Exhaust stack silencer design. using finite element analysis. Abstract

xhaust stack silencer design using finite element analysis Carl Q. Howard, Ben S. Cazzolato, and Colin H. Hansen Department of Mechanical ngineering, The University of Adelaide, South Australia 5005, Australia (Dated: November 2, 1999) Abstract Classical analytical models used for prediction of the performance of reactive silencers are limited to conditions where the dimensions of the duct and resonators are small compared to the wavelength of the sound. Finite lement Analysis does not suffer from such limitations and has therefore been used to analyze the design of a reactive silencer for the exhaust stack of a 980MW power station. To assist in the design process, resonators of various dimensions were analyzed using FA which has led to the derivation of expressions for the resonance frequencies of slot-type rhomboid shaped resonators as a function of the geometry. An important design issue is the influence that adjacent resonators have on the overall performance of the system. It was found that when resonators of similar resonance frequency are in close proximity, they can interact and lead to a decrease in the overall performance compared to that of a single resonator. PACS numbers: 43.50 Gf 1

2 I. INTRODUCTION The work presented here is a Finite lement based numerical and experimental investigation of a reactive silencer designed to reduce the noise emitted from the exhaust stack of a 980 MW coal fired power station. Conventional analytical techniques, such as transmission line theory, that are typically used to evaluate the performance of silencers were not suited to this application for two reasons; first, the wavelength of the sound was less than the resonator dimensions and second, non-plane wave conditions existed in the duct. The work also involved fundamental studies of the resonance frequencies of rhomboidal shaped resonators and the interaction between resonators that are in close proximity to each other and have similar resonance frequencies. Rhomboid shaped resonators (see Figure 3) were used in the design because they have several distinct advantages over rectangular shaped resonators: they have less tendency to become filled with debris [1]; they permit longer quarter wave tubes within a fixed lateral space and hence lower resonance frequencies may be achieved than with conventional rectangular tubes; and finally resonators with orifices that face away from the flow (ie inhibit inflow) tend to have lower self noise [2, 3]. A literature survey of classical analytical models for resonators revealed that most research deals with either circular or rectangular shaped cavities and principally with resonator dimensions that are small compared to the wavelength of sound. In addition most papers treat the throat/cavity junction as a flange (ie the throat does not intrude into the cavity) which meets at right angles to the cavity wall, unlike the current design where the throat enters into the cavity at an angle of 45. The limitation of current analytical models led to a FA investigation of the resonance frequencies of rhomboidal shaped resonators which accounted for their irregular geometry and higher order effects associated with large dimensions. Previous work on the application of finite element analysis to the design of mufflers has focused on automotive mufflers, where an acoustic disturbance travels along a narrow duct through a series of expansion chambers

3 [4 11]. These papers discuss the higher order effects that can occur inside the resonators. The problem under investigation differs from the previous research in that the side branch resonators are attached to a large rectangular duct, where plane wave conditions do not exist. It will be shown that non-plane wave conditions in the duct can permit the interaction between cells of similar resonance frequency which are in close proximity to each other. This interaction may decrease the TL compared to that achieved when the cells are placed well away from each other (> λ/4, where λ is the wavelength of sound). The investigation performed here involved calculating the Transmission Loss (TL) of resonators of various dimensions and performing a regression analysis on the collected data. From this data set, individual resonators or cells were selected (based on their resonance frequencies) and added to the silencer section to meet the desired silencer performance. II. TH PROBLM Two large Induced Draft (ID) fans are used to disperse exhaust gases from a power station s boiler through an exhaust stack. Figure 1 shows the arrangement of the fans, silencer and exhaust stack. Air Flow 7m Silencer Section Low Speed Fan High Speed Fan xhaust Stack FIG. 1: Arrangement of the silencer in the exhaust system. The fans generate a combination of broadband noise and strong tones at the fan s blade

