Optimization of internal geometry for maximum transmission loss in a reactive muffler

Abstract

Noise pollution from internal combustion engines in vehicles and industrial equipment poses an environmental challenge. Exhaust systems use mufflers to reduce noise by attenuating sound waves. This study focuses on optimizing the extended inlet length and diameter in a reactive muffler with a single expansion chamber and extended sections. The aim is to enhance transmission loss in a targeted frequency range for better noise reduction. Using the Taguchi Method, the study identified optimal design parameters for sound attenuation. Acoustic performance was assessed through two methods: (1) numerical simulations using COMSOL Multiphysics and the Finite Element Method (FEM) to analyze sound wave behavior, and (2) experimental validation via the two-load method to measure transmission loss in a prototype. Results highlight key design insights for optimizing reactive mufflers, offering a robust framework for achieving superior noise control in applications requiring frequency-specific attenuation.

Keywords

transmission loss reactive muffler finite element analysis Taguchi method experimental analysis

Introduction

The two-load method was used to validate the muffler design experimentally. Engine noise pollution, particularly from exhaust systems, is a significant concern because of its hazardous effects. Exhaust noise, in particular, is one of the most harmful noise levels, often exceeding 80 dB, posing a risk to human health. The effectiveness of various muffler types in attenuating exhaust noise is primarily contingent upon their design and functionality. The design of a muffler not only plays a crucial role in noise reduction but also significantly impacts engine performance and fuel efficiency.^1,2

Mufflers, also known as silencers, are acoustic devices integrated into exhaust systems to attenuate unwanted sound emissions generated by internal combustion engines. They play a pivotal role in maintaining regulatory compliance with noise standards and reducing the environmental and health impacts of noise pollution. Fundamentally, mufflers function based on principles of sound wave reflection, absorption, and interference.

In recent years, acoustic metamaterials have emerged as a transformative direction in muffler research, offering unprecedented control over sound propagation at subwavelength scales. These engineered structures utilize locally resonant mechanisms, negative effective parameters, and tailored geometries to achieve extraordinary low-frequency attenuation that is often unattainable with conventional absorptive or reactive designs. Ebrahimi-Nejad and Kheybari³ demonstrated the potential of honeycomb and square lattice metamaterials, showing distinct stop-band behavior that effectively suppressed low-frequency transmission. Similarly, Liu et al.⁴ explored membrane-type and plate-based acoustic metamaterials, highlighting how unit-cell geometry significantly influences the formation and width of bandgaps, enabling compact broadband noise suppression without increasing muffler size. Ghoudjani and Ravanbod⁵ advanced this field by integrating optimization algorithms with chiral honeycomb membrane metamaterials, achieving a 35% increase in sound transmission loss and ultra-wide bandgap coverage, thereby addressing the limitations of narrow-band resonant structures. More recently, Ravanbod and Ebrahimi-Nejad⁶ proposed auxetic metamaterial configurations with reentrant and perforated honeycomb geometries, further expanding tunability for low-frequency noise control while reducing structural mass. Collectively, these studies demonstrate that metamaterials, with their diverse geometrical and resonant designs, provide a versatile framework for next-generation mufflers, bridging gaps between compactness, broadband attenuation, and structural efficiency.

There are three primary types of mufflers—reactive, absorptive, and hybrid. Reactive mufflers operate by reflecting and destructively interfering with pressure waves, using a series of chambers and ducts to alter the phase of propagating sound waves. This phase cancellation significantly reduces noise across certain frequency bands. Absorptive mufflers, in contrast, rely on porous materials like fiberglass to convert acoustic energy into heat through friction and viscous loss, especially at higher frequencies. Hybrid mufflers combine both principles to achieve broadband noise suppression.

Among these, reactive mufflers—such as those utilizing a Single Expansion Chamber (SEC) design—are especially relevant for low-frequency noise control. Their performance is highly sensitive to geometric parameters such as chamber length, diameter, and the configuration of extended inlet and outlet ducts. These features directly influence the system’s acoustic impedance and resonance behavior. Hence, the design of a muffler becomes a multidimensional optimization problem that requires balancing noise attenuation, flow resistance, and engine performance metrics.

