# Measurement and simulation of acoustic load impedance for boilers.

INTRODUCTION

What we know as sound is a result of fluid density or pressure fluctuations. The distinctive feature of these fluctuations is that they propagate rapidly away from the source at a speed that depends on the type of fluid (Fahy 2001). The disturbance propagates as a wave with no net mass transport and the propagation speed is known as the speed of sound. The acoustic wavelength is the speed of sound (343 m/s or 1125 ft/s at room temperature for air) divided by the frequency of the disturbance.

If noise is produced in a duct or pipe system, the sound pressure will be constant across the duct cross-section at low frequencies. This is known as plane wave propagation and will occur if the cross-sectional dimensions are less than half an acoustic wavelength. For a square cross-section, the plane wave cutoff frequency (when the sound pressure is no longer uniform across the cross-section) is equal to c/2d where c is the speed of sound and d is a characteristic dimension (the larger of the length or width) of the duct cross-section. Similarly, Eriksson (1980) showed that the cutoff frequency for a circular duct is c/1.71d (d is the diameter in this case).

When a sound wave encounters an abrupt geometric change or an obstacle, the wave will at the very least be partially reflected. For example, a muffler uses cross-sectional area changes to reflect sound back towards the source. Sound is also reflected from the end of a pipe or duct due to the abrupt change in geometry. In the case of plane wave propagation, the sound field consists of an incident and a reflected wave. The superposition of these two waves results in a standing wave where the positions of high and low amplitude sound pressure inside a duct do not change (Fahy 2001; ASHRAE 2009).

Combustion oscillations are a common happening in boilers, furnaces, and water heaters. Oscillations in the burning rate result in a fluctuation of the mixture flow rate (i.e. an acoustic particle velocity). For the most part, the flame is a benign sound source. However, the sound reflected back from the combustion chamber produces a standing wave that will disturb the mixture flow rate or composition. At certain frequencies, a sympathetic resonance develops, resulting in a tone.

Changing the geometry of the system can eliminate these tones. Changes might include modifications to the combustion chamber, or the intake and vent pipe lengths. Additionally, problems have been solved by adding small holes into pipes (Baade, 2004; Baade and Tomarchio, 2008).

The acoustic metric that is most relevant to the combustion oscillation problem is the acoustic impedance. In fact, the acoustic impedance upstream and downstream of the flame is used as an input in the feedback loop model developed by Baade (1978, 2004; Baade and Tomarchio, 2008). In a companion paper (Zhou et al., 2013), the feedback loop stability model developed by Baade is applied to two boilers that exhibited combustion oscillation problems.

The acoustic impedance (in Rayl/[m.sup.2] or ([lb.sub.f]/[ft.sup.2])/([ft.sup.3]/s)) relates the particle velocity ([??]) directed away from the source to the acoustic pressure ([??]) and can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where

S = cross-sectional area ([m.sup.2] or [ft.sup.2])

Hence, a relationship between the acoustic pressure and particle velocity (i.e. flame) can be established by measuring or calculating the acoustic impedance at the location of the flame. For mathematical ease, the acoustic impedance is expressed as the ratio of the sound pressure to the volume velocity (i.e., the particle velocity multiplied by cross-sectional area). The unit for particle velocity is ft/s or m/s and the unit for volume velocity is [ft.sup.3]/s or [m.sup.3]/s. The unit for sound pressure is [lb.sub.f]/[ft.sup.2] or Pa.

The next section details how the upstream and downstream impedances are in parallel with each other at the source. The sections that follow describe how the acoustic impedance can be measured and simulated.

ACOUSTIC MODEL OF THE SYSTEM

In the following discussion, we have chosen to follow the line of reasoning in Elsari and Cummings (2003). The model assumes

* Plane wave propagation inside the duct.

* Low Mach number flow.

* The length of the flame is small compared to an acoustic wavelength.

