Transient growth of acoustical energy associated with mitigating thermoacoustic oscillations

Applied Energy 169 (2016) 481–490

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Transient growth of acoustical energy associated with mitigating thermoacoustic oscillations Xinyan Li a, Dan Zhao a,⇑, Xinglin Yang b, Huabing Wen b, Xiao Jin b, Shen Li b, He Zhao b, Changqing Xie c, Haili Liu c a b c

School of Mechanical and Aerospace Engineering, College of Engineering, Nanyang Technological University, Singapore 639798, Republic of Singapore School of Energy and Power Engineering, Jiangsu University of Science and Technology, Zhenjiang City, Mengxi Road 2, Jiangsu Province 212003, China Department of Energy and Electrical Engineering, Hunan University of Humanities Sciences and Technology, 410082 Loudi City, Hunan Province, China

h i g h l i g h t s
A thermoacoustic model with a linearly varied mean temperature is developed.
Pseudospectra analysis shows that the thermoacousic system is non-normal.
Heat-driven sound energy is defined and calculated.
Implementing LQG controller results in 76 dB sound pressure level reduction.
Transient energy growth is associated with mitigating thermoacoustic oscillations.

a r t i c l e

i n f o

Article history: Received 15 October 2015 Received in revised form 19 January 2016 Accepted 21 January 2016 Available online 22 February 2016 Keywords: Thermoacoustic oscillations Transient growth Thermoacoustic instability Active control Energy conversion

a b s t r a c t Energy conversion from heat to sound is desirable in some practical applications such as thermoacoustic heat engines or cooling systems. However, it is unwanted in gas turbine or aeroengine combustors. In this work, a Rijke-type thermoacoustic model with a linearly varied mean temperature configuration is developed. An acoustically compact heat source is confined and characterized by a modified form of King’s law. Unlike previous models available in the literature, the mean temperature is assumed to undergo not only a sudden jump across the heat source but also linearly increasing and decreasing in the pre- and afterheating regions respectively. Such mean temperature configuration is consistent with the experimental measurement. Coupling the heat source model with a Galerkin series expansion of the acoustic fluctuations provides a platform to gain insight on (1) the nonlinearity of the thermoacoustic system, (2) onset of limit cycle oscillations, (3) predicting its non-normality behaviors, (4) energy distribution and transfer between neighboring eigenmodes, and (5) evaluating the performance of feedback controllers. Pseudospectra and transient energy growth analyses are then performed. It reveals that the system is non-normal. And it is associated with transient growth of acoustical energy. The non-normality is found to be less intensified in comparison with that in a system with an invariant mean temperature from preand after-heating regions. To mitigate these limit cycle oscillations, the heat-to-sound coupling is interrupted by implementing multiple monopole-like actuators driven by a LQG (Linear Quadratic Gaussian) controller. For comparison, a pole-placement controller is also implemented. Approximately 76 dB sound pressure level reduction is achieved. However, implementing the LQG controller is shown to be associated with transient growth of acoustical energy, which has potential to trigger thermoacoustic instability. The present work opens up new applicable way to model thermoacoustic systems in the presence of a mean temperature gradient. Furthermore, it reveals new potential risk of applying active controllers to stabilize thermoacoustic systems. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction

⇑ Corresponding author. E-mail address: [email protected] (D. Zhao). http://dx.doi.org/10.1016/j.apenergy.2016.01.060 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.

In order to achieve low NOx emissions, combustion systems in ground-based gas turbines and aero-engines [1] tend to operate under lean premixed condition to meet stringent emission require-

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Nomenclature A
c Cp; Cv

coefficient matrix of the linearized governing equation speed of sound in air, m/s heat capacity ratio at constant pressure and volume, kJ/ kg K dw the diameter of the heated wires, m E1 ; E2 acoustical energy density at x1 and x2 , W/m2 Es total acoustical energy density, W/m2 G transient growth of acoustical energy J0 the Bessel function of order zero K control gain L the combustor length, m Lw the length of the heated wires, m m1 ; m2 the mean temperature gradient, K/m M mean flow Mach number N the number of actuators N interaction index N1 the number of eigenmodes p instantaneous pressure, Pa
; p0 p the mean and fluctuating pressure, Pa Q 0s unsteady heat release rate, kJ/s Q0a monopole-like sound source, kJ/s2 Rg the gas constant, J/mol K S cross-sectional area of the tube, m2 S k ; Rk actuation coefficients Tw the temperature of the heated wires, K T 1f ; T 2f initial mean temperatures in pre- and after-heating regions, K T the mean temperature, K u instantaneous flow velocity, m/s u control input
; u0 u the mean and fluctuating velocity, m/s wðtÞ; v ðtÞ white Gaussian noise

ments. However, under such lean premixed condition gas turbine combustors are more prone to thermoacoustic instability [1,2]. Such instability is generally caused by the coupling between unsteady combustion and acoustic disturbances. Unsteady heat release, which is an efficient monopole-like sound source, produces acoustic waves. When the sound waves propagate within the combustor, part of them are reflected from boundaries due to the impedance change, and return back to the combustion zone and further perturb the combustion process. If unsteady heat is added in phase with the pressure oscillations [3–5], acoustical energy is increased and finally ‘saturated’. And limit cycle thermoacoustic oscillations are generated. Such large-amplitude flow oscillations are wanted in thermoacoustic engines or cooling systems [6–9]. Efficient heat-to-sound conversion is desirable in some practical applications. When unsteady heat generates acoustic sound during the process of transferring from a high temperature source to a lowtemperature one, this sound could be used to produce electric power via piezoelectric generator [8] or linear alternator [4]. A variety of heat sources [8], such as solar energy and industrial waste heat, can be utilized. Thermoacoustic engines eliminate all the moving parts and are ‘pistonless’ [10]. The working gas undergoes compression and expansion processes in the form of sound waves to achieve the conversion of thermal energy to mechanical work [7,9]. So thermoacoustic engines can achieve high reliability and durability. There are two general types. One is a standing-wave one [8] and the other is traveling-wave one [9,7]. One typical example of standing-wave engines is Rijke-type thermoacoustic system [11]. It is a straight vertical tube with a mean flow and a

xf xak xs y Y0

axial location of the heat source, m axial location of the kth actuator, m axial location of the sensor, m vector of the state variables the Neumann function of order zero

