Experimental and Computational Acoustic Analysis of Recycled Automobile Dashboard Composites

Abstract

This study presents the development and optimisation of sustainable composite materials derived from automotive polymeric waste (dashboard crumbs). The influence of key formulation parameters on material performance was investigated using experimental analysis combined with model-based prediction. Acoustic behaviour was evaluated using an impedance tube method, while predictive modelling was performed using the Johnson–Champoux–Allard (JCA) model and a Padé approximation for efficient computation. Model performance was assessed using quantitative metrics, including root mean square error (RMSE), mean absolute error (MAE) and coefficient of determination (R²), demonstrating good agreement with experimental data across the investigated frequency range. The results show that catalyst concentration is a critical parameter, with an optimal value of 5 wt% yielding near-unity absorption within the mid-frequency range (1200–1800 Hz). Further increase in catalyst content resulted in reduced performance due to changes in pore structure and reaction kinetics. In contrast, particle size variation exhibited a limited effect on overall performance. The proposed modelling framework enables efficient prediction of material behaviour and supports optimisation of formulation parameters. This study highlights the potential of recycled polymeric materials for sustainable engineering applications and provides a practical approach for performance-driven material design.

1. Introduction

Acoustic materials are engineered to control, direct, or manipulate sound waves. Their performance is typically characterised by frequency dependent properties such as absorption coefficients and acoustic impedance. Modelling these properties accurately is essential for effective material design.

Conventional approaches rely on empirical models or numerical simulations such as finite element methods (FEM). However, these approaches can be computationally expensive and may lack generalisability. Rational function approximations, particularly Padé approximants, offer a compact and efficient way to represent complex acoustic responses. Meanwhile, AI techniques—especially machine learning (ML)—have shown promise in optimising parameter estimation and discovering patterns in high-dimensional datasets. Recent studies have explored the use of intelligent monitoring and data fusion techniques to enhance modelling accuracy and system performance prediction. For example, Wang et al. [1] employed a fractional-order multi-rate Kalman fusion approach for structural performance monitoring, demonstrating improved prediction accuracy in complex systems. Such approaches highlight the importance of integrating modelling efficiency with robust parameter estimation, which is also relevant in the context of acoustic material characterisation.

Polyurethane (PU) is a polymer characterised by elastomeric properties and demonstrating notable adhesion and longevity. PU exhibits a range of colours and demonstrates resilience to acidic and alkaline substances. The extensive utilisation of PU has resulted in a notable accumulation of these substances in landfills. Consequently, it can result in considerable worldwide plastic pollution. This increased environmental concern has prompted researchers and innovators to explore sustainable methods for recycling PU foam. Consequently, the recycling of PU waste is acknowledged as a crucial necessity because of its economic and environmental advantages [2].

In 2020, global output of virgin plastics was nearly 367 million tonnes, with Europe accounting for 55 million tonnes. In 2020, the European plastics industry generated a revenue above 330 billion euros. A total of 29.5 million tonnes of plastic waste were collected in the EU27 +3 for treatment. A total of 34.6% of this quantity was recycled, 42% was allocated for energy recovery, and 23.4% was disposed of in landfills. The automotive industry constitutes the third largest end-use market for plastics in Europe, accounting for around 9% of demand. In 2018, around 80% of recycled plastic generated in Europe was reintegrated into the European economy for the production of new products. Three percent of this amount went to the automobile industry [3].

The automobile sector has experienced substantial transformations over time, mostly driven by the quest for innovation, efficiency, and sustainability. The significant integration of plastics has critically impacted automotive design and production. Plastics provide numerous benefits that address the always advancing aspects of contemporary automotive manufacturing [4]. Historically, metals have predominated the automotive sector owing to their durability and resilience [5]. Nevertheless, automobile manufacturers have used lightweight materials due to increasing recognition of environmental issues, rigorous fuel efficiency standards, and the quest for improved automotive performance. Since the past century, these lightweight materials have been widely utilised in numerous vehicle components, including dashboards and bumpers, etc. [6]. Due to their remarkable strength-to-weight ratio, plastics have become a powerful alternative to conventional materials, significantly aiding in the decrease in overall automotive weight and the development of fuel efficiency [7]. The versatility of plastics in design is a significant factor contributing to their increasing use in the automobile sector. Plastics, capable of being shaped into diverse forms and dimensions, allow automakers to pursue new and visually appealing automobile designs, meeting consumer expectations for style and elegance [8].

In reaction to increasing environmental issues, the automotive sector is systematically implementing sustainable practices, particularly aimed at diminishing greenhouse gas emissions and improving recyclability. Plastics have significantly advanced in this field, with continuous research and development efforts concentrating on bio-based and recyclable plastics. These sustainable options reduce environmental impact and foster a circular economy by encouraging the reuse and recycling of materials. Nonetheless, despite the clear benefits, the integration of plastics in the automobile sector is not without difficulties. Automakers and researchers in this domain face challenges related to material deterioration in extreme environmental conditions, potential durability issues, and the necessity to guarantee fire safety in vehicles. Successfully confronting these problems is essential for maintaining the integrity and safety of plastic components in automobiles [9].