Figure 1 shows a diagram of a duct with heat release. The flame is assumed to be a volume velocity source (units of [ft.sup.3]/s or [m.sup.3]/s) having strength Q. Conservation of mass requires that

Q = [S.sub.1][[??].sub.1] + [S.sub.2][[??].sub.2] (2)

where

[[??].sub.1] = acoustic particle velocity in the upstream (ft/s 1 or m/s)

[[??].sub.2] = acoustic particle velocity in the downstream (ft/s or m/s)

[S.sub.1] = cross-sectional area in the upstream ([ft.sup.2] or [m.sup.2])

[S.sub.2] = cross-sectional area in the downstream ([ft.sup.2] or [m.sup.2])

Assuming that [DELTA] is small compared to an acoustic wavelength, the acoustic pressures [[??].sub.1] and [[??].sub.2] (indicated in Figure 1) should be equal to each other due to continuity of acoustic pressure (acoustic pressure cannot change abruptly with position). [Z.sub.u] and [Z.sub.d] are the upstream and downstream acoustic impedance (ratio of sound pressure to volume velocity) respectively. These impedances are temperature dependent since the speed of sound of a medium is proportional to the square root of the absolute temperature in Rankine or Kelvin.

Acoustic systems are analogous to electrical, mechanical and hydraulic systems. Accordingly, acoustic parallel and series impedances are analogous to their electrical counterparts. The upstream and downstream impedances ([Z.sub.u] and [Z.sub.d]) are in parallel with one another because the sound pressure (the analog to voltage for electrical systems) does not vary across the flame. The impedance at the flame (Z) can be written as

Z = [Z.sub.u][Z.sub.d]/[Z.sub.u] + [Z.sub.d] (3)

In the feedback loop model developed by Baade (1978, 2004; Baade and Tomarchio, 2008), the impedance at the flame and the reciprocal of the upstream impedance are utilized.

DETERMINATION OF ACOUSTIC IMPEDANCE

Measurement of Acoustic Impedance

Acoustic impedance is most commonly measured using the two microphone method (ASTM, 1998). The two micro phone method is shown schematically in Figure 2. A loud speaker is placed atone end of the tube and the sound pressure is measured at the two microphone locations. The microphone closest to the source is the reference and the transfer function between the two microphones (H12) is measured. The transfer function can be used to determine the sound pressure reflection coefficient (R) using the equation

R = [[H.sub.12] - [e.sup.-jks]/[e.sup.jks] - [H.sub.12]] [e.sup.2k(l+s)] (4)

where

k = acoustic wavenumber ([ft.sup.-1] or [m.sup.-1])

s = microphone spacing (ft or m)

l = spacing from load impedance position to microphone 2, (ft or m)

j = [square root of -1]

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The acoustic wavenumber is defined as

k = 2[pi]f/c (1 - j[eta]) = [omega]/c(1 - j[eta]) (5)

where

f = frequency (Hz)

[omega] = 2[pi]f = angular frequency (rad/s)

c = speed of sound (ft/s or m/s)

[eta] = loss factor normally selected as 0.0 for measurement or on the order of 0.01 for analysis purposes

The acoustic load impedance ([Z.sub.L]) can then be determined from the reflection coefficient via

[Z.sub.L] = [rho]c/[S.sub.t] 1 + R/1 - R (6)

where

[rho] = mass density of air (kg/[m.sup.3] or slugs/[ft.sup.3])

c = speed of sound (ft/s or m/s)

[S.sub.t] = area of tube ([ft.sup.2] or [m.sup.2])

Both the upstream and downstream impedances ([Z.sub.u] and [Z.sub.d]) shown in Figure 1 can be measured in this way.

Instrumentation is used to measure quantities that are harmonic in time. Accordingly, there will be a time lag between quantities. Accurate measurement of the phase (or time lag) is crucial to acquiring an accurate measurement of the phase of the impedance ([Z.sub.L]). Phase differences between the microphones, pre-amplifiers, and channels of the spectrum analyzer can lead to errors in the measurement of the load impedance. The phase between the microphones can be calibrated by switching the positions between the two microphones for an identical source and load. This procedure is described in ASTM E-1050 (1998).

The Spectronics impedance tube and software (Spectronics Inc., 2007) were used to acquire all the data in this paper. Equivalent instrumentation is available from other suppliers. The impedance tube is brass and the microphone holders are well sealed to prevent sound leakage. The source is a compression driver loudspeaker (JBL 2426J). For Boiler 2, the compression driver was replaced with a bookshelf loud-speaker in order to boost the source energy at low frequencies.

It is important to ensure that the signal to noise ratio is high. The field sound pressure should be substantially higher than the background noise in the pipe or tube. Standards recommend that the sound pressure level in the tube is at least 10 dB higher than the background noise, though 20-30 dB is preferred (Spectronics Inc., 2007).

