Abstract
This work treats the control of nonstationary pressure oscillations of combustion processes subject to external harmonic excitation. The low-order van der Pol model of nonstationary combustion processes with two degrees of freedom and a multiplicative feedback control function is considered. Nonstationary oscillations of the combustion process with asynchronous and synchronous excitation, respectively, are analyzed by means of the extended Lindstedt-Poincare (EL-P) method with multiple time scales. The steady-state amplitudes of competitive quenching with asynchronous excitation are not so much affected in comparison with steady-state amplitudes of an autonomous system, opposite to the synchronous excited system. The influence of control parameters on the quenching of the externally excited oscillations is investigated in details.
Keywords
- combustion processes
- pressure oscillations
- external harmonic excitation
- van der Pol model
- EL-P method with multiple time scales
- control
1. Introduction
The phenomenon of the instability of the combustion process is commonly explained by the nonstationary flame causing an acoustic wave, which reflects from the walls of the combustion chamber back to the combustion process. The thermoacoustic coupling between acoustic waves and unsteady heat release using a generalized energy equation, followed by feedback control concepts is presented by Dowling and Morgans [1]. Using theory of systems, this coupling can be explained alternatively by means of a feedback with a negative damping between the heat release process and acoustic wave propagation in the combustion chamber. The recent researches in the area of combustion process attempt to eliminate the unstable combustion and reduce the emission pollution, especially the emission of nitrogen-oxygen compounds at the same time. The reduction of pollution can be achieved by decreasing the fuel-to-air ratio; however, this causes the instability in the form of self-excited oscillations, which in general contain incommensurate frequencies. The problem of instability can be solved by the control of the fuel influx into combustion chamber, which can be realized by introduction of multiplicative control function performing the desired modulation of heat release rate. In order to obtain an efficient model of active control in this chapter, the reduced-order model of combustion instability process is proposed in the form of generalized, coupled van der Pol differential equations [2]. This model is built using a grey-box system approach [3], which is based on combination of physical knowledge determining the structure and identification of unknown parameter values, which are fitted to the measured input/output data of the system. In the proposed low-order model, the physics is present only in a feedback coupling between acoustic wave propagation and heat release giving rise to a negative damping as the cause of the combustion instability. The presented model of combustion instability is based on the linear acoustics and propagation of standing acoustic waves in the combustion chamber. Due to the presented nonlinear coupling between linear acoustics and nonlinear heat release rate, theoretically an infinite number of acoustic modes exist. Based on experimental results of Sterling [4] and Dunstan [3], the following analysis is considering the temporal variations of first two acoustic modes with corresponding natural frequencies. Unlike the models of Peracchio and Proscia [5], Dunstan and Murray et al. [6], respectively, natural frequencies in the present model can be incommensurate in general. From the system point of view, acoustic waves are modeled as pure linear oscillators and the heat release rate is modeled as the static nonlinearity, which can be accurately adapted to the real situation by identification of system parameters. On this basis, the model can be stated as a set of partial differential equations (PDEs) in terms of time scales, which are ideally suited to be attacked by the extended Lindstedt-Poincare (EL-P) method. The spatial scales are not present in the model, because it is assumed that all heat is released into the fluid at a point [5].
This chapter is organized as follows. In Section 2, the control structure and the governing coupled van der Pol equations of controlled combustion process with external harmonic excitation are presented. Section 3 applies the EL-P method with multiple time scales [7] to control problems of combustion process with asynchronous external harmonic excitation. Section 4 deals with control problems of combustion process with synchronous external harmonic force. Section 5 focuses on results and discussions of competitive quenching phenomenon. Section 6 discusses the validation of the proposed van der Pol model of combustion processes, and Section 7 outlines practical aspects of the model implementation. Section 8 contains conclusions.