Two primary parameters are commonly used to describe muffler performance: insertion loss (IL) and transmission loss (TL). TL is widely regarded as a key indicator of muffler effectiveness because it can be predicted using the known physical properties of the muffler. TL can be determined through analytical, numerical, or experimental approaches. However, analytical methods are often time-consuming due to the complexity of the governing equations, making them less practical for detailed design iterations. In contrast, numerical methods are more versatile and suitable for modeling complex geometries, offering a cost-effective alternative to experimental approaches.⁷

The literature, particularly the work of Munjal, offers a detailed theoretical framework for muffler modeling and acoustic performance evaluation. Typically, experimental data are used to validate both analytical and numerical results, ensuring the design meets target specifications. However, large-scale studies focused on optimizing the dimensions of reactive mufflers—particularly those using a Single Expansion Chamber (SEC) with extended inlet and outlet sections—remain limited, especially when employing optimization techniques such as the Taguchi Method.^8,9

This study utilized the Taguchi Method to optimize the dimensions of a reactive muffler with an extended inlet in an SEC configuration. In the initial design phase, the diameter and length of the extended inlet were selected as control variables. An L-9 orthogonal array was adapted to identify key factors influencing acoustic performance, enabling design optimization at specific target frequency ranges.

Literature review

The design and optimization of mufflers, particularly reactive types such as the Single Expansion Chamber (SEC), have been extensively researched due to their critical role in exhaust noise reduction and engine performance regulation. Reactive mufflers primarily attenuate sound through wave reflection and interference within structured chambers, making them highly effective for low-frequency noise control in automotive and industrial applications.

Munjal’s foundational work in muffler acoustics provided a theoretical framework for predicting transmission loss (TL) using one-dimensional wave propagation and transfer matrix techniques. Although efficient for simple geometries, these analytical models become less practical for complex multi-chamber designs, thereby necessitating numerical approaches such as Finite Element Analysis (FEA) and Boundary Element Methods (BEM) for higher fidelity predictions.⁸

Recent studies have investigated geometric and configuration-dependent improvements in muffler performance. Ebrahimi-Nejad et al. conducted a systematic analysis of inlet–outlet positioning, baffle design, and chamber geometry, demonstrating that careful arrangement of these features can significantly enhance TL while meeting stringent space constraints in vehicle applications.¹⁰ Similarly, Ebrahimi et al. optimized a double-chamber muffler using parametric and grid-based optimization methods, achieving up to 78 dB attenuation in the 0–500 Hz range while simultaneously reducing weight by 25%.¹¹ These results highlight the interplay between geometry, acoustic efficiency, and structural considerations.

Beyond conventional chamber-based designs, novel concepts such as acoustic metamaterials are also gaining prominence. Kheybari and Ebrahimi-Nejad proposed a dual-target-frequency stop-band muffler incorporating locally resonant acoustic metamaterial baffles, achieving up to 89 dB improvement in TL across dual frequency bands. Their work also validated aerodynamic performance using CFD simulations, demonstrating the practicality of hybrid acoustic–flow optimization.¹² Optimization methods have evolved significantly over the last two decades. The Taguchi method, a robust experimental design framework, has been extensively used to reduce the number of simulations while identifying influential parameters. Kalita and Singh¹ applied it to reactive mufflers, achieving improvements in TL and flow resistance. In parallel, evolutionary and stochastic approaches such as Genetic Algorithms (GA) and Random Search Methods have shown promise. Kulkarni and Ingle⁹ compared Transmission Loss for muffler geometries, reporting substantial gains over trial-and-error designs. Zuo et al.¹³ extended these efforts by integrating multi-objective GA with surrogate models, highlighting trade-offs between acoustics and aerodynamic penalties.

Despite these advancements, important research gaps remain:

Underrepresentation of practical SEC configurations with extended inlets/outlets, even though such designs are common in industry.

Limited experimental validation of optimization-based studies, undermining real-world applicability.

Insufficient low-frequency optimization, a critical band for combustion-driven systems.

Lack of systematic comparison between deterministic (Taguchi) and probabilistic (GA, PSO) frameworks in muffler dimension optimization.

Attempts to address these challenges include hybrid Taguchi–FEA methodologies, such as Kalita and Singh,¹⁴ who optimized absorption layer thickness but noted persistent low-frequency limitations, and Tanriver,¹⁵ who highlighted the limits of purely geometric optimization in CFD-driven studies.

Although advanced numerical models and even metamaterial-based designs show promise, the integration of simplified design frameworks like Taguchi with experimental validation remains scarce for SEC mufflers. This gap underscores the need for systematic yet practical optimization of fundamental geometric parameters such as extended inlet diameter and length.

