[EN] The influence of multimodal incident sound fields on the acoustic behaviour of large aftertreatment devices (ATD) is analysed in detail. The mode matching method is applied to the compatibility conditions of the three-dimensional (3D) acoustic fields at the device geometric discontinuities, leading to the computation of the complex wave amplitudes in all the subdomains involved and the corresponding transmission loss (TL). To have a realistic model, 3D propagation must be considered in the inlet/outlet ducts and chambers, while 1D wave propagation has to be assumed along the small capillaries of the catalytic converter/particulate filter monoliths of the ATD; therefore, these monoliths can be replaced by plane wave four pole transfer matrices from an acoustical point of view [1]. On the other hand, for large ATD inlet ducts such as those found in heavy-duty and off-road engines, the usual models with plane incident wave excitation are not accurate since the onset of higher order incident modes in the inlet duct is expected for the frequency range of interest. Therefore, a TL variation is likely to occur depending on these modes, similar to the results found in large dissipative silencers [2]. Results are presented for three different multimodal incident sound field hypotheses [3]: equal modal amplitude (EMA), equal modal power (EMP) and equal modal energy density (EMED). A relevant influence on the sound attenuation is found for the test problems considered in the current investigation.

References

[1]   Denia, F. D., Martínez-Casas, J., Carballeira, J., Nadal, E., Fuenmayor, F. J., Computational performance of analytical methods for the acoustic modelling of automotive exhaust devices incorporating monoliths. Journal of Computational and Applied Mathematics, 330: 995--1006, 2018.

[2]   Kirby, R., Lawrie, J. B., A point collocation approach to modelling large dissipative silencers. Journal of Sound and Vibration, 286: 313--339, 2005.

[3]   Mechel, F. P., Formulas of Acoustics. Berlin, Springer, 2008.The authors gratefully acknowledge Grants PID2020-112886RA-I00 and PID2020-118013RB-C21
funded by MCIN/AEI/10.13039/501100011033 and Project PROMETEO/2021/046 from Generalitat
Valenciana.Denia, FD.; Sánchez-Orgaz, EM.; Martínez Casas, J.; Carballeira, J.; Baeza González, LM. (2021). Acoustic modelling of large aftertreatment devices with multimodal incident sound fields. Universitat Politècnica de València. 208-215. http://hdl.handle.net/10251/19055620821

Baeza González, Luis Miguel

Carballeira, Javier

Denia, F. D.

