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Passive suppression of thermoacoustic instability in a Rijke tube
13th IFAC Workshop on Time Delay Systems 13th IFAC Workshop on Delay 13th Workshop on Time Time Delay Systems Systems JuneIFAC 22-24, 2016. Istanbul, Turkey 13th IFAC Workshop on Delay June 22-24, 2016. Turkey 13th Workshop on Time Time Delay Systems Systems JuneIFAC 22-24, 2016. Istanbul, Istanbul, Turkey Available online at www.sciencedirect.com June June 22-24, 22-24, 2016. 2016. Istanbul, Istanbul, Turkey Turkey
ScienceDirect IFAC-PapersOnLine 49-10 (2016) 059–064
Passive suppression of thermoacoustic instability in aa Rijke tube Passive suppression of thermoacoustic instability in tube Passive suppression of thermoacoustic instability in aa Rijke Rijke tube Passive suppression of thermoacoustic instability in Rijke Passive suppression of thermoacoustic instability in a Rijke tube tube
Umut Zalluhoglu*, Nejat Olgac** Umut Zalluhoglu*, Nejat Olgac** Umut Zalluhoglu*, Zalluhoglu*, Nejat Nejat Olgac** Olgac** Umut Umut Zalluhoglu*, Nejat Olgac** *University of of Connecticut, Connecticut, Storrs, Storrs, CT CT 06269-3139 06269-3139 USA USA (e-mail: (e-mail: [email protected]). [email protected]). *University *University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: [email protected]). ** University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: [email protected]) *University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: [email protected]). *University of Connecticut, Storrs, CT 06269-3139 USAUSA (e-mail: [email protected]). ** University of Connecticut, Storrs, CT 06269-3139 (e-mail: [email protected]) ** University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: ** University University of of Connecticut, Connecticut, Storrs, Storrs, CT CT 06269-3139 06269-3139 USA USA (e-mail: (e-mail: [email protected]) [email protected]) ** [email protected])
Abstract: This article investigates passive stabilization of thermoacoustic dynamics in Rijke tube by Abstract: This This article article investigates investigates passive passive stabilization stabilization of of thermoacoustic thermoacoustic dynamics dynamics in in aaa Rijke Rijke tube tube by by Abstract: using a Helmholtz resonator. Under certain assumptions, the mathematical model of the dynamics is Abstract: This article article investigates passive stabilization of the thermoacoustic dynamics in the Rijke tube by by Abstract: This investigates passive of thermoacoustic dynamics in aa Rijke tube using aa Helmholtz Helmholtz resonator. Under certainstabilization assumptions, mathematical model of of dynamics is using resonator. Under certain assumptions, the mathematical model the dynamics is shown to fall into the linear time-invariant multiple time delay systems class of neutral type. To study the using Helmholtz resonator. Under certain certain assumptions, thesystems mathematical model oftype. the To dynamics is using Under assumptions, the mathematical the dynamics is shownaato toHelmholtz fall into into the theresonator. linear time-invariant time-invariant multiple time delay delay class of ofmodel neutralof study the the shown fall linear multiple time systems class neutral type. To study thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic shown to the time-invariant time delay of neutral To shown to fall fall into into the linear linearphenomenon time-invariant multiple timesuppression, delay systems systems class of treatment neutral type. type. To study study the the thermoacoustic instability phenomenon andmultiple its passive passive suppression, the class cluster treatment of characteristic characteristic thermoacoustic instability and its the cluster of roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with roots (CTCR) is effort the of roots (CTCR) paradigm is implemented. implemented. This effort reveals reveals the placement placement of Helmholtz Helmholtz resonator along the Rijke Rijke tube paradigm for stabilizing stabilizing the unstable unstableThis dynamics. The analytically analytically obtained results are are resonator comparedalong with the tube for the dynamics. The obtained results compared with experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design parameters on its stabilization capability is investigated. experiments on Rijke experiments conducted on aa laboratory laboratory scale Rijke tube. tube. In In addition, addition, the the effect effect of of resonator’s resonator’s design design parameters on onconducted its stabilization stabilization capability is isscale investigated. parameters its capability investigated. parameters on its stabilization capability is investigated. parameters on its stabilization capability is investigated. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. resonators are used as acoustic dampers to abate the resonators are used as acoustic dampers to abate the 1. INTRODUCTION resonators are used as acoustic dampers to abate the 1. INTRODUCTION combustion-excited oscillations (Zhao and Morgans, 2009). resonators are used as acoustic dampers to abate the 1. INTRODUCTION resonators are used as acoustic dampers to abate the combustion-excited oscillations (Zhao and Morgans, 2009). 1. INTRODUCTION INTRODUCTION combustion-excited oscillations (Zhao and Morgans, 2009). 1. combustion-excited oscillations (Zhao and Morgans, 2009). Thermoacoustic instability (TAI) is a major problem for gas combustion-excited oscillations (Zhao and Morgans, 2009). Thermoacoustic instability instability (TAI) (TAI) is is aa major major problem problem for for gas gas Thermoacoustic turbines results in vibrations in the Thermoacoustic instability (TAI) is aa major major problem for for gas Thermoacoustic is problem turbines since since it itinstability results (TAI) in undesired undesired vibrations in gas the turbines since it results in undesired vibrations in the Neck combustor and raises fatigue concerns. TAI primarily results turbines since it results in undesired vibrations in the Neck turbines since it results inconcerns. undesired vibrations in the combustor and raises fatigue TAI primarily results Neck combustor and raises fatigue concerns. TAI primarily results Neck from the interaction of heat release and acoustic disturbances. combustor and raises fatigue concerns. TAI primarily results Neck combustor and raises fatigue concerns. TAI primarily results from the interaction of heat release and acoustic disturbances. from the interaction of heat release and acoustic disturbances. The regionally confined unsteady heat release drives the from the interaction of heat release and acoustic disturbances. from regionally the interaction of heatunsteady release and acoustic disturbances. The confined heat release drives the The regionally confined unsteady heat release drives the acoustic waves in the combustion chamber. They propagate The regionally confined unsteady heat release drives the The regionally confined unsteady heat release drives the acoustic waves in the combustion chamber. They propagate acoustic waves in the combustion chamber. They propagate along the combustor, reflect from the boundaries, return and acoustic waves in the combustion chamber. They propagate acoustic waves in the combustion chamber. They propagate Cavity along the combustor, reflect from the boundaries, return and along the combustor, reflect from the boundaries, return and Cavity meet with the heat source again with some time delays. This Cavity along the combustor, reflect from the boundaries, return and along the combustor, reflect from the boundaries, return and meet with the heat source again with some time delays. This Cavity meet with the heat source again with some time delays. This Fig. 1. Helmholtz resonator (black) Cavityand its analogy to a causes disturbances in heat release rate, which amplifies the meet with the heat source again with some time delays. This Fig. 1. Helmholtz resonator (black) and analogy meet with the heat source again with some time delays. This Fig. 1. Helmholtz resonator (black) and its its analogy to to aa causes disturbances in heat release rate, which amplifies the mechanical vibration absorber (red). causes disturbances in heat release rate, which amplifies the Fig. 1. Helmholtz resonator (black) and its its analogy to to aa oscillations when synchronized (Rayleigh, 1878). mechanical vibration absorber (red). Fig. 1. Helmholtz resonator (black) and analogy causes disturbances in heat release rate, which amplifies the mechanical vibration absorber (red). causes disturbances in heat release rate, which amplifies the oscillations when synchronized (Rayleigh, 1878). oscillations when synchronized (Rayleigh, 1878). mechanical vibration absorber (red). mechanical vibration absorber (red). Many researchers focus on simpler thermoacoustic devices oscillations when synchronized synchronized (Rayleigh, 1878). oscillations when (Rayleigh, 1878). Many researchers focus on simpler thermoacoustic devices Many researchers focus on simpler thermoacoustic devices A critical issue in this application is the placement of the such as aa Rijke tube, to have aa better understanding of the Many researchers focus on simpler thermoacoustic devices Many researchers focus on simpler thermoacoustic devices A critical issue in this application is the placement of the such as Rijke tube, to have better understanding of the A critical issue in this application is the placement of the such as a Rijke tube, to have a better understanding of the resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; A critical issue in this application is the placement of the such as a Rijke tube, to have a better understanding of the A critical issue in this application is the placement of the such as a Rijke tube, to have a better understanding of the resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; placements may result in amplification of pressure Olgac et al., 2014; Zalluhoglu et al., 2016). Similarly, we use resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; placements may result in amplification of pressure Olgac et al., 2014; Zalluhoglu et al., 2016). Similarly, we use placements The may result in amplification of pressure Olgac et al.,tube 2014; Zalluhoglu et al., 2016). Similarly, we use oscillations. contribution of in the Rijke as the platform to study and control may result in of Olgac et 2014; Zalluhoglu et 2016). Similarly, we use placements may main result in amplification amplification of is pressure oscillations. The The main contribution of this this paper paper ispressure in the the Olgac et al., al.,tube 2014; Zalluhoglu et al., al., 2016). Similarly, weTAI use placements the Rijke as the platform to study and control TAI oscillations. main contribution of this paper is in the the Rijke tube as the platform to study and control TAI development of an analytical guideline to resolve this issue. phenomenon in this paper. TAI control attempts can be oscillations. The main contribution of to thisresolve paper this is in in the the Rijke tube as the platform to study and control TAI oscillations. The main contribution of this paper is the development of an analytical guideline issue. the Rijke tube as the platform to study and control TAI phenomenon in this paper. TAI control attempts can be development of an analytical guideline to resolve this issue. phenomenon in this paper. TAI control attempts can be We first develop aa linear time-invariant neutral multiple time divided into two classes: active and passive methods. The development of an analytical guideline to resolve this issue. phenomenon in this paper. TAI control attempts can be development of an analytical guideline to resolve this issue. We first develop linear time-invariant neutral multiple time phenomenon in this paper. TAI control attempts can be divided into two classes: active and passive methods. The We first develop aa linear time-invariant neutral multiple time divided into two classes: and methods. The delay model that represents the former is usually by forming aa feedback loop that We first develop neutral multiple divided into two achieved classes: active active and passive passive methods. The We firstsystem develop(LTI-NMTDS) a linear linear time-invariant time-invariant neutral multiple time time divided into two classes: active and passive methods. The delay system (LTI-NMTDS) model that represents the former is usually achieved by forming feedback loop that delay system (LTI-NMTDS) model that represents the former is usually achieved by forming a feedback loop that dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to delay system (LTI-NMTDS) model that represents the former is is usually usually achieved by forming forming a feedback feedback loop that delay system (LTI-NMTDS) model