Effect on burner partitioning on system thermoacoustics

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Effect on burner partitioning on system thermoacoustics

Citation for published version (APA):

Hoeijmakers, P. G. M., Kornilov, V. N., Lopez, I., Nijmeijer, H., & Goey, de, L. P. H. (2011). Effect on burner partitioning on system thermoacoustics. In Proceeding of the 18th International Congress on Sound & Vibration (ICSV 18), 10-14 July 2011, Rio de Janeiro, Brazil

Document status and date: Published: 01/01/2011 Document Version:

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THERMO-ACOUSTICS

P.G.M. Hoeijmakers, V.N. Kornilov, I. Lopez, H. Nijmeijer, L.P.H. de Goey.

Department of Mechanical Engineering, Eindhoven University of Technology. e-mail: p.g.m.hoeijmakers@tue.nl

In this paper the thermo-acoustic properties of a system equipped with a segmented burner are investigated. Such segmentation can for example be realized by combining two different burners on the same deck. The study is conducted using finite element and network modelling techniques. It is found that even a small spatial separation of the burner’s sections can lead to significant alteration of the global acoustic behavior of the system. In particular, the location of the eigenmodes and hence the stability of the system can be affected. It is shown that the local modification of the acoustic field in the vicinity of the partitioned burner is the physical reason for this phenomena. This finding poses some limitations on to use of the lumped trans-fer function decomposition approach to estimate the global burner response of a composite flame/burner.

1. Introduction

Thermo-acoustic combustion instabilities usually manifest themselves by the autonomous gen-eration of acoustic tone(s) within a combustion system. This effect can arise in practically any com-bustor device where the burner/flame is confined in a vessel. Typical examples include gas turbines, rocket engines, industrial burners and domestic heating appliances. Regardless of the concrete ap-plication, instability is usually an undesirable effect which is difficult to foresee during the design process and to eliminate in the development stage.

Thermo-acoustic oscillations were first explained by Rayleigh [6] and consist of a feedback loop between the systems acoustics and the flame heat release rate. For many burner types the heat release fluctuation is a direct result of the acoustic velocity perturbations at the flame location. In this context, the thermo-acoustic response of a burner can be characterized via a flame/burner transfer function (F ).

Due to the coupled nature of the instability problem, the stability can be significantly influenced by either changing the system acoustics or the burner response. In what follows we concentrate on the latter approach. In this context, it is desirable to be able to design a burner with a specific response. It is well known that the actual transfer function of a burner is strongly dependent on properties such as the burner type, flow, and flame(s). However, due to the complex interplay between the different parameters the a-priori prediction of the transfer function remains a challenge. One way to circumvent this problem is by combining two or more burner segments with a known response. In this setting, a technique known as transfer function decomposition can be used to calculate the overall response based on the weighted transfer functions of the individual segments [2]. The weight factors in this approach are the ratios of any segment’s individual power to the total burner power. Thus,

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18th International Congress on Sound and Vibration, 10–14 July 2011, Rio de Janeiro, Brazil

the total response may be tuned by changing the relative power ratios. The method is based on the additive nature of the flame heat release and it assumes that the upstream acoustics for all segments of the burner(s) is the same. Under the assumption that the burners are acoustically compact and located at the same streamwise position one can expect that the acoustic oscillations close upstream of the burner are identical. In other words, the spatial non-uniformity of the transfer functions does not affect the acoustic field. However, in the present paper it is shown that this is not necessarily true. In order to illustrate the effects of a spatially distributed transfer function we will investigate the properties of a simple burner configuration composed out of two different burner segments. More specifically, the two sections are assumed to have equal powers where each section can be modelled by a different time delay transfer function. As illustrated in figure 1a, a practical burner with similar properties can easily be realized by a perforated burner deck with two fields of different diameter holes. Because the time-delay relates directly to the flame height, the larger diameter holes will lead to a larger time-delay due to the higher flames.

