# Recent Developments in Using a Modified Transfer Matrix Method for an Automotive Exhaust Muffler Design Based on Computation Fluid Dynamics in 3D

^{*}

## Abstract

**:**

**AVL FIRE**(process-safe 3D-CFD Simulations of Internal Combustions Engines), and the classic TMM for the exhaust muffler. For all the continuous and discontinuous sections of the exhaust muffler, the Mach number of the cross-section, the temperature, and the type of discontinuity of the exhaust gas flow were taken into consideration to evaluate the specific elements of the acoustic quadrupole that define the MTMM coupled with

^{TM}M Engine**AVL FIRE**for one given muffler exhaust. Also, the perforations of intermediary ducts were considered in the new MTMM (

^{TM}M Engine**AVL FIRE**linked with TMM) to predict the TL (transmission loss) of an automotive exhaust muffler with three expansion chambers. The results obtained for the TL in the frequency range 0.1-4 kHz agree with the experimental results published in the literature. The TMM was improved by adding the

^{TM}M Engine**AVL FIRE**as a valuable tool in designing the automotive exhaust muffler (AEM).

^{TM}M Engine## 1. Introduction

**AVL FIRE**[29] (that predicts the gas flow speeds and the temperatures in all elements of the internal ducts from the intake to the exhaust) with the classic TMM for the muffler exhaust gas flow, which is applied to a specific muffler configuration to predict transmission loss. This is the main purpose of the paper: the design of a more precise method for the prediction of TL based on the 3D-CFD

^{TM}M Engine**AVL FIRE**and the classic TMM. The method considers all the following aspects: the Mach number of the cross-section, the instant speed flow, the temperature, and the type of discontinuity of the muffler exhaust gas flow. The 3D-CFD

^{TM}M Engine**AVL FIRE**provides accurate physical–chemical models based on the internal combustion engine 3D-CFD, which, “backed up by AVL’s methodological knowledge, enable a broad user group to carry out simulations of the combustion process at the highest level” [29]. The 3D-CFD simulations of internal combustion engines generate meaningful results for each process element, such as velocity, temperature, and pressure gas flow.

^{TM}M Engine## 2. Modified Transfer Matrix Method

- -
- The 3D-CFD of AVL gives the first element [29] dedicated to the modeling, simulations, and calculus of internal combustion engine phenomena, starting from the initial air intake and finishing at the burn gas flow exhaust, computing all the velocities and temperatures of the gas flow through all the internal duct elements.
- -
- The second element is given by using all the values of the velocities and temperatures of the gas flow through the final muffler gas exhaust in the classic TMM to compute the TL of the AEM, which has a specific geometry and structure.

**AVL FIRE**[29] based on the following steps: good reproduction of realistic geometry by discretization, physically feasible boundary conditions, operating conditions applied to the model given by boundary conditions, and the best-suited physical models.

^{TM}M Engine**AVL FIRE**, respectively, the velocities and temperatures of the gas flow through the muffler.

^{TM}M Engine**AVL FIRE**to calculate the Mach numbers, the mean gas flow velocities, and the temperatures for each element of the muffler exhaust. After this stage, these values were inserted into relations of the classic TMM expressed by the abovementioned Equations (13)–(20) to determine the TL for the AME.

^{TM}M Engine## 3. Results and Discussions

**1D-CFD**(computation fluid dynamics in 1D), presented and developed in [31] (pp. 44–184, 240–290), the speed flow of the exhaust gases entering the muffler was calculated for an internal combustion engine of an Audi 1.4 TSI having the power 90 kW, with values in the range ${V}_{1}\in \left[60.0\u2013102.0\right]$ m/s and the temperature of the exhaust gases at the muffler in the range ${t}_{1}\in \left[60.0,135.0\right]$ °C. The geometric characteristics of the muffler are presented in Table 2, knowing that ${S}_{1}={S}_{5}={S}_{6i}={S}_{7}={S}_{11},$ ${S}_{3}={S}_{6e}={S}_{9},$ ${l}_{1}={l}_{11},$ ${l}_{2}={l}_{10},$ ${l}_{3}={l}_{9},$ ${l}_{4}={l}_{5},$ ${l}_{6in}={l}_{6out},$ ${l}_{7}={l}_{8}$ from geometric construction. Based on the dimensions in Table 2, Figure 7 illustrates the technical drawing of the muffler exhaust.

