Introduction

Currently, the noise pollution is considered as a major threat to the society which may affect life quality, efficiency and human physical and mental health and wellbeing (Dzhambov et al. 2017). Nonetheless, this problem can be relatively resolved by applying soundproofing techniques such as employing noise barriers or sound absorbers made of porous and fibrous materials (Arenas and Crocker 2010; Morandi et al. 2016). Sound absorbing materials used in indoor and outdoor environments are generally categorized as granular, cellular and fibrous absorbers (Cao et al. 2018). In their interaction with sound waves, fiber-based porous absorbers can alter the sound energy to thermal energy due to their network of intertwined pores (Jayamani and Hamdan 2013). Fibrous absorbers are mainly divided into two groups: synthetic (organic and inorganic fibers) and natural (animal based, mineral and cellulosic/lignocellulosic fibers) (Asdrubali et al. 2015; Jawaid and Abdul Khalil 2011). Recently, application of lignocellulosic fibers for sound absorption purposes has attracted special attention. Table 1 lists the comparison of noise reduction properties and the key findings of studies conducted on several natural fibrous materials.

Table 1 A summary of the findings from recent studies about the acoustic absorption properties of several natural fiber materials
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Several characteristics such as easier accessibility, biodegradability, high electric resistance, low cost as well as being nonabrasive and nontoxic have made them a reasonable alternative to synthetic fibers (Berardi and Iannace 2015; Tomyangkul et al. 2016). Labeled as “green technology,” such alternatives have also become popular among researchers due to increasing concerns about the safety and health issues, excessive generation of greenhouse gases as a result of increased emission of CO2 and massive amounts of waste produced through fabricating and utilization of the synthetic fibers (Li et al. 2007; Cao and Wu 2008). Moreover, sound absorbers and composites which are made of lignocellulosic fibers are relatively durable and high-performance alternatives because of their desirable physiochemical properties (Pickering et al. 2016; Cao et al. 2018; Tang and Yan 2017).

For instance, coir and oil palm fibers can be used to make potential porous sound absorbers since they can effectively dissipate the energy of incoming sound waves (Or et al. 2017; Hosseini Fouladi et al. 2010). Annually, a staggering amount of agricultural wastes is produced in the developing countries and due to the poor waste management systems, most of these wastes which contain lignocellulosic fibers are either burnt or buried without being utilized as alternative products. Such case is almost true about Iranian vast date palm plantations where to the best of our knowledge, the wastes, particularly from the date palm pruning, are disposed with no proper application.

On the other hand, there is a growing need for the development of empirical models with easy application to predict and measure the properties of acoustic absorbers fabricated from natural fibers. Delany–Bazley (DB) (Delany and Bazley 1970) and Miki (Miki 1990) are the most significant examples of empirical models frequently used to predict the acoustic behavior of natural sound absorbers. These models are widely employed owing to their quick application of results, explanation of the properties of waves’ impedance, predicting sound absorption mechanism along with modeling and calculating the parameters which are involved in the acoustic absorption of natural sound absorbers by initial physical properties.

Given the importance of this issue and global concerns over the complications associated to the use of synthetic fibers as well as the importance of identifying acoustic properties of new natural fibers, the present study compares the acoustic behavior of coir and data palm fibers as the two most identical fibers in relation to their diameter and texture. It also investigates the related empirical and phenomenological models for the prediction of sound absorption coefficient of samples fabricated from these fibers. Additionally, a critical analysis about the capability and accuracy of these models for the prediction of samples’ sound absorption coefficient and the level of accuracy obtained with changes in samples’ thickness and comparison of the related results with the experimental data is presented.

Materials and methods

Sample preparation

Coir and date palm fibers were first cut into small pieces after drying, and then, 5% polyvinyl alcohol (PVA) (Sigma-Aldrich) was applied to them as the binder in order to bind the fibers to one another. Next, the fibers were fitted into two stainless-steel molds with internal diameters of 30 and 100 mm, the same as the internal diameter of the impedance tube in the experiment. Therefore, the samples fabricated in this study had either 30 mm or 100 mm diameters. Having undergone the compression process by a manual hydraulic press machine, all samples were left at room temperature, so that they become dried completely. Finally, the prepared samples with a thickness of 20 mm or 40 mm and a constant density of 100 kg/m3 were sent to the acoustic laboratory for the measurement of sound absorption coefficient. Figure 1 illustrates the surface of the coir and date palm fibers samples imaged in a scanning electron microscope (SEM).