To calculate the eigenfrequencies of the system a finite element model is used. The results are compared to a network model for further validation. It will be demonstrated that in the vicinity of the spatially nonuniform burner the acoustic field can be significantly affected by the flame transfer function. As a result, the global eigen-mode frequencies, stability, and mode shapes can deviate significantly from the one calculated based on the lumped approach. To our knowledge, this effect was not treated before in literature and it is the main focus of the present study.

The remainder of this paper is organized as follows. In section 2 the system to which the burner is applied is shortly described. Next, section 3 is devoted to an overview of the available methods to model the composite burner deck and its interaction with the system acoustics. The main results are then presented in section 4. Finally, section 5 provides a discussion of the results and conclusions.

2. System

As mentioned before, the main interest is in the thermo-acoustic properties of the burner de-picted in figure 1a applied to a simple system. The specific configuration under investigation is depicted in figure 1b and consists of a closed end, supply duct, burner, and open ended exhaust tube.

The total length of the system is set at 1 [m]. Because the burner is placed in the middle this gives L1 = L2 = 0.5 [m]. In this case, the up- and downstream temperatures are chosen to be

T1 = 300 and T2 = 1600[K], yielding a density and speed of sound of ρ1 = 1.18, ρ2 = 0.22,

[kg/m3_{] and c}

1 = 347.24, c2 = 774.38 [m/s]. These values are typical for a practical combustion

configuration operating at u ≈ 1[m/s] and a lean mixture. For the two-dimensional finite element simulation the width of the ducts and burner is set to 0.1 [m].

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1

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Figure 1. The burner and system under interest. (a) Segmented burner, and (b) Simple thermoacoustic system (A)Acoustically closed end, (B) Intake tube, (C) Burner, (D) Exhaust tube, (E) Open end.

The burner itself is splitted in two parts where each part can be approached by a simple time-delay model. The transfer function of the first group of flames is F1 = e−iτ ω, with τ = 2×10−3, while

the second half of the burner is modelled to be F2 = e−2iτ ω. Thus, the larger flames are considered to

have twice the time delay of the smaller ones. Note that the areas A1and A2 in the figure refer to the

perforation areas and not the surface area of the complete burner. 2

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3. Thermo-acoustic model

A thermo-acoustic model can be obtained via a number of methods. In principle one can dis-tinguish between a distributed and a lumped approach. A distributed model is based on the governing equations and is typically solved in a finite element framework. Due to the space dependent nature of the burner of interest this is the main modelling method applied in this paper. The governing equations for the FEM model are described in section 3.1.

In order to get further insight into the results, an acoustic network model based on two-port transfer matrices is also used. A short description of the network model is provided in section 3.2.

In both the FEM and the Network model specific attention will be given to how the burner is represented in the model.

3.1 Finite element model

Notwithstanding the elegance of many academic and analytic test-cases most practical combus-tion devices are analytically intractable. Accordingly, a sizeable number of publicacombus-tions deal with the development and validation of thermoacoustic finite-element models, see for example [3, 5]. The derivations described in [5] and [1] forms the base for the model described below.

In any case, the development starts with a model describing the propagation of small pertur-bations, e.g. sound waves, in a inhomogeneous medium including heat release. More specifically, after the combination and linearization of the equations for (i) conservation of mass, (ii) conservation of momentum, (iiv) conservation of energy, and (iv) a state equation one obtains an inhomogeneous Helmholtz equation, ω2 γp0 p0+ ∇ · (1 ρ∇p 0 ) = −iωδ(x − b)γ − 1 γp0 ˙ Q0_v, (1)

where p0, ˙Q0_v, p, ρ and γ are the fluctuating pressure [P a], fluctuating heat release rate per unit volume [W/m3_{], mean pressure [P a], mean density [kg/m}3_{], and the ratio of specific heats respectively. This}

equation describes how fluctuating heat release acts as an acoustic monopole source. Note that at this stage such source is not directly coupled to the acoustic field.