There are primarily two categories of acoustic packaging materials: sound absorption materials and sound insulation materials. Sound insulation materials possess a high surface density and can reflect sound energy in the direction of incidence. However, the sound absorption materials are lightweight and exhibit considerable porosity, facilitating the easy penetration of sound waves into their interior [10]. The sound absorption and sound insulation capabilities of acoustic package materials cannot achieve their maximum values concurrently. Polyurethane (PU) foam serves as an efficient sound absorption material in the automotive industry, owing to its effective sound damping properties and low density. It has been extensively utilised in interior elements, including seats and inner dashboard mats as well as other acoustic trim components. Acoustic wave propagation in PU foams mostly dissipates through viscous friction within linked pores and thermal exchange at the solid–fluid interface [11]. Nevertheless, Pure PU foam exhibits significant sound absorption capabilities mostly in the high-frequency range due to its unique pore geometries. Prior research has demonstrated that the acoustic properties of PU foams can be altered by including functional particles or modifying the chemical compositions of the foams [12,13,14,15,16,17,18]. However, merely combining the materials not only fails to provide optimal acoustic performance but also results in material waste.

Several established acoustic models have been developed to predict the acoustical behaviour and pore characteristics of porous and granular materials using impedance tube measurements. Among the widely used approaches are the Johnson–Champoux–Allard–Lafarge (JCAL) model, [19,20], the empirical model proposed by Miki, and more recent non-uniform pore size distribution models developed for porous media analysis. These modelling approaches have been extensively applied for estimating important non-acoustical parameters, including airflow resistivity, porosity, and tortuosity, which significantly influence sound absorption behaviour in porous materials. Due to their reliability and predictive capability, these models have also been incorporated into several commercial and research-based acoustic simulation platforms, including COMSOL Multiphysics and AlphaCell [21].

The Johnson–Champoux–Allard (JCA) model has been widely applied for describing the dynamic density and bulk modulus of porous materials through parameters such as porosity, tortuosity, airflow resistivity, and viscous and thermal characteristic effects. However, direct implementation of some correction functions within the JCA framework may involve complex calculations, especially over broad frequency ranges. To improve computational efficiency and analytical tractability, Padé approximation techniques can be employed to approximate the frequency-dependent correction functions while maintaining acceptable prediction accuracy. In this study, experimentally obtained sound absorption data for recycled automobile dashboard composites were integrated with a JCA/Padé-based computational approach implemented in MATLAB to analyse the acoustic performance of the developed materials. The combined experimental and computational framework provides additional insight into the acoustic behaviour of sustainable automotive polymeric composites and supports the development of efficient predictive methodologies for porous acoustic materials.

Previous studies have investigated the acoustic behaviour of porous materials using impedance tube measurements under normal incidence conditions across a broad frequency range. For instance, materials derived from wood chips have been characterised by measuring sound absorption coefficients over frequencies typically spanning 63–6300 Hz and for varying sample thicknesses and particle sizes. In addition to acoustic measurements, key non-acoustic parameters such as porosity and airflow resistivity were obtained independently using standard methods. To model the acoustic response, other several analytical approaches for rigid porous media have been employed, including the slanted slit (SS) model, the Johnson–Champoux–Allard (JCA) model, the Johnson–Champoux–Allard–Lafarge (JCAL) model, and models based on non-uniform pore size distributions. In such studies, additional parameters required for JCA and JCAL are typically estimated through optimisation techniques, including evolutionary algorithms and finite element simulations, to achieve agreement between experimental and predicted absorption spectra [22]. In the present study however, emphasis is placed on the use of the Padé approximation to improve computational efficiency and predictive capability by importing the experimental frequency-dependent acoustic data into MATLAB, generating model response curves, and comparing the predicted and experimental sound absorption behaviour across the investigated frequency range. Statistical comparison metrics, including R², RMSE, and MAE, were used to evaluate the level of agreement between the experimental and model-predicted responses.

Park et al. also conducted a multi-scale numerical analysis of sound absorption by resolving flow issues in a representative unit cell (RUC) and the pressure acoustics equation, adopting the Johnson–Champoux–Allard (JCA) model; while [23] used the Grey Relational Analysis (GRA) method and Multi-objective Particle Swarm Optimisation (MOPSO) Algorithm to improve the acoustic performances of PUF composites, setting the average sound absorption coefficient and average transmission loss as optimisation objectives. This paper enhances the acoustic properties of PU foam composites through the optimisation of the synthetic formulation (catalyst and particle size variation) and using the JCA and Padé approximation models.

2. Materials and Methods

2.1. Materials

The binder, methylene diphenyl diisocyanate (MDI), catalyst (diisooctyl 2,2′-[(dioctylstannylene) bis(thio)] diacetate), and surfactant were supplied by Rosehill polymers, Halifax, UK. The blower, water, was obtained from the laboratory (University of Bradford) tap at room temperature and under standard atmospheric conditions. The filler (dashboard crumbs) was supplied by Rochdale granulators, Rochdale, UK.

