Rijke Tube Research Articles

Neural network PID control for combustion instability

Considerable research studies have been reported on developing effective active control means to mitigate combustion instability. In this work, a neural network PID (NN-PID) controller is proposed and demonstrated to suppress the oscillating pressure in a cylindrical Rijke tube, for which the combustion instability is modelled by a 1D acoustic network model together with linear and nonlinear flame models. The flame model is based on the classical n − τ model filtered by a first-order low pass filter. The nonlinearity of the flame model is realised using a gain function saturated with velocity perturbation. A fuel valve and a loudspeaker are adopted as two independent control actuators. Results show that system oscillation begins to attenuate when the NN-PID control starts. The attenuation rate using a fuel valve as the actuator is faster than that using a loudspeaker. For nonlinear combustion oscillations, the NN-PID controller could inhibit the reference pressure oscillation in the linear growth and saturation processes. A maximum overshoot of 2.3% could occur when the loudspeaker is used as the actuator to suppress system oscillation. The NN-PID controller is not affected by noise with a fuel valve or the loudspeaker and can effectively suppress and eliminate the system pressure oscillation in different oscillating stages with different flame models.

Research on the Mechanism of Rijke Tube Self-excited Sound

Combustion equipment generates self-excited vibration called combustion vibration. The noise generated by this ｒ vibration is undesirable because it can damage structures and degrade their performance. Because this vibration phenomenon is complex, it is very difficult to model. Therefore, the purpose of this study is to attempt modeling of Rijke tubes, which are prone to combustion vibration from the perspective of heat transfer engineering, and to reproduce the sound pressure waveform of Rijke tubes using the cellular automaton (CA) method. Since previous studies have summarized the properties of a Rijke tubes and the rules of the CA method, we referred to them and incorporated them into this study. Previous studies have shown that the particle velocity in the heat source section including time delay causes self-excited oscillation. In this study, we derived an equation for the particle velocity with time delay by using heat transfer engineering based on the expansion and contraction of air. The sound pressure waveform of the Rijke tube could be quantitatively reproduced by numerically calculating the time of passage through the heat source and determining the number of time delay from time to time．

Phase-reduction analysis of periodic thermoacoustic oscillations in a Rijke tube

Phase-reduction analysis captures the linear phase dynamics with respect to a limit cycle subjected to weak external forcing. We apply this technique to study the phase dynamics of the self-sustained oscillations produced by a Rijke tube undergoing thermoacoustic instability. Through the phase-reduction formulation, we are able to reduce these dynamics to a scalar equation for the phase, which allows us to efficiently determine the synchronisation properties of the system. For the thermoacoustic system, we find the conditions for which $m:n$ frequency locking occurs, which sheds light on the mechanisms behind asynchronous and synchronous quenching. We also reveal the optimal placement of pressure actuators that provide the most efficient route to synchronisation.

Stochastic bifurcations and its regulation in a Rijke tube model

This paper explores the stochastic dynamics in a Rijke tube model. Dynamical model is established to identify the bifurcation properties of the Rijke tube model. We gain the analytical expression of the local Hopf bifurcation and global saddle-node bifurcation of the limit cycle in the deterministic case. The stationary probability density function (PDF) of the model is attained base on the method of stochastic average in the case of stochasticity. The investigations indicate that the stationary PDF switches from unimodal shape to bimodal one, and then, from bimodal shape to unimodal one again, when noise intensity, fractional order, time delay monotonically increase which is the typical feature of stochastic P-bifurcation. Further, we conclude that the stochastic P-bifurcation can be induced or suppressed by modulating the time delay, the noise intensity, or the fractional order. These findings of the study will be helpful to the theoretical study of thermoacoustic instability and the preliminary design of thermoacoustic devices where thermoacoustic instability is a concern.