Accordingly, this study applies the Taguchi Method to optimize SEC muffler dimensions, focusing on extended inlet design. By combining structured experimental design, frequency-domain performance analysis, and experimental validation via the two-load method, this work addresses key gaps in muffler dimension optimization research.

Methodology

Taguchi method

The Taguchi Method is a robust statistical approach for systematically improving system performance by identifying the influence of design parameters under varying noise conditions. It has proven particularly effective in acoustic applications where multiple geometric and physical factors interact to affect sound attenuation. In the present study, the method was applied to optimize the Single Expansion Chamber (SEC) muffler with extended inlet. The primary control variables selected were the diameter of the extended inlet and the length of the extended inlet section, as these dimensions directly influence the wave reflection characteristics and thus the transmission loss (TL) across the target frequency range. These control factors were assigned discrete levels based on practical design constraints, manufacturability, and insights from prior studies on SEC mufflers.^16,17

Taguchi’s orthogonal array framework was used to minimize the number of simulation runs while ensuring balanced coverage of factor combinations. For performance evaluation, the signal-to-noise (S/N) ratio was employed with the higher-the-better criterion, reflecting the objective of maximizing TL. The evaluation of optimal performance was carried out using two key criteria: (1) achieving the highest possible TL values across dominant low-frequency bands typical of combustion-driven noise, and (2) ensuring that improvements were achieved without excessive flow resistance or structural complexity. Furthermore, analysis of variance (ANOVA) was conducted to quantify the relative contribution of each control variable to the overall acoustic performance. This structured methodology allowed the study not only to identify the most influential design parameters but also to determine an optimal configuration with practical applicability and potential for experimental validation.¹⁸

FEM analysis

The foundational process for the FEM begins with Computer-Aided Design (CAD) geometry. Within the scope of this research, the losses in the silencer’s gearbox were computed using a 3-D finite element methodology. The analysis assumed a Mach number of zero. COMSOL Multiphysics, which is a software tool for finite element analysis, was employed to conduct this analysis. Notably, the chosen tool does not encompass fluid-structure interaction capabilities. As outlined in the COMSOL Multiphysics User’s Manual by COMSOL A.B., both parametric and linear solvers are employed for the analysis.¹⁹ The Helmholtz equation as depicted by equation (1) was used to calculate the sound pressure.

\nabla • ((\frac{1}{ρ^{0}}) \nabla p - q) + (\frac{k^{2}}{ρ^{0}}) p = 0

(1)

Where, $k = \frac{2 π f}{c_{0}}$ is denoted as the wavelength, $ρ_{0}$ is denoted as the density of the fluid, and $c_{0}$ is denoted as the velocity of sound. In this study, the two-pole source term q, which represents the acceleration per unit volume, equals zero. A parameterized solver using this equation can find a solution in its frequency domain.²⁰

The muffler transmission loss was calculated using the following equation (2):

T L = 10 \log (\frac{p_{i n}}{p_{o u t}})

(2)

Where, $p_{i n}$ and $p_{o u t}$ indicates the acoustic impacts at the Inlet and Outlet, which are estimated as equations (3) and (4).

p_{i n} = \int_{Ω}^{1} \frac{{p_{0}}^{2}}{2 ρ c_{0}} dA

(3)

p_{o u t} = \int_{Ω}^{1} \frac{{| p_{c} |}^{2}}{2 ρ c_{0}} dA

(4)

The pressure value at the Inlet, $p_{0}$ is set to 1 bar.

Boundary conditions

The boundary conditions are categorized into three types

1. The model employs sound-hard (wall) boundary conditions at the solid boundaries, specifically at the outer walls of the resonator chamber and pipes. This boundary condition is shown by equation (5)

- (\nabla p / ρ) • n = 0

(5)

2. The boundary condition at the inlet is characterized by the interaction of incoming and outgoing plane waves, as delineated in equations (6) and (7).

n • (1 / ρ_{0}) \nabla p + i • k • (p / ρ_{0}) + (i / 2 k) • Δ_{r} = 0

(6)

p = {(i / 2 k) • Δ_{r} • p_{0} + [1 - (k • n)] • i • k • (p_{0} / ρ_{0})} • e^(- i • k • r)

(7)

Where,

$n$ : A unit normal vector pointing in a specific direction (often surface normal in acoustics).

$ρ_{0}$ : Ambient (or reference) fluid density [kg/m³]

$\nabla p$ : Gradient of the acoustic pressure, shows how pressure changes in space.