Calculation of Impedance Using Transfer Matrix Theory

The upstream and downstream acoustic impedances ([Z.sub.u] and [Z.sub.d]) can be determined using a model based on transfer matrix theory (Munjal 1987; Munjal et al. 2006; Abom and Elnady 2010). Transfer matrix theory relates the sound pressure and particle velocity at the inlet to that at the outlet of a component. The main assumption is that plane acoustic waves can be assumed at the inlet and outlet of each component, though sound waves need not be planar within the components. Provided that each component in the upstream or downstream piping system can be modeled as a transfer matrix, the impedance can be determined after multiplying the transfer matrices together.

The numerical computing software MATLAB[R] was used for all calculations. It should be borne in mind that impedances are complex quantities having both magnitude and phase.

A transfer matrix [T] is composed of four-pole parameters A, B, C, and D. These four-pole parameters relate the sound pressure and particle velocity at the inlet and outlet of a particular duct section. This can be expressed mathematically as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where

[[??].sub.1] = sound pressure at the inlet

[[??].sub.2] = sound pressure at the outlet

[[??].sub.1] = particle velocity at the inlet

[[??].sub.2] = particle velocity at the outlet

Figure 3 shows a schematic for a duct system. The transfer matrix for the complete system [T] can be found by multiplying each of the transfer matrices together. Thus,

[T] = [[T.sub.1]][[T.sub.2]][[T.sub.3]][[T.sub.4]][[T.sub.5]] (8)

for the case shown in Figure 3. The impedance at the left hand side can be determined from the four-pole parameters [A.sub.T], [B.sub.T], [C.sub.T], and [D.sub.T] for the system transfer matrix [T] and the radiation impedance at the end of the duct or piping system ([Z.sub.rad]) as shown in Figure 3. The impedance can be expressed as

Z = [A.sub.T][Z.sub.rad] + [B.sub.T]/[C.sub.T][Z.sub.rad] + [D.sub.T] (9)

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Transfer Matrices for Common Duct Elements

The most commonly used duct elements in boilers, furnaces and water heaters are shown in Figure 4 (Munjal 1987). The transfer matrix for a straight duct (Figure 4a) or tube can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The transfer matrix for a quarter wave tube (Figure 4b) or structural element modeled as a side branch (Figure 4d) can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

For the case of a quarter wave tube,

[Z.sub.p] = -j[rho]c cot(k[L.sub.B])/[S.sub.B] (12)

The impedance ([Z.sub.p]) for a vibrating plate which is modeled as a side branch can be found in the next section (Inclusion of Structural Vibration in Transfer Matrix).

The transfer matrix for a cone (Figure 4c) can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where

[z.sub.1] = linear cone dimension (ft or m)

[z.sub.2] = linear cone dimension (ft or m)

l = length of cone (ft or m)

In Figure 3, a straight duct is used to model [[T.sub.1]], [[T.sub.3]], and [[T.sub.5]]. A quarter wave tube is used to model [[T.sub.2]] and a cone to model [[T.sub.4]].

Figure 5 illustrates how a quarter wave tube can be configured as an extended outlet (or inlet). For an extended inlet or outlet, the Equation (10) can be used but [L.sub.B] must be adjusted to include near field effects at the flanged end (Karal, 1953). Accordingly,

[L.sub.B] = l + [[delta].sub.e] (15)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

and

[alpha] = [square root of S/[S.sub.B]] (17)

[FIGURE 5 OMITTED]

Inclusion of Structural Vibration in Transfer Matrix

It has been observed in practice that stiffening or adding damping to panels can eliminate combustion oscillations in some cases. This is especially the case for combustion chambers with thin flat plates. A simple model (Figure 4d) can be used to include the structural vibration of the plate. The plate can be treated as a side branch or parallel impedance similar to a quarter wave tube if it is assumed that the sound pressure is constant across the panel. This assumption should be appropriate at low frequencies since the acoustic wavelength is long compared to the panel dimension.