Experimental study of transition to instability in a Rijke tube with axially distributed heat source

Self-excited thermoacoustic oscillations are favorable in thermoacoustic engines, but unwanted in many combustion systems due to the detrimental outcomes, including the structure vibration and overloaded thermal flux to combustor wall. To explore some proper methods to modify thermoacoustic oscillations, understanding the mechanism of thermoacoustic instability is needed. In the present work, the transition to instability in a Rijke-type thermoacoustic system with axially distributed heat source is explored experimentally. A silicon/ceramic tube is used as the acoustic resonator, and the distributed heat source is designed by wounding electric wires over two ceramic rings. The influences of the characteristics of heat source (including heating power, heater length and heater location), mass flow rate, and tube materials on the nonlinear dynamic behaviors and stability boundaries of the Rijke-type thermoacoustic system are systematically evaluated. Large-amplitude limit cycle is observed, and subcritical and supercritical bifurcations are present in the thermoacoustic system. For the system with fixed mass flow rates, the critical heat power for transition to instability is found to be approximately linearly proportional to the heater length, while the strength of pressure oscillations responds nonlinearly. In addition, the system with a thin heater is more prone to undergo subcritical bifurcation. As for the influence of the mass flow rate, it has also been found that increasing the mass flow rate would enhance the nonlinearity of the coupling between the unsteady heat release rate and acoustic waves, which leads to a high possibility of occurrence of subcritical bifurcation. For all the heater length studied, there is an optimal mass flow rate where the critical heat power for the transition to instability is minimal. Last, the results show that the thermoacoustic system with ceramic tube has a narrower stability region at small air flow rate and is more prone to subcritical bifurcation than the silicon tube.

Different perspectives on predicting the thermoacoustic energy conversion response in a Rijke tube

In this work, six different perspectives on characterizing the thermoacoustic field in an open-ended Rijke tube are considered and discussed. These begin with a three-pronged approach consisting of theoretical, experimental, and numerical investigations of the Rijke tube's time-dependent field. It is followed by a discussion of effective techniques that rely on either Green's function or differential equation models. Finally, a perturbation expansion is introduced that leverages a naturally occurring small parameter in the open tube configuration. This approach is shown to produce accurate predictions of pressure modal shapes and frequencies for an arbitrary specified temperature distribution. It also leads to a set of linear partial differential equations that can be solved in conjunction with a Green's function expression for the thermoacoustic pressure, velocity, and heat oscillations. In this study, the underlying framework is presented and evaluated for the pressure disturbance only. Another fundamental result includes a similarity parameter, coined the Rijke number, which plays an essential role in driving thermoacoustic oscillations, namely, by relating heat-flux fluctuations to the acoustic velocity and pressure. In this context, we find that the peak value of the energy-flux vector modulus, which stands for the modular product of acoustic velocity and pressure, does indeed occur at the heat source location and increases with the heat power input.

Effect of position of radial air injection plane on control of thermo-acoustic instability using active closed-loop method

Thermo-acoustic instability occurs when self-excited oscillations are generated due to the coupling between unsteady heat release and acoustics. This phenomenon can result in an increased rate of vibration, structural damage, and produces unwanted emissions. Thermo-acoustic instability occurs in rocket engines, gas turbines, combustors, and furnaces. When thermo-acoustic instability occurs, many modes are developed naturally at a specific point. Some waves are unstable and some are stable. So, to study this phenomenon the most unstable waves are considered and a technique is developed to suppress these unstable waves. A radial air injector as a closed-loop active control method is used for breaking the coupling between the heat waves and acoustics inside the 1D combustion chamber. The distance between the burner and the air injector is varied for the fixed position of the burner with respect to the Rijke tube, that is, x/L = 0.01125, 0.0075, and 0.00375. This closed-loop method works based on the feedback acquired from a microphone. The control method is built using DAQ and Arduino with the LabVIEW as interface for Arduino (LIFA). An air flow rate controller setup is developed to control and measure air required for suppressing the thermo-acoustic instability. Thermo-acoustic instability is effectively suppressed with the help of radial injection in the form of micro-jets at the downstream of the burner as the closed-loop controlling method. It is concluded that when the radial micro-jet air injection plane is closer to the burner head, the thermo-acoustic instability gets suppressed in a short time and with a lesser quantity of air.