Greek letters aak the ratio of the cross-section area d the Dirac delta function c the ratio of specific heat jm the normalized coefficient for the mth mode k conduction coefficient, W/mK x oscillation frequency, rad/s wm basis function q
mean air density, kg/m3 tak the actuation signal sf time delay, s nm the damping coefficient Subscript a m k 1 2 f

actuator mth mode kth actuator before-heating after-heating heat source

Superscript 0 fluctuating value : time derivative : mean value approximation value ˆ

heat source placed in its lower half. Under certain conditions,the interaction between unsteady heat release and acoustic disturbances in the tube may give rise to self-sustained thermoacoustic oscillations. However, this heat-to-sound conversion (also known as thermoacoustic instability) is undesirable in gas turbines and aero-engines, since large-amplitude flow oscillations may result in unacceptable noise and structural vibration, even catastrophic engine failure. Stable operation is one of the main requirements for gas turbine combustors. Thus it is important to gain insight on the onset of thermoacoustic instability [12,13,2] and to develop mitigation/control approaches. Over last few decades, thermoacoustic instability has been intensively studied [14–20]. Generally, linear modal analysis is conducted via calculating the eigenfrequencies of thermoacoustic systems [3]. However, when the thermoacoustic eigenmodes are not orthogonal, the system is non-normal. And acoustic fluctuations may undergo transient growth. If the transient growth is large enough (in comparison with acoustic losses and damping), thermoacoustic instability might be triggered. To gain insight on the finite-time (transient) stability behaviors, transient growth analysis of acoustical energy in a thermoacoustic system receives more attention recently [16–18]. A choked thermoacoustic combustor is studied to gain insight on its nonnormality. It is found that the maximum transient growth can be 103 times greater than the initial disturbances energy. Experimental measurement of the transient growth was conducted on a premixed gas turbine combustor [15]. The combustor inlet is not acoustically open or closed. Theoretical non-normality investigations are conducted on modeled thermoacoustic systems

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[16–18]. However, most of thermoacoustics systems studied previously are open-ended one, of which pressure nodes are expected at both inlet and outlet [3]. Furthermore, the mean temperature in pre- and after-heating regions remains unchanged. The thermoacoustic system was shown to be non-normal and associated with the transient growth of acoustic disturbances. However, the assumption on the mean temperature configuration is neither justified nor consistent with experimental measurement [21]. In addition, the effect of such mean temperature gradient on predicting the thermoacoustic oscillation frequencies and modeshapes is not considered, even it greatly influences the dynamics and stability behaviors of the system. Lack of this investigation partially motivated the present study. Minimizing thermoacoustic oscillations can be achieved by using active control approaches [22,23]. Active control means [26–29,24,25] has been the research focus during the past few decades. It is generally applied in closed-loop configuration (also known as feedback control). Closed-loop control can actively modify the combustion system in response to a measured signal by introducing external forcing perturbations. And the actuation generally include two main categories: acoustic and fuel modulation. In acoustic modulation [24,25,29], monopole-like sound sources, such as loudspeakers, are used to modulate the pressure field in the combustion chamber. These acoustic actuators have been successfully implemented and demonstrated in annular [25] and swirl stabilized combustors [24]. For fuel modulation, a secondary fuel injector is introduced to modulate the unsteady heat release rate. The key component in closed-loop control is the design of an active controller, which determines the optimum actuation signal in corresponding to the measured signal. Unlike conventional feedback controller, strict dissipativity controllers are recently proposed and tested [32–34] to achieve unity maximum transient growth [31]. Various active controllers (also known as optimization algorithms) have been developed and tested. H1-based controller was applied on an industrial swirling combustor [26]. In addition, LQG and LQG/LTR controllers were implemented on a swirlstabilized spray combustor to mitigate thermoacoustic oscillations. To stabilize two unstable modes, Tierno and Doyle [27] developed a model-based controller by using H1 loop-shaping techniques to stabilize a Rijke tube. Based on pole placement argument, Yang et al. [28] developed a digital controller to suppress longitudinal pressure oscillations. Annaswamy et al. [29] developed a modelbased optimal controller to suppress thermoacoustic instability in a premixed laminar combustor. And its performance was experimentally evaluated. Recently, Hervas et al. [30] applied a linearized LQG controller (Linear-Quadratic-Gaussian) to stabilize a Rijke-type thermoacoustic system with an uniform mean temperature configuration. However, the effect of the acoustically compact heat source on stabilizing the system and its non-normality is not studied. Lack of this investigation partially motivated the present study. In this work, a thermoacoustic system in the presence of a linearly varied mean temperature configuration is considered. Emphasis is being placed on the effect of the mean temperature configuration on transient growth of acoustical energy with and without monopole-like actuators implemented. For this, a generalized thermoacoustic model with monopole-like actuators present is developed. This is described in Section 2. Heated wires are assumed to be the acoustically compact heat source. They are modeled by using a modified King’s law [35]. Coupling the heat source model with a Galerkin series expansion [32–34] of acoustic waves present enables the time evolution of flow disturbances to be calculated, thus providing a platform to gain insights on transient energy growth behaviors of the system. When the actuators are not actuated, the thermoacoustic system is in open-loop configura-