2.2. Sample Preparation and Formulation Design

• Sieve Analysis

Sieve analysis was conducted to characterise the particle size distribution of the dashboard crumbs (DBC) prior to composite fabrication. A representative mass of 200 g of the shredded DBC was subjected to mechanical sieving using a laboratory test shaker for a duration of 20 min. Seven sieve sizes were employed and arranged in descending order: 5 mm, 4 mm, 3.35 mm, 2 mm, 1.4 mm, 0.71 mm, and <0.71 mm. The mass retained on each sieve was recorded to quantify the particle size distribution as shown in

Table 1. Sieve analysis results showing particle size distribution, retained mass, percentage retained, and cumulative percentage passing of recycled automobile dashboard particles.

Sample Name	Sieve Size (mm)	mi (g)	%Ri	%Pi
PS1	<0.71	27.20	13.60	0.10
PS2	0.71	21.00	10.50	13.70
PS3	1.40	15.60	7.80	24.20
PS4	2.00	41.90	20.95	32.00
PS5	3.35	26.50	13.25	52.95
PS6	4.00	30.20	15.10	66.20
PS7	5.00	37.40	18.70	81.30
Total	* 199.80		* 99.90

* (Percentage loss = 0.1% (0.2 g)).

and the images are presented in Figure 1. Figure 2 presents samples prepared using particle sizes obtained from various sieve sizes in Table 1.

The mass retained on each sieve was determined using the standard relationship:

$$m_i=W_i$$

where $m_i=mass retained on sieve i \left(g\right)$

• $W_i=measured mass of material retained on sieve i \left(g\right)$

The total recovered mass after sieving is as follows:

$${\sum }_{i=1}^n{M}_r$$

(1)

where $n=total number of sieve (which is 7)$

• $M_r=total recovered mass \left(g\right)$

Percentage mass retained on sieve i is as follows:

$$\%R_i={\displaystyle \frac{W_i}{M_0}}\times 100$$

(2)

Cumulative percentage passing is as follows:

$${\%P}_i=100-\%R_i$$

(3)

(%P_i is calculated from the largest sieve size)

• $M_0=initial sample mass \left(200g\right)$

The results indicate a broad particle size distribution, with a higher proportion of material retained in the intermediate size ranges, suggesting favourable packing characteristics for composite formation. Following sieving, the classified fractions were recombined and used as filler in the preparation of composite samples to maintain a representative distribution of particle sizes. The composites were produced using a polyurethane binder system, with the binder content fixed at 100 wt%, blower fixed at 80 wt%, and filler fixed at 60 wt% while varying the catalyst and surfactant. The formulation variables were systematically varied: particle sizes (as shown in Table 1 above) and the catalyst and surfactant from 1 to 7 wt%, expressed relative to the binder.

B.

Sample formulation design

Composite samples were prepared using a polyurethane-based formulation in which the binder content was kept constant at 200 g (100 wt%) for all experiments. The preparation process was carried out in a controlled mixing procedure to ensure uniformity across all formulations. Initially, the binder was weighed into a mixing bowl, after which the catalyst was added in drops at varying concentrations ranging from 1 to 7 wt% relative to the binder. The mixture was then stirred to ensure adequate dispersion of all the components within the binder matrix. Subsequently, the blowing agent was introduced at a fixed level of 160 g (80 wt% relative to the binder) and thoroughly mixed until a homogeneous system was achieved. Following this, 120 g (60 wt%) of the polymeric filler (DBC) was gradually incorporated into the mixture and stirred to ensure uniform distribution. The mixtures were thoroughly homogenised to ensure uniform dispersion of the filler and additives and subsequently moulded into square-shaped specimens measuring 200 × 200 × 30 mm. The samples were allowed to cure for 24 h prior to testing.

Two categories of composite samples were produced based on the experimental focus of this study:

• Catalyst optimisation samples: seven formulations were prepared by varying the catalyst concentration (1–7 wt%) while keeping all other parameters constant;
• Particle size optimisation samples: seven additional samples were produced using filler materials corresponding to the different particle size fractions (PS1–PS7) obtained from the sieve analysis.

This experimental design allowed for the independent evaluation of catalyst concentration and particle size effects on composite performance, providing a structured basis for subsequent optimisation analysis. The formulation of the composite samples was defined relative to the binder content, which was taken as the reference component (100 wt%). A fixed mass of 200 g of binder was used for all the samples. The blowing agent was added at 80 wt% relative to the binder, while the filler was incorporated at 60 wt% relative to the binder. Catalyst content was varied between 1–7 wt% relative to the binder presented in

Table 2. Composition of catalyst-based samples.

Sample	Binder (g)	Blower (g)	Catalyst (g)	Catalyst (wt%)
Cat1	200.00	160.00	2.00	1.00
Cat2	200.00	160.00	4.00	2.00
Cat3	200.00	160.00	6.00	3.00
Cat4	200.00	160.00	8.00	4.00
Cat5	200.00	160.00	10.00	5.00
Cat6	200.00	160.00	12.00	6.00
Cat7	200.00	160.00	14.00	7.00

No surfactant added. Sample thickness = 30 mm.