Investigation of thermoacoustic oscillation attenuation by modified Helmholtz dampers with dual frequency bands

Two modified Helmholtz dampers with multi-adjustable frequency bands are proposed to overcome the drawback of Helmholtz dampers and Quarter-lambda tubes by which multi-irregular eigenfrequencies in gas turbine combustors cannot be simultaneously treated via one interface. Theoretical impedance models of these dampers are established and experimentally validated in an impedance tube. Enhanced thermoacoustic instability attenuation by modified dampers is validated in a Rijke tube. A Helmholtz-based method is combined with damper impedance models to predict the limit cycle behavior of a Rijke tube equipped with dampers. Results indicate that the impedance models can effectively calculate reflection coefficients of modified and traditional dampers with cooling purge air. The amount of the secondary tube influences the reflection coefficients of modified dampers. Experimental results indicate that amplitudes of the overall pressure fluctuation and each order fluctuation at the limit cycle are further attenuated by modified dampers compared with those by traditional ones. Larger purge flow rates for dampers lead to more pulsation attenuation caused by vortex shedding dissipation. The numerical method can predict the eigenfrequencies, velocity fluctuation ratios, and mode shapes at the limit cycle after using different types of dampers. When activating modified dampers, weaker fluctuations are successfully captured by this numerical approach.

Linear control of thermoacoustic oscillations with flame dynamics modeled by a level-set method

This article focuses on co-simulation of the coupled nonlinear dynamics of flame heat release rate, acoustics, and feedback control in an active thermoacoustic oscillation control system. This work is motivated by an extensive existing body of literature showing the potential of closed-loop active control to suppress thermoacoustic oscillation. Linear algorithms such as linear quadratic Gaussian (LQG) control are often used for thermoacoustic oscillation control, with the important caveat that flame heat release rate oscillation is often highly nonlinear. This creates a need for a tool that can co-simulate the coupled nonlinear thermoacoustic dynamics with linear control. The main contribution of this article is the development of a tool capable of co-simulating nonlinear heat release dynamics using a level-set formulation, together with linear acoustics plus a linear feedback controller. The article demonstrates this framework on an LQG-controlled wedge flame in a unidimensional Rijke tube. The nonlinear thermoacoustic model is formed in a modified level-set solver by connecting the linear acoustics to the original nonlinear flame dynamics via a velocity convection equation. This framework succeeds in preserving the nonlinearity of the thermoacoustics in the Rijke tube without the LQG control. For the flame-driven Rijke tube in this article, LQG control is successful in suppressing the nonlinear thermoacoustic oscillation. In addition, the simulation based on the framework is useful in elucidating the impact of factors such as flame location, flame temperature rise, and the timing of the onset of LQG control on the instability suppression performance and energy needs.

Comparison of strongly and weakly nonlinear flame models applied to thermoacoustic instability

Weakly nonlinear flame (or heater) dynamic models, only accounting for heat release rate disturbances from the flame (or heater) at forcing frequencies and omitting harmonic terms due to nonlinear mechanisms, are widely used in low-order tools for the analysis and prediction of thermoacoustic instabilities, because they have a numerical cost much cheaper than tools based on Navier–Stokes equations, and are easier to develop and validate. However, these models may lead to errors under certain conditions. The present work considers a Rijke tube model combustor, in which a classical third-order model is used to describe the flame dynamic response to the oncoming flow disturbance. We call this model the strongly nonlinear flame model. The weakly nonlinear flame model is then introduced. The wave-based approach is adopted as a low-order tool. The weakly and strongly nonlinear flame models are embedded in the low-order tool to reproduce the thermoacoustic instability of the model combustor. The natural frequency and growth rate of thermoacoustic instability are then determined by mode extracted methods. The differences between the results predicted by using the weakly and strongly nonlinear flame models are compared for a set of operating conditions, in order to find the conditions under which the weakly nonlinear flame model works. Short-time Fourier transform is adopted to analyze the extracted frequencies and growth rates of four selected cases. When the dominant acoustic mode strength is much stronger than the remaining modes, the weakly nonlinear models perform well. However, these models fail to capture the mode frequency and growth rate when multiple unstable modes are present.

Effect of nonlinear flame response on the design of perforated liners in suppression of combustion instability

This paper aims at investigating how to suppress combustion instability using perforated liners under nonlinear flame response. A theoretical model is thus presented to describe the interaction between the unsteady heat release, acoustics, and fluid fields, which takes both perforated liner and flame nonlinearity into account. To establish an eigenvalue problem to evaluate combustion instability, appropriate matching and boundary conditions are set up by applying conservation laws across the interfaces, consequently deriving the dispersion relation equation in the form of a matrix. In view of the nonlinearity of flame response function, it is found that the control strategy of combustion instability should be devoted to the reduction of the limit cycle amplitude to an acceptable level rather than the complete elimination of its onset. Based on the present model, we have studied the first longitudinal mode of a Rijke tube equipped with perforated liners. It is noted that the change of combustion instability evolution is sensitive to the design parameters of perforated liners, including cavity depth, installation position, and bias flow velocity. On the one hand, it is indeed possible to suppress the onset of combustion instability with elaborately designed parameters of the perforated liners. On the other hand, the limit cycle amplitude can be reduced as small as possible under various design restrictions of perforated liners in practice. Therefore, this study presents an alternative design criteria of perforated liners for the control of combustion instability.