$i$ : Imaginary unit (i² = −1) used due to time-harmonic wave representation (complex form).

$k$ : Acoustic wave number (k = ω/c), where:

ω: Angular frequency [rad/s]

c: Speed of sound in the medium [m/s]

$p$ : Acoustic pressure [Pa].

∆ _r : Some spatial differential operator or potential-like term (depends on context, possibly a source term or scattered field contribution).

$p_{0}$ : Reference or incident pressure amplitude.

$(p_{0} / ρ_{0})$ : Pressure amplitude divided by density — has units of particle velocity.

$e^(- i • k • r)$ : Complex exponential representing a plane wave traveling in direction k, at position r.

3. The model defines the outgoing wave at the boundary of the outlet. This boundary condition is shown by equation (8)

n • (1 / ρ_{0}) \nabla p + i • (k / ρ_{0}) p + (i / 2 k) • Δ_{r} • p = 0

(8)

Meshing

The model is set up with the needed boundary conditions, and then meshing is done. A physics-controlled mesh is used, and a finer size is chosen for meshing. Tetrahedral elements and automatic meshing are used in the process. The finite-element area is solved with at least 12 elements for each wavelength.

The model and meshed model of the SEC muffler is as shown in Figure 1.

The study encompassed the full frequency range from 1 to 1600 Hz, focusing specifically on the frequency spectrum between 650 and 850 Hz. The density of air was assumed to be 1.2 kg/m³, with the speed of sound considered to be 343 m/s.

Figure 2 illustrates the acoustic pressure and sound pressure levels generated by the muffler across a specified frequency range. The data show that the sound pressure level at the muffler’s inlet is approximately 90 dB, whereas that at the outlet is approximately 45 dB. This significant difference in sound pressure levels is due to the attenuation of sound waves as they pass through the internal components of the muffler.

Figure 3 shows the transmission-loss curve obtained from the preliminary experiment. The data show an average transmission loss of approximately 44 dB within the frequency range of 650–850 Hz.²¹ The main aim of this study is to improve the muffler transmission loss in this frequency range. This can be accomplished by carefully optimizing the dimensions of the Extended Inlet, including both its length and diameter.²²

Optimization of muffler

Model

A frequency range of 650–850 Hz was specifically selected for optimization in the context of a 4-cylinder engine operating at 1500 rpm. This frequency range corresponds to industry standards for evaluating transmission loss in engine applications.²³

Figure 4 provides a schematic of the single-expansion-chamber Muffler model, featuring both an Extended Inlet and Extended Outlet. The inlet and outlet pipes had diameters of d1 and 44 mm, respectively, whereas the main chamber had a diameter of 120 mm. To ensure uniformity, the lengths of both the inlet and outlet pipes were set to 95 mm, and the chamber of the muffler was 540 mm long.²⁴ For the design experimentation, two key parameters were chosen for optimization: the length of the extended inlet, denoted as L1, and the diameter of the extended inlet, represented by d1. These components are illustrated in Figure 4.^22,25

Selection of control parameters and levels

The methodology began with a preliminary experiment to select muffler parameters based on a literature review.^21,26 This review evaluates the effects of these parameters on transmission loss, providing the basis for the initial factor levels used in this study.²⁷ The experimental process was structured as follows: First, each factor was assessed at level 1 to determine the baseline and reference values under treatment condition 1.²⁸ The experiment proceeded by varying the length of the Extended Inlet (L1) to level 2 while keeping the other factors at level 1. In the subsequent treatment, the diameter of the Extended Inlet (d1) was increased to level 2, while the other factors remained at level 1. This iterative process continued until all factors at level 1 were evaluated.²⁹

Parameters L1 and d1 refer to the length and diameter of the extended inlet, respectively, for the SEC muffler and are considered control factors in this model.³⁰ The initial estimates in the preliminary experiment set L1 to 270 mm and d1 to 44 mm. The experiment involved adjusting L1 from 0 to 270 mm in 10 mm increments. With L1 fixed at 110 mm, the Extended Outlet Length (L2) was maintained at 105 mm and the diameter of the Extended Outlet was maintained at 44 mm. These models were tested in COMSOL using specified dimensions.³¹

The results show that the highest transmission loss occurred with L1 at 110 mm. For further optimization, L1 was maintained at 110 mm while d1 was varied from 29 to 64 mm in 5 mm increments. The maximum transmission loss was observed at d1 = 59 mm. Ultimately, the pilot experiment identified the optimized dimensions as L1 = 110 mm and d1 = 59 mm.³²

The levels in the Table 1 are decided on the basis of optimized dimensions obtained in the pilot experiment.

