1 Introduction

Because of urbanization and technological advancement, there is an increase in noise in the environment from transportation, construction, industrial, and commercial activities, which causes hearing loss, auditory-related diseases, stress, anxiety, speech interruption, sleeplessness, and so on. Thus, finding a solution to this problem is of the utmost importance [1,2,3]. Structural failure is caused by high noise levels, which causes vibration and decreases durability and serviceability. Increased noise levels also shorten the lifespan of industrial equipment. In control valves, for example, flow instability causes vibrations that occasionally degrade the response to the control system, resulting in severe oscillations. The government’s laws and regulations limiting environmental noise demonstrate the need for noise reduction. Human activities primarily cause noise pollution. Lightweight, eco-friendly, flexible, thin, minimal cost and excellent sound insulation in the lower and higher frequency ranges are desirable properties of a material for sound absorption. Synthetic fibers are currently mainly employed as sound insulating materials, which are harmful to humans and the environment [4]. As a result, a sustainable substance such as castor oil-based organic polyurethane foam could be used instead of hazardous petroleum-based foam. Polyurethane is commonly defined as “bridging the gap between rubber and plastic material,“ as Otto Bayer invented it in 1937. Polyol and isocyanate are required to produce PU foam [5]. Urethane bonds link together organic units to form polyurethane. Due to its excellent mechanical and physical features, Polyurethane foam also has a wide range of uses, accounting for almost 29% of the material market in the building materials of buildings, automotive, beds, and furniture [6, 7].

Thus, polyurethane foam is the preferred material matrix for hybrid polymer composites [8]. Lightweight, stretchability, damping, superior biocompatibility, resistance to abrasion, biostability, low-temperature flexibility, anti-aging, and environmental endurance are some of the qualities of polyurethane composites (PUC) [9,10,11,12,13,14,15]. PU foam has several drawbacks, including poor conductivity, elastic modulus, high flammability, low thermal conductivity, and poor adherence to metallic surfaces. The addition of nanoparticles and natural fibers to PU foams has compensated for the limitations of PU foam [16]. With a yearly production of 17 billion tonnes, PU foam ranks sixth in global plastic sales [17]. Zirconium oxides are an appealing adsorbent because they are non-poisonous, have excellent thermal stability, have low solubility in water, and have excellent resistance to acids, bases, and oxidants [18, 19]. Because of its porous and highly hydrated structure, amorphous zirconium oxide has a high adsorption capacity [20]. Generally, zirconium is available in the form of nanofluids and nanocrystals. Zirconium is used in making insulating and fire retarding materials as they can resist high temperatures up to 2715 °C. Graphene is a hexagonal lattice of carbon atoms obtained from graphite [21]. It is an allotrope of carbon [22]. It can be made by exfoliating graphite, which produces separated 2D aromatic sheets made up of sp2 hybridized carbon atoms [23]. It finds widespread use in sensing instruments, nanocomposite materials, battery packs, super capacitors, and hydrogen storage [24]. It has gained popularity due to its excellent thermal, optical, mechanical, electrical, and magnetic properties. Long range 2D-conjugation is responsible for physical properties such as large theoretical specific area [25], high Young’s modulus of elasticity, and unparalleled thermal and electric conductivity [26, 27], as well as electrical properties such as ambipolar electric field effect and ballistic conduction of charge carriers [28]. It has excellent electrical conductivity and chemical stability, making it ideal for next-generation electronics [29,30,31]. It has a large surface area, high charge carrier mobility, high transparency, and high flexibility [32,33,34]. Bamboo charcoal has almost double the surface area and four times the absorption rate of conventional charcoal. Bamboo charcoal has a porous structure that can absorb moisture, odors, and polluted air particles such as formaldehyde and ammonia due to its numerous minute holes. The negative ion charge of bamboo charcoal attracts odors and extra moisture in the air, and it is recycled in the environment [35, 36]. The literature [37, 38] investigates studies of the sound absorption capabilities of commonly used materials.

