Acoustic Attenuation Performance of Helicoidal Resonator Due to Distance Change from Different Cross-sectional Elements of Cylindrical Ducts

Transcription

1 Exerpt rom the Proeedings o the COMSOL Conerene 1 Paris Aousti Attenuation Perormane o Helioidal Resonator Due to Distane Change rom Dierent Cross-setional Elements o Cylindrial Duts Wojieh ŁAPKA* Division o Vibroaoustis and Systems Biodynamis, Institute o Applied Mehanis, Poznań University o Tehnology, Poland. *Corresponding author: W. Łapka, Piotrowo 3 Street, Poznań, Poland. wojieh.lapka@put.poznan.pl Abstrat: This work presents aousti attenuation perormane o helioidal resonator due to distane hange rom dierent rosssetional elements o ylindrial dut. The helioidal resonator properties are desribed mainly in ininite long ylindrial dut. For pratial appliations it is important to know the limits o use o this solution. This paper desribes how the helioidal resonator should be plaed inside aousti systems and what distane rom dierent ross-setional elements like round silener, expansion hamber, seond helioidal resonator, should be satisied or its proper work. Keywords: helioidal resonator, silener, ylindrial duts, sound propagation, sound attenuation. 1. Introdution Helioidal resonator [1-4] is a newly developed aousti band-stop ilter, whih an be onstruted by inserting a helioidal proile with proper dimensions to a ylindrial dut, in simple ormulation - Figure 1. proile g, diameter o a the mandrel d m, diameter o the ininite long ylindrial dut d), whih are shown in Figure. d d m s - lead o one helioidal turn helioidal proile mandrel ylindrial dut Figure. Basi elements o a helioidal resonator plaed inside a ylindrial dut. Also very important parameter o helioidal resonator is the number o helioidal turns n, whih strongly determines the harater o aoustial resonane. Figure 1. Helioidal resonator as a part o simple aousti system - ylindrial dut. Aoustial properties, mainly attenuation o sound, as presented example transmission loss harateristis in Figure 3, due to an aoustial resonane o helioidal resonators an be modiied by doing a hange in relations between its basi geometrial parameters (lead o the helioidal turn s, thikness o the helioidal Figure 3. Example transmission loss harateristis o helioidal resonators with dierent ratios d m /d [3].

2 This paper onsiders aousti attenuation perormane o helioidal resonator due to distane hange rom dierent ross-setional elements like round silener, expansion hamber and seond helioidal resonator plaed inside ylindrial dut.. Computational environment - basi equations, alulated parameters and boundary onditions The problem is solved in the requeny domain using the time-harmoni Pressure Aoustis appliation mode o COMSOL Multiphysis - Aousti Module [5]. The inal solving parameter is the aousti pressure p [Pa], whih an be omputed by the use o slightly modiied Helmholtz equation: p p s (1) where ρ is the density o air (ρ =1,3 kg/m 3 ), s is the speed o sound in air ( s =343m/s), and ω gives the angular requeny. As an aousti attenuation perormane parameter is used transmission loss (TL) [1]. For omputational needs, the TL is expressed as the dierene between the outgoing power at the outlet w o and the inoming power at the inlet w i, wi TL log 1 wo 1, [db] () Eah o omponent quantities in equation () are alulated as an integral over the orresponding surae Ω o inlet and outlet ylindrial duts ross-setions S n and S 1, respetively: w w i o p p s s ds ds (3) (4) where p is the soure aousti pressure at the inlet, [Pa], and p is the transmitted aousti pressure at the outlet, [Pa]. Also or better analysis the aousti attenuation perormane an be presented as inrease o transmission loss Δ as ollows: TL, [db] (5) TL di TL ini where: TL di - transmission loss o dierent ase o an aousti system with helioidal resonator, [db], TL ini - transmission loss o initial aousti system without helioidal resonator, [db]. For investigated models in this work the boundary onditions are o three types [5]. For aoustially hard walls at the solid boundaries, whih are the walls o the helioidal proile, mandrel, ylindrial dut, expansion hamber, outer walls o round silener the model uses sound hard (wall) boundary onditions: p n (6) The boundary ondition at the inlet surae (sound soure) o ylindrial duts is a ombination o inoming and outgoing plane waves: n 1 p i p ik k i k T p T p p ik r k n k 1 ik e (7) where Δ T denotes the boundary tangential Laplae operator, k=ω/ s is the wave number, n is the natural diretion vetor or investigated irular dut, and wave vetor is deined as k=kn k, where n k is the wave-diretion vetor. In equation (7), p represents the applied outer pressure, and i denotes the imaginary unit [6]. The inlet boundary ondition is valid as long as the requeny is kept below the uto requeny or the seond propagating mode in the ylindrial dut. At the outlet boundary is set as the radiation boundary ondition whih allows an outgoing