**AVL FIRE**, it was computed, for the same internal combustion engine of Audi 1.4 TSI with a power of 90 kW, the velocity flow of the exhaust gases entering the AEM (see Figure 7) and the velocities in all the continuity and discontinuity sections of the AEM, as well as the temperature at the AEM’s intake and the temperatures in all the continuity and discontinuity sections of the AEM. Figure 12 illustrates the 3D discretization of the exhaust tailpipe and the cylinder head of the internal combustion engine used by the software

^{TM}M Engine**AVL FIRE**to compute the exhaust velocities and the exhaust temperatures of the flow gas.

^{TM}M Engine**AVL FIRE**before using the classic TMM to compute the $T{L}_{AME}$ instead of 1D-CFD, presented and developed in [31] (pp. 44–184, 240–290), enlarges the frequency range from 1800–2500 Hz to 100–4000 Hz for the MTMM. A limitation of this study, however, which uses this final MTMM (3D-CFD

^{TM}M Engine**AVL FIRE**linked with the classic TMM) to compute the $T{L}_{AME}$, is the frequency range 100-4000 Hz, as this is a disadvantage. It is particularly needed to accurately predict the $T{L}_{AME}$ in the frequency range 100-8000 Hz. Concerning the accuracy of $T{L}_{AME}$ predicted with the MTMM, compared with the accuracy of the data published in the literature [33], several aspects of the data predicted and illustrated in Figure 13, Figure 14, Figure 15 and Figure 16 are noted:

^{TM}M Engine- -
- In the frequency range 0.1–4.0 kHz and for engine rotation speeds in the range 1500–2500 rpm, the predicted values of $T{L}_{AME}$ are 1–4% higher than the experimental ones mentioned in [33], while for engine rotation speeds in the range, 3500–4500 rpm, the predicted values are 4.5–8% higher than the experimental ones mentioned in [33].
- -
- For the frequency range over 4.0 kHz for engine rotation speeds in the range 1500-4500 rpm, the predicted values are at least 11% larger or even smaller by 14% than the experimental ones mentioned in [33].

## 4. Conclusions

**AVL FIRE**, to calculate in detail the velocities of gas flow and the temperatures of the gas flow in all the process phenomena of the internal combustion engine from the air intake to the gas exhaust through the AEM, coupled with the classic TMM for the AEM, to predict the $T{L}_{AME}$. For all the continuous and discontinuous sections of the AEM, the Mach number of the cross-section, the temperature, and the type of discontinuity of the exhaust gas flow were considered to evaluate the specific elements of the acoustic quadrupole that define the MTMM. The perforations of intermediary ducts were considered in the new MTMM to predict the TL of an automotive exhaust muffler with three expansion chambers. The flow chart of the MTMM consists of the following:

^{TM}M Engine- The use of 3D-CFD
**AVL FIRE**(based on FEM in 3D) to calculate the gas flow velocities and the gas flow temperatures of all the internal ducts of the internal combustion engine (taken into consideration, respectively, the internal combustion engine of Audi 1.4 TSI with a power of 90 kW) in all the process phenomena, from the initial air intake through the air filter to the cylinders, compression, ignition, detention, and burnt gas exhaust, from the cylinders to the exhaust manifold, gas flow through the catalytic muffler, and the exhaust through the AME.^{TM}M Engine

**AVL FIRE**before using the classic TMM to compute the $T{L}_{AME}$ instead of 1D-CFD, presented and developed in [31] (pp. 44–184, 240–290), enlarges the frequency range from 1800–2500 Hz to 100–4000 Hz for the MTMM.

^{TM}M Engine**AVL FIRE**.