The Johnson-Allard model uses a porous and rigid material as a frame and includes five non-acoustical parameters for calculating surface impedance: porosity, flow resistivity, tortuosity, viscous, and thermal characteristic length [39, 40]. It is a semi-phenomenological model popular for sound propagation modeling in porous media [41, 42]. Pennington modified the model for limp frames, which assumes resistance to fewer frames [43]. These non-acoustical parameters require laboratory facilities like ultrasonic measurement systems [44]. An inverse acoustic characterization method is used without such resources to reduce the difference between experimental and theoretically predicted data. Optimization techniques such as most minor square data fitting are employed. Porosity, flow resistivity, tortuosity, viscosity, and thermal characteristic length were determined by numerically solving an inverse problem defined in the time domain [45].

RSM (Response Surface Methodology) is a statistical technique for DOE that effectively identifies the link between many factors and one or more response variables. The primary goal of Response surface methodology in experimental design is to find the best response or output to a given input variable. It is an easy way to evaluate polynomial designs used as factorial experiments. RSM is essential in the real-time investigation as both the parameters are unknown. Also, it is beneficial to improve the experiment’s performance with a smaller number of trials [46,47,48]. COMSOL Multiphysics is a FEM analysis platform and multi-physics simulation software. It gives a wide range of solutions and provides a unified, integrated development environment in mechanical, fluid dynamics, electromagnetic interference, acoustics, and chemical applications. COMSOL can be defined as a versatile general-simulation analysis software that is mainly based on advanced numerical methods [49]. While compared with other Ansys tools like LS dyna, COMSOL is referred to as a user-friendly tool where we can combine multiple physics simultaneously, whereas ANSYS is considered quite difficult and tedious to solve a simple dynamic problem. More simply, COMSOL is much more oriented toward academia, whereas Ansys is industry oriented [50, 51].

The study aims to manufacture a smart PU foam using absorption and hydrothermal reduction techniques that are used in a wide range of applications, including automotive sound insulation, noise-producing walls and machinery, and aircraft and spaceship internal soundproofing. The nanocomposite PU foam is a unique advanced composite material due to the manufacturing technique of synthesizing the foam reinforced with a mixture of three distinct nanoparticles. The impacts of the weight% of Gr, ZrO2, and BC on SAC and NRC are being evaluated. The acoustic inverse characterization method has been used to obtain parameters like porosity, tortuosity, flow resistivity, viscous, and thermal characteristic length through SAC by MATLAB software. Further, response surface methodology, one of the DOE techniques for optimizing the best combination of nanoparticles that yields better NRC, is adopted. Additionally, the evaluation of experimental results of sound absorption coefficient was compared with inverse characteristics using Matlab code and with simulation using Comsol Multiphysics 5.5. After the best results from the comparative study, we recommend this smart polyurethane foam for a wide variety of applications.

2 Methodology

2.1 Materials

The Clear Castor Oil (CO) was bought from Jayant Agro-Organics Ltd. in Mumbai, India, with a hydroxyl value of 160 mg KOH/g, an acid value of 1.6 mg KOH/g, a peroxide value of 5.0, a saponification value of 178 and a refractive index of 1.476, respectively. This study used Voranol 3010, a triolpolyether polyol with viscosity (@25 ̊C) of 475 centistokes, a hydroxyl value of 56.5, and functionality of 3 from Jayant Agro-Organics Ltd., Mumbai, India. The foaming process was aided by using tin(II) 2-ethylhexanoate from Sigma-Aldrich as a catalyst. The diisocyanate used in this study was pTDI (polymeric toluene diisocyanate), with viscosity (@25 ̊C) of 3 centipoise, NCO content of 48.2%, and functionality of 2 bought from BASF under Lupranate T80. The catalyst for the foaming process was Niax B11 plus, while the surfactant for foam preparation was Niax L618 (Manali Petrochemicals Limited, Chennai, India). Distilled water was taken from the lab and employed as a chemical blowing agent for foam creation. The chemical composition of the components mentioned above is described in Table 1. Commercially available Zirconium Oxide, Bamboo Charcoal nanoparticles with an average size of 20–50 nm, and Graphene nanoparticles with a mean size of 5–10 nm (provided by Sigma Aldrich, India, 99.9% purity) were used as reinforcing materials. Graphene, Zirconium Oxide, and bamboo charcoal nanoparticles are described in detail in Table 2.