tion. This is described in Section 3. Pseudospectra and transient energy growth analyses of the open-loop system are then conducted. When the actuators are actuated via LQG and PP (poleplacement) controllers, the thermoacoustic system is in closedloop configuration. The implementation and performance of the controllers are discussed in Section 4. In addition, transient growth of acoustical energy associated with the implementation of such controller is discussed. Furthermore, the non-normality of the present system is compared with that in the absence of the mean temperature variation. The main findings are summarized in Section 5. 2. A simple thermoacoustic model with monopole-like actuators An open-ended thermoacoustic system with an acoustically compact heat source confined is considered. Schematic of the 1D system with multiple monopole-like actuators is shown in Fig. 1. The length of the system is L. Heated wires are used as an acoustically compact heat source. They are axially placed at xf . Following the previous work [35], it is modeled by using a modified form of King’s law. The heat source gives rise to not only the mean temperþ ature TðxÞ jumping from 0 6 x 6 x f to xf < x 6 L but also the non-uniform TðxÞ configuration in pre- and after-heating regions. Subscripts 1 and 2 are used to denote these regions. The instantaneous variables, including velocity uðx; tÞ, density qðx; tÞ and pressure pðx; tÞ consist of a mean and fluctuating part as given as

qðx; tÞ ¼ q
ðxÞ þ q0 ðx; tÞ; uðx; tÞ ¼ u
ðxÞ þ u0 ðx; tÞ;
þ p0 ðx; tÞ pðx; tÞ ¼ p

ð1Þ

The overbar denotes the mean value and the prime denotes the perturbation. Following the previous works [31,17], the governing equations for the 1D thermoacoustic system can be shown as

@u0 ðx; tÞ 1 @p0 ðx; tÞ þ ¼0
@t qðxÞ @x @p0 ðx; tÞ @u0 ðx; tÞ
þ cp ¼ ðc  1ÞQ 0s ðtÞdðx  xf Þ @t @x

ð2Þ ð3Þ


;
c and c denote the mean pressure, sound speed and the where p ratio of specific heat respectively. The Dirac delta function dðxÞ is used to describe the acoustically compact heat source. Q 0s ðtÞ denotes the unsteady heat release rate. And it is modeled as [35]

Q 0s ðtÞ

2Lw ðT w  TÞ pffiffiffi ¼ S 6

2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi3   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u0 ðt  sÞ 1 pkC m q
u
dw 4  þ f
  5 3 3 u

  @u0f ðtÞ  N u0f ðt  sÞ  N u0f ðtÞ  s @t

ð4Þ

Fig. 1. Schematic of closed-loop thermoacoustic system with multiple distributed monopole-like actuators. L ¼ 1 m, xf ¼ 0:25 m.

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where T w ; dw and Lw denote the temperature, diameter and the equivalent length of the heated source. k and C m are heat conductivity and the specific heat of air at constant volume. S denotes the cross-section area of the tube. s represents the time lag between the moment when the oncoming velocity perturbation acts and that when the corresponding heat release is felt. Linearization and the coefficient N are given later. Differentiating the momentum Eq. (2) with respect to x, the energy Eq. (3) with respect to t, and eliminating the crossderivative term lead to

673

Experimental data Curve Fitting

Temperature [K]

593

513

433

Heater


@p0 ðx; tÞ 1 @ 2 p0 ðx; tÞ 1  c @  0 @ 2 p0 ðx; tÞ 1 dq  ¼ 2  2 Q s ðtÞdðx  xf Þ 2 2


@x qðxÞ dx @x c @t c @t

353

ð5Þ 273

Q0a ðtÞ

If there is another source producing pressure waves, such as a loudspeaker, Eq. (5) can be modified as


ðxÞ @p0 ðx; tÞ 1 @ 2 p0 ðx; tÞ @ 2 p0 ðx; tÞ 1 dq  2  2
c
@x @x qðxÞ dx @t 2   1c @ ¼ 2 Q 0 ðtÞdðx  xf Þ þ Q0a ðtÞ
c @t s

ð6Þ

0

0.2

0.4

0.6

0.8

1

Normalized axial position x/L Fig. 2. Comparison of the measured and modeled mean temperature distribution in a Rijke-type thermoacoustic system. Eq. (10) is used to predict the variation of the mean temperature. Here m1 ¼ 200:18 K/m, m2 ¼ 144:37 K/m, T 1f ¼ 351:3 K, T 2f ¼ 609:15 K.

For a monopole-like sound source [32,33], Q0a ðtÞ can be described as

Q0a ðtÞ ¼

N1 cp
X


c2

k¼1

aak

@ tak ðtÞ dðx  xak Þ @t

!

ð7Þ

where aak denotes the ratio of the cross-sectional area Sak of the kth actuator to the cross-sectional area S of the tube [36], i.e. aak ¼ Sak =S. xak denotes the location of the actuators. It is worth noting that the actuators should be placed upstream of the heat source to avoid being damaged in high-temperature environment. N1 denotes the number of actuators. It is related to the number of thermoacoustic eigenmodes and determined by the property of the controller, as discussed in the following section. Following the previous works [32,33], the diaphragm velocity of the kth actuator tak is modeled as

u0 ðx ; tÞ p0 ðxak ; tÞ ; tak ðtÞ ¼ Rk
ak þ Sk u cMp


ð8Þ

where M denotes the mean flow Mach number. By choosing proper actuator parameters Rk and Sk , the acoustic field can be modified to lead to destructive interaction between unsteady heat release and acoustic waves. By assuming that the working gas is an ideal one, Eq. (6) can be simplified to 2 0

0

N1  cp
X 1c @  0 @t Q dðx  xf Þ þ 2 aak ak dðx  xak Þ
c
c2 @t s @t k¼1

(

T 1f þ m1 ðx  xf Þ 0 6 x 6 xf T 2f þ m2 ðx  xf Þ xf < x 6 L

N X g_ m ðtÞ m¼1

where

jm ¼

jm xm R L 0

wm ðxÞ

w2m dx

1=2

ð11Þ . xm is the eigenfrequency. wm ðxÞ is the

basis function. It can be obtained by solving the following wave equation

@ 2 p0 ðx; tÞ 1 dTðxÞ @p0 ðx; tÞ 1 @ 2 p0 ðx; tÞ  þ ¼0 @x2 @x TðxÞ dx cRg TðxÞ @t2

ð12Þ

Substituting Eq. (11) into Eq. (12) and then rewriting the equation in terms of TðxÞ gives

!2 2 ! dTðxÞ d wm ðxÞ 1 d dTðxÞ dwm ðxÞ x2m wm ðxÞ þ þ ¼0 TðxÞ dx dx cRg TðxÞ TðxÞ dx dT 2 ðxÞ dTðxÞ ð13Þ The basis function wm ðxÞ can be expressed as



wm ðxÞ ¼

w1m ðxÞ for 0 6 x 6 xf w2m ðxÞ for xf < x 6 L

ð14Þ

Substituting T 1 ðxÞ ¼ T 1f þ m1 ðx  xf Þ into Eq. (13) gives

!