2. Van der Pol model of combustion process with external harmonic excitation
In this section, the phenomenon of instability of the combustion process due to the feedback with negative damping is described by means of the low-order model, which consists of generalized two degrees of freedom (2-DOF) van der Pol oscillator with external harmonic excitation. The coupled van der Pol oscillator model is experimentally verified and it is possible to analyze the appearance of excited oscillations, which can evolve to limit cycles in accordance with experimental observations [8]. The dynamics of the generalized 2-DOF coupled nonautonomous van der Pol oscillator is governed by the following equation:
where the left-hand side of Eq. (1) models the linear acoustics of the combustion chamber represented by two harmonic oscillators with
The coupled van der Pol oscillator in Eq. (1) can be represented by the control structure, shown on Figure 1a, where the control plant consists of two harmonic oscillators in forward branches, which are harmonically excited and the feedback consists of derivative of the flame heat release rate multiplied by
The suitable candidates of control function
where
The stability analysis in the scope of the active control can be applied to reduce NOx pollution by extending the stability domain of computed pressure oscillations [9]. By applying the presented model, a realistic modeling of the flame heat release rate is quite possible, when parameters
In the next section, the EL-P method with multiple time scales [7] is applied for computation of pressure oscillations with external harmonic excitation arising in the coupled nonautonomous van der Pol oscillator. For the sake of compactness, we introduce new parameters:
and rewrite governing equation (2) with control function (3) in the form:
3. Application of the extended Lindstedt-Poincare method with multiple time scales in the combustion process with asynchronous external harmonic excitation
The instability problem of the combustion process, which is posed in the form of coupled van der Pol Eq. (5) with external harmonic excitation, can be analyzed by EL-P method with multiple time scales [2]. To provide a perturbation analysis of the generalized model of instability, it is assumed at first that interaction between both resonators and external harmonic excitation does not exist. In such a case, the natural frequencies
Following the EL-P method with multiple time scales, two fast time scales
Owing linear acoustics, following substitutions are proposed:
and by using differential operators (6a, 6b), governing equations of the coupled van der Pol oscillators [Eq. (5)] are rewritten into partial differential equations:
After performing indicated partial derivations, one obtains:
When the system is not subject of control,
The approximate solution of Eq. (9) in dependence on three time scales
Similarly, the nonlinear frequencies
where frequencies
for
The right-hand side of Eq. (13) is only a function of time scale
As mentioned earlier, in such circumstances we have deal with the asynchronous case of external excitation. When the excitation frequency is close or equal to one of the natural frequencies, the corresponding amplitude
The general solution of zeroth order, which allows two independent natural frequencies
from where it follows that the self-excited/driven oscillations of coupled autonomous van der Pol oscillator, which correspond to the zero-order solution, are foreseen as almost periodic solutions with slowly varying amplitude and phase. Inserting assumed solutions into zero-order PDE gives an important relation between linear and natural frequencies of the oscillator:
which reveals, that linear frequencies are equal to the natural frequencies of the coupled van der Pol oscillator under assumption that linear frequencies can be only positive. Consequently, it follows from Eq. (12) that nonlinear frequencies
Secular terms on the right-hand side appear, which causes an unbounded increase of the solution. To obtain an uniform solution, secular terms must be removed. The elimination of secular terms is achieved by assembling terms, which appear at functions cos[
for
Solvability conditions, given by means of Eqs. (20) and (22) for an autonomous case, agree perfectly with solvability conditions of first order as were derived in reference [2].
From Eq. (20) it follows that trivial solution
and frequencies
The solution of Eq. (21) tends to reach the asymptotic values, when
where following possibilities exist: (
From Eq. (25) it follows that by applying the control, the parameter
From Eqs. (25) and (26), respectively, it follows that all nontrivial steady-state values of the combustion process with external harmonic excitation are affected to some extent, when compared by asymptotic values of self-excited oscillations [asymptotic values of self-excited oscillations can be obtained by inserting value
To proceed with the perturbation procedure, generally the almost periodic solution of the first order is constructed, which contains all possible combination tones and has the form:
Coefficients
The second-order approximation is constructed in the same manner as the first-order approximation. To do this, one inserts solutions (17) and (27) into second-order PDE (15). The right-hand side of the obtained equation contains secular terms, which must be removed to prevent the unbounded increase of the solutions. The elimination of secular terms, which appear at functions cos[
After determination of solvability conditions of the second order, the perturbation procedure may be continued with the computation of the second-order solution
By considering the above result in solvability condition (20) again, it follows:
From Eq. (30), it follows that frequencies
From Eq. (31), one concludes that strictly speaking, phases are linear, slowly varying functions of the time scale
The obtained result simply means, that nonlinear frequencies
Now, the following important observation can be made. Substituting Eq. (32) into Eq. (20) and assuming
for
4. Application of the extended Lindstedt-Poincare method with multiple time scales in the combustion process with synchronous external harmonic excitation (ω close to ω n 2)
In this section, the combustion process with synchronous external harmonic excitation is treated. Although we will assume in the sequel, that excitation frequency
where
Equation (7.a) for the first mode,
where solution of PDEs (37.a,b) is sought by means of power series:
Performing perturbation steps, which are presented in details in the asynchronous case and finally assembling secular terms, which appear at functions cos[
Solvability conditions (40a) and (40c) for the first mode pressure oscillation,
5. Results
In this section we investigate self-excited as well as externally driven oscillations of coupled van der Pol model of combustion processes. Besides this, the effect of the control in order to suppress the nonzero amplitude of the limit cycles is analyzed. The control function is activated, when the switching time
The phenomenon of competitive quenching arises, when nonlinear frequencies
Characteristic for the competitive quenching phenomenon is excitation one of resonators, while oscillations of the other resonator are quenched in dependence on initial conditions. Such oscillations are analyzed in the case of natural frequencies
