Review of splitter silencer modeling techniques Mina Wagih Nashed Center for Sound, Vibration & Smart Structures (CVS3), Ain Shams University, 1 Elsarayat St., Abbaseya 11517, Cairo, Egypt. mina.wagih@eng.asu.edu.eg Tamer Elnady Center for Sound, Vibration & Smart Structures (CVS3), Ain Shams University, 1 Elsarayat St., Abbaseya 11517, Cairo, Egypt. tamer.elnady@eng.asu.edu.eg Mats Åbom Marcus Wallenberg Laboratory for Sound and Vibration Research, The Royal Institute of Technology (KTH), Teknikringen 8, 10044 Stockholm, Sweden. matsabom@kth.se Summary Dissipative silencers are commonly used in different applications. There are several configurations for dissipative silencers, but one of the most effective configurations is the Splitter (Baffle) Silencer. It has been used for many years for different applications such as HVAC, gas turbines, large diesel engines etc. However, these silencers need to be carefully designed to tune its effective frequency band of operation to the source excitation frequency. It must be also optimized with the amount of pressure drop they introduce to the system, and the amount of flow generated noise resulting from splitting the flow. Splitter Silencers are effective for high frequency sound attenuation and their modeling involves handling a lot of traveling modes. For years, there has been a lot of efforts made by several researches to develop models for splitter silencers, but usually these models can predict the acoustic behavior in the low and mid frequency ranges only. The characterization of the acoustic performance of these silencers is a challenge on its own, to measure the Insertion Loss for all frequencies. In this paper, reviews of models available in the literature are discussed, and a model recently developed for bar silencers is presented. This model is adapted to the splitter silencer and can use three different methods to calculate the acoustic performance; the least attenuated mode, global attenuation estimation, and mode matching. Also, Beranek model is implemented and used in this study. The results are compared to the literature, FEM simulation using COMSOL, and measurements listed in baffle silencer datasheet. 1. Introduction 1 The design of the splitter silencer has been a considerable issue in research field. The inaccurate design of the splitter silencer leads to less attenuation and/or undesirable high pressure drop. For many years, different algorithms were developed to predict the splitter silencer attenuation with acceptable accuracy. On the other hand, a requirement of relatively low computational cost was mandatory for industry. In 1992, Ramakrishnan developed design curves [1], that are generated from finite element scheme simulation of rectangular silencer. This model assumes equal power for all modes, isotropic absorbing material, and neglects the perforated effect. Basically, Ramakrishnan model is based on the same iterative algorithm presented by cummings [2]. By comparing his scheme with least attenuated mode results and measurements, one found that this model is suitable for mid-size silencer in low and mid frequency ranges, and low velocity applications such as HVAC systems. The model is generally underestimates the insertion loss due to perforate effect neglection, and ignorance of entrance and exit losses. Moreover, the model of lined duct model used in his algorithm, makes the algorithm valid when assuming incident plane Copyright 2018 EAA HELINA ISSN: 2226-5147 All rights reserved - 2285 -
waves only. In the same year, Beranek [3] presented another model that predicts the insertion loss of dissipative silencers. One of the discussed silencers was the splitter silencer type. Beranek model depends on assuming low velocity flow to neglect the flow generated noise. Beranek model calculates the insertion loss as sum of three terms: entrance loss, exit loss, and silencer attenuation. The entrance loss is estimated from simple curves, the exit loss is small and usually is neglected, and the silencer attenuation is estimated from curves based on flow resistivity and baffle width to airgap width ratio. This model is limited to baffle width to airgap width ratio from 0.5 to 2 only, and if one tried to extrapolate the data to narrower airgap the model will give unrealistic results. However, Beranek model is suitable more for simulation of low velocity duct silencer with one baffle section in the centre, and maximum baffle width to airgap width ratio equal to two. In 1993 Cummings [4] developed a model assumes rectangular splitter silencer with infinite length, arbitrary number of baffles and anisotropic absorbing material properties. Cummings assumed the sound field varies in x-y plane only neglecting variation in z direction. In the model, cummings neglected the perforated sheets effect which is acceptable in case of large open area. The model is based on least attenuated mode technique, and the algorithm starts from empty hard