2

ð9Þ

where Rg ¼ C p  C v is the gas constant. C p and C v are the specific heats at constant pressure and volume respectively. The heated wires generate not only unsteady heat release but also lead to the mean temperature being varied along the axial direction. As measured by Matveev [21], the mean temperature in a Rijke-type thermoacoustic is linearly varied in pre- and after-heating regions as shown in Fig. 2. By applying curve fitting, the mean temperature can be modeled as

TðxÞ ¼

p0 ðx; tÞ ¼

2 0

@ p ðx; tÞ 1 dTðxÞ @p ðx; tÞ 1 @ p ðx; tÞ þ  @x2 @x TðxÞ dx cRg TðxÞ @t2 ¼

The acoustic oscillation such as p0 ðx; tÞ can be expanded by using Galerkin series [31,17] as

ð10Þ

Subscript 1 and 2 denote the pre- and after-heating regions respectively. T 1f ; T 2f are the initial mean temperatures. m1 and m2 are the gradients of the mean temperature in axial direction.

d w1m ðxÞ dT 21

þ

1 dw1m ðxÞ T1

dT 1

þ

x2m w1m ðxÞ ¼0 cm21 Rg T 1

ð15Þ

Eq. (15) can be further simplified to be a Bessel differential equation as given as 2

d w1m ðxÞ dT 21

þ

1 dw1m ðxÞ þ w1m ðxÞ ¼ 0 T 1 ðxÞ dT 1

ð16Þ

where T 1 is related to T 1 as T 1 ¼ cRg ðm1 T 1 =2xm Þ2 . The solution to Eq. (16) can be shown as:

w1m ðxÞ ¼ C 1 J 0 ðT 1 Þ þ C 2 Y 0 ðT 1 Þ  xm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1f þ m1 ðx  xf Þ ¼ C 1 J0 a1  xm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C2Y 0 T 1f þ m1 ðx  xf Þ a1

ð17Þ

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X. Li et al. / Applied Energy 169 (2016) 481–490

pffiffiffiffiffiffiffiffi where a1 ¼ jm1 j cRg 2. J0 and Y 0 are the Bessel and Neumann functions of order zero. C 1 and C 2 are coefficients to be determined by using boundary conditions and conservation equations across the heat source. By conducting similar analysis, the basis function w2m ðxÞ can be shown as

w2m ðxÞ ¼ C 3 J 0 ðT 2 Þ þ C 4 Y 0 ðT 2 Þ  xm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2f þ m2 ðx  xf Þ ¼ C 3 J0 a2  xm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2f þ m2 ðx  xf Þ þ C4Y 0 a2

ð18Þ

pffiffiffiffiffiffiffiffi where a2 ¼ jm2 j cRg 2. C 3 and C 4 are coefficients, which are related to C 1 and C 2 . It is easy to show that the basis function wm RL is orthogonal, i.e. 0 wm ðxÞwn ðxÞ dx ¼ 0 for m – n. Pressure nodes are assumed to be at both open ends, i.e. x ¼ 0 and x ¼ L. These boundary conditions are described in mathematics as

w1m ð0Þ ¼ 0;

w2m ðLÞ ¼ 0

ð19Þ

Across the heat source, the pressure continuity and velocity jump occurs at x ¼ xf as given as

w1m ðxf Þ



w2m ðxþf Þ

¼ 0;

þ  1 dw1m ðxf Þ 1 dw2m ðxf Þ ¼0  dx q
1 q
2 dx

ð20Þ

These boundary and conservation conditions can be used to determine the four unknown coefficients C 1 ; C 2 ; C 3 and C 4 . To obtain non-trivial solutions, the determinant of the coefficient matrix in the homogenous wave equations needs to be set to zero. This enables the natural eigenfrequencies xm of the thermoacoustic system to be predicted. Substituting Eq. (11) into Eq. (9), multiplying it with wn ðxÞ and then carrying out an integration from x ¼ 0 to x ¼ L leads to N1 X jn d2 gn
aak tak wn ðxak Þ ð21Þ þ jn xn gn ¼ ðc  1ÞQ 0s ðtÞwn ðxf Þ þ cp 2 xn dt k¼1

where WTak ¼ ðwi1 ðxak Þ;wi2 ðxak Þ;;wiN ðxak ÞÞ, PTak ¼ ðuð1;xak Þ;uð2;xak Þ;...; qffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffi  pffiffiffiffiffi . x x uðN;xak ÞÞ, uðj; xÞ , ð1Þiþ1 RT i c C 2i1 J1 a j T i þ C 2i Y 1 a j T i i

The unsteady heat release Q 0s ðtÞ can be approximated by assum
j < 1=3, as ing ju0f ðt  sÞ=u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi# qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"  0  Lw ðT w  TÞ 0 1 þ u ðxf ; t  sÞ  1

6pkC m qudw ð26Þ Q s ðtÞ ¼ 3 
3S 3 u " # p ffiffiffi p ffiffiffi  0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 3 uf ðt  sÞ 3 3 uf ðt  sÞ Lw ðT w  TÞ

 6pkC m qudw  þ 

u u 2 3S 8 Further simplification leads to

Q_ s ðtÞ ¼ N u0f ðt  sÞ

jn d2 gn dg þ 2nn jn n þ jn xn gn xn dt2 dt N1 X
aak tak wn ðxak Þ ¼ ðc  1ÞQ 0s ðtÞwn ðxf Þ þ cp