Table 1 Chemical composition of CO-based flexible polyurethane foam
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Table 2 Details of nanoparticles reinforced in CO-based flexible polyurethane foam
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2.2 Preparation of PU foam nanocomposites

Voranol 3010, a triol polyether polyol, and castor oil (CO) were dehydrated individually in a revolving evaporator below a vacuum at 110 ̊C for 24 h before the reaction. In comparison, nanoparticles were dried at 110 ̊C for 2 h. Then, in a mold with dimensions of 120 × 120 × 2 cm, pre-weighted (50%) polyether polyol and castor oil were combined. As a catalyst, tin (II) 2-ethylhexanoate was added to this mixture and mixed well for 5 h at 3600 r/min in an inert atmosphere in an oil bath at 65 °C. Before adding a determined quantity of amine catalyst and surfactant, the mixture was stored in an oven for 12 h. Water was mixed into the solution as a chemical blowing agent and swirled at 3500 rpm for 18 s. Lastly, premeasured polymeric toluene diisocyanate was mixed, and mixing was maintained at 2500 r/min for another 18 s [52, 53]. The viscous liquid was rapidly and carefully placed into an equipped mold and allowed to rise freely. In this work, we have adopted the absorption, and hydrothermal reduction methods for fabricating nanoparticle reinforced flexible smart PU foam. The polyurethane foam sample is immersed in an ethanol container at room temperature for about 10 min. For homogeneous distribution and to avoid nanoparticles settling at the bottom of the container, the necessary weight% of zirconium oxide, bamboo charcoal, and graphene are blended with ethanol in another container for 20 min using an ultrasonic probe sonicator. The polyurethane foam is then entirely immersed in the nanoparticles solution for 20 min. The self-assembling nature of nanoparticles results in uniform nanoparticle dispersion throughout the foam. To vaporize the ethanol, the soaked foam is heated in a furnace at 80 °C for 60 min. The foam is removed carefully from the furnace once it has dried and is devoid of ethanol. The circular structure of size 10 cm in diameter and 2 cm in thickness is designed and fabricated to achieve a uniform, a symmetrical outer surface area so that the maximum amount of sound waves from the amplifier will be in contact with the material. The flow chart for the process of making reinforced foam is shown in Fig. 1 .

Fig. 1
figure 1

Experimental flow chart of nanoparticle reinforcement in flexible smart polyurethane foam

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2.3 Acoustic test setup

The experimental test setup comprises an impedance tube of length 900 mm and a diameter of 100 mm. A speaker with a frequency range of 200 to 2000 Hz is installed at the tube’s left end. A pair of mics are located in the tube’s middle. The speaker and microphone are controlled using a controller. A 100 mm diameter sample, cut from the manufactured foam, is positioned at the right end of the tube. The readings are controlled automatically by a computer running MATLAB software. The SAC of samples is calculated using MATLAB from the data collected. The test setup is shown in Fig. 4.

2.4 Response surface methodology (RSM)

Response surface methodology can investigate the relationship between numerous independent variables and one or many dependent variables. RSM’s main idea is to achieve an optimal solution through a sequence of well-planned tests. Statistical approaches like RSM could enhance operating parameters and improve material production. In contrast to traditional methodologies, statistical techniques can be used to determine the interaction between process factors. If major explanatory factors are suspected, a more elaborate design, such as a central composite design (CCD), can be used to estimate a second-degree polynomial model; however, this is still only an approximate approximation at best. Conversely, the second-degree model can be used to optimize the response variables of interest (maximize, minimize, or achieve a specific aim).