^{TM}M Engine## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bugaru, M.; Enescu, N. An overview of muffler modeling by transfer matrix method. Sci. Bull. Politeh. Univ. Timis. Trans. Mech.
**2005**, 1, 51–54. [Google Scholar] - Chung, J.Y.; Blaser, D.A. Transfer function method of measuring in-duct acoustic properties: I. Theory and II. Experiment. J. Acoust. Soc. Am.
**1980**, 68, 907–921. [Google Scholar] [CrossRef] - Davies, P.O.A.L. Realistic models for predicting sound propagation in flow duct systems. Noise Control Eng. J.
**1993**, 40, 135–141. [Google Scholar] [CrossRef] - Mechel, F.P. Formulas of Acoustics; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Munjal, M.L. Acoustics of Ducts and Mufflers, 1st ed.; John Wiley and Sons: New York, NY, USA, 1987. [Google Scholar]
- Sridhara, B.S.; Crocker, M.J. Review of theoretical and experimental aspects of acoustical modeling of engine exhaust systems. J. Acoust. Soc. Am.
**1994**, 95, 2363–2370. [Google Scholar] [CrossRef] - Kumar, S. Linear Acoustic Modelling and Testing of Exhaust Muffler. Master’s Thesis, The Royal Institute of Technology, Stockholm, Sweden, January 2007. Available online: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwik7fSMv6CEAxXH_7sIHauBCwgQFnoECBUQAQ&url=https%3A%2F%2Fwww.diva-portal.org%2Fsmash%2Fget%2Fdiva2%3A11878%2FFULLTEXT01.pdf&usg=AOvVaw36bDa-Lk1NGDXCyrbz8u9R&opi=89978449 (accessed on 10 February 2024).
- Pal, S. Design and Acoustic Analysis of Exhaust Mufflers for Automotive Applications. Master’s Thesis, Christ University, Bengaluru, India, March 2015. Available online: https://www.researchgate.net/publication/275408198 (accessed on 10 February 2024).
- Yasuda, T.; Wu, C.; Nakagawa, N.; Nagamura, K. Predictions and experimental studies of the tail pipe noise of an automotive muffler using a one dimensional CFD model. Appl. Acoust.
**2010**, 71, 701–707. [Google Scholar] [CrossRef] - Suwandi, D.; Middelberg, J.; Byrne, K.P.; Kessissoglou, N.J. Predicting the Acoustic Performance of Mufflers using Transmission Line Theory. In Proceedings of the ACOUSTICS 2005, Busselton, Australia, 9–11 November 2005; pp. 181–187. [Google Scholar] [CrossRef]
- Kalita, U.; Singh, M. Prediction of Transmission Loss on a Simple Expansion Chamber Muffler. J. Emerg. Technol. Innov. Res.
**2018**, 5, 1022–1031. Available online: https://www.researchgate.net/publication/347950563 (accessed on 10 February 2024). - Mohammad, M.; Muhamad, S.M.F.; Khairudin, M.H.; Kadir, M.K.; Dahlan, M.A.A.; Zaw, T. Complex geometry automotive muffler sound transmission loss measurement by experimental method and 1 D simulation correlation. In Proceedings of the Sustainable and Integrated Engineering International Conference SIE 2019, Putrajaya, Malaysia, 8–9 December 2019; p. 012098. [Google Scholar] [CrossRef]
- Olgar, T. Acoustical Analysis of Exhaust Mufflers for Earth-Moving Machinery. Master’s Thesis, Middle East Technical University, Ankara, Turkey, September 2009. Available online: https://open.metu.edu.tr/handle/11511/19024 (accessed on 10 February 2024).
- Fu, J.; Xu, M.; Zhang, Z.; Kang, W.; He, Y. Muffler structure improvement based on acoustic finite element analysis. J. Low Freq. Noise Vib. Act. Control
**2019**, 18, 415–416. [Google Scholar] [CrossRef] - Zhang, L.; Shi, H.M.; Zeng, X.H.; Zhuang, Z. Theoretical and Experimental Study on the Transmission Loss of a Side Outlet Muffler. Hindawi Schock Vib.
**2020**, 2020, 6927574. [Google Scholar] [CrossRef] - Patne, M.M.; Sentilkumar, S.; Stanley, J.M. Numerical Analysis on Improving Transmission Loss of Reactive Muffler using Various Sound Absorptive Materials. In Proceedings of the ICMECE 2020, Knacheepuram, India, 22 April 2020; Volume 993, p. 012150. [Google Scholar] [CrossRef]
- Fan, W.; Guo, L.-X. An Investigation of Acoustic Attenuation Performance of Silencers with Mean Flow Based on Three-Dimensional Numerical Simulation. Hindawi Schock Vib.
**2016**, 2016, 6797593. [Google Scholar] [CrossRef] - Puthuparampil, J.X. Aeroacoustic Noise Prediction and Acoustic Optimization of Mufflers. Master’s Thesis, University of Toronto, Toronto, ON, Canada, November 2018. Available online: https://tspace.library.utoronto.ca/handle/1807/91618 (accessed on 10 February 2024).
- Xie, X. Noise optimization design on the exhaust muffler of a special vehicle on the improved genetic algorithm. J. Vibroengineering