The impedance of the panel can be determined by considering the plate as being simply supported (Soedel 2004). The natural frequencies of the plate are then

[[omega].sub.mn] = [[pi].sup.2]([(m/a).sup.2] + [(n/b).sup.2]) [square root of (D/[[rho].sub.p]h])] (18)

where

m = number of harmonics along the x coordinate direction

n = number of harmonics along the y coordinate direction

a = length of the plate (ft or m)

b = width of the plate (ft or m)

h = thickness of the plate (ft or m)

[[rho].sub.p] = density of the plate (slugs/[ft.sup.3] or kg/[m.sup.3])

At low frequencies, the mode most likely to couple strongly with the acoustics is the first mode (m = 1 and n = 1). The flexural rigidity of the plate (D) is defined as

D = E[h.sup.3]/12(1 - [v.sup.2]) (19)

where

F = elastic modulus (N/[m.sup.2] or [lb.sub.f]/[ft.sup.2])

v = Poisson's ratio

The modal participation factors ([[eta].sub.mn]) can be determined using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

with

[[LAMBDA].sub.mn] = [F.sub.mn]/[[omega].sup.2.sub.mn][square root of [(1 - [([omega]/[[omega].sub.mn]).sup.2]).sup.2] + 4[[zeta].sup.2.sub.mn] [([omega]/[[omega].sub.mn]).sup.2]] (21)

where

[omega] = angular frequency

[[zeta].sub.mn] = modal damping coefficient

The modal damping coefficient is best determined experimentally. A frequency response function can be measured using an impact hammer and an accelerometer. The modal damping coefficient can be determined by identifying the peak amplitude and the half power frequencies from the frequency response function. See Ewins (2000) for more information.

[F.sub.mn] can be determined using

[F.sub.mn] = 4p(1 - cos (m[pi])) (1 - cos (n[pi]))/[rho]hmn[[pi].sup.2] (22)

where p is the pressure which can be set to 1 for determining the impedance. The phase lag is defined as

[[phi].sub.mn] = [tan.sup.-1] 2[[zeta].sub.mn]([omega]/[[omega].sub.mn])/1 - [([omega]/[[omega].sub.mn]).sup.2] (23)

The vibrational velocity for the first mode can then be expressed as

[??](x, y) = j [omega][[eta].sub.11] sin([pi]x/a) sin ([pi]y/b) (24)

where

x = position in x-direction along length a

y = position in y-direction along width b

Then, the branch impedance can be approximated as

[Z.sub.p] = -p/avg([??])[S.sub.p] (25)

where [S.sub.p] is the area of the plate.

The velocity of the plate ([??]) was averaged in a root mean square sense and the phase was averaged as well. The branch impedance ([Z.sub.p]) found in Equation (25) can then be inserted into Equation (11).

Calculation of Transfer Matrix using Acoustic Finite Element Method (FEM)

In some cases, the combustion chamber cannot be simulated using the transfer matrices described earlier because the geometry is too complex or the plane wave cutoff frequency has been exceeded. For example, sand-cast combustion chambers have a complicated geometry and it is difficult to model the chamber using plane wave approximations. More importantly, the plane wave cutoff frequency will be exceeded in most combustion chambers at a relatively low frequency. However, transfer matrix theory can still be used if it is assumed that plane waves exist at the inlet and outlet to the combustion chamber. However, analytical solutions for the transfer matrices above the plane wave cutoff frequency are not available in the literature for most geometries. Accordingly, the transfer matrix itself should be determined using a deterministic method like the acoustic finite element method (FEM) or boundary element method (BEM).

A schematic of a combustion chamber is shown in Figure 6. When using numerical methods, it is normally easier to first find modified four-pole parameters (Wu et al. 1998; Herrin et al. 2007; Herrin et al. 2007) which are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

To determine these modified four pole parameters, two BEM/FEM analyses need to be completed. The four pole parameters can be found by applying a unit velocity on the left end ([[??].sub.1] = 1). Then, a subsequent analysis should be performed with a unit velocity on the right end ([[??].sub.2] = -1). The first run is used to determine both [A.sup.*] and [C.sup.*]. The second run is used to find [B.sup.*] and [D.sup.*]. The BEM/FEM model should include a length of duct on both the inlet and outlet sides in order to ensure plane wave behavior so that the transfer matrix approach is valid.