. This approach ensures consistent formulation across samples while allowing systematic investigation of the effect of catalyst concentration and surfactant as presented in

Table 3. Composition of particle size variation samples.

Sample	Particle Size	Surf/Catalyst (g)	Surf/Catalyst (wt%)
PS1	<0.71	2.00	1.00
PS2	0.71	4.00	2.00
PS3	1.40	6.00	3.00
PS4	2.00	8.00	4.00
PS5	3.35	10.00	5.00
PS6	4.00	12.00	6.00
PS7	5.00	14.00	7.00

Binder, blower, and catalyst quantity are the same as in Table 3, except for the addition of surfactant. Sample thickness = 30 mm.

.

C.

Acoustic Absorption Measurement

The acoustic absorption of the developed composite samples was evaluated using an impedance tube method in accordance with standard procedures for normal incidence sound absorption measurement. A 44 mm diameter medium impedance tube was employed to determine the sound absorption coefficient of the samples over a frequency range 120–3200 Hz. Cylindrical specimens with a diameter of 44 mm and height 30 mm were cut from the centre of the square samples along the rising direction. The measurements were conducted under normal incidence conditions using a three-microphone transfer function method, where incident and reflected sound waves were analysed to determine the acoustic response of the material [24]. The sound absorption coefficient (α) was calculated as a function of frequency based on the ratio of absorbed to incident sound energy, $\alpha =1-|{R|}^2$. The measurements were performed across the specified frequency range and the resulting absorption spectra were obtained for each sample.

The measured acoustic parameters were subsequently used as responsive variables in the optimisation analysis to evaluate the influence of formulation variables on material performance.

D.

Airflow Resistivity (σ)

Airflow resistivity of the composite samples was determined to evaluate resistance to air movement through the porous structure. This parameter represents the ratio of pressure drop to airflow velocity across the material and is a key indicator of acoustic performance in porous media. It is based on forcing a steady or slowly oscillating airflow through a porous material. The picture in Figure 3B shows the equipment used for this experiment. Compressed air was passed through a compressor into the glass funnel and directed into the sample. The air pressure was maintained at 2 Pa. Measurements were conducted under controlled conditions, and the results were used to support the analysis of sound absorption behaviour. Flow resistivity can be defined as $\sigma =\left(P_2-P_1\right)S/vh$, where v (m³/s) represents the volumetric airflow rate through the material, S denotes the cross-sectional area of the material, and h indicates the thickness of the material. The unit of flow resistivity is (Pa.s/m²). This was measured according to ISO 9053 [25].

E.

Tortuosity (α∞)

Tortuosity was assessed to characterise the complexity of the pore pathways within the composite structure. It describes the degree to which air must follow a complex path through the material, influencing sound wave propagation and energy dissipation. Higher tortuosity is generally associated with improved acoustic absorption due to increased interaction between sound waves and the pore walls. Here, the sample size used was the 100 mm diameter size. The equipment used for evaluating the tortuosity of the samples can be seen in Figure 3D above. A high-frequency tweeter is above a PVC tube. The tweeter generates an ultrasonic wave at 48 kHz, which is received by two 1/4 microphones at the top and bottom of the sample, the centres of the microphones being aligned with the centre of both the sample and the tweeter. A separate measurement is performed without a sample, to determine the true sound velocity in air.

F.

Porosity (ϕ)

This was evaluated to quantify the fraction of void space within the composite samples. It is defined as the ratio of void volume to total volume and plays a critical role in determining acoustic performance. The porosity of the samples influences airflow resistivity and sound absorption by governing the accessibility and connectivity of the pore network. The device (Figure 3E) consists of two 70 mL chambers, a U-tube manometer, and pistons used to measure the water displacement and the air volume. The porosity measuring method starts with a calibration to balance the volumes of the reference chamber and measurement chamber without the sample, with the measurement pistons in their zero positions. The valves are opened and the two compartments are brought to the same atmospheric pressure. Then the valves are sealed and a minimum quantity of water is extracted. If the two chambers are equal in volume, then there is no pressure difference. Or the two volumes may be equalised by altering the calibration piston attached to the reference chamber. After calibration, the porous sample is introduced into the measurement chamber. At this point both chambers are exposed to the pressure of the air.

G.

Scanning Electron microscopy (SEM)

The surface morphology of the composite samples was characterised using a tabletop scanning electron microscope (SEM) (Hitachi TM3000), which provides high-resolution imaging of the microstructure of the material. Small specimens were taken from the central part of the foams along the path of rise.

H.

Johnson–Champoux–Allard (JCA)

The acoustic behaviour of the developed porous composites was further analysed using the Johnson–Champoux–Allard (JCA) model, which describes sound propagation in rigid-frame porous materials based on key microstructural parameters. The model incorporates airflow resistivity, porosity, tortuosity, and characteristic lengths to predict the effective dynamic density and bulk modulus of the material as functions of frequency.

I.