2.5 Inverse characterization using GA

The different methods for inverse characterization are the line search method, genetic algorithm, steepest descent, gradient search, and most minor curve fitting procedures. Each method has several advantages and disadvantages. For example, gradient-based techniques are utilized to address problems involving continuous variables. Gradient approaches, on the other hand, become locked in local minima for discrete problems, and their effectiveness depends entirely on the unique design variables. The best option is the stochastic nature genetic algorithm to evaluate objective functions quickly. According to Darwin’s theory, the genetic algorithm, also known as the evolution algorithm, understands challenges within the constraints and boundaries of the solution [16, 45]. In this study, design parameters were used as a genetic algorithm.

2.6 Computational simulation

Simulation software such as COMSOL Multiphysics is generally used in engineering, manufacturing departments, test laboratories, and scientific research to model designs, devices, and processes. Comsol Multiphysics with add-on modules provides specialized functionality for mechanical, fluid dynamics, electromagnetic interference, acoustics, and chemical applications.

3 Results and discussion

3.1 Field emission scanning electron microscope (FESEM)

The morphology of the nanoparticles reinforced flexible smart polyurethane foam is examined by a field emission scanning electron microscope, as shown in Fig. 2. Before being scanned, the specimens are gold-coated. Figure 2a shows the presence of open cellular porous honeycomb structure morphology with the average size of pore 300 μm and the thickness of the pore wall is 15 μm in fabricated bio-based polyurethane foam. It has been observed that plenty of nanoparticles are surrounded by honeycomb pores. The flake-like structure in Fig. 2b quickly reveals the presence of Gr nanoparticles. Under high Gr content, Gr overlapped each other to form an integral three-dimensional structure, which will help to improve the acoustical properties of the reinforced PU foam [52, 54]. From Fig. 2c, it was speculated that Gr played a role in refining crystal grains, promoting the aggregation and fusion of fine ZrO2. Thus, the spherical shape found in the FESEM denotes the existence of ZrO2. The accumulation of a large number of tiny particles confirms the presence of BC, as shown in Fig. 2d. The uniformity across the sample is seen in Fig. 2e due to the self-assembling characteristic of nanoparticles.

Fig. 2
figure 2

FESEM characterization of a Open cellular pores of bio-based flexible polyurethane foam, b Graphene nanoparticles, c Zirconium oxide nanoparticles, d Bamboo charcoal nanoparticles, e Nanoparticles reinforced flexible smart polyurethane foam

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3.2 Fourier-transform infrared spectroscopy (FTIR) characterization

The absorption, emission, and photoconductivity spectrum of solids, liquids, and gases are obtained using the Fourier transform infrared spectroscopy (FTIR) technique. Between 4000 and 400 cm− 1, the range is captured. The curves plotted between transmittance v/s wave number were used to identify the different functional groups in Poly-β-hydroxybutyrate (PHB) found in the various foams using a Nicolet NEXUS-670 FTIR instrument. Here, carboxylic acid OH stretch is confirmed in non-reinforced foam around 3000–2800 cm− 1 from stretching the peaks. N–H stretches of amines are present in the range of around 3300 cm− 1. The asymmetric and symmetric bending vibrations of the HO–H produce peaks at 1648 cm-1 and 1381 cm− 1. The carbonyl peak was found at 1724 cm− 1, with the isocyanate peak (–NCO) disappearing at around 2270–2250 cm− 1. No peak of hydroxyl at 3450 cm− 1, indicating that all isocyanates contained in the pre-polymer constituents were fully used. While comparing the graph from Fig. 3, the addition of nanoparticles to the smart polyurethane foam is readily seen. The medium-intensity sp2 hybridized C = C bonding in the 1680–1600 cm− 1 range validates graphene. Three strong absorption bands were seen at wave numbers of 3380, 1650, and 550 cm− 1 in tetragonal ZrO2 (Fig. 3, blue line). The tension vibration of the hydroxyl groups on the ZrO2 surface produced two bands at 3380 and 1650 cm− 1. The specimen contains other nanoparticles besides the Gr, as seen by the FTIR graph. The results are ambiguous, and there is a lot of noise due to the presence of alkanes and alkynes in the diverse compound mixture, indicating the presence of BC and Gr in the composite.