that represents the former achieved by a loop that dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, 2005; Banaszuk et al., dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, 2005; Banaszuk et al., Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, 2005; Banaszuk et al., locations the tube for an initially operation and 2004; Olgac al., 2015; and Olgac, 2015). In Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, Banaszuk et Then wealong analytically determine the unstable stabilizing resonator locations along the tube for an initially unstable operation and eliminate TAIet (Dowling andZalluhoglu Morgans, 2005; 2005; Banaszuk et al., al., 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In locations along the tube for an initially unstable operation and 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as locations along the tube for an initially unstable operation and 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In locations along the tube for an initially unstable operation and compare it with experimental results. In addition, the effect of 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In practice; however, active control has some drawbacks such as compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and investigated. potential to invite more instability, which make passive resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and investigated. potential to invite more instability, which make passive investigated. potential to more which make passive order to assess stability of LTI-NMTDS, numerical approach favourable. Among many passive approaches investigated. potential more to invite invite more instability, instability, which make passive In investigated. potential to invite more instability, which make passive In order to assess stability of LTI-NMTDS, numerical approach more favourable. Among many passive approaches In order to assess stability of LTI-NMTDS, numerical approach more favourable. Among many passive approaches algorithms that approximate the characteristic root locations such as altering the chamber design, installing resonators, In order to assess stability of LTI-NMTDS, approach more favourable. Among many passive approaches In order to assess stability of LTI-NMTDS, numerical approach more favourable. Among many passive approaches algorithms that approximate the characteristic root locations such as as altering altering the the chamber chamber design, design, installing installing resonators, resonators, algorithms that approximate the characteristic root numerical locations such (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovbaffles and acoustic liners to the combustor, we focus on the algorithms that approximate the characteristic root locations such as asand altering theliners chamber design, installing resonators, algorithms that approximate the characteristic root locations (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovsuch altering the chamber design, installing resonators, baffles acoustic to the combustor, we focus on the (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovbaffles and acoustic liners to the combustor, we focus on the Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of a Helmholtz resonator to attenuate TAI. (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovbaffles and acoustic liners to the combustor, we focus on the (Vyhlidal and Zitek, 2009) and methods utilizing LyapunovKrasovskii theory (Gu and Niculescu, 2006) are widely used. baffles and acoustic liners to the combustor, we focus on the utilization of a Helmholtz resonator to attenuate TAI. Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of aa Helmholtz resonator to attenuate TAI. Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of Helmholtz resonator to attenuate TAI. Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of a Helmholtz resonator to attenuate TAI. Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as demanding over large range of delays and the latter method shown in Fig. 1. Once mounted on a duct, it imparts noise Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as demanding over large range of delays and the latter method shown in Fig. 1. Once mounted on a duct, it imparts noise demanding over large results range offordelays and the latter method shown in Fig. 1. Once mounted on a duct, it imparts noise provides conservative stability. the reduction within aa frequency A single degree of demanding over range delays and the shown in 1. mounted on it noise demanding over large large results range of offor delays and Alternatively, the latter latter method method shown in Fig. Fig. 1. Once Once mounted band. on aa duct, duct, it imparts imparts noise provides conservative stability. Alternatively, the reduction within frequency band. A single degree of provides conservative results for stability. Alternatively, the reduction within a frequency band. A single degree of Cluster Treatment of Characteristic Roots (CTCR) paradigm freedom (SDOF) model of the resonator is derived in (Kinsler provides conservative results for stability. Alternatively, the reduction within a frequency band. A single degree of provides conservative results for stability. Alternatively, the Cluster Treatment of Characteristic Roots (CTCR) paradigm reduction within a frequency band. A single degree of freedom (SDOF) model of the resonator is derived in (Kinsler Cluster Treatment of Characteristic Roots (CTCR) paradigm freedom (SDOF) model of the resonator is derived in (Kinsler (Olgac et al., 2005) analytically declares the necessary and et al., 1996; Kim and Selamet, 2011) akin to a mechanical Cluster Treatment of Characteristic Roots (CTCR) paradigm freedom (SDOF) model of the resonator is derived in (Kinsler Cluster Treatment of Characteristic Roots (CTCR) paradigm (Olgac et al., 2005) analytically declares the necessary and freedom (SDOF) model of the resonator is derived in (Kinsler et al., 1996; Kim and Selamet, 2011) akin to aa mechanical (Olgac et al., 2005) analytically declares the necessary and et al., Kim and 2011) akin to mechanical sufficient conditions for stability of such systems. The vibration absorber Fig. mass of the et 2005) declares the and et al., 1996; 1996; Kim (see and Selamet, Selamet, 2011) akin to at mechanical (Olgac et al., al., 2005) analytically analytically declares the necessary necessary and sufficient conditions for stability of such systems. The et al., 1996; Kim and Selamet, 2011) to aa mechanical vibration absorber (see Fig. 1). 1). The The massakin of air air at the neck neck of of (Olgac sufficient conditions for stability of such systems. The vibration absorber (see Fig. 1). The mass of air at the neck of strength of this paradigm is that it can assess the stability the resonator oscillates against the volume of air in the cavity sufficient conditions for stability of such systems. The vibration absorber (see against Fig. 1). 1). the Thevolume mass of ofofair air atinthe the neck of strength sufficient conditions for stability of such systems. The vibration absorber (see Fig. The mass at neck of of this paradigm is that it can assess the stability the resonator oscillates air the cavity of this paradigm is that it can assess the and stability the resonator oscillates against the volume of air in the cavity cavity strength over very range of time delays exhaustively nonwhen excited by external pressure. The air in strength of this is it assess stability the resonator oscillates against the of air in strength ofbroad this paradigm paradigm is that that it can can assess the the and stability the resonator oscillates against the volume volume of the air larger in the the cavity over very broad range of time delays exhaustively nonwhen excited by external pressure. The air in the larger over very broad range of time delays exhaustively and nonwhen excited by external pressure. The air in the larger cavity conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper over very broad range of time delays exhaustively and nonwhen excited by external pressure. The air in the larger cavity over very broad range of time delays exhaustively and nonwhen excited by external pressure. The air in the larger cavity conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper tool throughout the paper. forces in mechanical vibration absorbers. Helmholtz conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper conservatively. It is used as the facilitating stability analysis tool throughout the paper. exerts a counter force resembling the spring and damper forces in mechanical vibration absorbers. Helmholtz throughout the paper. forces in mechanical vibration absorbers. Helmholtz tool forces tool throughout throughout the the paper. paper. forces in in mechanical mechanical vibration vibration absorbers. absorbers. Helmholtz Helmholtz tool
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acoustic waves f i (t ) and g i (t ) . The subscript i 1,...,6 denotes the cross-sections in Fig. 2a, where these functions are evaluated. The time delays 1 xu / c , 2 ( xd xr ) c
2. THE MATHEMATICAL MODEL The model described here carries the dynamics of the Rijke tube mounted with a Helmholtz resonator under commonly applied assumptions (Dowling and Morgans, 2005, Olgac et al., 2014): (i) the airflow is induced by natural buoyancy, therefore has negligibly small Mach number; (ii) the average (mean) flow quantities such as density and the speed of sound c are assumed as constant along the tube; (iii) the acoustic wave propagation is taken as 1-D event. ⑥
g6(t)
f6(t)
xd
F6 ( s )
Rd
e 3s g5(t)
⑤ f5(t) ④ f (t) 4
g4(t)
e 3s
Hr (s)
G4 ( s )
F4 ( s )
g3(t)
③
0
② g2(t)
f2(t)
G3 ( s ) Hh (s)
G2 ( s )
F2 ( s )
e 1s ①
-xu (a)
F1 ( s )
Ru
0 1 0 0
0 0 0 0 1 0 0 e 3 s
(4)
and resonator zone ( x xd xr ) , respectively, one obtains
G1 ( s )
G2 ( s ) F2 ( s ) F3 ( s ) G3 ( s ) H h G ( s ) H F ( s ) , H 02 4 4 F (s) G (s) 5 5
(b)
Fig. 2. (a) Schematic representation of Helmholtz resonator mounted Rijke tube dynamics, (b) its block diagram representation
The thermoacoustic dynamics is governed by the first principles, conservation of mass, momentum and energy, which are partial differential equations. When linearized under the listed assumptions, the pressure and velocity fluctuations within the tube obey the linear wave equation. Using the d’Alembert solution, they are expressed as
~ p( x, t ) f (t x / c ) g (t x / c ) ~ u ( x, t ) [ f (t x / c ) g (t x / c )] / c
(3)
Next, using the transfer matrices H h and H r that represent the interaction of acoustic waves at the heating zone ( x 0)
e 1s
g1(t)
f1(t)
G1 ( s ) G2 ( s ) e 1 s F3 ( s ) F3 ( s ) 0 T T , G (s) 1 G (s) 1 0 4 4 F (s) F (s) 0 6 5
e 2 s
F3 ( s )
0 0 1 0 0 0 0 Rd
0 0 1 0
in Laplace domain. Here, Ru and Rd are the reflection coefficients that characterize the boundary conditions at cross-sections ① and ⑥. The input vector in (3) can be expressed as
G5 ( s )
e 2 s f3(t)
Ru F1 ( s ) G1 ( s ) 0 G4 ( s ) F3 ( s ) , R R F (s) G ( s) 0 3 4 G ( s) F ( s) 6 6 0
G6 ( s )
F5 ( s )
xd - xr
HR
and 3 xr c correspond to the travel times of acoustic waves moving at the speed of sound. Utilizing the causal relationships between each variable in Fig. 2b, one can write
02 H r
(5)
Here, 0 2 represents a zero matrix of size 2 2 . More information on H h and H r matrices can be found in Appendices A and B, respectively. Similar to (4), the input vector in (5) can be expressed as
F2 ( s ) F1 ( s ) e 1 s 0 0 0 ( ) ( ) G s G s s 3 4 0 e 2 0 0 (6) F ( s ) T2 F ( s ) T2 0 0 e 2 s 0 4 3 G (s) G ( s) 0 0 e 3 s 5 6 0
(1) (2)
where overscript ~ denotes the non-steady (i.e., fluctuating) and represents the steady (mean) components of the quantity . f ( x, t ) and g ( x, t ) represent the acoustic waves traveling upwards and downwards in the tube, respectively. As shown in Fig. 2a, x is the position along the tube, where x 0 corresponds to location of the heater. At the tube ends x xu and x xd , where xu and xd denote the distances between ①-② and ③-⑥ (the upstream and downstream sides from the heating zone), respectively. The resonator is mounted at x xd xr (i.e. xr below the downstream end). The block diagram of the overall dynamics is given in Fig. 2b. In this representation, the system variables are the
Combining the transfer matrices (3), (4), (5) and (6) to close the loop in Fig. 2b, one gets
F1 ( s) F1 ( s) G4 ( s ) G4 ( s ) F ( s) RT1HT2 F ( s) 3 3 G (s) G ( s) 6 6
(7)
which yields 60
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F1 ( s ) G4 ( s ) M ( s, τ ) 0 , M(s, τ) I 4 RT1HT2 F (s) 3 G (s) 6
delay dependent stability analysis is required. We utilize the CTCR paradigm for this purpose, details of which can be found in Olgac and Sipahi (2005). Next, a brief review of the paradigm is presented.