4 passage frequency (BPF) and associated harmonics. They have 12 blades and rotate at either 735 RPM or 984 RPM, resulting in fundamental BPF s of 150 Hz and 200 Hz respectively. The dimensions of the exhaust duct were 7m high 3.5m wide, the gas temperature was 140 C and the volumetricflow rate was 300 m 3 /s. From these parameters it was determined that the speed of sound and fluid density were approximately c = 407 m/s and ρ =0.867 kg/m 3 respectively, resulting in a Mach number of M =0.03 in the main duct and M =0.06 in the airway between the resonators. While dissipative silencers have been used successfully in clean air systems, attempts to use dissipative silencers on the flue gas from coal fired power stations have repeatedly failed because of accumulation of fly ash and other particulates in the absorptive material and perforated facing plate. It has been shown that this can result in a rapid reduction in transmission loss of up to 50% in the space of 2 years [12, 13]. Reactive silencers provide a feasible alternative to the more conventional dissipative systems and do not suffer from a reduction in performance from build up of particulates. In the case of broadband silencers, the performance may even increase with contamination as the particulates act to increase the acoustic damping. In addition, reactive silencers offer better low frequency performance than dissipative silencers of similar dimensions. The reactive silencer proposed here consists of 5 parallel rows of resonator cells, referred to as baffles. Figure 2 illustrates the arrangement of the baffles. ach side of any given airway has a different resonator configuration. III. TH FINIT LMNT MTHOD Two dimensional finite element models using the software package Ansys ( c Ansys Inc.) were used to predict the TL of resonators of various dimensions. The models used Fluid29 acoustic elements [14] which are 2D and have 4 nodes per element, with each node having 3 degrees of freedom: translation along the X and Y axes, and pressure. The translational degrees of freedom were not used in the work described here, only the pressure degrees of

5 7102 214 B A B Quarter Wave Tube or Helmholtz Resonator A B A B A B A Direction of Air Flow 342 342 3512 342 342 450 342 236 FIG. 2: Schematic of the silencer. The numbers indicate dimensions in millimeters. The letters A and B indicate that a different configuration of resonators exists on each side of the baffles. freedom were used. A unit dynamicpressure was applied to the duct, upstream of the silencer. The effect of fluid flow cannot be model explicitly using the finite element model, but has been accounted for by adjusting the boundary absorption coefficient to provide damping that matches that predicted from classical theory. Anechoic end conditions were applied to the ends of the ducts by setting the specific acoustic impedance of the termination wall equal to the characteristic impedance ρc of the fluid. The use of a 2D model for this problem may not necessarily be accurate as cross modes could exist in the (out of plane) third dimension that is not modeled. The TL across the silencer is calculated by measuring the acoustic power which is lost through the silencer section. In this case, the TL is calculated by the ratio of the power that is incident on the silencer to that transmitted downstream of the silencer. The accepted test method [15] uses the two-microphone technique [16] to calculate the right traveling acoustic intensity at a discrete point in the duct. For the problem under investigation here, this method cannot be used as the duct examined here exhibits non-planar wave conditions and the acoustic intensity varies across cross section of the duct. The two-microphone technique can be adapted by integrating, across the cross sectional area of the duct, the acoustic

6 intensity of the right traveling pressure wave upstream and downstream of the silencer, to determine the acoustic power loss across the silencer [17]. This method is the most accurate for this problem but is also the most computationally demanding. Instead an approximate method was used to calculate the upstream and downstream acoustic power, by measuring the maximum squared pressure levels in the upstream and downstream sections. It should be noted that this method cannot be used to measure the TL for a single resonator in a duct because the pressure doubles on reflection (increase of 6 db) at the resonator, and would result in an incorrect measurement of the TL. However for this problem, where there are numerous resonators, a comparison of the last method with the exact method, which involved integrating the acoustic intensity across the cross sectional area of the duct, showed that the difference in TL results was less than 2dB. The approximate method was thus used to calculate the TL for subsequent analyzes. A Damping and nergy Dissipation As with any numerical acoustic model it is essential to account for the damping that occurs in real systems. The mechanisms through which the resonators dissipate energy are numerous and somewhat complicated and certainly beyond the scope of this paper. Kriesels et al. [18] provides a good discussion of the various mechanisms which can be summarized as follows; acoustic radiation, visco-thermal damping, structural damping, vortex generation and vortex shedding. The effect of mean flow on the resonators is to increase the damping. The quality factor, Q, of a resonator is related to the Mach number, M, by the following expression [19] Q = ω 0 l eff (1) cm As the resonance frequency for a resonator is approximately inversely proportional to the square root of the effective length of the neck, it can be shown that the quality factor is approximately proportional to the square root of the resonance frequency. This means that the damping associated with the mean flow decreases with increasing frequency.