Fig. 3
figure 3

FTIR graph of non-reinforced and nanoparticles reinforced flexible polyurethane foam

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3.3 Acoustic experimental analysis

Acoustics analysis of the sample was performed using an acoustic test setup per standard ASTM E 1050-12. The setup consists of two microphones, one speaker, an impedance tube, a DAQ system, MATLAB software, and a computer, as shown in Fig. 4. The 16 Ω speaker is connected at one end of the impedance tube, which can produce a sound wave in the frequency range between 200 and 2000 Hz. Two half-inch Microtech Gefell microphone placed in the impedance tube is connected to M + P Vibpilot DAQ as shown in Fig. 4. The samples are sliced into the spherical size of the diameter of 100 mm with the help of a cutter. It is placed on the other end of the impedance tube. Each test was carried out three times, and the average results were taken. MATLAB software is used to determine the Sound Absorption Coefficient using Eqs. 1,2,3. The time domain is collected using software M + P spectrum analyzer with 0.5 Hz frequency resolution and then fed into MATLAB® for determining the transfer function between two microphones. The transfer function may be estimated using the data acquired from the two microphones using Eqs. (1) and (2),

Wave number:

$${k}_{0}=\frac{2\pi f}{{c}_{0}}$$
(1)

where and \({c}_{0}\)are the frequency and speed of sound respectively.

$${H}_{12}=\frac{{p}_{2}}{{p}_{1}}=\frac{{e}^{j{k}_{0}{x}_{2}} + r{e}^{-j{k}_{0}{x}_{2}}}{{e}^{j{k}_{0}{x}_{2 }}+ {re}^{-j{k}_{0}{x}_{1}}}$$
(2)

where, \({H}_{12}\)- acoustic transfer function, 2 and 1- acoustic pressures of two microphones, x1 and x2 - distances between the location of the sample x = 0 and the two microphones. The reflection and absorption coefficients calculated using Eqs. (3) and (4).

$$R=\frac{{e}^{-jkS}-{H}_{12}}{{{H}_{12}-e}^{jkS}} \times {e}^{2jk(l+S)}$$
(3)
$$\alpha =1-{\left|R\right|}^{2}$$
(4)
$${\text{NRC }} = {\text{ }}(\alpha _{{250}} + \alpha _{{500}} + \alpha _{{1000}} + \alpha _{{2000}} )/4$$
(5)

α250, α500, α1000, α2000 are Sound absorption coefficient (SAC) at 250, 500, 1000 and 2000 Hz frequency.

$${\text{Error }}\left( \% \right){\text{ }} = {\text{ }}\left( {{\text{NRC}}_{{{\text{actual}}}} - {\text{NRC}}_{{{\text{predicted}}}} } \right)/{\text{NRC}}_{{{\text{actual}}}}$$
(6)

NRCactual and NRCpredicted are Actual and Predicted value of noise reduction coefficient obtained from the experimental statistical analysis.

The SAC can be calculated using Eq. 4, where R is the complex sound reflection coefficient calculated from Eq. 3, which is the ratio of reflected incident pressure. From Eq. 5, the NRC value can be obtained. Table 3 shows the SAC corresponding to 250 Hz, 500 Hz, 1000 Hz, and 2000 Hz frequencies. Figure 5a–c shows the similarity of the sound absorption coefficient of the samples with and without the nanoparticles. From Fig. 5b, it is seen that the sample A8 (0.2 wt%, 0.2 wt%, and 2 wt% for Gr, ZrO2, and BC, respectively) depicts an excellent SAC of 0.99 at a high frequency of 1400 Hz and 0.59 at a low frequency of 444 Hz, which is improved as compared to regular PU foam without reinforcement (A0).

