworth filter introduced in order to filter out high frequencies couples with the flame dynamics to create low frequency resonance. In Figure 19(b), a phase-lead component of the filter introduced so as to produce a larger phase at the unstable frequency, couples with the phase-shifter in the controller itself to produce resonance at a high frequency. In Figure 19(c), the high-pass filter action destabilizes the stable acoustic mode at a low frequency. The simplified : loops in Figure 19(b) and (c) illustrate the form of coupling which bring v c and yc in phase. y1 30
and y2 are filtered pressure variables which may be viewed as equivalent signals that induce the coupling. In Figure 19(d), the phase-lead component in the controller which essentially adds the right phase at the unstable frequency, couples with the antiresonant mechanism in the combustor to produce resonance at a high frequency. It should be noted that different gains kpi and kci are included in Figures 19(a)-(d). The gains kpi are different since W (s), the combustor transfer function has an amplitude which varies over the frequency range. The gains kci are distinct partly due to the same reason and partly since they correspond to different forms of the controllers. One can also interpret this as the occurence of resonances for different gains of the amplifier in the controller. That is, for a specific controller structure, secondary resonance can occur for certain values of the amplifer gains which in turn may be chosen for effective stabilization of the thermoacoustic resonance. A few comments regarding the contribution of the acoustic dynamics to secondary peaks in the two examples are worth making. As mentioned in Section 2, in the second example where the upstream end is closed, both the direct and indirect effects of the loudspeaker on the pressure output are present whereas in the first example, only the indirect effect is present. A consequence of this is that acoustic components do not play a role in the generation of secondary peaks in Example I (see Figure 19(a),(b)). In contrast, in Example 2, due to the direct power added from the loudspeaker motion to the acoustic pressure, the stable acoustic mode contributes to the low secondary peak (see Figure 19(c)) while the antiresonant frequency is responsible for the high secondary peak (see Figure 19(d)). The difference between the structure of the flame dynamics that contributes to the e f ) vs. (s + zf )=(s + ! e f )) is again due to the difference resonance in the two examples (i.e. 1=(s + ! between the phase contributions of the indirect effect and the direct effect (see Section 2 for more details).
In summary, secondary peaks appear because of the fact that the controller was designed while focusing primarily on the combustor behavior at the unstable frequency and not over the entire frequency range. The combustor as well as the controller respond dynamically over a range of frequencies not limited to the unstable mode. The combustor possesses low frequency dynamics due to the flame kinematics, since the flame stabilization mechanism is such that the flame characteristic frequency is significantly lower than the acoustic frequency. Stable acoustic modes can be present at a lower range of frequency which can sometimes induce secondary resonance. At frequencies higher than the unstable frequency, there can be antiresonance which can also be a destabilizing factor. As far as controllers are concerned, they are often synthesized in many investigations using analog components. These in turn introduce a gain and phase which is not limited to the unstable frequency but over a larger range. If one is not careful, the controller can couple with an appropriate mechanism in the combustor at frequencies other than the unstable frequency to induce resonance. It is therefore vital to characterize the combustor response over a wide range of frequencies and 31
v: c
P
+
6
yc
Combustor
6
v: c
P
Controller
1
s
-
+
,kc
+
1
2+
P
pe
- s +kp!e f 2b0 !b0 s + !b02
v: c
yc
6
2
1
s
pe
-
,kc
s
2
2 +
2s1 !s1 s + !s21
b
kp
y2
pe
- ,s + !e s2 + 2 ! s + !2 - s + z a a f f a
yc
kp
-
( )
Combustor
3
1
1
1
1
Controller
3
y1
Controller
s + !s0 1
a
kc s
-
( )
Combustor
P
+
6
yc
c
v: c
Combustor
pe
- s2 + 2k!p s + !2 z z z 4
Controller
,kc
s
4
2+
2s2 !s2 s + !s22
d
( )
( )
Figure 19: The feedback loops that represent the coupling responsible for the secondary peaks in Examples 1 and 2, with secondary resonance (a) at 135 Hz close to !b0 , (b) at 550 Hz close to !s1 , (c) at 50 Hz near !a1 , and (d) at 637 Hz close to !z . The forward loops and feedback loops represent, respectively, the dynamics of the combustor and the controller near the resonant frequencies. design a controller that generates the right gain and phase over this entire range. In this paper, we have provided such a model of the combustor.