ð22Þ

k¼1

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . where nn ¼ 0:7xn =x1 þ 0:07 x1 =xn 2p denotes the damping/ loss coefficient. It represents acoustic energy losses due to radiation from the open ends and dissipation in the acoustic viscous and thermal boundary layers at the duct walls. To study the transient growth of the thermoacoustic system and to minimize heat-to-sound conversion, the developed nonlinear governing equations are linearized and formulated in a state space form. If we define yT1 ¼ ðg1 ; g2 ; . . . ; gN Þ, yT2 ¼ ðg1 =x1 ; g2 =x2 ; . . . ; gN =xN Þ, Eq. (22) can be rewritten as

y_ 1 ¼ My2

ð23Þ 1

y_ 2 ¼ My1  DMy2 þ ðc  1ÞU

Q 0s ðtÞ

1

Wf þ U u

ð24Þ

where M ¼ diagðx1 ; x2 ; . . . ; xN Þ, D ¼ diagð2n1 ; 2n2 ; . . . ; 2nN Þ, U ¼   diagðj1 ; j2 ; . . . ; jN Þ, WTf ¼ w11 ðxf Þ; w12 ðxf Þ; . . . ; w1N ðxf Þ . u is the control input. It is defined as

u¼

N1 N1 X cp
X 1 aak Rk Wak PTak y1 þ aak Sk Wak WTak U1 y2


u

k¼1

k¼1

M

ð25Þ

ð27Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. pffiffiffiffiffiffi
. Assuming the time delay s
dw S 2u where N ¼ Lw ðT w  TÞ pkC m q is relatively small compared with the period of the fundamental mode, Taylor series expansion can be used to expand the velocity to first order in s as

u0f ðt  sÞ ¼ u0f ðtÞ  s

@u0f ðtÞ

ð28Þ

@t

The velocity u0f ðx; tÞ can be obtained by replacing p0 ðx; tÞ in the momentum Eq. (2) with Eq. (11) as

u0 ðx; tÞ ¼

N X uðn; xÞgn ðtÞ;

0 6 x 6 xf

ð29Þ

n¼1

Thus unsteady heat release Q 0s can be shown as

Q 0s ðt  sÞ  KPTf ðy1  sMy2 Þ

ð30Þ

Finally combining Eqs. (23) and (24) leads to

y_ ¼ Ay

ð31Þ

where y ¼ T

If acoustic damping and boundary loss [31,17] is considered, Eq. (21) can be rewritten as

i

jj p
, if xak 2 ½0; xf , i = 1, else i = 2.

 A¼

0 A1

ðyT1 ; yT2 Þ

 M ; A2

and the coefficient matrix A are given as

A1 ¼ M þ N ðc  1ÞWf U1 PTf ; A2 ¼ DM  N sðc  1ÞWf U1 PTf M

Since AA – A A, the thermoacoustic system is non-normal. The solution to y_ ¼ Ay can be shown as y ¼ expðAtÞyð0Þ. The thermoacoustic system’s non-normality can be then studied by using state-space Eq. (31) as discussed later. 3. Transient and nonlinear analysis of open-loop system Let’s first consider the thermoacoustic system in the absence of monopole-like actuators, i.e. Q0a ðtÞ ¼ 0 via setting aak ¼ 0. The governing Eq. (22) can be solved by using 4th order Runge–Kutta algorithm. Thus, the nonlinearity of the thermoacoustic system can be investigated. Fig. 3 shows the time evolution of acoustic pressure and velocity fluctuations. It can be seen that small acoustic disturbances grow exponentially into limit cycle oscillations resulting from the nonlinearity effect. To gain insight on the system stability and non-normality, pseudospectra analysis of the linearized thermoacousic system as described in Eq. (31) is conducted as shown in the contour plot of Fig. 4. Generally, the pseudospectra of non-normal operators are non-concentric circles centered on the eigenvalues corresponding to the maximum values. When part of the eigenvalues lies in the right half plane, the system is linearly unstable. This explains why the limit cycle oscillations are produced.

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X. Li et al. / Applied Energy 169 (2016) 481–490

0.08

x 10

(a)

2

0.04

−4

(b)

1

0

0

−1

−0.04 −2

−0.08 −0.6

−0.3

0

0.3

0.6

−0.002

−0.001

0

0.001

0.002

Fig. 3. Phase diagrams of velocity u0 ðxf ; tÞ and pressure oscillation p0 ðxf ; tÞ, as N ¼ 2; Lw ¼ 3:5 m, dw ¼ 0:5  103 m, T w ¼ 1680 K, xf ¼ 0:25 m, S ¼ 1:56  103 K,
¼ 1:01325  105 Pa, k = 0.0328 W/m K, c ¼ 1:3, C v ¼ 719, s ¼ 1:2 ms, u
1 ¼ 0:5 m/s. p

15

(a) |p,(ω , x)| 1

10

log10ξ 1.5

ℑ{Z}

5

T =T

→ Heater

1

1

2

T1≠T2

0.5

1

0

0

0.5

(b)

−5

−15 −10

2

|p,(ω , x)|

−10

−0.5 −5

0

5

→ Heater

1

0

0.5

−1

ℜ{Z} Fig. 4. Pseudospectra of the linearized thermoacoustic system, as N ¼ 2; Lw ¼ 3:5 m,
¼ 1:01325 dw ¼ 0:5  103 m, T w ¼ 1680 K, xf ¼ 0:25 m, S ¼ 1:56  103 K, p 105 Pa, k ¼ 0:0328 W/m K, c = 1.3, C v ¼ 719; s ¼ 1:2 ms, N ¼ 5:25  105 ;
1 ¼ 0:5 m/s. The contour values are calculated by using log10 n ¼ log10 jZI  Aj1 u and the maximum values correspond to eigenvalues.