3.1.1 Heat release

In order to fully describe the problem it is necessary to couple the fluctuating heat release and the acoustics. This can be achieved in two steps.

First, let us assume that the heat release ˙Q0_v takes place at a location L1 on the x-axis in an

infinitely small zone, and is of the form, ˙

Q0_v = ˙Q0_aδ(x − L1), (2)

where δ(x − L1) is the Dirac delta function, and ˙Q0ais the heat release rate per unit area [W/m2]. In

other words, the heat release is assumed to be concentrated in a plane. Indeed, for a practical laminar burner, the flame size is much smaller than the acoustical wavelength.

Next, one can couple the acoustic field to the heat release by using the flame transfer function: ˙ Q0_a = ˙ Q uA0 F u0, (3)

where ˙Q [W ] is the spatio-temporal mean heat release, u0 = −_iωρ1 ∂p0(L1)

∂x [m/s] the fluctuating velocity

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Substituting the latter expression in the acoustical source term of equation (1) and simplifying one obtains: ω2 γp0 p0 + ∇ · (1 ρ∇p 0 ) = F δ(x − L1) θ ρ ∂p0(L1) ∂x , (4)

where θ = T2/T1− 1 is the temperature ratio between the hot and cold parts of the flame.

3.1.2 Modelling the burner

In principle the above framework can be used to model the system depicted in figure 1. How-ever, one can seperate two approaches to model the heat-release zone. First, one can use the transfer function composition method to construct a single lumped flame transfer function for the complete burner.

Transfer function decomposition

Here a short review of the transfer function decomposition technique is given, for further details see reference [2]. Let us consider the burnerdeck as depicted in figure 1a. The individual Fi, i = 1, 2

and complete transfer function FΣcan be defined as,

FΣ =

Q0_Σ/Q_Σ

u0_/u (5) , Fi =

Q0_i/Q_i

u0_/u . (6)

Here the subscript Σ refers to the total transfer function and heat release, while the subscript i = [1, 2] denotes the individual parts.

It is important to note that the transfer function definition used here implicitly assumes that the fluctuating velocity is the same for both flame groups. Based on the property that energy is additive, one may write the total fluctuating and mean heat release as,

Q0_Σ =X i Q0_i =X i FiQi u0 u = q X i FiAiui u0 u (7) and Q_Σ =X i Q_i = qX i Aiui = qA0u, (8)

where q [J/m3_{] is the heat of combustion per unit volume, u}

i denotes the particular mean flow speed

throughout each burner deck portion, and A0 = A1+ A2 is the total area of the perforations.

Substi-tution of (7) and (8) in (5) gives,

FΣ = X i Fi Aiui A0u . (9)

Thus, the individual transfer functions are weighted according to the ratio of the individual to total mean heat release rates. As mentioned before, we will consider the specific case that the powers of each segment are equal. The transfer function to be used in equation (4) is then simply F = FΣ =

0.5F1+ 0.5F2. In this case, any y-dependence of heat release and acoustic velocity is neglected, and

hence one expects the same results as in a one dimensional network model. From now on, this shall be named method A.

Explicit space dependence

Alternatively, one can exploit the flexibility of the FEM framework to directly include the space dependence of the transfer function. More precisely, the heat release ˙Q0_ais then defined to be depen-dent on the y-direction,

˙ Q0_a(y) = ˙ Q uA0 F (y)u0 (y), (10) 4

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where both the y-dependence of the flame transfer function, as well as the possible variation of u0(y) in the y-direction are included. In case of the burner depicted in figure 1a the transfer function is defined to be F (y) = e−iωτ (y) with τ (y) [s] a piecewise continuous function,

τ (y) = (

2 × 10−3, for 0 ≤ y < W₂

4 × 10−3, for W₂ < y ≤ W . (11)

Note that the approach is not limited to such a simple distribution of transfer functions and in principle can be used to model arbitrary distributions of flame groups. In the following this is referred to as method B.