The combustion process is first analyzed without applied control by setting the control parameters
On the contrary, comparison of second mode oscillations
Combining zero-order solutions
On contrary, harmonically excited counterparts are computed by means of Eqs. (7a, 7b), where Eq. (11) is considered in approximation
While transient self-excited oscillation
As is clearly seen from Figure 2, the amplitudes of oscillations
The characteristic properties of the competitive quenching phenomenon depicted in Figure 3 indicate the effect of the applied fuel influx control on the combustion process. The control is activated in the switching time
Figure 3 shows the courses of slowly varying amplitudes
From Figure 3a, it follows that pressure oscillations
On contrary, Figure 3b reveals that quenching of pressure oscillations
6. Validation of the van der Pol model of combustion processes
In the combustion chamber, theoretically an infinite number of acoustic modes of pressure wave propagation can exist. However, experimental researches of combustion processes, done by Sterling [4] and Dunstan [3], indicate the existence of two dominant longitudinal acoustic modes with corresponding natural frequencies. This fact validates the choice of 2-DOF van der Pol model of combustion processes, proposed in this chapter. Another experimentally verified feature of acoustic modes is occurrence of incommensurate natural frequencies [8]. Considering this feature in the proposed van der Pol model leads to the multiple time scales concept. Due to the strong nonlinear coupling, various combination tones emanate in addition to dominant natural frequencies. As mentioned in Section 3, first-order solution in Eq. (27) contains all possible combination tones with 21 corresponding Fourier coefficients
The choice of the van der Pol model of combustion processes is justified especially due to the existence of limit cycles. Limit cycles in combustion process arise as a result of heat release saturation. This feature of the van der Pol model is particularly valuable because it allows to predict the highest level of the acoustic pressures by means of the calculation of the steady-state values, see Eq. (25). The static characteristic of the heat release process is described by means of the generalized nonlinearity of van der Pol type, which includes the saturation term. The generalized static characteristic of heat release rate contains three parameters
A further validation can be obtained by modification of van der Pol model to consider a transport time delay
In accordance with the proposed structure, the van der Pol model in Figure 4 is described by governing equations:
where
By performing the EL-P procedure as described in Section 3, one obtains the following set of PDEs:
Low pass filter is linear and the filter dynamics is assumed to be much faster than the slowly varying amplitudes and phases in dependence on the slow time scale
The low pass filter produces The gain
By means of zero-order solutions (44), one deduces the following solvability conditions for elimination of secular terms in Eq. (43):
Equations (46) and (47) determine the evolution of slowly varying amplitudes
The obtained results show that steady-state values of first- and second-mode pressure oscillations depend inversely on the values of natural frequencies
7. Implementation of the van der Pol model of combustion processes
To eliminate growing combustion oscillations, the coupling between acoustic waves and the unsteady heat release must be interrupted by applying the control. To reach this goal, passive and active control methods are enforced. The passive control is effective in a limited range of operating conditions and design changes are usually expensive and time consuming. Earlier concepts of active control of combustion instabilities used loudspeakers to increase surface losses in a combustor [1]. The lack of loudspeaker actuators is insufficient robustness for use in industrial applications and prohibitive power requirements at large-scale combustion systems. The recent development of active control seeks to achieve a stable operation regime of a full-scale combustion system. The proposed van der Pol model of combustion processes applies a recent concept of an active feedback control of the heat release by controlling the fuel flow rate. The control function is multiplicative and is realized with the modulation of the injection of the fuel flow. The properties of a good actuation can be reached by the use of fuel valve with a linear response characteristic, large bandwidth and operating range to achieve the effective control of two oscillation modes and to affect limit cycle oscillation. In addition, the fuel valve should have a fast response time, good robustness and durability; however, it should not have hysteresis behavior.
Sensors are generally provided to measure the dynamic quantities, which are related to the combustion oscillations. In the present model, the pressure of the acoustic wave and the fuel flow are measured quantities, which are acquired by means of the pressure transducers and sensors, which measure the light emission of some chemicals in the flame or by chemiluminescence sensors, respectively. The use of pressure transducers is advantageous, because they can be installed outside the area of high temperatures at the burning plane on account of acoustic wave propagation across the entire combustion chamber. Pressure transducers have large bandwidth and a good robustness. The placement of pressure transducers is important, because the location of the possible pressure node must be avoided and on other side, the signal-to-noise ratio must be maximized.
Recent development in the heat release sensing offers the use of diode lasers. Earlier, they had been used for temperature measurements, but recently they can directly measure fuel concentrations. A comprehensive review of this topic is given in [9].
The proposed van der Pol model of combustion processes is based on the grey-box principle. This means that the controller must be designed on system measurements. The instability of the combustion process may have severe consequences, which makes measurement a difficult task. Due to existence of two oscillation modes, which span over a wide frequency range, the controller must be able to control both modes. The combustion process almost always involves time delays caused by fuel convection and acoustic wave propagation. When such a delay becomes larger than an oscillation period, assumptions made in Section 6 are violated and provide a great challenge to the controller design [11].
8. Conclusions
In this chapter, the EL-P method with multiple time scales is successfully applied to the analysis of instability problem in the combustion process with external harmonic excitation, which is modeled by means of coupled van der Pol oscillator. By applying the control of the fuel influx, the asphyxiation of pressure oscillations is reached and studied in the phenomenon of the competitive quenching.
The solvability conditions of the first order in EL-P method are derived, which determines slowly varying amplitudes and phases as well as steady-state values. With external harmonic excitation, both limit cycle oscillations and compound oscillations can be successfully suppressed to the acceptable level by using the multiplicative feedback control.
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