Martínez Casas, José

Sánchez-Orgaz, Eva María

English

RiuNet

37Acoustic modelling of large aftertreatment deviceswith multimodal incident sound fieldsF. D. Denia[,1, E. M. Sánchez-Orgaz[, J. Mart́ınez-Casas[, J. Carballeira[ and L. Baeza[([) Instituto de Ingenieŕıa Mecánica y Biomecánica (I2MB),Universitat Politècnica de ValènciaCamino de Vera s/n, Valencia, Spain.1 IntroductionLarge silencers are commonly used to attenuate noise in many ventilation applications, gas turbinesystems, as well as large internal combustions engines (trucks, freight trains, etc.), among others.For their acoustic modelling and design, the well-established computational methods consideredfor the relatively small automotive exhaust silencers [1-8] can be extended. As these modelsfor small devices usually assume that plane waves are present in the inlet/oulet ducts (even ifhigher order acoustic modes are propagating inside the silencer), some problems may arise. Ifnumerical methods such as the finite element method (FEM) are applied [2, 3], a large increasein the number of degrees of freedom is required to achieve sufficiently accurate predictions athigher frequencies. When analytical approaches [4-8] are taken into account, the increase in thenumber of propagating higher order modes for large configurations significantly affects the silencerperformance [9]. This influence is associated not only with the modes inside the silencer, but alsowith the modes present in the incident sound field within the inlet port [10-13]. It is worth notingthat, for large inlet ducts, the incident plane wave excitation hypothesis no longer holds, andmultimodal excitation is necessary.The previous comments can be extended to a number of aftertreatment devices (ATD) suchcatalytic converters and particulate filters. While their acoustic attenuation performance can beassessed accurately for sizes typical of small automotive exhaust systems [14-19], further researchis required for large ATD since the ducts and chambers involved are big enough to allow higherorder modes to propagate inside the device as well as along the inlet/outlet ports. Accordingly,in this article the mode matching method is applied to the compatibility conditions of the three-dimensional (3D) acoustic fields at the ATD geometric discontinuities, leading to the computationof the complex wave amplitudes in all the subdomains involved and the corresponding transmissionloss (TL). To have a realistic model, 3D propagation must be considered in the inlet/outlet ductsand chambers, while 1D wave propagation has to be assumed along the small capillaries of thecatalytic converter/particulate filter monoliths of the ATD; therefore, these monoliths can bereplaced by plane wave four-pole transfer matrices from an acoustical point of view [17-19]. Onthe other hand, for large ATD inlet ducts such as those found in heavy-duty and off-road engines,the usual models with plane incident wave excitation are not accurate since the onset of higherorder incident modes in the inlet duct is expected for the frequency range of interest. Therefore,1fdenia@mcm.upv.esModelling for Engineering & Human Behaviour 2021a TL variation is likely to occur depending on these modes, similar to the results found in largedissipative silencers [9]. Results are presented for three different multimodal incident sound fieldhypotheses [20]: equal modal amplitude (EMA), equal modal power (EMP) and equal modalenergy density (EMED). A relevant influence on the sound attenuation is found for the testproblems considered in the current investigation.2 Overview of the acoustic problemFigure 1.1 shows a circular concentric ATD including a monolith. As shown in recent studies [19],the sound attenuation of an exhaust device incorporating a monolith can be properly predicted ifthe monolith is replaced by a plane wave four-pole matrix providing a relationship between theacoustic fields at both sides of the capillary region. Therefore, the presence of higher order modesin the cylindrical inlet/outlet regions is combined with one dimensional wave propagation withinthe capillary ducts of the central monolith.Figure 1: Scheme of a circular concentric ATD incorporating a monolith. The latter is replacedby a transfer matrix to model its acoustic behaviour.In all the rigid ducts involved (A, B, D and E), the acoustic pressure and velocity fields can bewritten in terms of a series expansion. For example, the solution of the wave equation in regionA is given by [4]PA(r, z) =∞∑n=0(A+n e−jkA,nz +A−n ejkA,nz)ΨA,n(r) (1)UA(r, z) =1ρ0ω∞∑n=0kA,n(A+n e−jkA,nz −A−n ejkA,nz)ΨA,n(r) (2)In the particular case of circular concentric ducts, the transversal pressure modes in Eqs. (1.1)and (1.2) are given by Bessel functions of the first kind and order zero, i.e. ΨA,n(r) = J0(αnr/R1),where