Padé Approximation

Padé Approximation (PA) was used to represent the frequency-dependent behaviour in a simplified analytical form. Padé is a non-acoustic model derived from four quantifiable parameters: flow resistivity, porosity, tortuosity, and the standard deviation of the log-normal distribution of pore size. The primary benefits of using a PA are as follows: (i) the capacity to consider the specific characteristics of the desired solution and articulate complex behaviour near internal and external boundaries, (ii) the expression of the interpolated solution through an analytical formula, and (iii) the potential for subsequent mathematical examination of the derived analytical expressions in the form of a Padé approximation [26,27,28]. The experimentally measured sound absorption coefficients were imported into MATLAB version 24a for computational analysis using the JCA framework and Padé approximation. The implemented model was used to predict the frequency-dependent acoustic behaviour of the developed composites, and the predicted responses were compared with experimental measurements.

The use of efficient approximation techniques, such as the Padé approximation, aligns with broader efforts in the literature to improve computational efficiency and predictive capability in complex systems. For instance, intelligent fusion-based approaches have been applied to enhance model accuracy in structural monitoring applications [1]. While the present study does not employ machine learning techniques, it focuses on improving predictive performance through efficient modelling strategies.

3. Results and Discussion

3.1. Particle Size Distribution

The particle size distribution of the DBC was quantified using mass-based sieve analysis. The results indicate that the material exhibits a broad distribution with a significant proportion of particles within the intermediate size range (2–5 mm) presented in Figure 4, accounting for over 50% of the total mass. Such a distribution is advantageous for composite fabrication as it promotes improved packing density and inter-particle contact, which can influence pore structure development and consequently acoustic performance.

3.2. Sound Absorption

Sound waves typically induce vibrations in cell walls and the air within cavities, with sound energy being emitted by the dampening of these vibrations in the cavity walls and air [29,30,31]. Furthermore, sound absorption performance can be enhanced through sound damping, which can be augmented by increasing the rigidity of cell walls [29,32,33]. Also, interconnectedness, along with the quantity, dimensions, and characteristics of pores, are significant considerations. These factors should be taken into account when examining sound absorption mechanisms in porous materials, as the penetration of acoustic waves into the porous structure governs the dissipation of acoustic energy through visco-thermal interactions [29,34]. The reduction in cavity size in an open pore structure often results in heightened airflow resistance and enhanced sound absorption performance [29,35,36]. In addition, the increase in interconnectivity within porous media may create irregular pathways for sound wave transmission. In this context, large porous cells or interconnected cells enhance sound absorption at lower frequencies.

The sound absorption behaviour of the composite samples with particle size variation and those with varying catalyst concentration is presented in Figure 5 and Figure 6. The results indicate that variations in particle size had a relatively limited influence on the acoustic performance of recycled materials. Across the different particle size fractions, the absorption curves followed a similar trend, with peak absorption coefficients of approximately 0.85 observed within the mid-frequency range of 1000–1400 Hz. This suggests that, within the investigated range, particle size distribution does not significantly alter the dominant sound absorption mechanisms, likely due to the preservation of overall pore connectivity and structural characteristics in the composites.

In contrast, catalyst concentration exhibited a pronounced effect on the acoustic performance of the materials. Increasing the catalyst content from 1 wt% to 5 wt% resulted in a progressive improvement in sound absorption, indicating enhanced pore structure development and more effective energy dissipation within the material. Notably, the sample with 5 wt% catalyst (Cat5) demonstrated near-complete absorption of incident sound waves, with absorption coefficients approaching unity within the frequency range of 1200–1800 Hz. This indicates that the system exhibits a well-defined optimum rather than a monotonic trend, which is characteristic of process-driven optimisation problems.

However, further increases in catalyst concentration beyond 5 wt% led to a decline in acoustic performance. This behaviour suggests the existence of an optimal catalyst concentration, beyond which the reaction kinetics become excessively rapid, potentially resulting in non-uniform cell structures, increased pore coalescence, or reduced structural integrity. Such effects can negatively impact airflow resistivity and tortuosity, thereby reducing sound absorption efficiency.

Overall, the results demonstrate that catalyst concentration is a critical parameter in the formulation of the recycled composites, with an optimal value of approximately 5 wt% identified for maximising acoustic performance. Similar trends were reported by A. Khan, 2017, where the sound absorption coefficient of recycled tyre shred residue (TSR) was optimised by varying the binder-to-TSR ratio [37]. This finding highlights the importance of controlled reaction kinetics in achieving desirable pore structures and provides a clear basis for formulation optimisation in subsequent analysis.

Figure 7 below shows the comparison between experimental and model-predicted sound absorption coefficients for the catalyst-based samples. Both the Padé approximation and JCA model generally show good agreement with the experimental results across the investigated frequency range. The models successfully capture the major absorption peaks and overall acoustic trends observed experimentally.

For lower catalyst concentrations (Cat1–Cat3), slight deviations are observed at higher frequencies, where the models tend to underpredict the experimental absorption behaviour. In contrast, stronger agreement is observed for Cat4–Cat6, particularly around the primary absorption peak region.

Although some discrepancies remain at higher frequencies, the overall results confirm the capability of both modelling approaches to predict the acoustic performance of the developed composites. The Padé approximation generally exhibits slightly lower error values, indicating improved predictive accuracy for several catalyst configurations.