7 For a typical resonator with the following dimensions: throat width d = 50mm, throat effective length l eff = 100mm, volume length L = 214mm, volume width D = 150mm, and ameanflowm = 0.06, it can be shown that the resonance frequency is approximately ω 0 = 200Hz and Q = 5. Based on q. (1), it can be seen that for a resonator with the same effective length but twice the resonance frequency (about 400Hz), the quality factor would be double (Q = 10). As the damping is frequency dependent and Ansys does not allow frequency dependent damping in the analysis, it is necessary to use a single quality factor that is representative over the whole frequency range. As the 250Hz and 500Hz octave bands were the most critical in this case, a conservative quality factor of 10 was chosen. A quality factor of 10 is not high considering that in practice, even without the additional damping provided by the mean flow, quality factors are rarely greater than 20. Munjal [17] states that, in practice, the end wall of the resonators is not completely rigid and generally has a reflection coefficient of 95%. The effect of higher damping would be to reduce the amplitude of the TL at resonance and increase the TL off-resonance. The overall effect on the octave or third-octave performance is generally to increase the TL. So the factor used here is believed to be somewhat conservative for all octave bands. The example resonator described above was modeled in Ansys with various boundary absorption coefficients, to simulate the effect of damping associated with the mean flow. The boundary absorption coefficient is equal to the acoustic resistance divided by the characteristic impedance of the fluid. It was found that a boundary absorption coefficient of 1% gave a quality factor of Q = 10.

8 IV. MULTI-DIMNSIONAL FFCTS A Prediction of resonance frequency : ffect of resonator and duct geometry on the resonance frequency For many silencer systems the assumption that plane wave conditions hold is valid because the critical dimensions of the system are significantly less than λ/2. However, when the frequency is increased, cross-modes occur and the performance of the resonators will be degraded and can no longer be evaluated using plane wave theory [20]. There exists a significant amount of literature on the effects of higher order modes and 3-D geometry on the performance of resonator type silencers [10, 21, 22]. The results obtained from studies incorporating higher order effects usually deviate from those obtained using classical theory due to the simplifications involved in reducing distributed, multi-dimensional phenomena to a lumped parameter formulation. If there are higher order modes in the duct, then the resonators are not able to reduce the power transmission by the same amount as they are able to for plane waves. In large industrial silencers, non-plane wave conditions generally occur above frequencies of the order of 500Hz when one of the duct cross sectional dimensions is equal to or more than half a wavelength. Because of the rhomboidal shape of the quarter wave tubes and the throat of the Helmholtz resonators, it was difficult to define the equivalent length and width of the throat. For long-narrow throats, the length and width are approximately equal to the line parallel and normal to the throat walls respectively. However, for short-wide throats, the relationship is less clear. For this reason, when deriving an empirical expression for the resonance frequencies of the resonators, it was necessary to have several higher order terms involving both the throat width and length to account for the uncertainty in the geometry of the throat. Panton and Miller [23] derived an expression for the resonance frequency of a Helmholtz resonator, which takes account of the impedance of the cavity, for a resonator with a long

9 narrow cavity as, f r = c S 2π Vl eff + 1 3 L2 S (2) which is applicable to resonators which satisfy kl < π/2; L < λ/4, where k is the wavenumber, l eff is the effective throat length, V is the volume of the cavity, S is the throat area and L is the depth as shown in Figure 3. D L l d g FIG. 3: The variables that describe the geometry of a Helmholtz resonator. The preceding expression formed the basis for a multiple regression analysis on the FA results for the 2D Helmholtz resonators. The finite element analysis of the 130 Helmholtz resonators with the geometricdimensions shown in Table I formed the data set. It was found that the effective length for a Helmholtz resonator with a narrow slot attached to an infinitely long duct is given by the following empirical relationship (with a correlation coefficient of 0.9998) l eff =0.0835 + 0.1358D +0.4334L 0.0502 ln(l)+0.0140 ln(g)+0.0493 ln(d) + l [3.8871 9.6155d +0.7403 ln(d)+0.3997 ln(l) 9.6110LD] (3) where all dimensions are in metres. quation (3) is only strictly correct for the case considered of a single resonator mounted in the wall of the duct when the fluid is air and the speed of sound is c = 407m/s. No effort was made to non-dimensionalize this expression. Note that Chanaud [24] showed that by placing the orifice that connects the side branch to the