[FIGURE 6 OMITTED]

The four-pole parameters in Equation (7) can then be obtained from the modified four-pole parameters using the expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Determination of the Termination Impedance

The termination impedance ([Z.sub.rad]) can be determined analytically for unflanged and flanged openings or terminations. The termination impedance for an unflanged opening (Levine and Schwinger 1948) is

[Z.sub.rad] = [rho]c (1 + R)/S(1 - R) (28)

The reflection coefficient (R) is given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

where

a = radius at the orifice

[R.sub.0] = amplitude of the reflection coefficient

[[zeta].sub.0] = end correction

The amplitude of the reflection coefficient ([R.sub.0]) is writ ten as

[R.sub.0] = 1 + 0.01336ka - 0.59079[(ka).sup.2] + 0.33576[(ka).sup.3] - 0.06432[(ka).sup.4], ka < 1.5 (30)

and the end correction ([[zeta].sub.0]) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

For the flanged or baffled opening (Pierce 1981), the radiation impedance ([Z.sub.rad]) is defined as

[Z.sub.rad] = [rho]c/S([R.sub.1](2ka) - j[X.sub.1](2ka)) (32)

where

[R.sub.1] = 1 - [J.sub.1](2ka)/ka [X.sub.1] = [H.sub.1](2ka)/ka (33)

[J.sub.1] and [H.sub.1] are the Bessel function and Struve function of first order, respectively.

IMPEDANCE COMPARISONS

The downstream impedance of three boilers was simulated and measured using the procedures described. Each of these boilers has distinguishing features that make them representative of the range of commercial boiler applications. For Boiler 1, the combustion oscillation problem was at high frequencies (around 2100 Hz at elevated temperatures) and the geometry of the combustion chamber was very complicated. Acoustic FEM had to be relied on in order to simulate the combustion chamber. For Boiler 2, the combustion oscillation problem occurred at low frequencies (around 10 Hz) and the geometry was simple. Consequently, plane wave methods were sufficient for the entire frequency range of interest. For Boiler 3, a panel resonance at approximately 230 Hz proved to be the source of the oscillation problem. Boiler 3 was simulated using plane wave methods with the thin panel included as a side branch.

Boilerl-Downstream Impedance

Boiler 1 was a 200,000 Btu/hour (59 kW) capacity propane gas boiler representative of relatively massive and rigid fire tube boilers. The heat exchanger was cast aluminum with numerous fingers in the lower chamber to facilitate heat transfer. The combustion oscillation problem occurred at approximately 2100 Hz when the boiler was running lean. The predicted and measured impedances at room temperature compare well up to 1300 Hz. The speed of sound is proportional to the square root of absolute temperature. Accordingly, low frequencies at room temperature directly correspond to high frequencies at elevated temperatures. When realistic operating temperatures (approximately 800[degrees]F or 425[degrees]C) are included, the model should be acceptable up to and above 2100 Hz.

The downstream impedance of Boiler 1 was measured using ASTM E1050 (1998). A 1.375-inch (35 mm) diameter impedance tube was used for the measurements and 0.5 inch (1.25 cm) condenser microphones (PCB 426E01) were used to measure the impedance. The microphone spacing was 1.35 inch (3.4 cm). All measurements were performed in a hemi-anechoic chamber in order to minimize background noise contamination.

The combustion oscillation occurred at approximately 2100 Hz. Consequently, it was desirable that the simulation model was valid above the plane wave cutoff frequency of the combustion chamber. Hence, an acoustic FEM model of the combustion chamber was created. As a first step, a simplified solid model of the combustion chamber was created and meshed. Alarge acoustic resistance of 82,500 rayls (525 lbf/[ft.sup.2]/[ft/s]) was added to the inside surface of the combustion chamber in order to add a small amount of acoustic absorption. The transfer matrix was determined by applying a unit velocity at the burner inlet and then the exhaust in sequential runs. See the section "Calculation of Transfer Matrix using Acoustic Finite Element Method (FEM)" for the methodology.

Plane wave theory was used for each of the other elements in the model. A schematic of the model is shown in Figure 7. Table 1 shows the specific dimensions that were used for each of the elements.

The magnitude and phase of the normalized impedance for a vent length of 1.0 m (39.4 in.) is compared in Figure 8. The normalized impedance is the impedance multiplied by S/[rho]c. The result is representative of other vent lengths. Vent length had little impact on the impedance. The predicted and measured impedances at room temperature compare well up to 1300 Hz at room temperature.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Boiler 2--Downstream Impedance

Boiler 2 was a much larger 500,000 Btu/hour (146 kW) capacity propane gas boiler with a stainless steel heat exchanger. Coils were arranged cylindrically in both the upper and lower parts of the heat exchanger. The combustion oscillation occurred at approximately 10 Hz and the cause of the oscillation was a fluctuating equivalence ratio.

The downstream impedance for Boiler 2 was measured using ASTM E1050 (1998) and simulated using plane wave theory (Munjal 1987). For Boiler 2, the instability occurred at close to 10 Hz, and plane wave theory was viable for the entire frequency range of interest.