Figure 8 also compares the experimental and model-predicted sound absorption coefficients for the particle size (PS) samples. Both the Padé approximation and JCA model.

These exhibit strong agreement with the experimental data across all particle size variations. The predicted curves closely follow the experimental absorption behaviour, particularly within the low- and mid-frequency regions. Minor deviations are observed at higher frequencies, where the models slightly smooth out some experimental fluctuations. Nevertheless, the overall acoustic trends and resonance behaviour are effectively captured by both models. Compared with the JCA model, the Padé approximation generally provides slightly improved agreement with the experimental data, particularly in terms of reduced deviation around the absorption minima and peak regions.

The close agreement observed across all PS samples supports the robustness and reliability of the modelling framework for predicting the acoustic behaviour of recycled polymeric composites with varying particle sizes.

The predictive capability of the developed acoustic models was evaluated by comparing experimentally measured sound absorption coefficients with those obtained from the Johnson–Champoux–Allard (JCA) model and Padé approximation across the investigated frequency range.

As shown in Figure 7 and Figure 8, both models successfully capture the overall trend of the acoustic response, characterised by a gradual increase in absorption at low frequencies, followed by a peak in the mid-frequence range (approximately 1000–1400 Hz), and a subsequent variation at higher frequencies. This behaviour is consistent with typical porous acoustic materials, where sound absorption is governed by viscous and thermal dissipation mechanisms within the pore structure.

The Padé approximation demonstrates close agreement with the experimental data across most of the frequency spectrum, particularly in the low- and mid-frequency regions. Its ability to accurately reproduce the peak absorption behaviour indicates that it effectively captures the dominant acoustic mechanisms while maintaining computational simplicity.

From an optimisation perspective, the Padé approximation offers a significant advantage due to its reduced computational complexity and strong agreement with experimental data. This makes it particularly suitable for iterative optimisation processes, where repeated evaluations of acoustic performance are required. The JCA model, on the other hand, provides valuable physical insight into the relationship between microstructural parameters and acoustic behaviour, supporting the interpretation of experimental trends.

Overall, the combined use of the JCA model and Padé approximation provides a robust framework for both understanding and optimising the acoustic performance of the developed composites. While the JCA model offers a physically informed basis for analysis, the Padé approximation enables efficient prediction, making it a practical tool for formulation optimisation. The table below shows the experimental data and the modelled data for the acoustic parameters.

The parameters presented in

Table 4. Experimental and modelled acoustic parameters of particle size (PS) samples, including porosity (ϕ), flow resistivity (σ, Pa.s/m²), and tortuosity (α∞).

Experimental				JCA			Padé
Sample	$\mathsf{\varphi }$	σ (Pa.s/m2)	${\mathsf{\alpha }}_{\mathbf{\infty }}$	$\mathsf{\varphi }$	σ (Pa.s/m2)	${\mathsf{\alpha }}_{\mathbf{\infty }}$	$\mathsf{\varphi }$	σ (Pa.s/m2)	${\mathsf{\alpha }}_{\mathbf{\infty }}$
PS1	0.81	73,700	1.33	0.87	63,962	1.78	0.69	58,122	1.75
PS2	0.80	53,300	1.33	0.85	87,996	1.00	0.67	44,376	1.06
PS3	0.80	63,900	1.17	0.85	72,523	1.00	0.71	43,771	1.52
PS4	0.79	88,000	1.45	0.85	86,675	1.00	0.68	42,064	1.00
PS5	0.79	119,000	1.45	0.85	103,606	1.00	0.70	61,623	1.04
PS6	0.78	94,300	1.29	0.85	121,822	1.00	0.69	66,772	1.00
PS7	0.77	108,200	1.33	0.85	103,606	1.00	0.77	97,754	2.22

and

Table 5. Experimental and modelled acoustic parameters of catalyst (Cat) samples, including porosity (ϕ), flow resistivity (σ, Pa.s/m²), and tortuosity (α∞).

Experimental				JCA			Padé
Sample	$\mathsf{\varphi }$	σ (Pa.s/m2)	${\mathsf{\alpha }}_{\mathbf{\infty }}$	$\mathsf{\varphi }$	σ (Pa.s/m2)	${\mathsf{\alpha }}_{\mathbf{\infty }}$	$\mathsf{\varphi }$	σ (Pa.s/m2)	${\mathsf{\alpha }}_{\mathbf{\infty }}$
Cat1	0.79	73,067	1.21	0.87	54,904	1.00	0.63	32,052	1.46
Cat2	0.80	60,067	1.33	0.85	96,321	1.00	0.68	29,939	1.04
Cat3	0.82	56,700	1.45	0.85	56,628	1.84	0.71	23,602	1.58
Cat4	0.82	56,700	1.49	0.85	41,672	1.30	0.78	19,943	1.58
Cat5	0.83	56,757	1.29	0.85	46,103	1.00	0.81	26,921	1.78
Cat6	0.84	95,327	1.41	0.85	56,718	1.00	0.80	49,944	2.03
Cat7	0.84	73,067	1.29	0.85	52,094	1.00	0.78	42,934	1.82

represent both experimentally measured and model-predicted values used in the acoustic modelling framework. Porosity (ϕ), airflow resistivity (σ), and tortuosity (α_∞) were determined experimentally. The JCA and Padé values correspond to model predictions based on the measured input parameters. The results from Table 4 and Table 5 show the effect of the formulation on the acoustic and non-acoustic behaviour of the recycled granulates. Increasing the particle size of the formulation tends to reduce the porosity and increase the airflow resistivity. Conversely, increasing the catalyst was observed to increase the porosity and reduce the airflow resistivity up to Cat5 (5 wt%).