wall duct with no flow, then increase the absorbing material resistivity step by step till reaches the aimed flow resistivity; then starts to add the flow effect in the same way. This technique gives the model the ability to trace the wave number in the silencer from empty hard wall duct to the required silencer. Although the model results showed good agreement with measurements in frequency range from 100 Hz to 2 khz, but the algorithm suffers from jumping modes that leads to wrong prediction in some cases and there is no evidence that the algorithm can be used above 2 khz. In the following years, many articles has been published [5 8] that address algorithms to predict the acoustics behaviour in ducts with absorbing material that some of them can be applied to splitter type. Another interesting work is Benoit Master thesis [9]. Benoit thesis work intended to predict the attenuation of large dissipative silencer (splitter and bar types) in a frequency range up to 10 khz. His work was based on Rayleigh-Ritz method that computes the eigenmodes and their respective axial wave numbers in a duct with absorbing material, considering the mean flow effect. Benoit introduced three ways to calculate the attenuation: least attenuation mode, global estimation, and mode matching technique. In this paper, Beranek model and Benoit model are discussed in brief, then both models are compared with Finite Element Simulation (FEM) using COMSOL Multiphysics [10], and with splitter silencer datasheets. 1. Benoit model Benoit model is based on Rayleigh-Ritz method to predict the eigenmodes of the duct. His work is inspired mainly by Cummings work [11]. Starting from defining the pressure field as a series of axial modes. Then splitting the pressure field equation to follow different regions inside the silencer. Finally, using Rayleigh-Ritz method for projecting the eigenmodes of the duct based on interpolation polynomial of degree three on a finite element mesh. After calculating the eigenmodes, the algorithm should be smart enough to remove hydrodynamic modes and the left eigenmodes will be used to calculate the transmission loss (TL). Benoit present three different ways to calculate TL: First, using the least attenuated mode. This method is used frequently in calculating the performance of silencers. it provides the worst-case prediction which is the minimum value of the actual TL. Equation (1) shows the TL calculation based on least attenuated mode. Im( ) TL 20. k n (1) ln(10) where n is the eigenmode and k is the wave number. The least attenuated mode gives relatively accurate results in plane wave mode, but as the frequency increases, this method failed to predict the TL. The second method is the global attenuation estimation. This method assumes equal acoustical energy over different modes. By choosing the number of modes involved in the calculation N, the TL can be estimated using the following equation. 1 2Im( i ) kl 0 TL 10log10 e (2) N i 1, N Global estimation method is more accurate than least attenuated mode method, because it is considering more than one mode. The third method is the mode matching technique. In this method Benoit assume equal sound power over all propagating modes in the incoming field, - 2286 -
and by combing these modes with random phases, the pressure distribution pattern is created in the inlet side. Then a mode-matching is carried out and the TL is calculated using equation (3). 2Im( i ) kl 0 BW i ie i 1, M' TL 10log BW (3) i i i 1, M' More details and equations proofs are found in reference [9]. 2. Beranek model Beranek assumes that the insertion loss of the silencer is divided to total silencer attenuation and flow generated noise. in case of very high attenuation silencers the flow generated noise is comparable with attenuation, and cannot be neglected. However, in case that the flow velocity in the silencer passages is sufficiently low, the flow generated noise can be neglected and the silencer insertion loss can be described by the following equation. IL Ls LENT LEX (4) Where Ls is the silencer attenuation, LENT is the entrance loss, and LEX is the exit loss. The entrance loss LENT can be neglected if the incident acoustics energy is in form of plan waves which it is the case in low frequency range in straight ducts. However, in high frequency range where the cross dimensions of the duct are much larger than the wave length a large number of higher order modes exist and the sound field become a semi diffuse field. In this range the entrance loss cannot be neglected, hence it may be assumed to have values from 3-6 db. It is important that Beranek leave the estimation of the entrance loss to the designer estimation based on prior experience. The exit loss is usually present when the silencer is located at the opening end of a duct and the opening cross dimension is small compared to wave length. In this case the exit loss is calculated as end reflection. In case that the