αn are the roots satisfying the rigid wall boundary condition [4-7].The complete acoustic field of the system requires the computation of the wave amplitudes A±n ,B±n , D±n and E±n in all the ducts. The continuity conditions of the acoustic fields at the interfacesbetween ducts A and B (inlet expansion), as well as ducts D and E (outlet contraction) are209Modelling for Engineering & Human Behaviour 2021taken into account. The mode matching method [4-7] is here applied to the previous continuityconditions of the acoustic fields at all the interfaces to couple the solutions of the wave equation inthe corresponding ATD subcomponents [19]. Regarding the capillary ducts, the acoustic couplingbetween both sides of the monolith (at the interfaces SB ≡ SD, see Figure 1.1) can be expressedas [14-19]PB(r, z = LB) = Tm11PD(r, z′ = 0) + Tm12UD(r, z′ = 0) on SB ≡ SD (3)UB(r, z = LB) = Tm21PD(r, z′ = 0) + Tm22UD(r, z′ = 0) on SB ≡ SD (4)In matrix form yields{PB(r, z = LB)UB(r, z = LB)}=[Tm11 Tm12Tm21 Tm22]{PD(r, z′ = 0)UD(r, z′ = 0)}on SB ≡ SD (5)2.1 Mode matching method. Equations associated with the monolithThe equations and integrals associated with the inlet expansion and outlet contraction are omittedhere for the sake of brevity. It is worth noting that advantage can be taken from the orthogonalityproperties of the rigid duct transversal modes, thus reducing the computational effort of themode matching approach. Computational details of the corresponding integrals can be foundin references [1, 4-7, 19]. Here, the application of the mode matching method focuses on themonolith. Thus, Eqs. (1.3) and (1.4) are multiplied by the transversal mode ΨB,s(r) = ΨD,s(r) =J0(αnr/R2), with s = 0, 1, 2, . . . , Nm (a suitable series truncation is considered). Integrating overSB ≡ SD, yields the following equations for s = 0, 1, 2, . . . , NmB+s e−jkB,sLB +B−s ejkB,sLB = Tm11(D+s +D−s)+ Tm12kD,sρ0ω(D+s −D−s)(6)kB,sρ0ωB+s e−jkB,sLB −B−s ejkB,sLB = Tm21(D+s +D−s)+ Tm22kD,sρ0ω(D+s −D−s)(7)As indicated in [19], it is worth noting here that very simple algebraic expressions have beenobtained, where neither integrations nor modal summations appear, relating directly wave coeffi-cients with equal modal number. These equations do not depend on the transversal cross section,provided that its geometry is axially uniform.2.2 Multimodal excitationAs indicated previously, plane wave propagation in the inlet/outlet ducts of large ATD no longerholds. Thus, a procedure to account for multimodal excitation is described here. This can besummarized in five main steps. Step 1 is associated with the computation of the number ofpropagating modes in each duct for a given excitation frequency. For the inlet, this includesthe plane wave mode (always present) and the non-planar excitation modes up to N , obtainedthrough the conditionkA,n =√k20 − (αn/R1)2 ≥ 0 for n = 0, 1, ..., N (8)Note that N can vary during an ATD acoustic computation, increasing as the excitation frequencyexceeds the cut-on frequencies of the successive transverse higher order modes. Step 2 requires210Modelling for Engineering & Human Behaviour 2021the computation of the excitation wave amplitudes A+n of the inlet duct [9, 20]. If only a planewave (PW) is included in the analysis, this yieldsA+0 = 1;A+n = 0 for n > 0 (9)Alternative mode mixtures include equal modal amplitude (EMA), equal modal power (EMP) andequal modal energy density (EMED), the corresponding expression being as follows [20]:A+n = 1 for n ≤ N (EMA) (10)A+n =√(δ2n/N) (1/Re (kA,n/k0)) for n ≤ N (EMP) (11)A+n =√√√√δ2n/ N∑jRe (kA,j/k0) for n ≤ N (EMED) (12)where δn depends on the integration of the transversal modes over the cross section [20]. Oncethe excitation wave amplitudes A+n are known, the mode matching system of equations is solvedin step 3. While the treatment of the monolith has been outlined previously in Section 1.2.1, thereader is referred to reference [19] for details of the complete mode matching equations. Then,step 4 implies the computation of the acoustic incident/transmitted power at the end sections ofthe inlet/outlet ducts:Potinc =∫S12Re (PincU∗inc)dS =πρ0ωN∑n=0kA,n|A+n |2∫ R10J0 (αnr/R1)2 rdr (13)Pottrans =∫S12Re (PtransU∗trans)dS =πρ0ωN∑n=0kE,n|E+n |2∫ R30J0 (αnr/R3)2 rdr (14)And finally, in step 5 the transmission loss is evaluated asTL(dB) = 10 log(PotincPottrans)(15)2.3 Acoustic modelling of the ATD monolithFigure 1.2 shows examples of monoliths and schemes of the associated flow pattern. Catalyticconverter capillaries have upstream and downstream ends open, while particulate filters capillariesare characterized by channel ends alternatively plugged to force the flow through the porous walls[15].Figure 2: Monolith and flow pattern: (a) Catalytic converter (CC); (b) Particulate filter (PF).211Modelling for