3.1.3 Solution

Because the flame transfer function F gain and phase depend nonlinearly on the frequency this means that equation (4) leads to a nonlinear eigenvalue problem [5]. The commercial finite-element package Comsol Multiphysics was used to solve equation 4. Using COMSOL the nonlinear eigenvalue problem can be solved in an iterative manner.

The solution procedure is as follows. First, a second order approximation of the transfer-function is made around some linearization point w0, this leads to a quadratic eigenvalue problem

which can be readily solved using standard methods [5]. Next, the linearization point is updated to the solution w1 of the quadratic eigenvalue problem around the first point w0. This procedure is

re-peated until the difference between the linearization point and solution is lower than a convergence limit of 1×10−5. The whole procedure is then repeated for a range of initial points w0in the frequency

range of interest.

3.2 Network model

A complete model of the system might also be obtained using the common acoustic network modelling method. This technique is based on the so-called two-port acoustic transfer matrices of each system element [4]. For each of the elements the output variables, namely pressure p0 and velocity u0are linked to the input variables by linear equations. A complete system description can be obtained by matching the subsequent in- and outputs of the elements. The eigenfrequencies are then determined from the resulting system of equations.

Because the network model only serves as a validation tool here the description will be limited to the different network structures one might choose to model the configuration of interest.

3.2.1 Modelling the burner

In close analogy to the the finite element approach, one can again separate two basic approaches, as depicted in figure 2.

First, one can lump response of the complete burner to one transfer function using transfer function composition, F = 0.5F1 + 0.5F2. Using this method one can then proceed to model the

system using the simple network shown in figure 2a.

Second, one can approximate the two different parts of the burner by two branches each contain-ing it’s appropriate transfer function, F1, and F2respectively, as shown in figure 2b. The choice of the

length of the intermediate ducts ’Duct-2’ and ’Duct-3’ is nontrivial and will be justified a-posteriori based on the FEM results.

Note that the same name convention is used as for the FEM case, were method A refers to the lumped approach and method B for the branched layout. For both methods, the temperature, speed of sound, and density down- and upstream of the flame were set according to the values stated in section 2.

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Figure 2. The two possible network topologies to model the burner. (a) Simple one branch model, method A. (b) Branched network model, method B.

4. Results

In this section the stability and frequencies of the eigenmodes of the burner in the system are investigated. In particular, we compare the results for the described methods A and B for both the FEM and the network models.

Figure 3a shows a direct comparison of the eigenmode locations between the lumped (A) and distributed (B) modeling techniques in case of the finite element model. Note that frcorresponds to

the actual eigenfrequency, and fi to imaginary value of the eigenvalue, with negative unstable. For

clarity, all the modes are denoted by a lowercase letter.

0 100 200 300 400 500 600 −50 −40 −30 −20 −10 0 10 20 30 40 50 f_r [Hz] fi a b1 b b2 c d e FEM−A FEM−B (a) 0 100 200 300 400 500 600 −50 −40 −30 −20 −10 0 10 20 30 40 50 f_r [Hz] fi NM−A NM−B (b)

Figure 3. Results from the (a) FEM model method A vs B, and (b) Network model method A vs B.

The lumped approach (A) leads to 3 unstable modes at 187, 363 and 590 [Hz] respectively, fur-ther referenced to as mode b, d, and e. The calculation based on the space dependent transfer function on the other hand (B) shows a significantly different picture. More precisely, mode b disappears and instead two unstable modes b1 and b2 appear at 154 and 252 [Hz]. This is remarkable difference, and indicates that the space dependence cannot be neglected. Furthermore, the mode c at 328 [Hz] shifts to a higher and slightly less stable location for method B. In order to get further insight into the problem, let us now evaluate the results from the network model.