(8)
For a linear time-invariant time delay system characterized by (10) to switch the stability posture, the characteristic roots should cross the imaginary axis, say at i for some delays
where I 4 is 4 4 identity matrix, τ (1, 2 , 3 ) and M(s) represents the overall system matrix. Its determinant gives the characteristic equation of the system as CE(s, τ) det(M) 0
τ 3 . We use
S s i Ω
CEs, τ
l
Pijkr e
( j 1 k 2 r 3 ) s i
s 0
2 2 2 j, 2 k , 3 r (1 j , 2k , 3r ) 1
where j , k and r are positive integers. These trajectories divide the delay domain into encapsulated regions in which the number of unstable roots, NU, is fixed. Stability reversal can only occur at the boundaries of these regions and the system is declared stable when NU 0 . Therefore, for a complete stability declaration, one needs to determine the boundaries of these infinitely many regions exhaustively. The following two propositions bring a discipline to this chaotic looking picture (Sipahi and Olgac, 2006): Proposition 1. There are only a manageably small number of
3 respectively. Notice that the highest order term in (10) involves delays, thus it is representative of a neutral timedelay system. Next, we give two theorems that assess the stability of this class of systems. Theorem 1: The stability of a neutral time delay system requires the dynamics governed by its associated difference equation to be stable. Therefore as a necessary condition, all the zeros of m
l
trajectories in τ 3 space called the kernel hypercurves. 2 0 τ τ, , τ 3 , Ω, 0 τ
p
1
2
3
(13)
(10)
where n is the order of the characteristic equation, m, l and p are the highest order of commensuracy of delays 1 , 2 and
Pnjkr e ( j k r )s 0
(12)
corresponding to τ 3 will be repeated infinitely many times as
i 0 j 0k 0r 0
D ( s, τ )
One can observe from (10) that an imaginary root s i
The Rijke tube dynamics falls into the class of linear time invariant multiple time delay systems with three delays, as discussed in Sec. 2. Its asymptotic stability is completely determined by the roots of its characteristic equation in (9). In general form, it can be written as m
notation to indicate their causal
Ω CE( s i, τ) 0, τ 3 ,
3. STABILITY ANALYSIS METHODOLOGY
n
τ,
correspondence. In order to assess the stability of the system exhaustively, all delay compositions for which (10) has an imaginary root should be determined. We denote the complete set of such imaginary root frequencies with Ω , and the corresponding root set with S as
(9)
It is a quasi-polynomial equation involving three independent delays (1, 2 , 3 ) , which are related to xu , xd and xr . They vary with the tube length, heater and Helmholtz resonator locations.
p
61
(11)
(14)
Notice that 0 represents the loci of the smallest 1 , 2 and 3 combinations corresponding to each imaginary root frequency Ω . All the remaining hypercurves are created from this set, 0 , utilizing the point-wise translation
j 0 k 0 r 0
have to lie in the left half of the complex plane. This is known as the strong stability condition (Hale and Verduyn Lunel, 2002).
property in (13) for j, k , r 0 . They are called the offspring
Theorem 2: The necessary and sufficient condition for the asymptotic stability of (10) is that all its infinitely many zeros should (i) lie in the open left half plane and (ii) be bounded away from the imaginary axis (Hale and Verduyn Lunel, 2002). The proof is again suppressed here.
hypercurves, and denoted by jkr where j, k and r identify the jth, kth and rth generation offspring in 1 , 2 and 3 , respectively. Consequently, the set of kernel and offspring hypercurves becomes 0 j 1 k 1 r 1 jkr . The set
Theorem 1 is a precondition to Theorem 2. It guarantees the condition (ii) in Theorem 2. If the associated difference equation (11) is stable, we only need to determine the conditions for which roots of (10) are in the open left half plane in order to assess its asymptotic stability. For this, a
constitutes the complete distribution of (1, 2 , 3 ) where the characteristic equation (10) has root sets containing at least one pair of imaginary roots, S .
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The root tendency (RT) for an imaginary characteristic root s S , along a delay axis (say 1 ) is defined by
RT
s Re sgn s i 1 s i
1
Once the resonator is mounted at a certain location on the Rijke tube, it alters the system dynamics and the characteristic equation in (9) becomes CE ( s, τ ) (1 0.865e 2( τ1 τ2 τ3 ) s )s 3 (2094.0 189.7e 2 τ1s
(15)
189.7e 2( τ 2 τ3 ) s 1287.0e 2( τ1 τ 2 ) s 1287.0e 2 τ3s 582.1e 2( τ1 τ 2 τ3 ) s ) s 2 (3.49 10 6 2.64 10 5 e 2 τ1s
This property indicates the evolution direction of the imaginary root i (to the left or the right-half of the complex plane) as we pass through a hypercurve by increasing one delay ( 1 ) infinitesimally.
2.82 10 5 e 2 τ 2 s 9.06 10 5 e 2 τ3s 9.06 10 5 e 2( τ1 τ2 ) s 2.44 10 5 e 2( τ1 τ3 ) s 2.61 10 5 e 2( τ2 τ3 ) s 1.33 10 6 e 2( τ1 τ 2 τ3 ) s ) s 1.77 10 9 4.76 108 e 2 τ1s
Proposition 2. At s S as one delay (say 1 ) increases at a point on a kernel and its corresponding offspring, the root
4.76 108 e 2( τ 2 τ3 ) s 1.53 10 9 e 2( τ1 τ 2 τ3 ) s 0
First, the strong stability condition in Theorem 1 should be checked. Notice that both associated difference equations corresponding to (16) and (17) are:
tendency RT s1 , remains invariant so long as the other delays 2 and 3 are kept fixed. This feature, in essence, declares the stabilizing (or destabilizing) transitions across the regional boundaries defined by . Based on these propositions of CTCR, one can establish the stability outlook of the system by generating its stability map in the domain of system parameters. To determine the complete set of kernel and offspring hypercurves various methods can be used such as the Rekasius transformation, Spectral Delay Space and Kronecker Summation (Sipahi and Olgac, 2006; Fazelinia et al., 2007; Ergenc et al., 2008; Gao et al., 2014). Once the complete set is obtained, RT is evaluated and the stable/unstable regions on the stability map are revealed.
D(s, τ) 1 0.865e 2( τ1 τ2 τ3 ) s 0
lower end of the tube) and xr c 3 (resonator distance from the upper end of the tube) domain is similarly generated using (17) and displayed in Fig. 3b. In both figures, the frequency information is imbedded on the kernel and offspring hypercurves. The shaded region is declared stable. The cross-hatched regions correspond to xu L in Fig. 3a
In this section, first the stability of the Rijke tube is assessed using the CTCR paradigm. Then, for the unstable operations the stabilizing Helmholtz resonator locations are revealed. These analytically obtained results are compared with the experimental findings from a laboratory scale Rijke tube. The setup consists of 0.508m long cylindrical tubes with electrical coil inside. Identical bottle shaped air cavities (Helmholtz resonators) are mounted on each tube at a different location. The resonant frequency of the resonator (252 Hz) is selected slightly off from the frequency at which TAI is observed in the Rijke tube (337 Hz). Since the absorber is not perfectly tuned, the correct placement of the resonator is of vital importance. The operational parameters regarding the Rijke tube are taken as Ru Rd 0.93 ,
A 7.07 cm 2 ,
(18)
which implies | e (1 2 3 ) s | 1.156 1 , meaning that all the infinitely many roots of (18) have Re(s) < 0. Thus, the strong stability condition is satisfied. The stability map of the plain Rijke tube in the space of xu c 1 and L c (1 2 3 ) is obtained by implementing CTCR paradigm on (16) and displayed in Fig. 3a. For the fixed tube length at L 0.508m , the stability map of the resonator mounted Rijke tube in xu (heater distance from the
4. MAIN RESULTS
1.4 ,
(17)
xu xr L in Fig. 3b, which are physically infeasible. The black line at L 0.508m in Fig. 3a represents the tube length in our experiment. The experimental results are marked with “ ” for stable and “ ” for unstable observed operations in both figures. There is good agreement between the analytical and experimental results in Fig. 3a. However, the analytical stability map in Fig. 3b appears to be more conservative. The predicted regions corresponding to stabilizing resonator locations are in fact larger as observed in experiments. This discrepancy may be overcome by better identification of the resonator parameters (Kim and Selamet, 2011). Nevertheless, the trends are in agreement between the analytical and experimental findings in Fig. 3b. The vertical line “L1” in Fig. 3 shows that by correctly placing the resonator, the unstable operation can be stabilized. On the other hand, initially stable Rijke tube can be destabilized by wrongly placing the resonator (see vertical line “L2”). The same feature is observed in a more advanced thermoacoustic structure in Cora et al., (2014).