Table 1.

Control parameters and levels.

Sr. no.	Control parameters	Level 1	Level 2	Level 3
1	Extended inlet length, L1	105	110	115
2	Extended inlet diameter, d1	54	59	64

Taguchi introduced a versatile approach using fractional factorial experiment (FFE) matrices that can be applied to a range of experimental conditions. For this study, a 9-Trial Orthogonal Array (OA), often referred to as the L9 matrix, is of particular importance. One of the key benefits of using an orthogonal array is its efficiency in evaluating multiple factors, while minimizing the number of experimental trials required.

Table 2 lists the L9 orthogonal array employed in this study. The degrees of freedom are represented by Figure 9 in the notation “L9,” which also relates to the number of conditions of treatment and rows in the matrix. The upper section of the array indicates that it supports up to two factors. The levels are indicated by the numbers 1 and 2, while arrays designed for additional levels might use numbers such as 3, 4, and 5. Alternative symbols, such as −, 0−, and +, can also denote different levels.

Table 2.

L9 orthogonal array.

Treatment condition	Extended inlet length, L1	Extended inlet diameter, d1	Response transmission loss (dB)
1	105	54	90.214
2	105	59	99.583
3	105	64	89.92
4	110	54	66.748
5	110	59	65.264
6	110	64	63.831
7	115	54	60.569
8	115	59	59.26
9	115	64	57.864

The L9 orthogonal array is generated using software, such as Minitab, based on these control parameters, and the results are displayed in tabular form, as shown in Table 3.

Table 3.

L9 table of Taguchi for frequency range (1–1600 Hz).

Treatment condition	Extended Inlet Length, L1	Extended Inlet Diameter, d1	Response TL in dB	Square of TL
1	105	54	90.214	8138.566
2	105	59	99.583	9916.774
3	105	64	89.92	8085.606
4	110	54	66.748	4455.296
5	110	59	65.264	4259.39
6	110	64	63.831	4074.397
7	115	54	60.569	3668.604
8	115	59	59.26	3511.748
9	115	64	57.864	3348.242
			Total = 653.253	Total = 49,458.62

Using this table, the effect of individual parameter on Transmission Loss can be analyzed.

ANOVA analysis

To determine the relative influence of control factors on the transmission loss (TL), an Analysis of Variance (ANOVA) was performed on the Taguchi L9 orthogonal array results. The outcomes of ANOVA revealed that the extended inlet length (L1) is the most significant parameter, contributing approximately 96.66% of the total variance in TL. In contrast, the extended inlet diameter (d1) contributed only 1.27%, while the interaction effect (L1 × d1) accounted for 2.07%. Since the Taguchi design was conducted without replicates, no residual error was available for statistical testing, which limited the computation of F-values and p-values. Nevertheless, the dominance of the inlet length highlights its critical role in muffler performance optimization. These findings suggest that even small variations in extended inlet length can substantially influence acoustic attenuation, whereas changes in diameter play a relatively minor role.

The ANOVA results mentioned in Table 4 clearly indicate that extended inlet length (L1) is the most dominant factor influencing muffler transmission loss, while inlet diameter (d1) and interaction effects contribute only marginally. This emphasizes that dimensional variations in length have a substantially greater effect on acoustic attenuation compared to diameter changes.

Table 4.

ANOVA results for transmission loss (TL).

Source	Degree of freedom	Sum of squares	% Contribution
Extended inlet length (L1)	2	1974.796	96.66%
Extended inlet diameter (d1)	2	26.033	1.27%
Interaction (L1 × d1)	4	42.295	2.07%
Error	0	0.000	0.00%
Total	8	2043.124	100%

Result for Taguchi analysis

The Taguchi analysis was conducted using Minitab software. Figure 5 shows the Main Effects plot, which demonstrates how individual muffler parameters influence the transmission loss. This analysis utilized an L9 orthogonal array, as detailed in Table 2. The plot indicates that as the length of the extended inlet (L1) increased, the average transmission loss tended to decrease. In contrast, the mean transmission loss initially increased and reached a maximum before decreasing, as the diameter of the extended inlet (d1) increased.