Measuring the impedance at very low frequencies is challenging for a few reasons. Most loudspeakers do not put out sufficient sound energy below 100 Hz and condenser microphones do not measure sound pressure accurately below 20 Hz. Additionally, background noise levels are high even in a hemi-anechoic chamber below 50 Hz.

Several steps were taken to improve the measurements so that the simulation model could be compared. A bookshelf loudspeaker was used rather than a compression driver to increase the sound energy at low frequencies, and the spacing between the microphones in the impedance tube was increased to 0.54 m (21.4 in) to improve the accuracy of the measurement at low frequencies.

Additionally, the exhaust pipe was either removed entirely or only a short length was used. Though this does not match the boiler in operation, the first resonance frequency (also known as the Helmholtz frequency) of the combustion chamber was moved higher in frequency with a shorter exhaust pipe. Moving the first acoustic resonance higher in frequency (above 20 Hz) enables the measurement to capture the first acoustic resonance of the combustion chamber. The measured first resonance can then be more easily correlated with plane wave simulation.

The downstream impedance of Boiler 2 was modeled as shown in Figure 9. Detailed dimensions are shown in Table 2. For good agreement between measurement and simulation, it was important to model Element 4 as a quarter wave tube or extended outlet.

[FIGURE 9 OMITTED]

The magnitude and phase of the normalized impedance for a vent length of 0 m (0 in) is shown in Figure 10. As the result shows, there is good agreement from 20 to 230 Hz. Below 20 Hz, the measurement is suspect due to the microphones used, background noise, and insufficient source strength. Above 230 Hz, acoustic FEM should be used since the plane wave cutoff frequency is exceeded in the combustion chamber.

Boiler 3--Downstream Impedance

A combustion chamber was selected which had a thin and nearly flat panel on one side and is referred to as Boiler 3. Since the plate was thin, it was believed that the vibration of the plate would affect the measured impedance at certain frequencies. The downstream impedance of Boiler 3 was measured and simulated to determine if this was indeed the case. The impedance was measured in a manner identical to the measurements for Boilers 1 and 2. A vent pipe was added to one end of the chamber.

A schematic of the plane wave model is shown in Figure 11. The thin plate was modeled as a side branch. The theoretical approach for dealing with a thin plate is detailed in the section "Inclusion of Structural Vibration in Transfer Matrix". Table 3 provides details about the dimensions and individual transfer matrix elements. Table 4 summarizes the plate dimensions and material properties.

The magnitude and phase of the normalized impedance is compared in Figure 12 for the no vent case. Measured results are compared with simulation with and without Element 4 (the thin plate in Table 3). The results demonstrate that the resonance at 220 Hz is due to the thin plate and will be present regardless of the vent length. Additionally, the good agreement with simulation demonstrates that the effect of structural vibration can be incorporated into the plane wave model.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

CONCLUSION

Combustion oscillations are produced when sound produced by the flame is reflected from the combustion chamber towards the mixture chamber. The amount of reflection depends primarily on the geometry of the combustion chamber and attached ductwork. The acoustic impedance is the ratio of sound pressure to the volume velocity at a given position. Accordingly, it is directly applicable to characterizing the sound pressure at the flame produced by the volume velocity fluctuations of the flame. Acoustic impedance has been used to characterize the acoustics both upstream and downstream to the flame in the model developed by Baade (Baade, 1978; Baade, 2004; Baade and Tomarchio, 2008).

The acoustic impedance of boilers can be determined at room temperatures by experimentation. However, simulation is required to determine the impedance at elevated temperatures if the temperature is varied from element to element. In this study, simulation models for three combustion chambers have been correlated with measured results. The simulation models were based on transfer matrix theory, which assumes plane wave behavior. In the case of Boiler 1, acoustic finite element analysis was used to characterize a combustion chamber since the combustion oscillation occurred above the plane wave cutoff frequency.

This paper has detailed the most commonly used duct elements and their transfer matrices. In addition, a transfer matrix element has been developed for combustion chambers consisting of thin and flat plates. Simulation of the acoustic impedance has been compared to measurement with good agreement for each combustion chamber. These results are used in a companion paper (Zhou et al. 2013) where the feedback loop stability model developed by Baade (Baade, 1978; Baade, 2004; Baade and Tomarchio, 2008) is used to predict combustion oscillations.