Numerical simulation of thermoacoustic instability in Rijke tube

The influence of three different states on the thermoacoustic instability characteristics of Rijke tube was compared in order to reseach the influencing factors of thermoacoustic oscillation by using the Rijke tube model with stack as the heat source. The thermoacoustic oscillations are numerically simulated from the start-up to the saturation state, and the effects of the temperature on the dynamic viscosity and the thermal conductivity are compared. The results show gravity has a greater influence than the thermoacoustic oscillation caused by thermal buoyancy, and it is related to the inner balance of the tube after the gravity and the temperature gradient caused by the protrusion and the temperature gradient caused by the reduction of the amplitude dissipation. For the comprehensive comparison of the two variable parameters, it is found that when the viscosity coefficient changes with temperature and the thermal conductivity is a fixed value, both of them decrease by 49.5% with the temperature change rate. This result far exceeds the viscosity coefficient itself influences.

Transfer Learning for Detection of Combustion Instability Via Symbolic Time-Series Analysis

Abstract Transfer learning (TL) is a machine learning (ML) tool where the knowledge, acquired from a source domain, is “transferred” to perform a task in a target domain that has (to some extent) a similar setting. The underlying concept does not require the ML method to analyze a new problem from the beginning, and thereby both the learning time and the amount of required target-domain data are reduced for training. An example is the occurrence of thermoacoustic instability (TAI) in combustors, which may cause pressure oscillations, possibly leading to flame extinction as well as undesirable vibrations in the mechanical structures. In this situation, it is difficult to collect useful data from industrial combustion systems, due to the transient nature of TAI phenomena. A feasible solution is the usage of prototypes or emulators, like a Rijke tube, to produce largely similar phenomena. This paper proposes symbolic time-series analysis (STSA)-based TL, where the key idea is to develop a capability of discrimination between stable and unstable operations of a combustor, based on the time-series of pressure oscillations from a data source that contains sufficient information, even if it is not the target regime, and then transfer the learnt models to the target regime. The proposed STSA-based pattern classifier is trained on a previously validated numerical model of a Rijke-tube apparatus. The knowledge of this trained classifier is transferred to classify similar operational regimes in: (i) an experimental Rijke-tube apparatus and (ii) an experimental combustion system apparatus. Results of the proposed TL have been validated by comparison with those of two shallow neural networks (NNs)-based TL and another NN having an additional long short-term memory (LSTM) layer, which serve as benchmarks, in terms of classification accuracy and computational complexity.

Transfer learning of deep neural networks for predicting thermoacoustic instabilities in combustion systems

The intermittent nature of operation and unpredictable availability of renewable sources of energy (e.g., wind and solar) would require the combustors in fossil-fuel power plants, sharing the same grid, to operate with large turn-down ratios. This brings in new challenges of suppressing high-amplitude pressure oscillations (e.g., thermoacoustic instabilities (TAI)) in combustors. These pressure oscillations are usually self-sustained, as they occur within a feedback loop, and may induce severe thermomechanical stresses in structural components of combustors, which often lead to performance degradation and even system failures. Thus, prediction of thermoacoustic instabilities is a critical issue for both design and operation of combustion systems. From this perspective, it is important to identify operating conditions which can potentially lead to thermoacoustic instabilities. In this regard, data-driven approaches have shown considerable success in predicting the instability map as a function of operating conditions. However, often the available data are limited to learn such a relationship efficiently in a data-driven approach for a practical combustion system. In this work, a proof-of-concept demonstration of transfer learning is provided, whereby a deep neural network trained on relatively inexpensive experiments in an electrically heated Rijke tube has been adapted to predict the unstable operating conditions for a swirl-stabilized lean-premixed laboratory scaled combustor, for which data are expensive to obtain. The operating spaces and underlying flow physics of these two combustion systems are different, and hence this work presents a strong case of using transfer learning as a potential data-driven solution for transferring knowledge across domains. The results show that the knowledge transfer from the electrically heated Rijke tube apparatus helps in formulating an accurate data-driven surrogate model for predicting the unstable operating conditions in the swirl-stabilized combustor, even though the available data are significantly less for the latter.