Discussion on Taguchi analysis

When optimizing a single-objective response, the Taguchi method generally uses the S/N ratio.^33,34 In this study, however, the Taguchi approach was tailored to evaluate quality parameters that deviate from their target values rather than focusing solely on the S/N ratio.^35–37

This methodology involves computing S/N ratios from the experimental results for various responses. The “higher the better” characteristic is applied to responses that should be maximized to achieve optimal performance, while the “lower the better” characteristic is used for responses that need to be minimized. This approach aids in assessing and improving the desired outcomes by measuring the extent to which the quality parameters deviate from their ideal values.

The different types of S/N ratios include the following:

For the “Higher the better” strategy, the S/N ratio is determined as

η = - 10 \log [(1/n) * Σ ({1/yi}^{2})]

The “Lower the better” strategy’s S/N ratio is calculated as

η = - 10 \log [(1/n) * Σ ({yi}^{2})]

The “nominal-the-better” strategy’s S/N ratio is calculated as

η = 10 \log ((y^{2}) / σ^{2})

where n is the number of observations, y is the appropriate response, and η is the resulting SN ratio.

The analysis of the S/N ratio plot, presented in Figure 6 and created using MINITAB software,³⁸ demonstrates that both L1 (the length of the extended inlet) and d1 (the diameter of the extended inlet) significantly influence the muffler performance. This study employs the “larger is better” criterion, which indicates that superior performance is associated with higher values of these parameters.³⁹ Thus, the goal is to maximize the S/N ratio to achieve optimal transmission loss characterized by lower sound levels.⁴⁰

Figure 6 shows that the maximum S/N ratio was achieved when L1 was 105 mm and D1 was 59 mm. Therefore, the optimal dimensions for the extended inlet length (L1) and diameter (d1) of the muffler were identified as 105 and 59 mm, respectively.^41,42 This approach aligns with the objective of enhancing muffler performance by improving the transmission loss and reducing noise levels.

The optimum configuration of the muffler is shown in Figure 7. This optimal setup entailed an Extended Inlet Length of 105 mm and an Extended Inlet diameter of 59 mm.

Figure 8 displays the results obtained from the SEC Muffler Model over the frequency range of 650–850 Hz. A detailed examination revealed that the TL curve from the initial pilot experiment showed notable troughs at 651, 721, and 851 Hz, with corresponding peaks at 691 and 761 Hz.^43,44

These troughs can be explained by impedance mismatches occurring at frequencies of 651, 721, and 851 Hz. Such mismatches cause destructive interference of sound waves, leading to the significant attenuation of sound at these frequencies.^45,46 This indicates that the muffler is effective at reducing sound propagation at these points, suggesting efficient sound wave cancellation within the muffler design.⁴⁷

For the given SEC muffler configuration, the highest TL recorded was 77.21 dB at 691 Hz, with an average TL of 43.4 dB throughout the tested frequency range.⁴⁸ Although this reflects a consistent performance, further optimization is necessary.

After applying the optimized design, the TL curve shows a notable improvement, with a broad-spectrum increase in the transmission loss across the frequency range. The optimized model achieved a peak TL of 104.46 dB at 751 Hz, and the average TL increases to 45.23 dB.⁴⁹ This substantial enhancement indicates the effectiveness of the adjustments in reducing the acoustic emissions and improving the overall muffler performance.

Experimental analysis

The optimized model was confirmed in the experimental analysis using the two-load approach. Figure 9 shows the experimental setup.

In Figure 9, the microphones are mentioned at 1, 2, 3, and 4, whereas A and B are diffusers.

The experimental setup comprised three primary subsystems: noise generation, sound propagation, and noise measurement.^50,51

The main parts of this setup are the sound source, amplifier, FFT analyzer, and impedance tube. These elements constituted the core of the experimental apparatus. Microphone placements were designated as 1, 2, 3, and 4 in the accompanying diagram.⁵²

A rigid impedance tube featuring evenly spaced measurement points was essential for this configuration. It serves as a conduit for sound transmission, with one end attached to the test muffler and the other end connected to the sound source.⁵³ To capture both incident and transmitted waves, two impedance tubes were positioned on either side of the muffler.⁵⁴

The FFT analyzer plays a crucial role in data acquisition. It collects pressure data from microphones, which are then transferred to a data storage system for recording.⁵⁵ In addition, the FFT analyzer includes an output channel linked to the speaker. The speaker receives a random noise signal from the analyzer, which is amplified before being emitted. A white noise signal was employed to maintain a uniform power density across the frequencies. The sound source can generate an intensity of up to 120 dB.⁵⁶