Fig. 4
figure 4

Experimental setup for the Acoustic test

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Table 3 Experimental results of sound absorption coefficient (α)
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3.4 Optimization using response surface methodology

The experimental design was built using the Design of Experiments methodology. The experiment was then analyzed using RSM, with a central composite design being used to obtain the optimized weight% of nanoparticles. In the Central Composite Design, the input parameters are the weight% of graphene, zirconium oxide, and bamboo charcoal, and the output parameter is the NRC (dB) of 20 samples. Table 4 shows the minimum and maximum values of nanoparticles used as input for CCD. Table 5 shows how the RSM of central composite design L20 was used for the design of experiment (DOE) using Minitab software, the density and standard deviation of each sample is also determined [55]. The same software was used to analyze and develop numerical models for acoustic tests using input and output data from the response surface methodology experimental design. A quadratic model was statistical significance for all output parameters, as per a fit summary of the fit investigation. Table 5 shows that the sample S8 with 0.2 wt% of Gr, 0.2 wt% of ZrO2, and 2 wt% of BC has the highest noise reduction coefficient (NRC) of 0.457, and according to the results of the experiment, specimen S8 is indeed one of the perfect samples for the low-frequency range. As a result, the required weight% of the specified fillers provides adequate and desired acoustic shielding. The model’s dependability was tested using analysis of variance (ANOVA), as shown in Table 6. The F-value indicates that almost all of the findings are statistical significance, showing that the model fits the experimental data statistically. According to the results, the model could predict outputs with a confidence level greater than 90%.

Table 4 Different levels of nanoparticles worn in flexible smart polyurethane foam
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Table 5 Density and NRC data for various weight% in the experimental design (L20)
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Table 7 shows the real and statistical results from the RSM models for acoustic shielding, utilizing regression Eq. 7 as given below. The error percentage is calculated using Eq. 6. Table 7 shows that the largest error% between the real and RSM projected NRC was less than 7%, implying that the regression Eq. 7 produced from the analysis using the response surface methodology model is correct; consequently, the actual and anticipated results are in good accord. The regression equation is shown below.

$${\text{Noise reduction coefficient }}\left( {{\text{dB}}} \right){\text{ }} = {\text{ }}0.{\text{679 }} - {\text{ 1}}.0{\text{75 A }} - {\text{ }}0.{\text{972 B }} + {\text{ }}0.0{\text{12}}0{\text{ C }} + {\text{ }}0.{\text{379 A}} \times {\text{A }} + {\text{ }}0.{\text{914 B}} \times {\text{B }} + {\text{ }}0.0{\text{232 C}} \times {\text{C }} + {\text{ 2}}.{\text{637 A}} \times {\text{B }} + {\text{ }}0.0{\text{5}}0{\text{ A}} \times {\text{C }} - {\text{ }}0.{\text{288 B}} \times {\text{C}}$$
(7)
Table 6 ANOVA for noise reduction coefficient (dB)
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Table 7 NRC error estimation between real and statistically anticipated values
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Figure 6 shows the contour plots for the interaction of the specified parameters. It is obvious from Fig. 6a–c that combinations of Gr of 0.45–0.5 wt% and ZrO2 of 0.45–0.5 wt%, Gr of 0.1–0.18 wt% and BC of 2–2.5 wt%, ZrO2 of 0.1–0.15 wt% and BC of 1.25–2.5 wt% offer improved sound absorption with a noise reduction coefficient above 0.6. A comparison of the actual and predicted NRC is shown in Fig. 7. The actual and anticipated values were found to exhibit a linear regression relationship. This supports the statistical significance of the regression Eq. 7 obtained graphically from the Response surface methodology model, and this is used to find the best range of nanoparticles to use. The model’s correctness and adequacy are demonstrated by the fact that most of the relevant values in Fig. 6 seem to be on or close to the regression line. The optimum weight% ratios of Gr, ZrO2, and BC to obtain the maximum NRC value are shown in Table 8; Fig. 8. The mixture of 0.1 wt% of graphene, 0.1 wt% of zirconium oxide, and 2.5 wt% of bamboo charcoal exhibits the highest noise reduction coefficient value of 0.628 dB, which is within the acceptable range of significance.