1.2 kg/m3 ,
c 340 m/s a 200 and b 0.002 . The cavity volume of the resonator is V 265 ml , whereas the length and cross-sectional area of its neck are, l n 2.5cm and S n 2.27cm 2 respectively. The Rijke tube dynamics without the Helmholtz resonator can be calculated by simply taking H r [0 1; 1 0] . The resulting characteristic equation becomes CE( s, τ) (1 0.865e 2( τ1 τ2 τ3 ) s ) s 704.0 189.7e 2 τ1s (16) 189.7e 2( τ2 τ3 ) s 608.9e 2( τ1 τ2 τ3 ) s 0
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design parameter value used in the experiments, which led to the stability map in Fig. 3b. As observed in Fig. 5, larger stable regions can be obtained by decreasing V and ln or
A
increasing d n . These changes bring the undamped natural frequency of the resonator, r c S n / V leff , closer to the
(a)
frequency at which the Rijke tube (see Fig. 3a) is unstable. Figs. 5a, b and c can be used as a guideline to alter the Helmholtz resonator design to enhance stabilization capability. L1
L2 0.12
A A B B D C
0.1 Normalized Amplitude
B
C
0.08 0.06 0.04 0.02
(b)
0 0
Fig. 3. The stability maps of (a) plain Rijke tube and (b) Helmholtz resonator attached Rijke tube.
200 300 400 Frequency [Hz]
500
600
Fig. 4. FFT results of experimentally unstable observed operations.
Three experimentally observed unstable operations are marked with points A, B and C in Figs. 3a and b. A is an unstable operating point even when there is no resonator mounted on the Rijke tube. When the resonator is attached at the locations corresponding to B and C, TAI cannot be prevented. The pressure oscillations during these unstable operations are recorded and fast Fourier transform of these signals at the onset of instabilities are performed. The resulting resonant frequency information is provided in Fig. 4. Notice that attachment of the resonator changes the frequency at which instability occurs in the Rijke tube. The experimentally captured resonant frequencies are in agreement with the color-coded frequencies on the stability boundaries in Fig. 4, which are in the close vicinity of points A, B and C.
5. CONCLUSIONS In this paper, passive control of thermoacoustic instability in a Rijke tube is investigated using a Helmholtz resonator. A mathematical model of the dynamics in LTI-NMTDS form is derived and using the CTCR paradigm, stabilizing resonator locations are revealed to suppress thermoacoustic instability. The analytical findings are compared with experimental results from a laboratory scale Rijke tube. Furthermore, the effect of resonator’s design parameters on its stabilization capability is explored. The analytical guideline developed in this study is expected to contribute to futuristic designs of gas turbine combustors, where TAI is avoided. 6. ACKNOWLEDGEMENTS
Next, the effect of the Helmholtz resonator design parameters (cavity volume V , neck length ln and neck diameter d n ) on the stabilization capability of the resonator is investigated. Fixing the tube length to L 0.508m and the heater location to xu 0.152m , the stabilizing range of xr is studied. Utilizing the CTCR paradigm, the stability maps in the domain of ( xr ,V ) ( xr , ln ) and ( xr , d n ) are obtained as in Figs. 5a, b and c respectively. The dashed lines represent the
(a)
100
The authors would like to express their appreciation to the National Science Foundation grant CMMI-1462301 and UCONN (University of Connecticut) research excellence program for financial support. REFERENCES Dowling, A.P. and Morgans, A.S. (2005). Feedback control of combustion oscillations, Annu. Rev. Fluid Mech., 37, 151-
(b)
(c)
Fig. 5. Effect of the Helmholtz resonator design parameters on stable regions for L 0.508 m and xu 0.152m 63
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182. Rayleigh, J.W.S. (1878). The explanation of certain acoustic phenomena, Nature 18, 319-321. Gelbert, G., Moeck, J.P., Paschereit, C.O., King R., (2012). Feedback control of unstable thermoacoustic modes in an annular Rijke tube, Control Eng. Prac. 20, 770-782. Olgac, N., Zalluhoglu, U., Kammer, A.S. (2014). Predicting thermoacoustic instability; a novel analytical approach and its experimental validation, Journal of Propulsion and Power, 30, 4, 1005-1015. Zalluhoglu, U., Kammer, A.S., Olgac, N., (2016) Delayed feedback control laws for Rijke tube thermo-acoustic instability, synthesis and experimental validation, IEEE Transactions on Control Systems Technology (in press), accessed January 14, 2016. doi:10.1109/TCST.2015.2512938 Banaszuk, A., Ariyur, K.B., Krstić, M., and Jacobson, C.A. (2004). An adaptive algorithm for control of combustion instability, Automatica, 40, pp. 1965-1972. Olgac, N., Zalluhoglu, U., Kammer, A.S. (2015). A New Perspective in Designing Delayed Feedback Control for Thermo-Acoustic Instabilities (TAI), Combustion Science and Technology, 187, 5, 697-720. Zalluhoglu U., and Olgac N. (2015) Deployment of time delayed integral control for suppressing thermoacoustic instabilities, Journal of Guidance, Dynamics and Control (in press), accessed September 23, 2015. doi: http://arc.aiaa.org/doi/abs/10.2514/1.G001362. Kinsler, L.E., Frey, A.R., Coppens, A.B. and Sanders, J.V. (1996), Fundamentals of acoustics, 4th ed., John Wiley & Sons, Inc., New York, pp. 284-291. Kim H. and Selamet, A. (2011) Acoustic performance of a Helmholtz resonator with flow, International Journal of Vehicle Noise and Vibration, 7, pp. 285-305. Zhao, D., Morgans, A.S. (2009) Tuned passive control of combustion instabilities using multiple Helmholtz resonators, Journal of Sound Vibration, 320, pp. 744-757. Cora, R., Martins, C.A., and Lacava, P.T. (2014) Acoustic instabilities control using Helmholtz resonators, Applied Acoustics, 77, pp. 1-10. Vyhlídal, T., Zítek, P., (2009). Mapping based algorithm for large-scale computation of quasi-polynomial zeros, IEEE Trans. Automat. Control, 54, 171-177. Gu, K., Niculescu, S. I. (2006). Stability Analysis of Timedelay Systems: A Lyapunov Approach. In Advanced Topics in Control Systems Theory, Springer London, 4, 139-170. Olgac, N., Sipahi, R. (2005). The cluster treatment of characteristic roots and the neutral type time-delayed systems, ASME J. Dyn. Sys. Measur. Control, 127, 88-97. Hale, J.K., Verduyn Lunel, S.M., (2002) Strong stabilization of neutral functional differential equations, IMA J. Math. Control Info. 19, 5-23. Sipahi, R., Olgac, N. (2006). A unique methodology for the stability robustness of multiple time delay systems, Sys. Control Lett., 55, 819-825. Fazelinia, H., Sipahi, R., Olgac, N. (2007). Stability Robustness Analysis of Multiple Time Delayed Systems Using “Building Block” Concept, IEEE Trans. Automat. Control, 52, 799-810.
Gao, Q., Zalluhoglu, U., Olgac, N. (2014). Investigation of local stability transitions in the spectral delay space and delay space, ASME J. Dyn. Sys. Measur. Control, 136, 051011-1. Ergenc, A.F., Olgac, N. and Fazelinia, H., (2008). Extended Kronecker summation for cluster treatment of LTI systems with multiple delays, SIAM Journal on Control and Optimization, 46, pp. 143-155. APPENDIX A The transfer matrix H h represents the effect of heat injection on the incoming and outgoing acoustic waves as
G2 ( s) F ( s) H h ( s) 2 F3 ( s) G3 ( s)
(A1)
Utilizing the development in (Olgac et al., 2014), H h can be derived as follows
H h ( s)
Z 1 s 2Z W s Z W
s W Z
(A2)
where Z a( 1) /(2 Abc 2 ) and W 1/ b . Here, a and b are the gain and time constant of the so called “flame transfer function” between the velocity and heat release fluctuations. is the heat capacity ratio and A represents the cross-sectional area of the Rijke tube. APPENDIX B The transfer matrix H r represents the effect of the mounted Helmholtz resonator on the incoming and outgoing acoustic waves as
G4 ( s) F ( s) H r ( s) 4 F5 ( s) G5 ( s)
(B1)
Benefitting from (Kinsler et al., 1996; Kim and Selamet, 2011), H r can be derived as follows
H r ( s)
1 s
2
(Cr Tr ) s r2
Tr s 2 s Cr s r2
s 2 Cr s 2r Tr s (B2)
where Cr C / M , r K / M and Tr S n2 c /(2MA) . Here
M leff S n ,
C S n 3 c /(2 V leff )
and
K c 2 S n2 / V represent the equivalent mass (of the neck), damping constant and stiffness of the Helmholtz resonator, analogous to a mechanical vibration absorber. S n and l eff l n 0.85d n ( d n is the neck diameter) are the crosssectional area and the effective length of the resonator’s neck. V is the volume of the resonator’s cavity.