The experimental setup also incorporates the Transfer Function technique using two microphones to ensure precise measurements within the framework of the experiment.⁵⁷

Theory of two load method

In their experimental investigation of mufflers, Tao and Seybert applied the two-load method, also known as the two-load approach, which is based on the transfer matrix method.⁵⁸ This technique involves solving four-pole equations derived from acoustic pressure measurements taken at specific microphone positions, enabling efficient calculation of the transmission loss (TL) for any muffler using transfer matrix theory.⁵⁹

A critical requirement of this method is that the two acoustic loads applied at the outlet of the test muffler must differ sufficiently to ensure numerical stability. In the study conducted by Tao and Seybert, two distinct load conditions were generated by employing the outlet tube both with and without additional termination elements.⁶⁰ In the present study, the same principle is adopted: measurements were performed for two different outlet configurations, as illustrated in Figure 10, which forms the basis of the two-load method used here.⁶¹

Implementation and measurement setup

The experimental arrangement consists of an acoustic impedance tube rig equipped with a loudspeaker as the broadband noise source, positioned upstream of the test muffler. Four high-sensitivity ¼-inch condenser microphones (with preamplifiers and calibration certificates) are flush-mounted at predetermined locations along the tube to capture incident and transmitted sound pressures. The signals are acquired using a multi-channel data acquisition system (NI PXI platform) and processed with LabVIEW-based software for real-time Fast Fourier Transform (FFT) analysis.

The two-load condition is generated by modifying the termination of the muffler outlet. In Load 1, the muffler outlet is left open to the free field, while in Load 2, an extended outlet pipe is attached to alter the acoustic impedance. This variation ensures adequate load difference, which is essential for solving the governing equations.

Measurement technique

The transfer functions between microphone pairs are obtained using the two-microphone random excitation technique, in which a pseudo-random broadband signal drives the loudspeaker to excite the system across the frequency range of interest. The transfer functions HijH_{ij}Hij are defined as the ratio of acoustic pressures at microphones Pj/PiP_j / P_iPj /Pi , and they form the inputs for the four-pole parameter matrices. To minimize bias and improve repeatability, each test was conducted three times, and the results were averaged.

The transfer matrix approach can be employed to compute the transmission loss of a muffler by solving a system of four-pole equations using four designated microphone positions.^62,63 Specifically, when there is no airflow, the four poles corresponding to elements 1–2 can be represented as in equation (9).

[\begin{array}{l} A_{12} B_{12} \\ C_{12} D_{12} \end{array}] = [\begin{array}{l} c o s k l_{12} j ρ c s i n k l_{12} \\ \frac{j s i n k l_{12}}{ρ c} c o s k l_{12} \end{array}]

(9)

The following equation (10) represents four poles of elements 2–3

[\begin{array}{l} A_{23} B_{23} \\ C_{23} D_{23} \end{array}]

(10)

Where,

A_{23} = \frac{Δ_{34} (H_{32 a} H_{32 b} - H_{32 b} H_{34 a}) + D_{34} (H_{32 b} - H_{32 a})}{Δ_{34} (H_{34 b} - H_{34 a})}

B_{23} = \frac{B_{34} (H_{32 a} - H_{32 b})}{Δ_{34} (H_{34 b} - H_{34 a})}

C_{23} = \frac{(H_{31 a} - A_{12} H_{32 a}) (Δ_{34} H_{34 b} - D_{34}) - (H_{31 b} - A_{12} H_{32 b}) (Δ_{34} H_{34 a} - D_{34})}{B_{12} Δ_{34} (H_{34 b} - H_{34 a})}

The four poles for elements 3–4 can be expressed as shown in the equation (11)

[\begin{array}{l} A_{34} B_{34} \\ C_{34} D_{34} \end{array}] = [\begin{array}{l} c o s k l_{34} j ρ c s i n k l_{34} \\ \frac{j s i n k l_{34}}{ρ c} c o s k l_{34} \end{array}]

The term $H_{i j}$ signifies transfer function between $P_{i}$ and $P_{j}$ , where

H_{i j} = \frac{P_{j}}{P_{i}}

A final Transfer matrix appears as followed after cascading these matrices as shown in equation (12)

(\begin{array}{l} A_{14} B_{14} \\ C_{14} D_{14} \end{array}) = (\begin{array}{l} A_{12} B_{12} \\ C_{12} D_{12} \end{array}) (\begin{array}{l} A_{23} B_{23} \\ C_{23} D_{23} \end{array}) (\begin{array}{l} A_{34} B_{34} \\ C_{34} D_{34} \end{array})

(12)

The Transmission Loss is calculated as using equation (13).