Fig. 5
figure 5

Evaluation of SAC values for different flexible smart polyurethane foam

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Fig. 6
figure 6

Contour plots displaying the impact of a Graphene & ZrO2 on NRC effect, b Graphene & BC on NRC effect, c BC & ZrO2 on NRC effect

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Fig. 7
figure 7

Actual vs. Predicted NRC value

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Table 8 Optimum wt% of Graphene, Zirconium oxide and Bamboo charcoal
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3.5 Analysis of inverse characterization

The optimized sample’s acoustic analysis is simulated in COMSOL software. To simulate acoustic analysis in COMSOL Multiphysics, non-acoustical properties of the foam are needed. So, we used a genetic algorithm in MATLAB software to get the non-acoustical properties using the equations from the JCA model as explained by [16]. The inverse characteristics can be performed from the MATLAB program with JCA model equations, and porosity, tortuosity, flow resistivity, viscous, and thermal characteristic length can be obtained.

$${F}_{m}=\sum _{n}\left|{\alpha }_{exp}{(\omega }_{n})-{\alpha }_{mod}({\omega }_{n})\right|$$
(8)
$${\rho }_{eff}=\frac{{\alpha }_{\infty }{\rho }_{air}}{\phi }\left[1+\frac{\sigma \phi }{j\omega {\alpha }_{\infty }{\rho }_{air}}\sqrt{1+j\frac{4{\alpha }_{\infty }^{2}{\eta \rho }_{air}\omega }{{\sigma }^{2}{\varLambda }^{2}{\phi }^{2}}}\right]$$
(9)
$$K_{{eff}} = \frac{{{\raise0.7ex\hbox{${\gamma P_{0} }$} \!\mathord{\left/ {\vphantom {{\gamma P_{0} } \phi }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\phi $}}}}{{\gamma \left( {\gamma - 1} \right)\left[ {1 + \frac{{\sigma \phi }}{{jP_{{r\omega }} \alpha _{\infty } \rho _{{air}} }}\sqrt {1 + j\frac{{4\alpha _{{\infty ~}}^{2} \eta \rho _{{air}} \omega P_{r} }}{{\sigma ^{2} \Lambda ^{2} \phi ^{2} }}} } \right]}}$$
(10)
$$Z =\sqrt{{\rho }_{eff}*{k}_{eff}}$$
(11)
$$\text{K}= \omega \sqrt{{\rho }_{eff}/{k}_{eff}}$$
(12)

ϕ, ∝∞, σ, Λ, Λ| are the porosity, tortuosity, flow resistivity, viscous characteristics length, and thermal characteristic length, Where, Density of air in kg/m3 is ρair = 1.21, ω is the angular frequency. Viscosity of air in Ns/m2 is η = 1.84 × 10−5, Prandtl number of air is Pr = 0.702, Specific heat ratio of air, γ = 1.14, Atmospheric pressure in N/m2 is P0 = 101,320.

3.6 Simulation analysis

The simulation analysis has been done using the basic properties and the properties obtained from inverse characterization. Acoustic simulation for the best sample (A8) is done in COMSOL Multiphysics software. Here, the JCA model is used to model acoustic analysis. For this JCA model, five parameters are needed (porosity, tortuosity, flow resistivity, viscous, and thermal characteristic length). As explained above, these parameters are predicted using inverse characterization using a genetic algorithm. The values of inverse characterization are shown in Table 9, which are given as input for the COMSOL simulation. The dimensions of the model in this study consist of 600 mm in height and 100 mm in width and a porous layer of 30 mm in height. In this COMSOL model, the sound wave hits the porous absorber at theta = 0 radians. An air pressure domain is applied to model the incident wave. The pressure acoustic model is used for the air domain of the model. Perfectly matched layers are applied on both sides of the model to absorb all the pressure fields. Then the model was meshed using a physics-controlled mesh with a free triangle fine size element. Finally, the sound absorption coefficient graph is plotted and compared with the experimental graph, as shown in Fig. 9.