To further demonstrate the applicability of the modelling framework in acoustic analysis, the modelling approach considered key physical parameters influencing porous acoustic behaviour, including material thickness, airflow resistivity, porosity, tortuosity, and particle size distribution (PSD)-related structural characteristics. These parameters are closely associated with the microstructural and transport behaviour of porous media and influence the frequency-dependent acoustic response of the developed composites. Experimentally obtained sound absorption data were imported into MATLAB, where the Johnson–Champoux–Allard (JCA) framework and Padé approximation was implemented to analyse and predict the acoustic behaviour over the investigated frequency range. The Padé approximation was employed to express the dynamic density and bulk modulus in a simplified analytical form, thereby improving computational efficiency during the modelling process. The combined experimental and computational approach provides additional insight into the relationship between porous structure and acoustic performance while supporting efficient acoustic analysis of recycled automotive polymeric composites. The model helps to optimise composite formulations by including quantifiable properties mentioned earlier, making it possible to find the best combinations of parameters for sound absorption performance. This shows that the Padé-based framework could be a useful tool for predicting and designing sustainable acoustic materials.

To further evaluate the predictive performance of the modelling approaches, quantitative validation was conducted across all PS and Cat samples using mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R²) as presented in

Table 6. Quantitative validation of model performance for PS samples using MAE, RMSE, and R², comparing the predictive accuracy of the Padé approximation and the JCA model against experimental sound absorption data.

Padé				JCA
Sample	MAE	RMSE	R2	MAE	RMSE	R2
PS1	0.02	0.04	0.98	0.04	0.04	0.99
PS2	0.02	0.02	0.99	0.03	0.04	0.99
PS3	0.02	0.02	0.99	0.03	0.03	0.99
PS4	0.02	0.02	0.99	0.03	0.01	0.99
PS5	0.02	0.02	0.99	0.03	0.04	0.99
PS6	0.02	0.02	0.99	0.03	0.03	0.99
PS7	0.02	0.02	0.99	0.03	0.04	0.99

and

Table 7. Quantitative validation of model performance for Cat samples using MAE, RMSE, and R², comparing the predictive accuracy of the Padé approximation and the JCA model relative to experimental measurements.

Padé				JCA
Sample	MAE	RMSE	R2	MAE	RMSE	R2
Cat1	0.02	0.03	0.95	0.02	0.02	0.99
Cat2	0.02	0.03	0.96	0.02	0.03	0.99
Cat3	0.03	0.05	0.94	0.03	0.05	0.99
Cat4	0.03	0.04	0.97	0.04	0.05	0.99
Cat5	0.04	0.06	0.99	0.05	0.05	0.99
Cat6	0.01	0.02	0.96	0.05	0.05	0.99
Cat7	0.03	0.05	0.94	0.03	0.04	0.99

. The results show that both JCA and Padé models exhibit strong predictive capability with consistently high R² values ranging from 0.94 to 0.99 across all samples. This confirms that both models effectively capture the overall trend of the experimental data. In terms of prediction accuracy, Padé approximation of PS samples show lower MAE and RMSE with values around 0.02 and 0.02–0.04, respectively. In comparison, the JCA model shows slightly higher error values for these samples, indicating marginally lower predictive accuracy, although the high R² values confirm that it still provides a strong representation of the experimental trends.

For the Cat samples, the performance of the two models is more comparable. While the Padé approximation exhibits improved accuracy for certain samples like Cat6, the JCA model performs similarly or slightly better in others as shown in Cat1 and Cat2. This suggests that the relative performance of the models is influenced by formulation parameters such as catalyst concentration.

Overall, the combined use of MAE, RMSE, and R² provides a robust validation framework, demonstrating that both models are suitable for predicting acoustic performance. However, the Padé approximation offers a slight advantage in predictive accuracy and computational efficiency, particularly for particle size variations, making it a practical tool for optimisation and material design applications.

Although quantitative validation metrics (RMSE, MAE, and R²) were evaluated for all investigated samples, Cat5 was selected for detailed graphical comparison because it exhibited one of the highest acoustic performances and demonstrated strong agreement between the experimentally measured and model-predicted absorption coefficients. The graphical representation therefore serves as a representative example illustrating the predictive capability of the implemented JCA/Padé-based computational framework, while the broader validation results for all samples are summarised in Table 6 and Table 7.

The comparison between model predictions and experimental data observed in this study is consistent with findings reported in the literature for porous materials. Previous studies have shown that models such as the JCA and JCAL generally provide good agreement with measured absorption behaviour, particularly when model parameters are obtained through fitting procedures. However, it has also been noted that such agreement may be influenced by the fitting process itself, which can introduce non-uniqueness in parameter estimation [22].