silencer is located away from opening end; the exit loss can be either considered as safe margin or neglected. In the case studies considered here, the exit loss is neglected. The silencer attenuation of the splitter type is proportional to the perimeter-area ratio, the length of the silencer, and normalized attenuation L h. This makes Equation (4) be as follows l IL Lh LENT (5) h Where l is the silencer length, and h is the half width of the airgap. Beranek computed L h for various baffle width to airgap ratios d/ h, and for various normalized flow resistance R Rf d / c, where R f is the flow resistivity of the porous material. The results were presented with three graphs. Each graph contains curves for the same baffle width to airgap ratio and draw relation between normalized frequency and L h for various R. Figure 1 shows one of normalized attenuation curves computed by Beranek. The limitation of Beranek model appears when one tries to calculate either different d/ h ratio or different R than included in the calculated curves. This limitation raises strongly when trying to interpolate curves data as shown in the results section. Figure 1. Normalized attenuation-versus-frequency curves for d/ h 1 [3] 3. Finite Element model COMSOL Multiphysics software [10] is used with its Acoustics module to simulate the transmission loss of a splitter silencer using FEM. Figure 2 shows the geometry setup of COMSOL simulation. The model consists of the silencer under investigation, outlets empty ducts for smooth wave propagations, Source sector which simulates acoustic noise source, and PML domains (Perfectly Matched Layer). The PML are used to eliminate end reflections and damping from both sides to simulate the case for long empty ducts connected to the silencer from both sides. By default, the boundary between the absorbing material and the air passage - 2287 -
is considered as internal boundary between two different material. Outlets empty ducts The boundaries between the absorbing material and the airways are considered as perforated sheets with different porosity percentages. 1.1. Figures PML Source Splitter silencer PML combinations between the three parameters, the total number of cases will equal to 18 cases. The next section shows the results of all cases. 5. Simulation results Comparisons made first between Benoit and Beranek models to show the differences between their models. Figure 2. FEM simulation geometry in order to cover the frequency range of interest, the simulation is run over a frequency range from 50 Hz up to 10 khz. Then the sound power is calculated at the inlet and the outlet of the silencer. Then it is used to calculate the transmission loss of the investigated silencer at each frequency. Figure 3 shows a sample of sound pressure level distribution along the silencer of at frequency equal to 1 khz. Figure 3. SPL in db along the silencer at 1kHz Figure 4. Benoit vs Beranek for d/h=0.66, l=0.91m, and R is 1000, 2000, and 5000 rayl/m respectively from top. 4. Case studies The case studies considered in this paper investigate the effect of different parameters on Benoit and Beranek models. the main investigated parameters are: baffle width to airgap ratio, flow resistivity of the absorbing material, and the length of the silencer. The following table shows the values of the investigated parameters. Table I. Values of investigated parameters. Baffle width (m) / airgap (m) Flow resistivity (rayl/m) Silencer length (m) 0.2/0.3 1000 0.91 0.2/0.2 2000 2.24 0.22/0.08 5000 The baffle width to airgap ratio are selected to cover the three possibilities. However, the last ratio is equal to silencer dimensions in the datasheet which will be used in the comparison. Moreover, the selected silencer lengths match the silencer lengths in the datasheet used. By making all the possible Figure 5. Benoit vs Beranek for d/h=1, l=0.91m, and R is 1000, 2000, and 5000 rayl/m respectively from top. Figures 4-6 show the silencer transmission loss of relatively short silencer, for three different baffle width to airgap ratios, and for different flow resistivity. In these three figures one can observe that in case of low R, the Beranek model estimation is much higher in the mid frequency range than three methods represented by Benoit. This difference in estimation decreases as R - 2288 -