Further modal analysis is conducted to predict the eigenfrequencies and mode shapes. For comparison, we consider two thermoacoustic systems with different mean temperature configurations. One is associated with an invariant mean temperature along the tube, i.e. T 1 ¼ T 2 ¼ Const. The other is with a mean temperature gradient, i.e. T 1 – T 2 (see Eq. (10)). Fig. 5 depicts the first and second pressure mode shapes. It can be seen that there is an obvious shift in the pressure antinodes. And the shift became significant at higher frequency. Table 1 summarizes the first and second eigenfrequencies obtained from the two systems. Comparison is then made between the predicted solutions and the experimental measurements [21]. It can be seen that the eigenfrequencies calculated from the present thermoacoustic system with a mean temperature gradient are much closer to the experimental measurements than that in the absence of a mean temperature gradient, i.e. T 1 ¼ T 2 . This confirms that the mean temperature configuration plays an important role in determining the system dynamics. In order to predict the transient stability behavior of the thermoacoustic system, transient growth analysis of acoustic distur-

0 0

0.2

0.4

0.6

0.8

1

Normalized axial position xf /L Fig. 5. The first and second normalized pressure mode shape. m1 ¼ 200, m2 ¼ 144:37, T 1f ¼ 351:3 K, T 2f ¼ 609:15 K. Table 1 Comparison between predicted and measured oscillation frequencies.

T1 ¼ T2 T1 – T2

Theoretical data (Hz)

Experimental data (Hz)

Error (%)

x1 =2p

x2 =2p

x1 =2p

x2 =2p

x1 =2p

x2 =2p

215.09 190.65

430.18 406.66

192.72 192.72

384.10 384.10

11.61 1.07

12.00 5.87

bances is then conducted. The total acoustic energy Eðxf ; tÞ per unit cross-sectional area is chosen as a measure to characterize the transient growth of the acoustic disturbances

Z

L

Eðxf ; tÞ ¼ 0

p02 q
u0 þ
2cp 2

2

!

dx ¼

N  1 X g_ 2n 2 g þ n
2cp x2n

ð32Þ

n¼1

Since y is defined as y ¼ ðg1 ; g2 ; . . . ; gN ; g_ 1 =x1 ; g_ 2 =x2 ; . . . ; g_ N =xN Þ in the governing Eq. (31), it can be easily shown that the 2-norm kyk2 is given as

kyk2 ¼

N  X

g2n þ

n¼1



g_ 2n ¼ k expðAtÞyð0Þk2 x2n

ð33Þ

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X. Li et al. / Applied Energy 169 (2016) 481–490

By comparing Eq. (32) with Eq. (33), the acoustical energy can be rewritten as

1 Eðxf ; tÞ ¼ kyðtÞk2
2cp

−3

ð34Þ

The maximum amplification of the total acoustical energy is defined as

Gðxf ; tÞ ¼ maxEðxf ;0Þ

Eðxf ; tÞ kyðtÞk2 ¼ k expðAtÞk2 ¼ maxyð0Þ Eðxf ; 0Þ kyð0Þk2

4

(a)

x 10

(b)

1

2

0

0

−1

−2

−2

ð35Þ

−4

x 10

2

0

0.02

0.04

0.06

−4

0

E −3

that the non-normality is less intensified in a system with a nonuniform mean temperature than that with a constant mean temperature configuration. Further increasing N leads to Gmax being infinitely large, as shown in Fig. 6(c). And the thermoacoustic systems become linearly unstable. Phase diagrams of the acoustical energy is shown in Fig. 7. It can be seen that the fundamental and first harmonic eigenmodes are simultaneously triggered. The acoustical energy grows exponentially and finally saturates due to the effect of nonlinearity. However, the fundamental mode at x1 plays a dominant role in comparison with its harmonics. This could due to the larger damping associated with higher frequency modes.

x 10

2

(c)

0

ent values. Obviously, when N is small ðN ¼ 3:0  102 Þ and the unsteady heat release is low, the transient growth Gðxf ; tÞ is decreased monotonously and the maximum of transient growth Gmax is unity, as shown in Fig. 6(a). This means that the thermoacoustic system is stable. Increasing N to 1:5  105 as shown in Fig. 6(b) leads to Gðxf ; tÞ being increased first and then decreased in both configurations of thermoacoustic systems. Gmax ðxf ; tÞ  1:50. It indicates that the acoustical energy is increased by 50% due to the non-normality. Furthermore, Gmax jT 1 ¼T 2 > Gmax jT 1 – T 2 . This reveals

0.002

E2

1

where the growth factor Gðxf ; tÞ is the maximal value over all possible initial conditions. In order to calculate the Gðxf ; tÞ, singular value decomposition method (SVD) is used to determine the singular values of expðAtÞ. Fig. 6 shows the variation of transient growth G(xf ¼ 0:25, t) for the two mean temperature configurations, as N is set to three differ-

0.001

−2

0

0.02

0.04

0.06

Es Fig. 7. Time evolution of acoustical energy. (a) acoustical energy E1 at x1 , (b) acoustical energy E2 at x2 , (c) Es for the total acoustic energy, as N = 2, L = 1.0 m,
¼ 1:01325  105 Pa and xf ¼ 0:25 m, s = 1.2 ms, Lw ¼ 3:5 m, N ¼ 5:25  105 , p
1 ¼ 0:5 m/s. Arrow denotes time increasing. u

4. Mitigating thermoacoustic oscillations Self-sustained thermoacoustic oscillations are unwanted in gas turbine and aeroengine systems. In order to eliminate these oscillations, monopole-like actuators can be applied. For this, LQG and pole-placement controllers are implemented to the modeled thermoacoustic system to evaluate their damping performance. With the controllers implemented, the developed model (i.e. Eq. (22)) is formulated into a state-space form as given as

y_ ¼ Ay þ Bu

ð36Þ

where B ¼ diagð0; U1 Þ, U ¼ diagðj1 ; j2 ; . . . ; jN Þ. When white Gaussian system noise wðtÞ and measurement one v ðtÞ are considered, Eq. (36) can be rewritten as 1.2