[FIGURE 12 OMITTED]

This paper is based on findings resulting from ASHRAE Research Project RP-1517.

ACKNOWLEDGEMENTS

This work reported in this paper was supported by RP1517. The authors gratefully acknowledge the assistance of TC 6.10, and especially the invaluable guidance and tutelage of Dr. Peter Baade.

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Zhou, L., Herrin, D. W. and Li, T. 2013. Assessing the Causes of Combustion Driven Oscillations in Boilers using a Feedback Loop Stability Model (RP-1517). ASHRAE Transactions 119(2).

L. Zhou

Student Member ASHRAE

D.W. Herrin, PE, PhD

Member ASHRAE

T. Li, PhD

L. Zhou is a PhD candidate, D. W. Herrin is an associate professor, and T. Li is a associate research professor at the University of Kentucky, Lexington, KY.
```Table 1. Dimensions and details of simulation model for
Boiler 1 downstream impedance.

No.   Description   Element Type     Area,        Area,
[cm.sup.2]   [in.sup.2]

1      Impedance        Duct          9.6          1.5
tube end
2     Combustion    Acoustic FEM    46 to 20    7.1 to 3.1
chamber
3      Pipe from        Duct           20          3.1
chamber to
elbow
4        Elbow        Quarter          28          4.3
Wave Tube
5      Vent pipe        Duct           20          3.1
6     Termination    Unflanged         20          3.1

No.   Description   Length,   Length,
cm        in

1      Impedance      2.2       0.9
tube end
2     Combustion      N/A       N/A
chamber
3      Pipe from      12        4.7
chamber to
elbow
4        Elbow        10        3.9

5      Vent pipe      90       35.4
6     Termination

Table 2. Dimensions and details of simulation model for
Boiler 2 downstream impedance.

No.    Description      Element      Area,        Area,
Type      [cm.sup.2]   [in.sup.2]

1       Impedance        Duct         9.6          1.5
tube end
2       Short duct       Duct          44          6.8
3       Combustion       Duct         1180        182.9
chamber
(burner side)
4       Combustion      Quarter       926         143.5
chamber       Wave Tube
(exhaust side)
5     Outlet chamber     Duct         254          39.4
6          Vent          Duct         103          16.0
7      Termination     Unflanged      103          16.0

No.    Description     Length,   Length,
cm        in

1       Impedance        2.2       0.9
tube end
2       Short duct       3.0       1.2
3       Combustion      36.0      14.2
chamber
(burner side)
4       Combustion      31.0      12.2
chamber
(exhaust side)
5     Outlet chamber    31.0      12.2
6          Vent         11.5       4.5
7      Termination

Table 3. Dimensions and details of simulation model for
Boiler 3 downstream impedance.

No.    Description       Element       Area,        Area,
Type       [cm.sup.2]   [in.sup.2]

1       Impedance         Duct          9.6          1.5
tube end
2       Short duct        Duct           43          6.7
3       Combustion        Duct           60          9.3
chamber
(near burner)
4       Thin plate     Side branch      380          58.5
5       Combustion        Duct           60          9.3
chamber
(between thin
plate and
baffle)
6     Region outside      Duct           39          6.0
baffle
7       Combustion        Duct           60          9.3
chamber (after
baffle)
8       Short duct       Quarter         13          2.0
Wave Tube
9       Vent pipe         Duct           18          2.8
10     Termination      Unflanged        18          2.8

No.    Description     Length,   Length,
cm        in

1       Impedance        1.8       0.7
tube end
2       Short duct       2.2       0.9
3       Combustion       8.7       3.4
chamber
(near burner)
4       Thin plate
5       Combustion       4.7       1.9
chamber
(between thin
plate and
baffle)
6     Region outside     1.4       0.6
baffle
7       Combustion       2.7       1.1
chamber (after
baffle)
8       Short duct       0.7       0.3

9       Vent pipe       1.51       0.6
10     Termination

Table 4. Dimensions and material properties for
the thin plate.

Young's        Poisson's    Density     Thickness
Modulus          Ratio

19.3 Gpa          0.3       8000 kg/     1.8 mm
[m.sup.3]
2800 kips/        0.3      0.289 lbs/   0.071 in
[in.sup.2]               [in.sup.3]

Young's        Length   Width    Damping
Modulus                           Loss
Factor

19.3 Gpa       0.20 m   0.19 m    0.01

2800 kips/     7.8 in   7.5 in    0.01
[in.sup.2]
```
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