Engineering & Human Behaviour 2021As indicated previously, the acoustic modelling assumes plane wave propagation in the capillaries,that can be computed through transfer matrices involved in Eqs. (1.3)-(1.7), relating the acousticfields at both sides of the monolithic region. For further details of the models implemented in thecurrent investigation to obtain these matrices, the reader is referred to works [14, 15, 20].3 Results and discussionTwo axisymmetric configurations with a catalytic converter monolith are considered for validationand computation purposes. In the first case, geometry 1 has inlet and outlet ducts defined bythe radii R1 = R3 = 0.125 m, while the chambers and monolith are characterized by the radiusR2 = 0.25 m and the lengths LB = LD = 0.1475 m and LC = 0.175 m (see Figure 1.1 fordetails). In the second case, geometry 2 is defined by the dimensions R1 = R3 = 0.25 m, R2 = 0.5m, LB = LD = 0.475 m and LC = 0.75 m. It is worth noting that in this latter case radiihave been doubled and thus more higher order modes will propagate. Cold flow hypotheses withT = 25◦C are retained through the computations, the properties for the air being c0 = 346.1m/s and ρ0 = 1.184 kg/m3. Regarding the monolith acoustic model, the following values areassumed: resistivity R = 1500 Pa·s/m2, porosity φ = 0.88, geometrical factor αg = 1.14, dynamicviscosity µ = 1.837 · 10–5 Pa·s, thermal conductivity κ = 0.0255 W/(m·K) and specific heatCp = 1006.4 J/(kg·K). The detailed model for the four-pole matrix computation can be foundin the bibliography [14, 20].3.1 Geometry 1: N = 1For the conditions associated with this case, the incident modal field has a maximum of onehigher order mode (N = 1), the cut-on frequency being 1688.6 Hz [1]. As it can be seen in Figure1.3, the mode mixture rules modify the attenuation above this frequency value; in general, whenincident energy propagates in higher order modes this seems to have a positive impact on theperformance of the aftertreatment device. As observed by Kirby and Lawrie for large dissipativesilencers [9], the three different multimode excitations considered for the incident acoustic fieldyield comparable results in general, although EMA predictions are lower than both EMED andEMP predictions.500 1000 1500 2000 2500 3000Frequency (Hz)051015TL (dB)PWEMAEMPEMEDFigure 3: TL of an ATD (catalytic converter) for different modal excitations.212Modelling for Engineering & Human Behaviour 20213.2 Geometry 2 (double radii): N = 4In this case, the incident acoustic field has a maximum of four higher order modes (N = 4),the cut-on frequencies being 844.3 Hz, 1545.8 Hz, 2241.6 Hz and 2935.7 Hz respectively [1]. Asshown in Figure 1.4, the main trends observed in the previous figure are also found here, the samecomments being applicable in general. Anyway, the TL discrepancies among the different modelsseem to have decreased slightly in the high frequency range.500 1000 1500 2000 2500 3000Frequency (Hz)051015202530354045TL (dB)PWEMAEMPEMEDFigure 4: TL of an ATD (catalytic converter) for different modal excitations and double radii.4 ConclusionsAn accurate and efficient analytical method has been presented for the acoustic modelling of largeATD with a monolith. The model is based on the mode matching method and assumes 3D wavepropagation in the inlet/outlet regions and 1D wave propagation in the monolith (these hypotheseshave been validated experimentally in previous works). In large aftertreatment devices, plane wave(PW) propagation in the inlet/outlet ducts no longer holds for the frequency range of interest,and energy incident on the device propagating in higher order modes may have a strong impacton the acoustic attenuation performance. The specification of the incident modal field requires aknowledge of the source characteristics, but this is currently an open field of research and thereis a lack of accurate representations as well as experimental validation. In addition to the PWmodel, several mode mixtures have been implemented: equal modal amplitude (EMA), equalmodal power (EMP) and equal modal energy density (EMED). As expected, a strong influence ofthe mode mixture technique on the attenuation has been found. The three different multimodeconditions adopted for the incident sound field can yield predictions that are consistent with theliterature for some particular configurations, in the sense that at higher frequencies the acousticperformance of the exhaust device generally improves when energy from the source is carried byhigher order modes (especially if dissipative effects are present in the analysis). However, theopposite trend has been also found for other geometries analysed in the current work (not shownfor the sake of brevity). Thus, a plausible conclusion is that an accurate TL estimation in largeaftertreatment devices requires the consideration of multimodal excitation, but the results arestrongly dependent on the particular configuration under analysis and the mode mixture rulechosen. Literature seems to suggest that EMED is probably the most suitable model, althoughthis depends on the noise source characteristics and further work is required.213Modelling for Engineering & Human Behaviour 20215 AcknowledgementsThe authors gratefully acknowledge Grants PID2020-112886RA-I00 and PID2020-118013RB-C21funded by MCIN/AEI/10.13039/501100011033 and Project PROMETEO/2021/046 from Gener-alitat Valenciana.References[1] Munjal, M. L., Acoustics of Ducts and Mufflers. New York, Wiley, 2014.