1.8

(a)

1

(b)

y_ ¼ Ay þ Bu þ wðtÞ;

G (x ,t) f

1.5

Gmax>1.0

G(x ,t)=1 f

y ¼ Cy þ v ðtÞ;

0.9

G 0

=1.0

G(x ,t)=1

max

0

0.005

0.01

0.6

f

0

0.005

0.01

Time (s) 8

(c)

G (x ,t) f

6

G

→ ∞

4

0

G(x ,t)=1 f

0

0.005

0.01

ð38Þ

^_ ¼ fðy ^Þ þ Bu þ LðtÞðy  y ^ðtÞÞ y ^ u ¼ Ky

ð39Þ ð40Þ

The Kalman gain L can be obtained from L ¼ PC T V 1 by using the linear approximation as

max

2

ð37Þ

For simplicity, a pressure transducer is used to measure the reference signal by setting C ¼ ½0; 0; . . . ; 0; wi1 ðxs Þ; wi2 ðxs Þ; . . . ; wiN ðxs Þ. xs denotes the sensor location. To make our analysis more generalized, an extended Kalman filter [37] is applied to estimate the state variable:

1.2

0.5

wðtÞ Nð0; WðtÞÞ

v ðtÞ Nð0; VðtÞÞ

T1=T2 T1≠T2

_ PðtÞ ¼ APðtÞ þ PðtÞAT þ W  LVLT

The state feedback gain K is determined by minimizing the cost function

Z

0.015

Time (s)

ð41Þ

J ¼

1



yT y þ uT Ru dt

ð42Þ

0

Fig. 6. Variation of Gðxf ; tÞ with time t, as N = 10, L = 1.0 m, xf ¼ 0:25 m, s = 1.2 ms,
¼ 1:01325  105 Pa and u
1 ¼ 0:5 m/s: (a) N ¼ 3:0  102 , (b) N ¼ 1:50  105 , and p (c) N ¼ 3:0  105 .

where y denotes the output of the system. R ¼ sI and s is a scaling factor. It describes the cost between transient states and the control

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X. Li et al. / Applied Energy 169 (2016) 481–490

effort. Partition the state feedback gain K as K ¼ ½K1 ; K2  enables the control input u to be rewritten as

u ¼ Ky ¼ K1 y1  K2 y2

ð43Þ

Comparing Eq. (43) with Eq. (25) leads to

K1 ¼ 

N1 N1 X cp
X 1 T a K2 ¼  aak Sk Wak WTak U1 ak Rk Wak Pak ;


u

k¼1

k¼1

M

ð44Þ

Eq. (44) reveals that there are 2N 2 linear equations in terms of the control parameters Rk and Sk . Thus N 2 actuators are needed to stabilize N unstable modes. The number of actuators N1 is related with that of the eigenmodes as N 1 ¼ N 2 . The LQG controller is first implemented on the thermoacoustic system with two eigenmodes. The feedback gain and the Kalman filter gain are chosen as given in Appendix A. The performance of the controller in terms of preventing onset of limit cycle oscillations is shown in Fig. 8. It can be seen that implementing the controller leads to initial pressure perturbation being quickly eliminated. And the system becomes stable. For comparison and validation of the proposed LQG controller, a pole placement controller is also implemented. It can be seen from Fig. 8(a) that LQG controller can achieve a better damping performance with the proper actuation signals as shown in Fig. 8(b). Largeamplitude actuation is needed initially. However, as the thermoacoustic oscillations become less intensified, the actuation signals also decay. Fig. 8(c) and (d) show the performance of the extended Kalman filter by comparing the estimated difference and the measured difference. The estimated difference denotes the deviation from the signal estimated by the observer and the simulated actual signal without noise. The measured difference means the deviation from the sensor’s signal and the simulated actual signal. It is obvious that implementing the Kalman filter leads to the measured noise being dramatically decreased. Compared with the extended Kalman filter, the pole placement observer does not perform well.

The capability of implementing these controllers to mitigate limit cycle oscillations are further evaluated. Fig. 9(a) and (b) shows time evolution of pressure and velocity fluctuations, as the controller is implemented at t ¼ 0:3 s. When the actuators are off (t 6 0:3 s), acoustic perturbations grow exponentially and become ‘saturated’ at t ¼ 0:1 s. Periodic limit cycle oscillations are produced. As the actuators are actuated at t P 0:3 s, the limit cycle oscillations are quickly dampened by both LQG and pole placement controllers. However, LQG controller takes a shorter time to achieve the optimal mitigation performance than that of the pole placement controller. The performance of the LQR control on reducing the acoustical energy is shown in Fig. 9(c)–(e). It can be seen that the acoustical energy E1 and E2 at x1 and x2 are increased first and then become ‘saturated’ at t  0:1 s. However, the energy of the first fundamental mode is found to play a dominant role by comparing the magnitude of the energy oscillation with that of the total energy (see Fig. 9(e)). When the actuators are actuated, i.e. t P 0:3 s, the acoustical energies E1 and E2 are simultaneously decreased. So thermoacoustic systems with the multiple unstable modes can be stabilized by implementing LQG controller. Stabilizing such systems is more challenging than that typically found in practice, which is associated with a dominant mode. The reduction of the sound pressure level is shown in Fig. 10(a). It can be seen that approximate 76 dB sound reduction is achieved. Therefore, the LQG controller is not only able to prevent the onset of limit cycles, but also to mitigate limit oscillations. Further insights are gained on the non-normality of the actuated thermoacoustic system by studying the variation of the transient growth of acoustical energy, as the LQG controller is implemented. Fig. 10(b) illustrates time evolution of Gðxf ; tÞ. It can be seen that the acoustical energy is slightly increased before eventually decreased. And the maximum transient growth is greater than unity, i.e. Gmax > 1:0. This confirms that thermoacoustic oscillations can be mitigated by implementing LQG controller. However, the actuated system is associated with transient growth of acoustical energy,