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Passive suppression of thermoacoustic instability in aa Rijke tube Passive suppression of thermoacoustic instability in tube Passive suppression of thermoacoustic instability in aa Rijke Rijke tube Passive suppression of thermoacoustic instability in Rijke Passive suppression of thermoacoustic instability in a Rijke tube tube
Umut Zalluhoglu*, Nejat Olgac** Umut Zalluhoglu*, Nejat Olgac** Umut Zalluhoglu*, Zalluhoglu*, Nejat Nejat Olgac** Olgac** Umut Umut Zalluhoglu*, Nejat Olgac** *University of of Connecticut, Connecticut, Storrs, Storrs, CT CT 06269-3139 06269-3139 USA USA (e-mail: (e-mail: [email protected]). [email protected]). *University *University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: [email protected]). ** University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: [email protected]) *University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: [email protected]). *University of Connecticut, Storrs, CT 06269-3139 USAUSA (e-mail: [email protected]). ** University of Connecticut, Storrs, CT 06269-3139 (e-mail: [email protected]) ** University of Connecticut, Storrs, CT 06269-3139 USA (e-mail: ** University University of of Connecticut, Connecticut, Storrs, Storrs, CT CT 06269-3139 06269-3139 USA USA (e-mail: (e-mail: [email protected]) [email protected]) ** [email protected])
Abstract: This article investigates passive stabilization of thermoacoustic dynamics in Rijke tube by Abstract: This This article article investigates investigates passive passive stabilization stabilization of of thermoacoustic thermoacoustic dynamics dynamics in in aaa Rijke Rijke tube tube by by Abstract: using a Helmholtz resonator. Under certain assumptions, the mathematical model of the dynamics is Abstract: This article article investigates passive stabilization of the thermoacoustic dynamics in the Rijke tube by by Abstract: This investigates passive of thermoacoustic dynamics in aa Rijke tube using aa Helmholtz Helmholtz resonator. Under certainstabilization assumptions, mathematical model of of dynamics is using resonator. Under certain assumptions, the mathematical model the dynamics is shown to fall into the linear time-invariant multiple time delay systems class of neutral type. To study the using Helmholtz resonator. Under certain certain assumptions, thesystems mathematical model oftype. the To dynamics is using Under assumptions, the mathematical the dynamics is shownaato toHelmholtz fall into into the theresonator. linear time-invariant time-invariant multiple time delay delay class of ofmodel neutralof study the the shown fall linear multiple time systems class neutral type. To study thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic shown to the time-invariant time delay of neutral To shown to fall fall into into the linear linearphenomenon time-invariant multiple timesuppression, delay systems systems class of treatment neutral type. type. To study study the the thermoacoustic instability phenomenon andmultiple its passive passive suppression, the class cluster treatment of characteristic characteristic thermoacoustic instability and its the cluster of roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with roots (CTCR) is effort the of roots (CTCR) paradigm is implemented. implemented. This effort reveals reveals the placement placement of Helmholtz Helmholtz resonator along the Rijke Rijke tube paradigm for stabilizing stabilizing the unstable unstableThis dynamics. The analytically analytically obtained results are are resonator comparedalong with the tube for the dynamics. The obtained results compared with experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design parameters on its stabilization capability is investigated. experiments on Rijke experiments conducted on aa laboratory laboratory scale Rijke tube. tube. In In addition, addition, the the effect effect of of resonator’s resonator’s design design parameters on onconducted its stabilization stabilization capability is isscale investigated. parameters its capability investigated. parameters on its stabilization capability is investigated. parameters on its stabilization capability is investigated. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. resonators are used as acoustic dampers to abate the resonators are used as acoustic dampers to abate the 1. INTRODUCTION resonators are used as acoustic dampers to abate the 1. INTRODUCTION combustion-excited oscillations (Zhao and Morgans, 2009). resonators are used as acoustic dampers to abate the 1. INTRODUCTION resonators are used as acoustic dampers to abate the combustion-excited oscillations (Zhao and Morgans, 2009). 1. INTRODUCTION INTRODUCTION combustion-excited oscillations (Zhao and Morgans, 2009). 1. combustion-excited oscillations (Zhao and Morgans, 2009). Thermoacoustic instability (TAI) is a major problem for gas combustion-excited oscillations (Zhao and Morgans, 2009). Thermoacoustic instability instability (TAI) (TAI) is is aa major major problem problem for for gas gas Thermoacoustic turbines results in vibrations in the Thermoacoustic instability (TAI) is aa major major problem for for gas Thermoacoustic is problem turbines since since it itinstability results (TAI) in undesired undesired vibrations in gas the turbines since it results in undesired vibrations in the Neck combustor and raises fatigue concerns. TAI primarily results turbines since it results in undesired vibrations in the Neck turbines since it results inconcerns. undesired vibrations in the combustor and raises fatigue TAI primarily results Neck combustor and raises fatigue concerns. TAI primarily results Neck from the interaction of heat release and acoustic disturbances. combustor and raises fatigue concerns. TAI primarily results Neck combustor and raises fatigue concerns. TAI primarily results from the interaction of heat release and acoustic disturbances. from the interaction of heat release and acoustic disturbances. The regionally confined unsteady heat release drives the from the interaction of heat release and acoustic disturbances. from regionally the interaction of heatunsteady release and acoustic disturbances. The confined heat release drives the The regionally confined unsteady heat release drives the acoustic waves in the combustion chamber. They propagate The regionally confined unsteady heat release drives the The regionally confined unsteady heat release drives the acoustic waves in the combustion chamber. They propagate acoustic waves in the combustion chamber. They propagate along the combustor, reflect from the boundaries, return and acoustic waves in the combustion chamber. They propagate acoustic waves in the combustion chamber. They propagate Cavity along the combustor, reflect from the boundaries, return and along the combustor, reflect from the boundaries, return and Cavity meet with the heat source again with some time delays. This Cavity along the combustor, reflect from the boundaries, return and along the combustor, reflect from the boundaries, return and meet with the heat source again with some time delays. This Cavity meet with the heat source again with some time delays. This Fig. 1. Helmholtz resonator (black) Cavityand its analogy to a causes disturbances in heat release rate, which amplifies the meet with the heat source again with some time delays. This Fig. 1. Helmholtz resonator (black) and analogy meet with the heat source again with some time delays. This Fig. 1. Helmholtz resonator (black) and its its analogy to to aa causes disturbances in heat release rate, which amplifies the mechanical vibration absorber (red). causes disturbances in heat release rate, which amplifies the Fig. 1. Helmholtz resonator (black) and its its analogy to to aa oscillations when synchronized (Rayleigh, 1878). mechanical vibration absorber (red). Fig. 1. Helmholtz resonator (black) and analogy causes disturbances in heat release rate, which amplifies the mechanical vibration absorber (red). causes disturbances in heat release rate, which amplifies the oscillations when synchronized (Rayleigh, 1878). oscillations when synchronized (Rayleigh, 1878). mechanical vibration absorber (red). mechanical vibration absorber (red). Many researchers focus on simpler thermoacoustic devices oscillations when synchronized synchronized (Rayleigh, 1878). oscillations when (Rayleigh, 1878). Many researchers focus on simpler thermoacoustic devices Many researchers focus on simpler thermoacoustic devices A critical issue in this application is the placement of the such as aa Rijke tube, to have aa better understanding of the Many researchers focus on simpler thermoacoustic devices Many researchers focus on simpler thermoacoustic devices A critical issue in this application is the placement of the such as Rijke tube, to have better understanding of the A critical issue in this application is the placement of the such as a Rijke tube, to have a better understanding of the resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; A critical issue in this application is the placement of the such as a Rijke tube, to have a better understanding of the A critical issue in this application is the placement of the such as a Rijke tube, to have a better understanding of the resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; placements may result in amplification of pressure Olgac et al., 2014; Zalluhoglu et al., 2016). Similarly, we use resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; resonator. As observed in Cora et al., (2014), wrong problem (Dowling and Morgans, 2005; Gelbert et al., 2012; placements may result in amplification of pressure Olgac et al., 2014; Zalluhoglu et al., 2016). Similarly, we use placements The may result in amplification of pressure Olgac et al.,tube 2014; Zalluhoglu et al., 2016). Similarly, we use oscillations. contribution of in the Rijke as the platform to study and control may result in of Olgac et 2014; Zalluhoglu et 2016). Similarly, we use placements may main result in amplification amplification of is pressure oscillations. The The main contribution of this this paper paper ispressure in the the Olgac et al., al.,tube 2014; Zalluhoglu et al., al., 2016). Similarly, weTAI use placements the Rijke as the platform to study and control TAI oscillations. main contribution of this paper is in the the Rijke tube as the platform to study and control TAI development of an analytical guideline to resolve this issue. phenomenon in this paper. TAI control attempts can be oscillations. The main contribution of to thisresolve paper this is in in the the Rijke tube as the platform to study and control TAI oscillations. The main contribution of this paper is the development of an analytical guideline issue. the Rijke tube as the platform to study and control TAI phenomenon in this paper. TAI control attempts can be development of an analytical guideline to resolve this issue. phenomenon in this paper. TAI control attempts can be We first develop aa linear time-invariant neutral multiple time divided into two classes: active and passive methods. The development of an analytical guideline to resolve this issue. phenomenon in this paper. TAI control attempts can be development of an analytical guideline to resolve this issue. We first develop linear time-invariant neutral multiple time phenomenon in this paper. TAI control attempts can be divided into two classes: active and passive methods. The We first develop aa linear time-invariant neutral multiple time divided into two classes: and methods. The delay model that represents the former is usually by forming aa feedback loop that We first develop neutral multiple divided into two achieved classes: active active and passive passive methods. The We firstsystem develop(LTI-NMTDS) a linear linear time-invariant time-invariant neutral multiple time time divided into two classes: active and passive methods. The delay system (LTI-NMTDS) model that represents the former is usually achieved by forming feedback loop that delay system (LTI-NMTDS) model that represents the former is usually achieved by forming a feedback loop that dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to delay system (LTI-NMTDS) model that represents the former is is usually usually achieved by forming forming a feedback feedback loop that delay system (LTI-NMTDS) model that represents the former achieved by a loop that dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, 2005; Banaszuk et al., dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to dynamics in the Helmholtz resonator mounted Rijke tube. modulates the acoustic pressure or fuel injection rates to Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, 2005; Banaszuk et al., Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, 2005; Banaszuk et al., locations the tube for an initially operation and 2004; Olgac al., 2015; and Olgac, 2015). In Then we analytically