T L = 20 l o g_{10} [\frac{1}{2} (| A_{14} + \frac{B_{14}}{ρ c} + ρ c C_{14} + D_{14} |)]

(13)

Transmission Loss can be determined experimentally with two microphones and the random excitation technique.

Procedure for experimental analysis

To derive the transfer functions H31, H32, and H34 and their corresponding locations, a microphone was positioned at location 3, and additional microphones were placed at positions 1, 2, and 4. The configuration used for calculating the transmission loss is shown in Figure 11 .⁶⁴ In figure there are total 6 microphone ports. Three ports namely 1′,1 and 2 are located on impedance tube with speaker and remaining three ports namely 3,4 and 4′ are located on opposite impedance tube. The two load method is employed using two different loads. For “with load” end cap is applied and for “without load” the end cap is removed.

The experimental setup followed the guidelines established by the International Standard ISO 10534-2 to measuring the transmission loss.⁶⁵ The procedure involves configuring the analyzer and processing data to quantify transmission loss (TL) across a frequency range of 1–1600 Hz.⁶⁶

Microphone positions 1, 2, 3, and 4 were used to measure the sound pressure in the lower frequency range of 1–400 Hz. For the higher frequency range of 400–2000 Hz, microphones placed at 1′, 2, 3, and 4′ were employed.⁵⁹ This arrangement ensures a comprehensive evaluation of the sound characteristics throughout the frequency spectrum.⁶⁷

The precise determination of the transfer functions H31, H32, and H34 based on these microphone placements is crucial for accurate transmission loss calculations. This setup, depicted in Figure 11, allows for a detailed analysis and assessment of the acoustic performance of the muffler.^68–70

Results and discussion

Figure 12 compares the TL results obtained from finite element analysis with those derived from the experimental methods within the targeted frequency range. The experimental TL curve revealed minor troughs at 671, 701, 731, 811, and 841 Hz, and significant peaks at 661, 721, 751, and 821 Hz.

The observed peaks in the TL curve indicate the maximum transmission losses at specific frequencies. These peaks are linked to impedance mismatches, which result in destructive interferences. At frequencies of 661, 721, 751, and 821 Hz, impedance mismatches cause opposing sound waves to cancel each other out, leading to substantial attenuation of the sound propagation. This suggests that the sound waves entering the muffler at these frequencies were nearly fully absorbed.

For the SEC muffler model, the recorded peak TL was 101.89 dB at 750 Hz. On an average, the experimental TL was 44.64 dB, whereas the FEM analysis yielded a TL of 45.23 dB. The close match between the experimental and FEM results underscores the effectiveness of the modeling approach, although minor discrepancies may arise owing to factors such as the surface finish of the impedance tube, challenges in generating white noise with FFT, and potential sound leakage into the surrounding environment.

Conclusion

This study employed the Taguchi method to optimize the transmission loss of a single expansion chamber reactive muffler by adjusting the extended inlet length and diameter. Finite Element Analysis (FEA) via COMSOL Multiphysics was validated experimentally using the two-load method, showing good agreement between numerical and measured data. The optimization identified inlet dimensions of 105 mm length and 59 mm diameter as optimal, achieving improved transmission loss across the target frequency range.

The study highlights the Taguchi method as an effective tool for acoustic optimization in muffler design, particularly for geometric parameters influencing reactive performance. The integration of simulation and experimental techniques provides a reliable framework for performance enhancement in noise control applications.

However, the findings are limited to a specific muffler configuration and steady-state conditions. Future work could explore alternative geometries, the use of advanced materials, and the effects of varying flow and thermal conditions to broaden applicability. These directions may support the development of more efficient and adaptable muffler systems for diverse acoustic environments.

Footnotes

Acknowledgements

The authors are thankful to Dr Vishwanath Karad MIT World Peace University, Pune for providing facility for performing experimental analysis at NVH lab.

ORCID iD

Mahesh Kulkarni

Ethical considerations

This article does not contain any studies with human or animal participants.

Author contributions

All authors contributed to the study conception and design. All authors read and approved the final manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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