Dynamics of Coupled Thermoacoustic Oscillators Under Asymmetric Forcing

Quenching of limit cycle oscillations (LCO), either through mutual coupling or external forcing, has attracted wide attention in several fields of science and engineering. However, the simultaneous utilization of these coupling schemes in quenching of LCO has rarely been studied despite its practical applicability. We study the dynamics of two thermoacoustic oscillators simultaneously subjected to mutual coupling and asymmetric external forcing through experiments and theoretical modeling. We investigate the forced response of both identical and non-identical thermoacoustic oscillators for two different amplitudes of LCO. Under mutual coupling alone, identical thermoacoustic oscillators display the occurrence of partial amplitude death and amplitude death, whereas under forcing alone, asynchronous quenching of LCO is observed at non-resonant conditions. When the oscillators are simultaneously subjected to mutual coupling and asymmetric forcing, we observe a larger parametric region of oscillation quenching than when the two mechanisms are utilized individually. This enhancement in the region of oscillation quenching is due to the complementary effect of amplitude death and asynchronous quenching. However, a forced response of coupled non-identical oscillators shows that the effect of forcing is insignificant on synchronization and quenching of oscillations in the oscillator which is not directly forced. Finally, we qualitatively capture the experimental results using a reduced-order theoretical model of two coupled Rijke tubes which are coupled through dissipative and time-delay coupling and asymmetrically forced. We believe that these findings offer fresh insights into the combined effects of mutual and forced synchronization in a system of coupled nonlinear oscillators.

Different Experimental Approaches for Characterization of Thermo-acoustic Instabilities

The coupling between pressure fluctuations and heat release variations during the combustion process is responsible for occurrence of thermo-acoustic instability. These instabilities will cause many adverse effects in power-producing devices. To control thermo-acoustic instabilities, its characterization is essential for design and implementation of control technique. The characterization of thermo-acoustic instabilities can be done using pressure amplitude and its frequency. In this study, the authors have presented two different approaches for thermo-acoustic instability characterization. In the first approach, the wall pressure measurements were taken at eleven locations, along the longitudinal directions, using differential pressure transducer. In the second approach, acoustic pressure outside the Rijke tube is measured. To check the consistency in the results, measurements are taken at different equivalence ratios and total mass flow rates. The result shows that the sound pressure level is higher in case of wall pressure measurement as compared to acoustic measurement outside the Rijke tube. The peak frequency is similar in both the cases, over a wide range of equivalence ratios and total flow rates. The captured standing wave pattern is similar to third mode of instabilities, but it was affected by the pulling force due to use of suction pump at the other end of Rijke tube. The suggested approaches can be effectively used to design the control techniques for suppression of thermo-acoustic instabilities. The acoustic pressure measured outside the Rijke tube is able to give the exact frequency of thermo-acoustic instabilities and also free from disruptions of flow. The exact prediction of mode of instability is essential, to identify location for mounting the pressure transducer on the wall of combustor, which was the major drawback of wall pressure measurements. The study further concluded that the intensity of thermo-acoustic instability is dependent on equivalence ratio.

Shape Optimization of Thermoacoustic Systems Using a Two-Dimensional Adjoint Helmholtz Solver

Abstract Thermoacoustic instabilities, which arise due to the interaction between flames and acoustics, are sensitive to small changes to the system parameters. In this paper, we apply adjoint-based shape optimization to a 2D finite element Helmholtz solver to find accurately and inexpensively the shape changes that most stabilize a 2D thermoacoustic system in the linear regime. We examine two cases: a Rijke tube and a turbulent swirl combustor. Both systems exhibit an unstable longitudinal mode and we suppress the instability by slightly modifying the geometry. In the case of the turbulent swirl combustor, the sensitivities are higher in the plenum and in the burner than in the combustion chamber, mainly due to the effect of the mean temperature. In the cooler regions, the local wavelength is shorter, which means that geometry changes of a given distance have more influence than they do where the local wavelength is longer. This is the first time adjoint-based shape optimization is applied to 2D Helmholtz solvers in thermoacoustics, after being previously applied to low-order thermoacoustic networks. But Helmholtz solvers have an intrinsic advantage: they can handle complex geometries. The easy scalability of this method to complex 3D geometries makes this tool a strong candidate for the iterative design of thermoacoustically stable combustors.