determine the stabilizing resonator eliminate TAI (Dowling and Morgans, Banaszuk et Then wealong analytically determine the unstable stabilizing resonator locations along the tube for an initially unstable operation and eliminate TAIet (Dowling andZalluhoglu Morgans, 2005; 2005; Banaszuk et al., al., 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In locations along the tube for an initially unstable operation and 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as locations along the tube for an initially unstable operation and 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In locations along the tube for an initially unstable operation and compare it with experimental results. In addition, the effect of 2004; Olgac et al., 2015; Zalluhoglu and Olgac, 2015). In practice; however, active control has some drawbacks such as compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as compare it with experimental results. In addition, the effect of practice; however, active control has some drawbacks such as resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and investigated. potential to invite more instability, which make passive resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and resonator’s design parameters on its stabilization capability requirement of high actuation power, limited bandwidth, and investigated. potential to invite more instability, which make passive investigated. potential to more which make passive order to assess stability of LTI-NMTDS, numerical approach favourable. Among many passive approaches investigated. potential more to invite invite more instability, instability, which make passive In investigated. potential to invite more instability, which make passive In order to assess stability of LTI-NMTDS, numerical approach more favourable. Among many passive approaches In order to assess stability of LTI-NMTDS, numerical approach more favourable. Among many passive approaches algorithms that approximate the characteristic root locations such as altering the chamber design, installing resonators, In order to assess stability of LTI-NMTDS, approach more favourable. Among many passive approaches In order to assess stability of LTI-NMTDS, numerical approach more favourable. Among many passive approaches algorithms that approximate the characteristic root locations such as as altering altering the the chamber chamber design, design, installing installing resonators, resonators, algorithms that approximate the characteristic root numerical locations such (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovbaffles and acoustic liners to the combustor, we focus on the algorithms that approximate the characteristic root locations such as asand altering theliners chamber design, installing resonators, algorithms that approximate the characteristic root locations (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovsuch altering the chamber design, installing resonators, baffles acoustic to the combustor, we focus on the (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovbaffles and acoustic liners to the combustor, we focus on the Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of a Helmholtz resonator to attenuate TAI. (Vyhlidal and Zitek, 2009) and methods utilizing Lyapunovbaffles and acoustic liners to the combustor, we focus on the (Vyhlidal and Zitek, 2009) and methods utilizing LyapunovKrasovskii theory (Gu and Niculescu, 2006) are widely used. baffles and acoustic liners to the combustor, we focus on the utilization of a Helmholtz resonator to attenuate TAI. Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of aa Helmholtz resonator to attenuate TAI. Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of Helmholtz resonator to attenuate TAI. Krasovskii theory (Gu and Niculescu, 2006) are widely used. utilization of a Helmholtz resonator to attenuate TAI. Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as demanding over large range of delays and the latter method shown in Fig. 1. Once mounted on a duct, it imparts noise Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as Nevertheless, the former approach becomes computationally Helmholtz resonator is a simple encapsulated air cavity as demanding over large range of delays and the latter method shown in Fig. 1. Once mounted on a duct, it imparts noise demanding over large results range offordelays and the latter method shown in Fig. 1. Once mounted on a duct, it imparts noise provides conservative stability. the reduction within aa frequency A single degree of demanding over range delays and the shown in 1. mounted on it noise demanding over large large results range of offor delays and Alternatively, the latter latter method method shown in Fig. Fig. 1. Once Once mounted band. on aa duct, duct, it imparts imparts noise provides conservative stability. Alternatively, the reduction within frequency band. A single degree of provides conservative results for stability. Alternatively, the reduction within a frequency band. A single degree of Cluster Treatment of Characteristic Roots (CTCR) paradigm freedom (SDOF) model of the resonator is derived in (Kinsler provides conservative results for stability. Alternatively, the reduction within a frequency band. A single degree of provides conservative results for stability. Alternatively, the Cluster Treatment of Characteristic Roots (CTCR) paradigm reduction within a frequency band. A single degree of freedom (SDOF) model of the resonator is derived in (Kinsler Cluster Treatment of Characteristic Roots (CTCR) paradigm freedom (SDOF) model of the resonator is derived in (Kinsler (Olgac et al., 2005) analytically declares the necessary and et al., 1996; Kim and Selamet, 2011) akin to a mechanical Cluster Treatment of Characteristic Roots (CTCR) paradigm freedom (SDOF) model of the resonator is derived in (Kinsler Cluster Treatment of Characteristic Roots (CTCR) paradigm (Olgac et al., 2005) analytically declares the necessary and freedom (SDOF) model of the resonator is derived in (Kinsler et al., 1996; Kim and Selamet, 2011) akin to aa mechanical (Olgac et al., 2005) analytically declares the necessary and et al., Kim and 2011) akin to mechanical sufficient conditions for stability of such systems. The vibration absorber Fig. mass of the et 2005) declares the and et al., 1996; 1996; Kim (see and Selamet, Selamet, 2011) akin to at mechanical (Olgac et al., al., 2005) analytically analytically declares the necessary necessary and sufficient conditions for stability of such systems. The et al., 1996; Kim and Selamet, 2011) to aa mechanical vibration absorber (see Fig. 1). 1). The The massakin of air air at the neck neck of of (Olgac sufficient conditions for stability of such systems. The vibration absorber (see Fig. 1). The mass of air at the neck of strength of this paradigm is that it can assess the stability the resonator oscillates against the volume of air in the cavity sufficient conditions for stability of such systems. The vibration absorber (see against Fig. 1). 1). the Thevolume mass of ofofair air atinthe the neck of strength sufficient conditions for stability of such systems. The vibration absorber (see Fig. The mass at neck of of this paradigm is that it can assess the stability the resonator oscillates air the cavity of this paradigm is that it can assess the and stability the resonator oscillates against the volume of air in the cavity cavity strength over very range of time delays exhaustively nonwhen excited by external pressure. The air in strength of this is it assess stability the resonator oscillates against the of air in strength ofbroad this paradigm paradigm is that that it can can assess the the and stability the resonator oscillates against the volume volume of the air larger in the the cavity over very broad range of time delays exhaustively nonwhen excited by external pressure. The air in the larger over very broad range of time delays exhaustively and nonwhen excited by external pressure. The air in the larger cavity conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper over very broad range of time delays exhaustively and nonwhen excited by external pressure. The air in the larger cavity over very broad range of time delays exhaustively and nonwhen excited by external pressure. The air in the larger cavity conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper tool throughout the paper. forces in mechanical vibration absorbers. Helmholtz conservatively. It is used as the facilitating stability analysis exerts a counter force resembling the spring and damper conservatively. It is used as the facilitating stability analysis tool throughout the paper. exerts a counter force resembling the spring and damper forces in mechanical vibration absorbers. Helmholtz throughout the paper. forces in mechanical vibration absorbers. Helmholtz tool forces tool throughout throughout the the paper. paper. forces in in mechanical mechanical vibration vibration absorbers. absorbers. Helmholtz Helmholtz tool
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acoustic waves f i (t ) and g i (t ) . The subscript i 1,...,6 denotes the cross-sections in Fig. 2a, where these functions are evaluated. The time delays 1 xu / c , 2 ( xd xr ) c
2. THE MATHEMATICAL MODEL The model described here carries the dynamics of the Rijke tube mounted with a Helmholtz resonator under commonly applied assumptions (Dowling and Morgans, 2005, Olgac et al., 2014): (i) the airflow is induced by natural buoyancy, therefore has negligibly small Mach number; (ii) the average (mean) flow quantities such as density and the speed of sound c are assumed as constant along the tube; (iii) the acoustic wave propagation is taken as 1-D event. ⑥
g6(t)
f6(t)
xd
F6 ( s )
Rd
e 3s g5(t)
⑤ f5(t) ④ f (t) 4
g4(t)
e 3s
Hr (s)
G4 ( s )
F4 ( s )
g3(t)
③
0
② g2(t)
f2(t)
G3 ( s ) Hh (s)
G2 ( s )
F2 ( s )
e 1s ①
-xu (a)
F1 ( s )
Ru
0 1 0 0
0 0 0 0 1 0 0 e 3 s
(4)
and resonator zone ( x xd xr ) , respectively, one obtains
G1 ( s )
G2 ( s ) F2 ( s ) F3 ( s ) G3 ( s ) H h G ( s ) H F ( s ) , H 02 4 4 F (s) G (s) 5 5
(b)
Fig. 2. (a) Schematic representation of Helmholtz resonator mounted Rijke tube dynamics, (b) its block diagram representation
The thermoacoustic dynamics is governed by the first principles, conservation of mass, momentum and energy, which are partial differential equations. When linearized under the listed assumptions, the pressure and velocity fluctuations within the tube obey the linear wave equation. Using the d’Alembert solution, they are expressed as
~ p( x, t ) f (t x / c ) g (t x / c ) ~ u ( x, t ) [ f (t x / c ) g (t x / c )] / c
(3)
Next, using the transfer matrices H h and H r that represent the interaction of acoustic waves at the heating zone ( x 0)
e 1s
g1(t)
f1(t)
G1 ( s ) G2 ( s ) e 1 s F3 ( s ) F3 ( s ) 0 T T , G (s) 1 G (s) 1 0 4 4 F (s) F (s) 0 6 5
e 2 s
F3 ( s )
0 0 1 0 0 0 0 Rd
0 0 1 0
in Laplace domain. Here, Ru and Rd are the reflection coefficients that characterize the boundary conditions at cross-sections ① and ⑥. The input vector in (3) can be expressed as
G5 ( s )
e 2 s f3(t)
Ru F1 ( s ) G1 ( s ) 0 G4 ( s ) F3 ( s ) , R R F (s) G ( s) 0 3 4 G ( s) F ( s) 6 6 0
G6 ( s )
F5 ( s )
xd - xr
HR
and 3 xr c correspond to the travel times of acoustic waves moving at the speed of sound. Utilizing the causal relationships between each variable in Fig. 2b, one can write
02 H r
(5)
Here, 0 2 represents a zero matrix of size 2 2 . More information on H h and H r matrices can be found in Appendices A and B, respectively. Similar to (4), the input vector in (5) can be expressed as
F2 ( s ) F1 ( s ) e 1 s 0 0 0 ( ) ( ) G s G s s 3 4 0 e 2 0 0 (6) F ( s ) T2 F ( s ) T2 0 0 e 2 s 0 4 3 G (s) G ( s) 0 0 e 3 s 5 6 0
(1) (2)
where overscript ~ denotes the non-steady (i.e., fluctuating) and represents the steady (mean) components of the quantity . f ( x, t ) and g ( x, t ) represent the acoustic waves traveling upwards and downwards in the tube, respectively. As shown in Fig. 2a, x is the position along the tube, where x 0 corresponds to location of the heater. At the tube ends x xu and x xd , where xu and xd denote the distances between ①-② and ③-⑥ (the upstream and downstream sides from the heating zone), respectively. The resonator is mounted at x xd xr (i.e. xr below the downstream end). The block diagram of the overall dynamics is given in Fig. 2b. In this representation, the system variables are the
Combining the transfer matrices (3), (4), (5) and (6) to close the loop in Fig. 2b, one gets
F1 ( s) F1 ( s) G4 ( s ) G4 ( s ) F ( s) RT1HT2 F ( s) 3 3 G (s) G ( s) 6 6
(7)
which yields 60
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F1 ( s ) G4 ( s ) M ( s, τ ) 0 , M(s, τ) I 4 RT1HT2 F (s) 3 G (s) 6
delay dependent stability analysis is required. We utilize the CTCR paradigm for this purpose, details of which can be found in Olgac and Sipahi (2005). Next, a brief review of the paradigm is presented.