In addition, discrepancies between predicted and experimental absorption spectra are often observed at higher frequencies, where models tend to exhibit more pronounced resonance behaviour than measured data. This difference has been attributed to factors such as bandwidth differences between measurements and predictions, as well as additional damping mechanisms not captured by the models. Surface characteristics, including roughness and structural irregularities, may also contribute to enhanced sound absorption in real materials compared to idealised model predictions [38,39].

These observations support the trends obtained in the present study, where strong overall agreement is achieved between model predictions and experimental results, with some deviations attributable to modelling assumptions and material heterogeneity.

3.3. Image Analysis

The microstructure of the developed composite as observed using Hitachi TM3000 (Figure 9) reveals a heterogeneous porous structure characterised by a combination of open and partially open cells. The presence of interconnected pores and voids of varying sizes indicates a well-developed cellular network which is essential for effective sound absorption. Pore sizes observed in the micrograph are on the order of hundreds of micrometers with some larger voids and smaller dispersed pores distributed throughout the matrix. This multi-scale porosity contributes to enhanced acoustic performance by facilitating sound wave penetration and increasing viscous and thermal dissipation within the material. The observed structural features are consistent with the measured acoustic behaviour. In particular, the more uniform and interconnected pore structure observed in the optimal sample (Cat5) supports its superior sound absorption performance, while the increased irregularity and partial pore closure in other samples may contribute to reduced efficiency.

Overall, the observed microstructure supports the acoustic results, as the combination of pore connectivity, size distribution, and structural heterogeneity is favourable for improved airflow resistivity and tortuosity, thereby enhancing sound absorption performance. The SEM images of Cat5 and Cat6 are presented in Figure 10.

3.4. Pore Size Distribution Analysis

Additional structural interpretation of the acoustic behaviour observed during testing was provided by pore size distribution analysis for further investigation into the porous morphology of the developed samples. Figure 11 and Figure 12 show the probability density function (PDF) plots of the pore area distribution of the PS modified and the catalyst modified sample groups, respectively. The distributions show a significant difference in the pore size properties for the formulations investigated. Some samples in the PS series exhibited wider pore distributions and higher pore density in particular pore size ranges, suggesting variations in internal pore structure and pore connectivity. Similarly, differences were also observed across the catalyst-modified samples where changes in catalyst concentration seem to affect pore formation and distribution behaviour. Negative values are due to the log transformation and indicate pore areas of less than 1 mm².

The differences in pore morphology observed may contribute to the differences in sound absorption performance by affecting the interaction with airflow, energy dissipation, and propagation of sound waves within the porous structure. Samples with more distributed and interconnected pore structures generally showed better acoustic absorption behaviour in some frequency ranges. However, the relationship between pore morphology and acoustic performance is still complex and the present analysis is intended primarily as supportive structural characterisation rather than a direct quantitative prediction of acoustic behaviour. The pore size distribution analysis provided further insight into the microstructural characteristics of the recycled polymeric composites and supported the qualitative observations attained from the SEM analysis.

While the pore size distribution analysis gave more structural insight into the materials created, advanced picture segmentation, comprehensive quantitative microstructural modelling, and direct pore–acoustic correlation analysis were outside the scope of the present investigation. These characteristics are proposed for future studies in order to improve further the understanding of the relationship between pore structure and acoustic performance.

4. Conclusions

This study investigated the development and optimisation of sustainable acoustic composites developed from automotive polymeric waste. The influence of formulation parameters, particularly catalyst concentration and particle size distribution, on the acoustic performance of the materials was systematically evaluated through experimental analysis and modelling.

The results demonstrated that particle size variation had a limited effect on sound absorption behaviour, with all samples exhibiting similar trends and peak absorption within the mid-frequency range. In contrast, catalyst concentration was found to significantly influence acoustic performance, exhibiting a non-linear relationship with sound absorption. An optimal catalyst content of 5 wt% was identified, where the composite achieved near unity absorption within the frequency range of 1200–1800 Hz. Beyond this point, further increase in catalyst content resulted in reduced performance, attributed to changes in pore structure and kinetics. The integration of experimental measurements with acoustic modelling using JCA and Padé approximation provided a robust framework for understanding and predicting material behaviour. The Padé approximation demonstrated strong agreement with experimental data and proved particularly suitable for optimisation due to its computational efficiency. The minor deviations observed at higher frequencies where both models tend to underestimate the absorption may be attributed to the simplifying assumptions of ideal pore structure and material homogeneity, as well as increased sensitivity of acoustic response to microstructural variations at higher frequencies. Overall, the models demonstrate reliable predictive capability for optimisation purposes. The findings highlight the importance of controlled formulation in achieving high-performance acoustic materials from recycled polymeric waste. The optimised composite developed in this study shows strong potential for application in noise control and building acoustics, contributing to sustainable material innovation and circular economy objectives. Future work may explore the integration of advanced data-driven and intelligent fusion approaches, such as those reported by Wang et al. [1], to further enhance prediction accuracy and enable real-time optimisation of acoustic material performance.