10 TABL I: Dimensional limits of the resonators in the data set (with all dimensions in metres). Helmholtz Resonator Quarter Wave Tube Minimum Maximum Minimum Maximum g 0.050 0.800 0.050 0.800 l 0.005 0.070 0.100 0.500 d 0.050 0.150 0.050 0.100 L 0.080 0.236 N/A N/A D 0.100 0.350 N/A N/A duct at the upstream or downstream edge of the resonator face (see Figure 3) rather than in the centre of the resonator face, the resonance frequency can be lowered. The placement of the orifices at the downstream edge of the resonators in the current design assisted in achieving the necessary low frequency performance and helped negate the increase in the resonance frequency that occurs when a slot orifice is used instead of a square orifice. The expression for the fundamental resonance frequency f r of a quarter wave tube is given by f r = c 4 l eff (4) The above expression formed the basis for a multiple regression analysis on the FA of the quarter wave tubes. Finite element analysis was conducted on 45 quarter wave tubes, bounded by the geometricdimensions listed in Table I. The variables that describe the geometry of a quarter wave tube are shown in Figure 4. The effective length for a quarter wave tube attached to an infinitely long duct which satisfies equation (4) empirically is given by l eff = 0.0569 + 1.5845l 0.1432l 2 +0.2028d + 0.0205g l 0.0040l g (5) where c = 407m/s. It should be noted that equations (3) and (5) for the effective lengths are

11 l d g FIG. 4: The variables that describe the geometry of a quarter wave tube resonator. by no means definitive. The calculation of the resonance frequencies (and effective lengths) used a rigid walled infinite duct with no other elements attached to the duct wall. When additional cells are added to the system, subsequent modification of the duct geometry (hence impedance) as well as modal coupling will alter the resonance frequency of the cells. Hence, equations (3) and (5) will provide only an estimate of the effective length of a particular resonator when a series of resonators are installed in a duct. It was found that adjacent resonators may alter the resonance frequency of a single resonator in a duct by up to 15%, depending on the proximity of the adjacent resonator and the difference in resonance frequencies of the two resonators. B Influence of adjacent resonators on transmission loss When designing a series of resonators to be inserted into a silencer, it is essential that resonators with similar resonance frequencies are not placed too closely together, otherwise the benefit of additional resonators may not be realized. The effect of resonator separation can be seen with classical transmission line theory which assumes plane wave conditions. When two identical resonators are placed adjacent to each other, the transmitted power is halved, ie an increase of 6dB in TL is observed. However, when the resonator centres are placed at distances much greater than λ/4 thetl in db is cumulative. In addition, the acoustic coupling between two closely spaced resonators produces a shift

12 in the resonance frequencies, producing two distinct frequencies; one lower and one higher than the original frequency. This de-tuning of the resonators may have a detrimental effect on the overall performance of the silencer by creating holes in the TL spectrum and is particularly apparent when the resonators have similar resonance frequencies. Figure 5 shows qualitatively the sound pressure level at 300Hz in a duct with two rhomboidal quarter wave tubes attached to the duct wall and separated by 80mm. 1 D D D I III MX I III H HHH G GG H HH F G F G F F F F G G F F F F F F F D D D C D D B C D C MN AB D C AB D C D C D D D D ANSYS 5.5.3 SP 29 1999 10:04:32 PLOT NO. 1 AVG LMNT SOLUTION STP=1 SUB =16 TIM=300 NMIS4 (AVG) SMN =70.05 SMX =115.608 A =72.581 B =77.643 C =82.705 D =87.767 =92.829 F =97.891 G =102.953 FIG. 5: Sound Pressure Level contour plot of two closely spaced rhomboid quarter wave tubes at 300Hz. The dimensions of the tubes are d = 100mm, l = 214mm, g = 430mm. The figure shows that both cells are resonating. The maximum sound pressure level occurs inside the resonator on the right hand side and is labeled MX, and the minimum sound pressure level occurs mid way between the two resonators in the duct. From the phase plots of the pressure it was found that there was a 180 phase difference between the two cells. Figure 6 shows the pressure gradient (or particle velocity) around the two resonator cells, from which it may be concluded that the fluid is moving out of the resonator on the left and into the resonator on the right. When the two cells are excited 180 out of phase and have a spacing of less than a quarter of a wavelength, the TL is significantly reduced because the cells tend to cancel one another s