**2015**, 17, 4625–4639. Available online: https://www.semanticscholar.org/paper/Noise-optimization-design-on-the-exhaust-muffler-of-Xie/f338124725c16ed33bf4774e4d4cebbd42d11a3c#citing-papers (accessed on 10 February 2024). - Bowden, D.R. Development of a large experimental acoustic transmission loss test bench suitable for large marine diesel exhaust system components. In Proceedings of the ACOUSTICS 2016, Brisbane, Australia, 9–11 November 2016; Available online: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjO_eCB_KKEAxW2_rsIHYCxBioQFnoECA0QAQ&url=https%3A%2F%2Facoustics.asn.au%2Fconference_proceedings%2FAASNZ2016%2Fpapers%2Fp54.pdf&usg=AOvVaw0YkzRvANLZayLyaMfRf_Uh&opi=89978449 (accessed on 10 February 2024).
- Mohamad, B. A review of flow acoustic effects on a commercial automotive exhaust system-methods and materials. J. Mech. Energy Eng.
**2019**, 3, 149–156. [Google Scholar] [CrossRef] - Babu, S.; Akhildev, V.P.; Sabu, J. Design optimization of hybrid muffler and acoustic transmission loss prediction. Int. Res. J. Eng. Technol.
**2020**, 7, 2721–2727. Available online: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwiahqyrhaOEAxWn7rsIHTeYCJcQFnoECA8QAQ&url=https%3A%2F%2Fwww.irjet.net%2Farchives%2FV7%2Fi7%2FIRJET-V7I7482.pdf&usg=AOvVaw1ApSTc3ld9e3kSk79k_cLH&opi=89978449 (accessed on 10 February 2024). - Liu, L.; Zheng, X.; Hao, Z.; Qiu, Y. A time-domain simulation method to predict insertion loss of a dissipative muffler with exhaust flow. Phys. Fluids
**2021**, 33, 067114. [Google Scholar] [CrossRef] - Damyar, N.; Mansouri, F.; Khavanin, A.; Jafari, A.J.; Asilian-Mahabadi, H.; Mirzaei, R. Acoustical Performance of a Double-Expansion Chamber Muffler: Design and Evaluation. Health Scope
**2022**, 11, e103226. [Google Scholar] [CrossRef] - Das, S.; Das, S.; Mondal, K.; Ahmad, A.; Shuayb, S.A.; Faizan, M.; Ameen, S.; Pandey, A.; Vadiraja, B.R. A novel design for muffler chambers by incorporating baffle plate. Appl. Acoust.
**2022**, 197, 108888. [Google Scholar] [CrossRef] - Cheng, Y.; Yuan, W.; Fu, J.; Ma, Y.; Zheng, W. Research on the Influence of Characteristics of the Annular Connecting Pipe, on the Transmission Loss of the Expanded Exhaust Muffler. Hindawi Schock Vib.
**2024**, 2024, 3404328. [Google Scholar] [CrossRef] - Cui, Z.; Huang, Y. Boundary Element Analysis of Muffler Transmission Loss With LS-DYNA. In Proceedings of the LS-DYNA 12th International Conference, Detroit, MI, USA, 3–5 June 2012; Available online: https://www.dynamore.de/en/training/conferences/past/12.-internationale-ls-dyna-konferenz (accessed on 18 March 2024).
- Vasile, O. Theoretical and experimental analysis of acoustic performances on the multi-chamber muffler. In Proceedings of the 21st International Congress on Sound and Vibration, Beijing, China, 13–17 July 2014; Available online: https://www.academia.edu/94861050/Theoretical_and_Experimental_Analysis_of_Acoustic_Performances_on_the_Multi_Chamber_Muffler?uc-g-sw=30082541 (accessed on 18 March 2024).
- AVL FIRE
^{TM}M Engine. Available online: https://www.avl.com/en/engineering/vehicle-engineering/vehicle-systems-development-and-integration (accessed on 18 March 2024). - Beranek, L.L.; Istvan, L. Noise and Vibration Control Engineering: Principles and Applications; John Wiley & Sons: New York, NY, USA, 1992. [Google Scholar]
- Grünwald, B. Theory, Computation and Design of Internal Combustion Engines for Automotive; Didactic & Pedagogical Publishing House: Bucharest, Romania, 1980. (In Romanian) [Google Scholar]
- Abdul Rani, M.N.; Mat Isa, A.A.; Rahman, Z.A.; Ali Al-Assadi, H.M.A. Dynamic characterization of an exhaust system. J. Mech. Eng.
**2011**, 8, 41–55. Available online: https://www.researchgate.net/publication/289188425_Dynamic_characterisation_of_an_exhaust_system (accessed on 10 February 2024). - Tao, Z.; Seybert, A.F. A review of current techniques for measuring muffler Transmission Loss. SAE Trans.
**2003**, 112, 2096–2100. Available online: https://www.jstor.org/stable/44745590 (accessed on 19 March 2024).