References Annaswamy, A. M., El-Rifai, O., Fleifil, M., and Ghoniem, A. F., “A model-based self-tuning controller for thermoacoustic instability,” in Proceedings of the 16th International Colloquium on the Dynamics of Explosions & Reactive Systems, (Cracow, Poland), August 1997a. Annaswamy, A. M., Fleifil, M., Hathout, J. P., and Ghoniem, A. F., “Impact of linear coupling on the design of active controllers for thermoacoustic instability,” Combustion Science and Technology, vol. 128, pp. 131–180, 1997b. Annaswamy, A. M., Fleifil, M., Rumsey, J. W., Hathout, J. P., and Ghoniem, A. F., “An input-output model of thermoacoustic instability and active control design,” IEEE Control System Thechnology, submitted 1997c. Billoud, G., Galland, M. A., Huu, C. H., and Candel, S., “Adaptive active control of combustion instabilities,” Combust. Sci. and Tech., vol. 81, pp. 257–283, 1992.
32
-
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A
A General Resonance Condition with a Feedback Controller
We consider an actively controlled combustor shown schematically in Figure 20 where W (s) and G(s) represent the dynamics of combustor and controller, respectively. ua and ya are the inputs
applied to combustor and controller, while yg and ug are the outputs generated by the combustor and controller, respectively. Assuming that the controller input is driven directly from the combustor output and that there is positive feedback, we obtain that
ua = ug
y a = yg :
and
(A.1)
The closed-loop transfer function is therefore given by
T (s)
=
W (s) : 1 , W (s)G(s)
(A.2)
The closed-loop system can exhibit peaks in its frequency response if the denominator is close to zero. That is, if at any frequency !s ,
W (j!s)G(j!s) 1;
(A.3)
resonance can occur. Eq. (A.3) is equivalent to
6
W (j!s)G(j!s)
0
and
Gain
,
W (j!s)G(j!s)
1
(A.4)
Thus, if the total phase between ya and yg is close to zero, then the closed-loop system can be driven into resonance by adjusting only the gain of the controller. We note that this resonance condition is independent of the behavior of W (s) and G(s) at other frequencies. Hence, even if the phase of G is such that it provides a stabilizing effect at the thermoacoustic resonant frequency, it can destabilize the combustor at another frequency. This is essentially the mechanism responsible for secondary peaks. When the phase between the input into the combustor and the output from the controller is close to zero at !s , we have Z 0
T
yg (ya) dt >
0
for any T:
(A.5)
Equation (A.5) can be viewed as a general resonance condition, and the Rayleigh’s criterion for 34
ua
P
+
6
-
yg
W (s)
G(s)
ug
-
+ ? P
ya
Figure 20: Resonance in a feedback loop. thermoacoustic instability as a special case. In the latter case,
W (s) corresponds to the acoustic
dynamics, i.e, the dynamic relation between the pressure oscillation and heat release rate as a source, while G(s) corresponds to the flame dynamics, i.e, the dynamic relation between rate of
heat release and pressure oscillation. In this case,ya is the pressure acting on the flame, ug is the corresponding response of the heat release rate, ua is the input energy into the acoustic field and yg
is the pressure oscillation generated due to the addition of heat to the acoustic field. At conditions corresponding to thermoacoustic instability, ua is essentially equal to ug . Moreover, if the flame is not actuated by any external driver, ya is equal to yg . That is, ua = ug = qf0 ; ya = yg = p0 . Since yg
=
W (s)ua, one can restate the resonance condition as follows: T 0 p W (s)qf0 dt 0
Z
>
0
for any T:
(A.6)
It happens that at the frequencies of the acoustic modes, the phase of W (s) is zero, i.e, the pressure oscillation generated due to heat addition from the flame is in phase with the heat release rate. Therefore Eq. (A.6) will reduce to the well known Rayleigh’s criterion: Z
T 0 0 p qf dt 0
>
0
35
for any T:
(A.7)