13 FIG. 6: Pressure gradient plot of two closely spaced rhomboid quarter wave tubes at 300Hz. effectiveness by a dipole effect. Figure 7 shows the TL of the system in figure 5 as a function of frequency. 30 25 Single Cell 2 Cells - 80mm Transmission Loss (db) 20 15 10 5 0 250 260 270 280 290 300 310 320 330 340 350 Frequency (Hz) FIG. 7: Transmission loss for two closely spaced rhomboid quarter wave tubes. The figure clearly shows that the TL afforded by two adjacent resonators is significantly less than that provided by even a single cell. The spacing shown here is the worst case situation and spacings of greater than λ/4 provide better attenuation than shown here. This result is different to that obtained using the classic transmission line analysis because non-planar conditions exist in the duct shown in figure 5. The analysis highlights the need for careful consideration of the location of side

14 branch resonators in relation to others of similar resonance frequency. V. DSIGN CHARTS The silencer for the power station was designed to attenuate broadband noise through the use of Helmholtz resonators and tonal noise through the use of quarter wave tubes. To tune a quarter wave tube, it is essential that the gas temperature and hence the speed of sound, remains constant. Small variations in the gas temperature change the wavelength of the sound, which means that the quarter wave tubes will be significantly less effective at the desired frequency, thus degrading the overall performance. In general, if gas temperature variations can be expected, then it is wiser to use Helmholtz resonators to meet the acoustic specifications than quarter wave tubes, as the former have broader TL versus frequency characteristics. The design process described here involved creating series of design charts derived from equations (3) and (5), with resonance frequency as a function of the dimensions of the resonators. Figure 8 shows an example of a design chart for a Helmholtz resonator with a duct width g = 430mm, throat width d = 50mm, resonator depth L = 214mm. The figure shows the resonance frequency of the resonator for varying throat lengths l along the abscissa, and the lines on the graph indicate the resonator width D. From these curves individual resonators or cells were selected (based on their resonance frequencies) and added to the silencer section to meet the desired silencer performance. Because the silencer was constructed from individual resonators, deficiencies in the TL were identified. Additional cells were added to increase the TL in regions of poor performance. This procedure was repeated until the silencer met the required performance.

15 Resonance Frequency (Hz) 280 260 240 220 200 180 160 140 120 100 Helmholtz Resonator : Throat Width d = 50mm, Resonator Depth L = 214mm Resonator Width D 100 mm 150 mm 200 mm 250 mm 300 mm 350 mm 80 0 10 20 30 40 50 60 70 80 90 100 Throat Length l (mm) FIG. 8: A typical design chart for a Helmholtz resonator. VI. FINAL DSIGN Once a general design was found, ten variations of this design were investigated to optimize the overall performance, each design taking approximately 12 hours to evaluate on a Silicon Graphics R8000. Figure 9 shows an outline of the 58 resonators used in the finite element model. Approximately 16,000 nodes and elements were used. FIG. 9: The final design of the reactive silencer. The Ansys examples manual [25] recommends a minimum of 15 elements per wavelength to obtain accurate results. The model described here had approximately 60 elements per metre which indicates that the results might have significant errors at frequencies above 1500 Hz. Figure 10 shows the SPL within the duct at 95 Hz. The effect of a resonating cell can be seen quite clearly in the figure. The SPL plots

16 FIG. 10: SPL plot of silencer section and duct. Scale: 86dB (white) - 108dB (black). were used to identify resonator combinations that were destructively coupled. This was achieved by finding cells that were resonating and in close proximity to one another, and then examining the phase of the sound pressure level inside the resonators. If there were a 180 phase difference between the cells, then the cells were destructively coupling. To overcome the destructive coupling the cells were re-positioned further from each other. A Fabrication & Structural Compliance In the previous sections it was assumed that the walls of the baffles were rigid. In reality, walls of finite thickness such as those fabricated from sheet metal are compliant. The compliance of the wall is generally not an issue for most reactive silencers such as automobile mufflers which have very stiff walls arising from the high curvature and the large thickness to length ratios [18]. However, if the walls of the baffles are thin compared with their length then the structure may not be sufficiently rigid to prevent significant coupling between the structure and the cavity, especially when the resonance frequencies of the acoustic modes are close to the resonance frequencies of the low order panel modes. This structural-acoustic coupling may severely limit the performance that can be obtained in reactive devices. Three mechanisms act to reduce the TL. First, a compliant structure will limit the impedance change that can be developed across a reactive element, thereby limiting the TL across the element; second, the modal coupling that occurs between the structure and cavity tends to de-tune the resonators; and finally, once the energy enters