(8)
For a linear time-invariant time delay system characterized by (10) to switch the stability posture, the characteristic roots should cross the imaginary axis, say at i for some delays
where I 4 is 4 4 identity matrix, τ (1, 2 , 3 ) and M(s) represents the overall system matrix. Its determinant gives the characteristic equation of the system as CE(s, τ) det(M) 0
τ 3 . We use
S s i Ω
CEs, τ
l
Pijkr e
( j 1 k 2 r 3 ) s i
s 0
2 2 2 j, 2 k , 3 r (1 j , 2k , 3r ) 1
where j , k and r are positive integers. These trajectories divide the delay domain into encapsulated regions in which the number of unstable roots, NU, is fixed. Stability reversal can only occur at the boundaries of these regions and the system is declared stable when NU 0 . Therefore, for a complete stability declaration, one needs to determine the boundaries of these infinitely many regions exhaustively. The following two propositions bring a discipline to this chaotic looking picture (Sipahi and Olgac, 2006): Proposition 1. There are only a manageably small number of
3 respectively. Notice that the highest order term in (10) involves delays, thus it is representative of a neutral timedelay system. Next, we give two theorems that assess the stability of this class of systems. Theorem 1: The stability of a neutral time delay system requires the dynamics governed by its associated difference equation to be stable. Therefore as a necessary condition, all the zeros of m
l
trajectories in τ 3 space called the kernel hypercurves. 2 0 τ τ, , τ 3 , Ω, 0 τ
p
1
2
3
(13)
(10)
where n is the order of the characteristic equation, m, l and p are the highest order of commensuracy of delays 1 , 2 and
Pnjkr e ( j k r )s 0
(12)
corresponding to τ 3 will be repeated infinitely many times as
i 0 j 0k 0r 0
D ( s, τ )
One can observe from (10) that an imaginary root s i
The Rijke tube dynamics falls into the class of linear time invariant multiple time delay systems with three delays, as discussed in Sec. 2. Its asymptotic stability is completely determined by the roots of its characteristic equation in (9). In general form, it can be written as m
notation to indicate their causal
Ω CE( s i, τ) 0, τ 3 ,
3. STABILITY ANALYSIS METHODOLOGY
n
τ,
correspondence. In order to assess the stability of the system exhaustively, all delay compositions for which (10) has an imaginary root should be determined. We denote the complete set of such imaginary root frequencies with Ω , and the corresponding root set with S as
(9)
It is a quasi-polynomial equation involving three independent delays (1, 2 , 3 ) , which are related to xu , xd and xr . They vary with the tube length, heater and Helmholtz resonator locations.
p
61
(11)
(14)
Notice that 0 represents the loci of the smallest 1 , 2 and 3 combinations corresponding to each imaginary root frequency Ω . All the remaining hypercurves are created from this set, 0 , utilizing the point-wise translation
j 0 k 0 r 0
have to lie in the left half of the complex plane. This is known as the strong stability condition (Hale and Verduyn Lunel, 2002).
property in (13) for j, k , r 0 . They are called the offspring
Theorem 2: The necessary and sufficient condition for the asymptotic stability of (10) is that all its infinitely many zeros should (i) lie in the open left half plane and (ii) be bounded away from the imaginary axis (Hale and Verduyn Lunel, 2002). The proof is again suppressed here.
hypercurves, and denoted by jkr where j, k and r identify the jth, kth and rth generation offspring in 1 , 2 and 3 , respectively. Consequently, the set of kernel and offspring hypercurves becomes 0 j 1 k 1 r 1 jkr . The set
Theorem 1 is a precondition to Theorem 2. It guarantees the condition (ii) in Theorem 2. If the associated difference equation (11) is stable, we only need to determine the conditions for which roots of (10) are in the open left half plane in order to assess its asymptotic stability. For this, a
constitutes the complete distribution of (1, 2 , 3 ) where the characteristic equation (10) has root sets containing at least one pair of imaginary roots, S .
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The root tendency (RT) for an imaginary characteristic root s S , along a delay axis (say 1 ) is defined by
RT
s Re sgn s i 1 s i
1
Once the resonator is mounted at a certain location on the Rijke tube, it alters the system dynamics and the characteristic equation in (9) becomes CE ( s, τ ) (1 0.865e 2( τ1 τ2 τ3 ) s )s 3 (2094.0 189.7e 2 τ1s
(15)
189.7e 2( τ 2 τ3 ) s 1287.0e 2( τ1 τ 2 ) s 1287.0e 2 τ3s 582.1e 2( τ1 τ 2 τ3 ) s ) s 2 (3.49 10 6 2.64 10 5 e 2 τ1s
This property indicates the evolution direction of the imaginary root i (to the left or the right-half of the complex plane) as we pass through a hypercurve by increasing one delay ( 1 ) infinitesimally.
2.82 10 5 e 2 τ 2 s 9.06 10 5 e 2 τ3s 9.06 10 5 e 2( τ1 τ2 ) s 2.44 10 5 e 2( τ1 τ3 ) s 2.61 10 5 e 2( τ2 τ3 ) s 1.33 10 6 e 2( τ1 τ 2 τ3 ) s ) s 1.77 10 9 4.76 108 e 2 τ1s
Proposition 2. At s S as one delay (say 1 ) increases at a point on a kernel and its corresponding offspring, the root
4.76 108 e 2( τ 2 τ3 ) s 1.53 10 9 e 2( τ1 τ 2 τ3 ) s 0
First, the strong stability condition in Theorem 1 should be checked. Notice that both associated difference equations corresponding to (16) and (17) are:
tendency RT s1 , remains invariant so long as the other delays 2 and 3 are kept fixed. This feature, in essence, declares the stabilizing (or destabilizing) transitions across the regional boundaries defined by . Based on these propositions of CTCR, one can establish the stability outlook of the system by generating its stability map in the domain of system parameters. To determine the complete set of kernel and offspring hypercurves various methods can be used such as the Rekasius transformation, Spectral Delay Space and Kronecker Summation (Sipahi and Olgac, 2006; Fazelinia et al., 2007; Ergenc et al., 2008; Gao et al., 2014). Once the complete set is obtained, RT is evaluated and the stable/unstable regions on the stability map are revealed.
D(s, τ) 1 0.865e 2( τ1 τ2 τ3 ) s 0
lower end of the tube) and xr c 3 (resonator distance from the upper end of the tube) domain is similarly generated using (17) and displayed in Fig. 3b. In both figures, the frequency information is imbedded on the kernel and offspring hypercurves. The shaded region is declared stable. The cross-hatched regions correspond to xu L in Fig. 3a
In this section, first the stability of the Rijke tube is assessed using the CTCR paradigm. Then, for the unstable operations the stabilizing Helmholtz resonator locations are revealed. These analytically obtained results are compared with the experimental findings from a laboratory scale Rijke tube. The setup consists of 0.508m long cylindrical tubes with electrical coil inside. Identical bottle shaped air cavities (Helmholtz resonators) are mounted on each tube at a different location. The resonant frequency of the resonator (252 Hz) is selected slightly off from the frequency at which TAI is observed in the Rijke tube (337 Hz). Since the absorber is not perfectly tuned, the correct placement of the resonator is of vital importance. The operational parameters regarding the Rijke tube are taken as Ru Rd 0.93 ,
A 7.07 cm 2 ,
(18)
which implies | e (1 2 3 ) s | 1.156 1 , meaning that all the infinitely many roots of (18) have Re(s) < 0. Thus, the strong stability condition is satisfied. The stability map of the plain Rijke tube in the space of xu c 1 and L c (1 2 3 ) is obtained by implementing CTCR paradigm on (16) and displayed in Fig. 3a. For the fixed tube length at L 0.508m , the stability map of the resonator mounted Rijke tube in xu (heater distance from the
4. MAIN RESULTS
1.4 ,
(17)
xu xr L in Fig. 3b, which are physically infeasible. The black line at L 0.508m in Fig. 3a represents the tube length in our experiment. The experimental results are marked with “ ” for stable and “ ” for unstable observed operations in both figures. There is good agreement between the analytical and experimental results in Fig. 3a. However, the analytical stability map in Fig. 3b appears to be more conservative. The predicted regions corresponding to stabilizing resonator locations are in fact larger as observed in experiments. This discrepancy may be overcome by better identification of the resonator parameters (Kim and Selamet, 2011). Nevertheless, the trends are in agreement between the analytical and experimental findings in Fig. 3b. The vertical line “L1” in Fig. 3 shows that by correctly placing the resonator, the unstable operation can be stabilized. On the other hand, initially stable Rijke tube can be destabilized by wrongly placing the resonator (see vertical line “L2”). The same feature is observed in a more advanced thermoacoustic structure in Cora et al., (2014).