3 wave to leave the modeling domain with no or minimal reletions: n 1 k i p i p T p k (8) The numerial model is omputed by the use o inite element method (FEM) by the terms o the element size [7] and maximum element size equals h e =,( s / max ), where max is the value o maximum investigated requeny (in this paper max =khz). The longitudinal dimension o irular duts is alulated as ininite by the use o radiation boundary onditions. Aousti attenuation properties o absorptive materials inside round silener are estimated by the use o the well-known model o Delany and Bazley [8] or omplex impedane Z and omplex wave number k : irst requeny range rom 1Hz to khz with omputational step o 5Hz, and the seond range rom 1,1kHz to 1,5kHz with the omputational step o 1Hz. The helioidal resonators ratios are onstant in all ases and equals: s/d=, g/d=,4, d m /d=,4. (1) 4.1 Expansion hamber In Figure 4 is presented example view on investigated expansion hamber, whih is m long and it has 3m in diameter, with helioidal resonator at the inlet. The inlet and outlet ylindrial duts have the same diameter d=1,5m. Dierent ases o distane hange between helioidal resonator and expansion hamber are exeuted. Z C6 C8 s 1 C5 ic (9) 7 R R k C C4 1 C 1 ic (1) 3 s R R with original oeiients: C 1 =,978, C =,7, C 3 =,189, C 4 =,595, C 5 =,571, C 6 =,754, C 7 =,87, C 8 =,73. Whereas, or glass-wool-like materials the empirial orrelation o Bies and Hansen [9] R 3,18 1 d 9 av 1,53 ap Pa, m s (11) Figure 4. Example view on expansion hamber (5m in diameter and m long) with helioidal resonator at the inlet. In Figure 5 is presented TL harateristis o expansion hamber, whih is the initial harateristis (TL ini ) to obtain inrease o TL or dierent ases o aoustis systems with helioidal resonator. is used to ahieve low resistivity R, where ρ ap [kg/m 3 ] denotes apparent density o absorptive material, and d av [m] is average iber diameter. 4. Results Three aousti systems were exeuted to obtain aousti attenuation perormane o helioidal resonator due to distane hange rom dierent ross-setional elements o ylindrial dut: with expansion hamber, round silener and seond helioidal resonator. The results are presented as the and Δ in the Figure 5. Transmission loss harateristis o expansion hamber - m long and 3m in diameter.

4 In Figure 6 are presented ΔTL harateristis or distanes rom m to 9m with the step o 1m between helioidal resonator and expansion hamber. 4. Round silener The aousti attenuation perormane o helioidal resonator due to distane hange rom round silener is investigated. In Figure 7 is presented example view on round silener (58m long, 34m o external diameter, 1,5m o internal diameter) with helioidal resonator at the inlet. The inlet and outlet duts have diameter d=1,5m. Also the internal diameter o round silener and diameter o inlet and outlet duts are idential. Figure 7. Example view o round silener with helioidal resonator at the inlet. Figure 8 shows the initial transmission loss harateristis o a round silener with glasswool like material (illed in volume between diameters 1,5m and 34m) o apparent density ρ ap =75 kg/m 3 and average iber diameter d av =8*1-6 m. The aousti attenuation o this porous absorptive material were obtained by the use o equations (9), (1) and (11). Figure 6. Inrease o transmission loss harateristis or dierent distanes rom m to 9m with the step o 1m between helioidal resonator at the inlet and expansion hamber. Figure 8. Initial transmission loss harateristi o a round silener. In ase that the main hanges o TL harateristis obtain near the resonane requeny r [Hz] o helioidal resonator the

5 range between 1,1kHz and 1,5kHz with the step o 1Hz is investigated here. The TL harateristis o aousti systems with distane hange between helioidal resonator and round silener are presented in Figure m 1m m 3m 4m 4. Seond helioidal resonator The aousti attenuation perormane o aousti systems onsisting two helioidal resonators due to distane hange between them is investigated. In Figure 1 is presented example view o two helioidal resonators plaed inside ylindrial dut o diameter d=1,5m m 6m 7m 8m 9m 1m 11m 1m 13m 14m m 16m 17m 18m 19m m 3m 4m Figure 9. Transmission loss harateristis o aousti systems with distane hange in [m] between helioidal resonator at the inlet and round silener. Figure 1. Example view o two helioidal resonators plaed inside ylindrial dut. In Figure 11 are presented TL harateristis o aousti systems with two helioidal resonators due to distane hange between them in [m] ,6m 1m m 3m m 5m Figure 11. Transmission loss harateristis o aousti systems with two helioidal resonators inside ylindrial dut due to distane hange between them in [m].