Figure 3b depicts the results in case of the network model. Comparing the eigenmode location for method A and B it is obvious that the network model reproduces the transition from mode b to b1 and b2 . Such result is surprising, as it suggests that the behavior of the FEM model with the space dependent transfer function approaches the branched configuration. This is a fundamental result and 6

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shows that a burner deck as modeled here can induce a significant different behavior compared to one with a homogeneous distribution of flames. However, it is intriguing that only mode b is affected significantly and d and e are at approximately the same location for both method A and B.

In order to extend our understanding of the aforementioned effects, it is illustrative to consider the eigenmode shapes of a and b1 for method B. To clarify the influence of the space dependence we consider cross-sections of the pressure-profile along the length direction at two different y-locations; [0.025, 0.075] [m]. It is important to note that the eigenmodes are in general not pure standing waves and hence the pressure profile varies during the cycle due to the traveling wave component. As to get insight into the influence of the variation of the transfer function in the y-direction, the shapes were plotted at the angle for which the greatest difference occurs between the two y-locations.

Figure 4a depicts the pressure distribution at the two y-locations for mode a. Clearly, only a minor difference in pressure between the two y-location exists. This result is in contrast to the significant difference in the y-locations for mode b1, as shown in figure 4b. In this case, there is a major phase difference between the two y-positions around the flame position at x = 0.5 [m]. Note that the actual dissimilarity is visible from a distance of 0.1 [m] up- and downstream of the flame. This observation leads to the a-posteriori justification of the choice of the intermediate duct lengths for the network model as mentioned in section 3.2.1.

Given the minor influence of the space dependence for mode a it is not surprising that the frequency and stability of this mode are virtually the same as for the space independent case. A similar reasoning can be followed for modes d and e. For mode b1 on the other hand, the large difference between the two y-locations suggest that here the behaviour is close to the branched duct system. Although not shown here, this is also the case for mode b2. As a consequence, these eigenmodes are only possible when the space-dependence is explicitly included.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 x [m] p [pa] y=0.025 y=0.075 (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 x [m] p [pa] y=0.075 y=0.025 (b)

Figure 4. Modes shapes from the FEM model using method B. (a) Mode a, @60 [deg] and (b) Mode b1, @110 [deg]

5. Discussion and conclusions

5.1 Discussion

The analysis presented in the current paper suggests that the space dependence of the flame transfer function for this particular burner deck can play an important role in the overall thermoacous-tic behaviour of the system. The fact that both the finite-element and network model show exactly the same result further strengthens this observation. Overall, the results indicate that one has to be careful with applying the transfer function decomposition method.

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Notwithstanding the importance of this result, it is unclear at this stage how general the pre-sented results are in terms of changes in the transfer functions, burner deck layout, and other system properties. This question deserves further attention.

Another interesting issue is the experimental verifiability of the results. First and foremost, the chosen pure time-delay transfer functions can hardly be achieved in practice because any practical flame has a frequency dependent gain. With this in mind, a forced response measurement of a burn-erdeck as shown in figure 1a is more likely to give concrete answers. In particular, based on the results presented here, one expects a strong acoustic velocity variation in the burner deck plane at certain frequencies. Clearly, this is another issue worthwhile to investigate.

Finally, it is noteworthy that if the observed effects can be confirmed in practice they might be exploited to yield a more desirable combustor behavior. From this perspective, the geometrical layout of the burner deck with regard to the location of the perforations can become a useful design parameter.

5.2 Conclusions

In the present paper a burner deck with a strongly space-dependent transfer function is investi-gated. In particular, the influence of such space dependence on the eigenmode locations, and shape for a simple thermo-acoustic system were considered. In order to model the system, a finite element model and a network model were used.

The response of the system was calculated with and without explicitly taking into account the space dependent nature of the problem. For both approaches, the network model and the finite-element case give the same result.

Based on the results presented in section 4, it can be concluded that the space dependency of the flame transfer function for composite burner decks cannot always be neglected. Because of the very specific properties of the considered setup, it is not clear if this conclusion can be generalized to a larger class of systems and burner configurations. This is an excellent topic for further research.

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