1.2 kg/m3 ,
c 340 m/s a 200 and b 0.002 . The cavity volume of the resonator is V 265 ml , whereas the length and cross-sectional area of its neck are, l n 2.5cm and S n 2.27cm 2 respectively. The Rijke tube dynamics without the Helmholtz resonator can be calculated by simply taking H r [0 1; 1 0] . The resulting characteristic equation becomes CE( s, τ) (1 0.865e 2( τ1 τ2 τ3 ) s ) s 704.0 189.7e 2 τ1s (16) 189.7e 2( τ2 τ3 ) s 608.9e 2( τ1 τ2 τ3 ) s 0
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design parameter value used in the experiments, which led to the stability map in Fig. 3b. As observed in Fig. 5, larger stable regions can be obtained by decreasing V and ln or
A
increasing d n . These changes bring the undamped natural frequency of the resonator, r c S n / V leff , closer to the
(a)
frequency at which the Rijke tube (see Fig. 3a) is unstable. Figs. 5a, b and c can be used as a guideline to alter the Helmholtz resonator design to enhance stabilization capability. L1
L2 0.12
A A B B D C
0.1 Normalized Amplitude
B
C
0.08 0.06 0.04 0.02
(b)
0 0
Fig. 3. The stability maps of (a) plain Rijke tube and (b) Helmholtz resonator attached Rijke tube.
200 300 400 Frequency [Hz]
500
600
Fig. 4. FFT results of experimentally unstable observed operations.
Three experimentally observed unstable operations are marked with points A, B and C in Figs. 3a and b. A is an unstable operating point even when there is no resonator mounted on the Rijke tube. When the resonator is attached at the locations corresponding to B and C, TAI cannot be prevented. The pressure oscillations during these unstable operations are recorded and fast Fourier transform of these signals at the onset of instabilities are performed. The resulting resonant frequency information is provided in Fig. 4. Notice that attachment of the resonator changes the frequency at which instability occurs in the Rijke tube. The experimentally captured resonant frequencies are in agreement with the color-coded frequencies on the stability boundaries in Fig. 4, which are in the close vicinity of points A, B and C.
5. CONCLUSIONS In this paper, passive control of thermoacoustic instability in a Rijke tube is investigated using a Helmholtz resonator. A mathematical model of the dynamics in LTI-NMTDS form is derived and using the CTCR paradigm, stabilizing resonator locations are revealed to suppress thermoacoustic instability. The analytical findings are compared with experimental results from a laboratory scale Rijke tube. Furthermore, the effect of resonator’s design parameters on its stabilization capability is explored. The analytical guideline developed in this study is expected to contribute to futuristic designs of gas turbine combustors, where TAI is avoided. 6. ACKNOWLEDGEMENTS
Next, the effect of the Helmholtz resonator design parameters (cavity volume V , neck length ln and neck diameter d n ) on the stabilization capability of the resonator is investigated. Fixing the tube length to L 0.508m and the heater location to xu 0.152m , the stabilizing range of xr is studied. Utilizing the CTCR paradigm, the stability maps in the domain of ( xr ,V ) ( xr , ln ) and ( xr , d n ) are obtained as in Figs. 5a, b and c respectively. The dashed lines represent the
(a)
100
The authors would like to express their appreciation to the National Science Foundation grant CMMI-1462301 and UCONN (University of Connecticut) research excellence program for financial support. REFERENCES Dowling, A.P. and Morgans, A.S. (2005). Feedback control of combustion oscillations, Annu. Rev. Fluid Mech., 37, 151-
(b)
(c)
Fig. 5. Effect of the Helmholtz resonator design parameters on stable regions for L 0.508 m and xu 0.152m 63
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182. Rayleigh, J.W.S. (1878). The explanation of certain acoustic phenomena, Nature 18, 319-321. Gelbert, G., Moeck, J.P., Paschereit, C.O., King R., (2012). Feedback control of unstable thermoacoustic modes in an annular Rijke tube, Control Eng. Prac. 20, 770-782. Olgac, N., Zalluhoglu, U., Kammer, A.S. (2014). Predicting thermoacoustic instability; a novel analytical approach and its experimental validation, Journal of Propulsion and Power, 30, 4, 1005-1015. Zalluhoglu, U., Kammer, A.S., Olgac, N., (2016) Delayed feedback control laws for Rijke tube thermo-acoustic instability, synthesis and experimental validation, IEEE Transactions on Control Systems Technology (in press), accessed January 14, 2016. doi:10.1109/TCST.2015.2512938 Banaszuk, A., Ariyur, K.B., Krstić, M., and Jacobson, C.A. (2004). An adaptive algorithm for control of combustion instability, Automatica, 40, pp. 1965-1972. Olgac, N., Zalluhoglu, U., Kammer, A.S. (2015). A New Perspective in Designing Delayed Feedback Control for Thermo-Acoustic Instabilities (TAI), Combustion Science and Technology, 187, 5, 697-720. Zalluhoglu U., and Olgac N. (2015) Deployment of time delayed integral control for suppressing thermoacoustic instabilities, Journal of Guidance, Dynamics and Control (in press), accessed September 23, 2015. doi: http://arc.aiaa.org/doi/abs/10.2514/1.G001362. Kinsler, L.E., Frey, A.R., Coppens, A.B. and Sanders, J.V. (1996), Fundamentals of acoustics, 4th ed., John Wiley & Sons, Inc., New York, pp. 284-291. Kim H. and Selamet, A. (2011) Acoustic performance of a Helmholtz resonator with flow, International Journal of Vehicle Noise and Vibration, 7, pp. 285-305. Zhao, D., Morgans, A.S. (2009) Tuned passive control of combustion instabilities using multiple Helmholtz resonators, Journal of Sound Vibration, 320, pp. 744-757. Cora, R., Martins, C.A., and Lacava, P.T. (2014) Acoustic instabilities control using Helmholtz resonators, Applied Acoustics, 77, pp. 1-10. Vyhlídal, T., Zítek, P., (2009). Mapping based algorithm for large-scale computation of quasi-polynomial zeros, IEEE Trans. Automat. Control, 54, 171-177. Gu, K., Niculescu, S. I. (2006). Stability Analysis of Timedelay Systems: A Lyapunov Approach. In Advanced Topics in Control Systems Theory, Springer London, 4, 139-170. Olgac, N., Sipahi, R. (2005). The cluster treatment of characteristic roots and the neutral type time-delayed systems, ASME J. Dyn. Sys. Measur. Control, 127, 88-97. Hale, J.K., Verduyn Lunel, S.M., (2002) Strong stabilization of neutral functional differential equations, IMA J. Math. Control Info. 19, 5-23. Sipahi, R., Olgac, N. (2006). A unique methodology for the stability robustness of multiple time delay systems, Sys. Control Lett., 55, 819-825. Fazelinia, H., Sipahi, R., Olgac, N. (2007). Stability Robustness Analysis of Multiple Time Delayed Systems Using “Building Block” Concept, IEEE Trans. Automat. Control, 52, 799-810.
Gao, Q., Zalluhoglu, U., Olgac, N. (2014). Investigation of local stability transitions in the spectral delay space and delay space, ASME J. Dyn. Sys. Measur. Control, 136, 051011-1. Ergenc, A.F., Olgac, N. and Fazelinia, H., (2008). Extended Kronecker summation for cluster treatment of LTI systems with multiple delays, SIAM Journal on Control and Optimization, 46, pp. 143-155. APPENDIX A The transfer matrix H h represents the effect of heat injection on the incoming and outgoing acoustic waves as
G2 ( s) F ( s) H h ( s) 2 F3 ( s) G3 ( s)
(A1)
Utilizing the development in (Olgac et al., 2014), H h can be derived as follows
H h ( s)
Z 1 s 2Z W s Z W
s W Z
(A2)
where Z a( 1) /(2 Abc 2 ) and W 1/ b . Here, a and b are the gain and time constant of the so called “flame transfer function” between the velocity and heat release fluctuations. is the heat capacity ratio and A represents the cross-sectional area of the Rijke tube. APPENDIX B The transfer matrix H r represents the effect of the mounted Helmholtz resonator on the incoming and outgoing acoustic waves as
G4 ( s) F ( s) H r ( s) 4 F5 ( s) G5 ( s)
(B1)
Benefitting from (Kinsler et al., 1996; Kim and Selamet, 2011), H r can be derived as follows
H r ( s)
1 s
2
(Cr Tr ) s r2
Tr s 2 s Cr s r2
s 2 Cr s 2r Tr s (B2)
where Cr C / M , r K / M and Tr S n2 c /(2MA) . Here
M leff S n ,
C S n 3 c /(2 V leff )
and
K c 2 S n2 / V represent the equivalent mass (of the neck), damping constant and stiffness of the Helmholtz resonator, analogous to a mechanical vibration absorber. S n and l eff l n 0.85d n ( d n is the neck diameter) are the crosssectional area and the effective length of the resonator’s neck. V is the volume of the resonator’s cavity.
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