20

υ

a

0

−10

−20

a1

60

υ

30

υ

a2 a3

υ

a4

0

0

0.1

0.2

0.3

−30

0

0.025

0.05

3

25

(c)

(d)

measured

measured LQG estimation

LQG estimation 15

1.5

P−P estimation

P−P estimation

f

Δ P′ (Pa)

υ

(b)

Pole placement

10

p′f (Pa)

90

LQG

(a)

5

−5

0

0

0.02

Time (s)

0.04

−1.5 0.46

0.48

0.5

Time (s)

Fig. 8. (a) Time evolution of pressure oscillation in the actuated thermoacoustic system. (b)–(d) Time evolution of estimated pressure difference from LQG and pole
1 ¼ 1:01325  105 Pa, u
1 ¼ 0:5 m/s, placement controllers and measured difference, as N = 2, xf ¼ 0:25 m, xs ¼ 0:4 m, s ¼ 1:2 ms, N ¼ 5:25  105 ; c1 ¼ 0:7, c2 ¼ 0:07, p s ¼ 1:0  105 , W = 100, V = 0.1.

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X. Li et al. / Applied Energy 169 (2016) 481–490

(a)

(b)

100

p′ (Pa)

u′ (m/s)

0.2

0

f

f

0

→ on

−0.2 0

0.1

0.2

0.3

−100

→ on 0

0.4

0.1

0.2

x 10

0.06

(c)

0.4

Time (s)

Time (s)

→ on

−3

(d)

2

→ on

E

2

E1

0.04

0.3

0.02 0

0

0.1

0.2

0.3

1

0

0.4

0

0.1

0.2

0.3

0.4

Time (s)

Time (s) 0.06

(e) → on

Pole placement LQG

E

s

0.04 0.02 0

0

0.1

0.2

0.3

0.4

Time (s) Fig. 9. (a) Time evolution of velocity u0 ðxf ; tÞ, (b) acoustic pressure p0 ðxf ; tÞ, (c) time evolution of acoustic energy (a) E1 for the first mode acoustic energy, (d) E2 for the second
1 ¼ 1:01325  105 Pa, u
1 ¼ 0:5 m/s and mode acoustic energy, and (e) Es for the total acoustical energy. as N = 2, xf ¼ 0:25; s ¼ 1:2 ms, N ¼ 5:25  105 ; c1 ¼ 0:7, c2 ¼ 0:07, p the controller is turned on at t ¼ 0:3 s.

(a)

1.2

0.1≤ t ≤ 0.3 s 0.35≤ t ≤ 0.5 s

(b)

Closed−loop

1

100

0.8

G(t)=1

Δ≈76 dB G (t)

Sound pressure level (dB)

150

0.6 0.4

50

0.2 0

0

200

400

600

800

Frequency (Hz)

0

0

1

2

3

Time (s)

4

5

x 10

−3

Fig. 10. (a) Variation of sound pressure level, as the LQG controller is actuated at t ¼ 0:3 s, (b) variation of transient growth GðtÞ of the total energy, as N = 2, xf ¼ 0:25, s ¼ 1:2 ms, N ¼ 5:25  105 , p
1 ¼ 1:01325  105 Pa, u
1 ¼ 0:5 m/s.

which may have the potential to trigger thermoacoustic instability. The present study reveals new potential risk of applying active controllers to stabilize thermoacoustic systems.

5. Discussion and conclusions The present work considers a Rijke-type thermoacoustic system with multiple monopole-like actuators implemented. Acoustically compact heated wires are assumed to be enclosed and modeled by using a modified King’s Law. Unlike previous models available

in the literature, the mean temperature is assumed to undergo not only a sudden jump across the heat source but also linearly increasing and decreasing in the pre- and after-heating regions respectively. Such mean temperature configuration is consistent with the experimental measurement. However, for comparison, thermoaocustic systems with a uniform mean temperature configuration is also studied in this work. Coupling the heat source model with a Galerkin series expansion of acoustic fluctuations provides a state-space platform on which to gain insights on (1) the nonlinearity of the thermoacoustic system, (2) onset of limit cycle oscillations, (3) predicting its

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non-normality behaviors, (4) energy distribution and transfer between neighboring eigenmodes, and (5) evaluating the performance of feedback controllers. By solving the nonlinear governing equations, the eigenfrequencies and mode-shapes are predicted. They are found to agree well with the experimental measurements. Pseudospectra and transient energy growth analyses of the thermoacoustic systems are then performed by linearizing the heat source model into the classical time-delay n  s formulation. It reveals that the system is non-normal. And it is associated with transient growth of acoustical energy. The non-normality is found to be less intensified in comparison with that in a system in the absence of mean temperature variation from pre- and afterheating regions. In order to mitigate these limit cycle oscillations, the heat-tosound energy conversion is interrupted by implementing multiple monopole-like actuators driven by a LQG (linear quadratic Gaussian) and pole-placement controllers. An extended Kalman filter is developed to estimate the system status. With the modelbased control strategy implemented, the thermoacoustic system is successfully stabilized. Approximately 76 dB sound pressure level reduction is achieved. However, implementing the LQG controller is shown to be associated with transient growth of acoustical energy. The present study reveals new potential risk of applying active controller to stabilize thermoacoustic systems. Acknowledgement The work is supported by Singapore Ministry of Education AcRF-Tier1 Grant RG91/13-M4011228 and the National Natural Science Foundation of China under Grant No. 51506079. This financial support is gratefully acknowledged. The authors thank Dr. Jaime Rubio and Prof. Mahmut Reyhanoglu from Embry-Riddle Aeronautical University for helpful discussion. Appendix A. Kalman filter coefficients

2

35:08

6 202:12 6 KT ¼ 6 4 873:60 33:24

140:70

3

7 7 7; 33:24 5 9:33

31:50

2

25:60

3

6 138:32 7 7 6 L¼6 7 4 390:84 5

ðA:1Þ

22:59

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