Fig. 8
figure 8

Optimization curvature of factor for different nanoparticles influencing NRC value

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Table 9 Porosity, Tortuosity, Flow resistivity, VSL and TCL by Inverse characteristic
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3.7 Sound pressure level (SPL) simulation

The prepared nanoparticles PU foam effect inside a car cabin is simulated by obtaining the sound pressure level using COMSOL Multiphysics software. The car cabin dimensions are (length × height × depth) (3.0 m × 1.4 m × 1.7 m). The CAD model also resembles the actual car cabin dimensions; the windshield’s lower portion is 0.8 m above the floor with 32° of inclination. Finally, the entire volume of the CAD model is 6.5 m3. Here, geometry is modeled in the front part of the car that separates the car’s engine from the inside of the car’s cabin. This separator is modeled as a prepared sample material by assigning it as a porous absorber with the acoustic and non-acoustic parameters as input. At the top portion of the cabin, a monopole point flow source of strength QS = 105 m3/s is selected. The next steps are as follows: First, the pressure acoustic frequency domain module was used on the entire CAD model, and then poro-acoustic material sample foam was defined with five parameters of the johnson champoux allard model selected from a dropdown menu. Then the model was meshed using a physics-controlled mesh with a free triangle fine size element. Then the frequency domain of 200–2000 Hz is computed. Figure 10a and b shows the CAD model of the car cabin for SPL simulation recorded at a random point inside the car cabin at P1 = (1.3, 1.2, and 0.8). The COMSOL results of the car cabin acoustic simulation with an empty cavity and with nanoparticles foam are shown in Figs. 11 and 12. Figure 11 shows the SPL simulation plot for the 200 to 2000 Hz frequency range. Figure 12 shows a 1/3rd octave band SPL simulation. The octave plot was plotted using the results tab in COMSOL, which can be selected from the more plots option. It can be seen from Fig. 12 that in the car cabin, the SPL has been reduced to 5-31 dB with nanoparticles reinforced PU foam inside the car cavity.

Fig. 9
figure 9

Comparison of sound absorption coefficient between experimental and numerical

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Fig. 10
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Car cabin simulation model

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Fig. 11
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Sound pressure level simulation plot for multilayer configuration

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Fig. 12
figure 12

Comparison of SPL plot in one-third octave band

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4 Conclusion

The flexible smart PU foam samples were easily produced using absorption and hydrothermal reduction methods, and they were shown to be dependable and delicate with precise control. In total, 15 samples were made using different compositions determined by the RSM central composite design technique. Among which, Sample A8 (0.2 wt%, 0.2 wt%, and 2 wt% for Gr, ZrO2, and BC, respectively) depicts a maximum noise reduction coefficient of 0.457 and also an excellent sound absorption coefficient of 0.99 at a high frequency of 1400 Hz and 0.59 at a low frequency of 444 Hz, which is improved as compared to regular PU foam without reinforcement. The presence of nanoparticles in the PU foam has been authenticated through FESEM and FTIR. Influential parameters have been discovered for statistical computation, and their amounts have been set using ANOVA. The regression equation was resolved, and the noise reduction coefficient was obtained using the iterative results of the different samples and the ANOVA for NRC. The difference between the regression equation’s predicted NRC values and those measured in the experiment is less than 7%. The ideal weight% for attaining the highest NRC value is 0.1 wt%, 0.1 wt%, and 2.5 wt% for Gr, ZrO2, and BC, respectively, through the optimization curve.

The simulation of the sound pressure level inside a car cabin was done in COMSOL software. Finally, the sound pressure level (SPL) for empty cavity and nanoparticles reinforced polyurethane foam was compared. The SPL plot shows an average 5 to 31 dB SPL reduction for the best sample (A8). The nanocomposite flexible smart PU foam is biodegradable, light-weight, flexible, and low cost. As a result, it can be utilized as a long-term replacement for traditional PU foam in sound-absorbing material for automobiles, walls of buildings and machinery that causes loud sounds, and internal insulation of airplanes and spacecraft, etc. Moreover, with increasing demands for suitable nanomaterials, this research further opens doors to future work with the reinforcement of different nanoparticles integrated with computational results produced with Comsol Multiphysics targeting the essential nature of the application.