increases. On the other hand, both modeling techniques have very good agreement in low and high frequency ranges. Figure 8. Benoit vs Beranek for d/h=1, l=2.13m, and R is 1000, 2000, and 5000 rayl/m respectively from top. Figure 6. Benoit vs Beranek for d/h=2.75, l=0.91m, and R is 1000, 2000, and 5000 rayl/m respectively from top. When trying to simulate longer silencer, one will get almost the same trend of the results. Figures 7-9 show results of silencers with length equal to 2.13m. it is important to point that mode matching technique and global estimation technique models have a very good agreement, while the least attenuated mode failed to agree with them as the frequency increases. This imply that the least attenuated mode is not always suitable for attenuation prediction in the high frequency range. Figure 9. Benoit vs Beranek for d/h=2.75, l=2.13m, and R is 1000, 2000, and 5000 rayl/m respectively from top. Figure 7. Benoit vs Beranek for d/h=0.66, l=2.13m, and R is 1000, 2000, and 5000 rayl/m respectively from top. The silencer used in cases showed in Figure 6 is simulated using FEM. The three cases are simulated with the same flow resistivity showed in Figure 6. The simulation results are compared with Beranek and Benoit models as shown in Figure 10. Although FEM results showed good agreement with other models in the low frequency range and part of the mid frequency range, but it shows deviation after that. Also, it is showed that as the flow resistivity increases, the attenuation peak of FEM simulation will have better agreement with other models. Moreover, FEM simulation suffer from noise and leak of accuracy in the higher frequencies. This is because that when frequency increases, the wave length decreases, and more elements are needed in order to have acceptable - 2289 -
accuracy. Then is one of limitations of using FEM for this type of simulation. Figure 10. Comparion between COMSOL, Benoit, and Beranek for d/h=2.75, l=0.91m, and R is 1000, 2000, and 5000 rayl/m respectively from top. The last comparison is made between both models, and datasheet of splitter silencer. The silencer dimensions are the same in Figure 10, and the flow resistivity is equal to 1500 Rayl/m. Figure 11 shows the comparison with three silencer types from the datasheet. In general, the Benoit model prediction shows good agreement in the mid and high frequency range, and under estimation of the attenuation in low frequency range. However, the deviation in the low frequency is may be due to especial treatment of the compared silencer to enhance the attenuation in the low frequency range. Figure 11. Benoit and Beranek model vs. silencer datasheet On the hand, Beranek model shows over estimated attenuation which is predictable as it fails to calculate the attenuation when d/ h 2. 6. Conclusions Splitter silencer one of the most common types of silencer that used in HVAC systems and large machines silencers, for its cheap price and high performance. However, it is critical to have good design for the silencer to achieve desired attenuation with minimum pressure drop. This article reviews modeling techniques of baffle silencer, and focus on Beranek and Benoit models. the two concerned models are implemented and their simulation results are compared. The comparison showed that the Beranek model usually gives higher attenuation than Benoit, and it is limited to a specified baffle width to airgap ratio. Then, when the two models are compared with FEM using COMSOL Multiphysics, the results showed good agreement with both models in the low frequency range, but this agreement becomes better in higher frequencies as the flow resistivity increases. Lastly, the two models compared with splitter silencer datasheet. The results showed a good agreement with Benoit models. This concludes that Benoit model technique - especially when using global estimation or mode matching is more accurate, and can be used for preliminary design of this type of splitter silencers. References [1] R. Ramakrishnan, W.R. Watson, Design curves for rectangular splitter silencers, Appl. Acoust. 35 (1992) 1 24. doi:10.1016/0003-682x(92)90033-o. [2] A. Cummings, Sound attenution in ducts lined in two opposite walls with porous material, with some applications to splitters, J. Sound Vib. 49 (1976) 9 35. [3] I. Vér, L. Beranek, Noise and Vibration Control Engineering, (2005). doi:10.1002/9780470172568. [4] A. Cummings, N. Sormaz, Acoustic Atenuation in Dissipative splitter Silencers Containing Mean Fluid Flow, J. Sound Vib. 168 (1993) 209 227. [5] A. Cummings, HIGH FREQUENCY RAY ACOUSTICS MODELS FOR DUCT SILENCERS, J. Sound Vib. 221 (1999) 681 708. [6] R. Kirby, Transmission loss predictions for dissipative silencers of arbitrary cross section in the presence of mean flow, J. Acoust. Soc. Am. 114 (2003). doi:10.1121/1.1582448. [7] R. Kirby, P.T. Williams, J. Hill, R. Kirby, P.T. - 2290 -
Williams, A three dimensional investigation into the acoustic performance of dissipative splitter silencers, J. Acoust. Soc. Am. 135 (2014). [8] R. Kirby, K. Amott, P.T. Williams, W. Duan, On the acoustic performance of rectangular splitter silencers in the presence of mean flow, J. Sound Vib. 333 (2014) 6295 6311. doi:10.1016/j.jsv.2014.07.001. [9] B. FARVACQUE, Modeling of of Large Dissipative Silencers, 2003. [10] C. Inc., Acoustics Module User s Guide, version 5.3, (n.d.). [11] A. Cummings, Finite element computation of attenuation in bar silencers and comparison with measured data, J. Sound Vib. 196 (1996) 351 369. - 2291 -
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