**Figure 1.**Plane wave propagation [1].

**Figure 2.**Real muffler [1].

**Figure 4.**Triple-chamber muffler with perforations [1].

**Figure 5.**The 6th region of the muffler of the triple-chamber muffler with perforations [1].

**Figure 6.**Geometry of the perforations [1].

**Table 1.**Parameter values of the transition elements for discontinuity [30].

Element Type | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | $\mathit{K}$ |
---|---|---|---|

−1 | −1 | $\frac{\left(1-\frac{{S}_{1}}{{S}_{3}}\right)}{2}$ | |

−1 | 1 | ${\left(\frac{{S}_{1}}{{S}_{3}}-1\right)}^{2}$ | |

1 | −1 | ${\left(\frac{{S}_{1}}{{S}_{3}}\right)}^{2}$ | |

1 | −1 | −0.5 |

${\mathit{S}}_{1}$ [m^{2}] | ${\mathit{S}}_{3}$ [m^{2}] | ${\mathit{l}}_{1}$ [mm] | ${\mathit{l}}_{2}$ [mm] | ${\mathit{l}}_{3}$ [mm] | ${\mathit{l}}_{4}$ [mm] | ${\mathit{l}}_{6\mathit{i}\mathit{n}}$ [mm] | ${\mathit{l}}_{6\mathit{p}}$ [mm] | ${\mathit{l}}_{7}$ [mm] | ${\mathit{d}}_{\mathit{h}}$ [mm] | C [mm] |
---|---|---|---|---|---|---|---|---|---|---|

0.003318 | 0.031416 | 80 | 45 | 120 | 50 | 65 | 50 | 50 | 3 | 10 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bugaru, M.; Vasile, C.-M.
Recent Developments in Using a Modified Transfer Matrix Method for an Automotive Exhaust Muffler Design Based on Computation Fluid Dynamics in 3D. *Computation* **2024**, *12*, 73.
https://doi.org/10.3390/computation12040073

**AMA Style**

Bugaru M, Vasile C-M.
Recent Developments in Using a Modified Transfer Matrix Method for an Automotive Exhaust Muffler Design Based on Computation Fluid Dynamics in 3D. *Computation*. 2024; 12(4):73.
https://doi.org/10.3390/computation12040073

**Chicago/Turabian Style**

Bugaru, Mihai, and Cosmin-Marius Vasile.
2024. "Recent Developments in Using a Modified Transfer Matrix Method for an Automotive Exhaust Muffler Design Based on Computation Fluid Dynamics in 3D" *Computation* 12, no. 4: 73.
https://doi.org/10.3390/computation12040073