[2] Antebas, A. G., Denia, F. D., Pedrosa, A. M., Fuenmayor, F. J., A finite element approach for theacoustic modeling of perforated dissipative mufflers with non-homogeneous properties. Mathematicaland Computer Modelling, Volume(57):1970–1978, 2013.[3] Denia, F. D., Sánchez-Orgaz, E. M., Mart́ınez-Casas, J., Kirby, R., Finite element based acoustic anal-ysis of dissipative silencers with high temperature and thermal-induced heterogeneity. Finite Elementsin Analysis and Design, Volume(101):46–57, 2015.[4] Selamet, A., Ji, J. L., Acoustic attenuation performance of circular expansion chambers with offsetinlet/outlet: I. Analytical approach. Journal of Sound and Vibration, Volume(213):601–617, 1998.[5] Kirby, R., Denia, F. D., Analytic mode matching for a circular dissipative silencer containing meanflow and a perforated pipe. Journal of the Acoustical Society of America, Volume(122):3471–3482,2007.[6] Denia, F. D., Selamet, A., Mart́ınez, M. J., Fuenmayor, F. J., Sound attenuation of a circular multi-chamber hybrid muffler. Noise Control Engineering Journal, Volume(56):356–364, 2008.[7] Denia, F. D., Antebas, A. G., Selamet, A., Pedrosa, A. M., Acoustic characteristics of circular dissi-pative reversing chamber mufflers. Noise Control Engineering Journal, Volume(59):234–246, 2011.[8] Denia, F. D., Sánchez-Orgaz, E. M., Baeza, L., Kirby, R., Point collocation scheme in silencerswith temperature gradient and mean flow. Journal of Computational and Applied Mathematics,Volume(291):127–141, 2016.[9] Kirby, R., Lawrie, J. B., A point collocation approach to modelling large dissipative silencers. Journalof Sound and Vibration, Volume(286):313–339, 2005.[10] Willimas, P. T., Kirby, R., The effect of higher order modes on the performance of large diameterdissipative silencers. Kraków, Forum Acusticum, 2014.[11] Willimas, P. T., Åbom, M., Kirby, R., Hill, J., The influence of higher order incident modes on theperformance of a hybrid reactive-dissipative splitter silencer. Boston, Acoustics’17, 2017.[12] Kirby, R., Mimani, A., Attenuating sound in large ductwork using reactive and dissipative silencers.Ibiza, The 6th Conference on Noise and Vibration Emerging Methods, 2018.[13] Willimas, P. T., Kirby, R., Hill, J., Åbom, M., Malecki, C., Reducing low frequency tonal noise inlarge ducts using a hybrid reactive-dissipative silencer. Applied Acoustics, Volume(13):61–69, 2018.[14] Selamet, A., Easwaran, J. M., Novak, J. M., Kach, R. A., Wave attenuation in catalytic converters:reactive versus dissipative effects. Journal of the Acoustical Society of America, Volume(13):935–943,1998.[15] Allam, S., Åbom, M., Sound propagation in an array of narrow porous channels with application todiesel particulate filters. Journal of Sound and Vibration, Volume(291):882–901, 2006.[16] Denia, F. D., Antebas, A. G., Kirby, R., Fuenmayor, F. J., Multidimensional acoustic modelling ofcatalytic converters. Krákow, The Sixteenth International Congress on Sound and Vibration, 2009.[17] Jiang, C., Wu, T. W., Xu, M. B., Cheng, C. Y. R., BEM modeling of mufflers with diesel particulatefilters and catalytic converters. Noise Control Engineering Journal, Volume(58):243–250, 2010.214Modelling for Engineering & Human Behaviour 2021[18] Denia, F. D., Mart́ınez-Casas, J., Baeza, L., Fuenmayor, F. J., Acoustic modelling of exhaust deviceswith nonconforming finite element meshes and transfer matrices. Applied Acoustics, Volume(73):713–722, 2012.[19] Denia, F. D., Mart́ınez-Casas, J., Carballeira, J., Nadal, E., Fuenmayor, F. J., Computational perfor-mance of analytical methods for the acoustic modelling of automotive exhaust devices incorporatingmonoliths. Journal of Computational and Applied Mathematics, Volume(330):995–1006, 2018.[20] Mechel, F. P., Formulas of Acoustics. Berlin, Springer, 2008.215

Acoustic modelling of large aftertreatment devices with multimodal incident sound fields

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Acoustic modelling of large aftertreatment devices with multimodal incident sound fields

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