6 m 7m m 9m Figure 11. Continuation. It an be observed rom Figure 11 that the helioidal resonators an't be plaed lose to eah other beause they an lost their aousti attenuation properties. I the distane between them grows up the sound attenuation beomes higher. But also we an observe the speii inluene on sound attenuation or dierent distanes between helioidal resonators. Also there an be observed the inluene on low requeny sound attenuation. When the distane between helioidal resonators is small we observe small hanges, lower than 5dB, in low requenies. When the distane grows up we observe more hanges o attenuation at lower requenies. It an be ompared to results in irst ase, expansion hamber, where we observed similiar hanges in low requenies due to distane hange between ross-setional elements o ylindrial dut. 5. Conlusions The aousti attenuation perormane o helioidal resonator due to distane hange rom three dierent ross-setional elements o ylindrial dut was onsidered: irst ase - expansion hamber, seond ase - round silener, third ase - seond helioidal resonator. In irst ase, it an be observed in Figure 5 that the TL o investigated expansion hamber in requeny range rom 9Hz to 16Hz is small. The helioidal resonator plaed at the inlet o expansion hamber mainly inreases the sound attenuation. This inrease is high espeially in range o requeny rom about 1Hz to about 14Hz. Also the resonane requeny r [Hz] o helioidal resonator hanges due to distane hange rom expansion hamber. The highest r =1315Hz an be ahieved with the distane o m, due to hange o 1m it equals 175Hz. In distane m it is 17Hz and there is visible small hange till 9m (about 165Hz). It an be also observed, in Figure 6, that the distane hange o helioidal resonator rom expansion hamber inluenes on attenuation in lower requenies, but the attenuation hange is not higher than ±5dB, and the longer distane it is the more hanges in attenuation o lower requenies are obtained. At the seond ase, the aousti attenuation perormane o round silener with helioidal resonator at the inlet due to distane hange between them strongly diers at the distane rom m to 5m. There is visible in Figure 9 that the helioidal resonator an't strongly resonate i plaed diretly at the inlet o a round silener. But in this ase the range o attenuated sounds in the requeny domain is larger than or example m beore the silener. Also it depends on what is more needed, stronger resonane or wider range o attenuated sounds. For third ase, there is visible strong minimization o sound attenuation o two helioidal resonators when plaing them one by one. The range o attenuated sounds in requeny domain doesn't hange a lot in at that two helioidal resonators are plaed inside the dut. But it ould hange when the helioidal resonators would have dierent s/d ratios. Presented results shows that helioidal resonator an be an eetive additional sound attenuation element or duted systems. This researh work doesn't present all spetrum o possible use o this solution, also urther researh work should be undertaken. 6. Reerenes 1. Łapka W., Helioidal resonator, Proeedings o the INTER-NOISE 1, the 39 th International Congress and Exposition on Noise Control Engineering, June 1, Lisbon, Portugal, 9 pages in CD-ROM (1). Łapka W., Substitutional transmittane untion o helioidal resonator, Vibrations in Physial Systems, 4, 65-7 (1) 3. Łapka W. Inluene o hange o mandrel

7 diameter o helioidal resonator on its aousti attenuation perormane, Proeedings o the 57 th Open Seminar on Aoustis, -4 September 1 Gliwie, Poland, (1) 4. Łapka W. The Eet o Plaing a Helioidal Proile in the Round Silener with Varying Properties o an Absorptive Material, Proeedings o the 1 st European Congress on Sound and Vibration EAA Euroregio, September 1 Ljubljana, Slovenia, Abstrats p. 59, 4 pages in CD-ROM (1) 5. COMSOL Multiphysis version 3.4, Aousti Module, User s Guide and Model Library Doumentation, COMSOL AB, Stokholm, Sweden (7) 6. Givoli D., Neta B., High-oreder non-releting boundary sheme or time-dependant waves, J. Comp. Phys., 186, 4-46 (3) 7. Marburg S., Nolte B., Computational Aoustis o Noise Propagation in Fluids Finite and Boundary Element Methods, 578, Springer-Verlag, Berlin, Germany (8) 8. Delany M. A., Bazley E. N., Aousti properties o ibrous absorbent materials, Applied Aoustis, 3, (197) 9. Bies D. A., Hansen C. H., Aoustial properties o ibrous absorbent materials, Applied Aoustis, Vol.14, (198) 1. Munjal M.L., Aoustis o Duts and Mulers with Appliation to Exhaust and Ventilation System Design, 38, John Wiley & Sons In., Calgary, Canada (1987) 7. Aknowledgements Sientii work partially inaned by the Polish Ministry o Siene and Higher Eduation rom the budget or siene in the years 1-13 as a researh projet. Author grateully aknowledges the partial inanial support rom the researh projet 1-337/1 DS.