Recurrence network analysis exploring the routes to thermoacoustic instability in a Rijke tube with inverse diffusion flame.

Inverse diffusion flame (IDF) is a reliable low NOx technology that is suitable for various industrial applications including gas turbines. However, a confined IDF may exhibit thermoacoustic instability, a kind of dynamic instability, which is characterized by catastrophically large amplitude pressure oscillations. Transition to such instability for an inverse diffusion flame is less explored compared to other types of flame. In the present study, thermoacoustic instability in a Rijke tube with IDF is achieved by varying air flow rate and input power independently, and the onset of thermoacoustic instability is examined using the framework of recurrence network (RN). During the transition to thermoacoustic instability, we find new routes and two new intermediate states, here referred to as "amplitude varying aperiodic oscillations" and "low amplitude limit cycle-like oscillations." Furthermore, we show that recurrence network analysis can be used to identify the dynamical states during the transition to thermoacoustic instability. We observe an absence of a single characteristic scale, resulting in a non-regular network even during thermoacoustic instability. Furthermore, the degree distributions of RN during combustion noise do not obey a single power law. Thus, scale-free nature is not exhibited during combustion noise. In short, recurrence network analysis shows significant differences in the topological information during combustion noise and thermoacoustic instability for IDF with those for premixed flames, reported earlier.

Assimilation of Experimental Data to Create a Quantitatively Accurate Reduced-Order Thermoacoustic Model

Abstract We combine a thermoacoustic experiment with a thermoacoustic reduced order model using Bayesian inference to accurately learn the parameters of the model, rendering it predictive. The experiment is a vertical Rijke tube containing an electric heater. The heater drives a base flow via natural convection, and thermoacoustic oscillations via velocity-driven heat release fluctuations. The decay rates and frequencies of these oscillations are measured every few seconds by acoustically forcing the system via a loudspeaker placed at the bottom of the tube. More than 320,000 temperature measurements are used to compute state and parameters of the base flow model using the Ensemble Kalman Filter. A wave-based network model is then used to describe the acoustics inside the tube. We balance momentum and energy at the boundary between two adjacent elements, and model the viscous and thermal dissipation mechanisms in the boundary layer and at the heater and thermocouple locations. Finally, we tune the parameters of two different thermoacoustic models on an experimental dataset that comprises more than 40,000 experiments. This study shows that, with thorough Bayesian inference, a qualitative model can become quantitatively accurate, without overfitting, as long as it contains the most influential physical phenomena.

Optimizing Thermoacoustic Characterization Experiments for Identifiability Improves Both Parameter Estimation Accuracy and Closed-Loop Controller Robustness Guarantees

ABSTRACT This article examines the degree to which optimizing a Rijke tube experiment can improve the accuracy of thermoacoustic model parameter estimation, thereby facilitating robust stability control. We use a one-dimensional thermoacoustic model to describe the combustion dynamics in a Rijke tube. This model contains two unknown parameters that relate velocity perturbations to heat release rate oscillations, namely, a time delay and amplification factor . The parameters are estimated from experiments where the system input is the acoustic excitation from a loudspeaker and the output is the pressure response captured by a microphone. Our work is grounded in the insight that optimizing an experiment’s design for higher Fisher identifiability leads to more accurate parameter estimates. The novel goal of this paper is to apply this insight in the laboratory using a flame-driven Rijke tube setup. For comparison purposes, we conduct a benchmark experiment with a broadband chirp signal as the excitation input. Next, we excite the Rijke tube at two frequencies optimized for Fisher identifiability. Repeats of both experiments show that the optimal experiment achieves parameter estimates with uncertainties at least one order of magnitude smaller than the benchmark. With smaller parameter estimate uncertainties, an LQG controller designed to attenuate combustion instabilities is able to achieve stronger robustness guarantees, quantified in terms of closed-loop structured singular values that account for parameter estimation uncertainty.