17 the structure it may be re-radiated from the structure downstream from the reactive element, effectively short circuiting the resonator. It appears from modeling that the first mechanism dominates. Figure 11 shows the effect that varying the wall thickness has on the calculated TL of the silencer. 70 Transmission Loss (db) 60 50 40 30 20 1mm Steel 1.5mm Steel 2mm Steel 4mm Steel Rigid Wall 10 0 50 63 80 100 125 160 200 250 315 400 1/3 Octave Band Centre Frequency (Hz) FIG. 11: The calculated effect of wall thickness on silencer performance. The results are presented for the 50Hz to 400Hz third octave bands. The structural FM had insufficient nodes to adequately model above 400Hz due to node and element limits of the software license. It should be noted that the TL calculated using a fully coupled model is highly dependent on the structural boundary conditions and the results shown in Figure 12 simply demonstrates that the wall compliance has a significant influence on the insertion loss of the system. B Measured Performance A full-scale section of the silencer was tested at the NATA registered acoustics chambers at the CSIRO division of Building, Construction and ngineering, Melbourne. Figure 13 shows a schematic of the testing apparatus.

18 70 60 50 Measured - Flow Measured - No Flow Theory - Rigid Wall No Flow Theory - 1mm Steel No Flow IL (db) 40 30 20 10 0 63 125 250 500 1000 2000 1/3 Octave Band Centre Frequency (Hz) FIG. 12: Predicted versus measured insertion loss for the final silencer design. Fan nclosure Termination Duct Flexible Coupling Fan Discharge Duct Test Silencer Microphone Relief Silencer Loud Speakers Silencer Inlet Section Termination Duct Reverberation Chamber FIG. 13: Schematic of the experimental test apparatus to measure the IL of the silencer.

19 The test silencer of figure 13 was constructed using two different baffle configurations on two opposite walls as shown in figure 10. One of these baffles is shown in figure 14 and was made from sheet metal and measured 900mm high, 1190mm wide and 4000mm in length. FIG. 14: Silencer baffle made from sheet metal. White noise, filtered into 1/3 octave bands, was introduced into the duct by loudspeakers mounted upstream of the silencer section. The sound power emitted into the reverberation chamber was measured by a microphone in the room. The sound power in the chamber was also measured with the orifices in the test silencer blocked. The insertion loss was thus calculated from the difference between these two sound power measurements. Insertion loss measurements were also taken when the fan was forcing air through the system. The predicted insertion loss of the silencer for the system installed in the power station was also determined by adding a model of the exhaust stack to the end of the downstream duct section. The model of the exhaust stack was simply a 2D vertical duct with the same dimensions as the real stack. The free-space at the exit of the stack was modeled by a circular region (with a diameter of 5m) to ensure the boundary was not in the near field of the stack radiation. The boundary conditions on the perimeter of the circular area were set to provide infinite absorption. The IL was calculated by taking the difference in the sound power radiated from the exhaust stack with and without the baffles installed in the duct. The sound power was calculated by integrating the squared sound pressures over the perimeter of the circular area. Figure 12 shows the predicted and measured insertion loss for the final design. During testing of the silencer with air flow some of the third octave band measurements suffered from

20 a poor signal to noise ratio so the measured results shown have been adjusted to remove the effects of the background noise levels [19]. The measured and predicted results show good correlation, although the measured IL is slightly less than predicted for a 1mm fabrication, particularly in the 125Hz octave band. The lower measured values were not unexpected as the structural model had a low element density (due to software limitations) that artificially stiffened the structural model, thereby increasing the predicted IL results. VII. CONCLUSIONS Finite element methods were used successfully to design and optimize a reactive silencer for the exhaust stack of a coal fired power station. Classical analytical models were not suitable for the design process. This led to the development of a technique using FA to predict the transmission loss for rhomboid shaped resonators attached to a main airway. Separate regression analyzes were performed for Helmholtz and quarter wave tube resonators to provide an expression for the effective length as a function of the system geometry. When these resonators were placed in series, interactions between resonators of similar resonance frequency reduced the transmission loss compared to that obtained when the same resonators were separated by more than λ/4. Compliant duct walls were found to severely degrade the performance of the silencer, compared to that predicted for rigid duct walls. Two-dimensional Finite lement Analysis provides a technique suitable for the design and evaluation of large industrial reactive silencers and should suitable computing resources be available then the design may be further optimized by using automated linear optimization routines.

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