Fig. 1
figure 1

SEM image of samples made of date palm (a) and coir fiber (b)

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Laboratory data

Generally, there are various methods to measure acoustic absorption coefficients for sound absorbing materials. In the present study, sound absorption coefficient of the prepared samples was measured with an impedance tube (SW422 and SW477, BSWA, China) based on transfer function method (ISO10534-2) (ISO, 2001). The impedance tube system was equipped with a larger diameter tube (100 mm) for measuring sound absorption at lower frequencies (63–1600 Hz) and a smaller diameter tube (30 mm) for measuring sound absorption at higher frequencies (1000–6300 Hz). Figure 2 illustrates the location of the sample and the loudspeaker at the two ends of the impedance tube.

Fig. 2
figure 2

Schematic view of impedance tube system for sound absorption measurement

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Empirical models

Empirical models usually take into account the porous material from a macroscopic point of view and ignore the characteristics of wave propagation in the pores of the material. The first serious discussions and analyses of these models emerged in 1969 based on the empirical relations and a large number of measurements that Delany and Bazley had conducted on several types of samples made of wool with various densities (Delany and Bazley 1970). The fundamental principle underlying this model is the power-law relations as well as the fact that the characteristic impedance and the wavenumber are related to the static airflow resistivity of the porous material. The empirical relations of the model are therefore as follows:

$$Z_{\text{c}} = \rho_{0} c_{0} \left[ {1 + 0.0571\left( {{\raise0.7ex\hbox{${\rho_{0} f}$} \!\mathord{\left/ {\vphantom {{\rho_{0} f} \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.754} - i0.087\left( {{\raise0.7ex\hbox{${\rho_{0} f}$} \!\mathord{\left/ {\vphantom {{\rho_{0} f} \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.732} } \right]$$
(1)
$$K_{\text{c}} = {\raise0.7ex\hbox{$\omega $} \!\mathord{\left/ {\vphantom {\omega {c_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${c_{0} }$}}\left[ {1 + 0.0978\left( {{\raise0.7ex\hbox{${\rho_{0} f}$} \!\mathord{\left/ {\vphantom {{\rho_{0} f} \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.7} - i0.189\left( {{\raise0.7ex\hbox{${\rho_{0} f}$} \!\mathord{\left/ {\vphantom {{\rho_{0} f} \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.595} } \right]$$
(2)
$$Z_{\text{s}} = - jZ_{\text{c}} \cot \left( {k_{\text{c}} .d} \right)$$
(3)
$$R = \left| {\frac{{\frac{{Z_{\text{s}} }}{{\rho_{0} c_{0} }}{ \cos }\left( \theta \right) - 1}}{{\frac{{Z_{\text{s}} }}{{\rho_{0} c_{0} }}{ \cos }\left( \theta \right) + 1}}} \right|$$
(4)
$$\alpha = 1 - \left| R \right|^{2}$$
(5)

where \(Z_{\text{c}}\) is characteristic impedance, \(K_{\text{c}}\) is characteristic wave number, \(C_{0}\) is sound speed in porous environment, \(\rho_{0}\) is air density, f is frequency, \(\sigma\) is resistance to air flow, \(\theta = 0\) wave incidence angle, \(\omega\) is angular frequency, R is reflection coefficient and \(\alpha\) is sound absorption coefficient.

Another empirical model employed in this study was the model presented by Miki in 1989. He improved the parameters of impedance and propagation constant based on the relations introduced by Delany–Bazley (Miki 1990) and adapted the regression coefficients to achieve more precise results. As the negative values related to the real part of surface impedance are not considered in this model, the results are accurate for both real and fictitious parts of the equation. Therefore, the improved propagation constant and characteristic impedance are as the following:

$$Z_{\text{c}} = \rho_{0} c_{0} \left[ {1 + 0.070\left( {{\raise0.7ex\hbox{$f$} \!\mathord{\left/ {\vphantom {f \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.632} - i0.107\left( {{\raise0.7ex\hbox{$f$} \!\mathord{\left/ {\vphantom {f \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.632} } \right]$$
(6)
$$K_{\text{c}} = {\raise0.7ex\hbox{$\omega $} \!\mathord{\left/ {\vphantom {\omega {c_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${c_{0} }$}}\left[ {1 + 0.160\left( {{\raise0.7ex\hbox{$f$} \!\mathord{\left/ {\vphantom {f \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.618} - i0.109\left( {{\raise0.7ex\hbox{$f$} \!\mathord{\left/ {\vphantom {f \sigma }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\sigma $}}} \right)^{ - 0.618} } \right]$$
(7)

Phenomenological models

Unlike the empirical models, the phenomenological models consider the wave propagation inside the pores and cavities of the materials microscopically. Moreover, these models yield accurate results for samples with randomly and arbitrarily shaped pores. The phenomenological model that was employed in the present study was Johnson–Champoux–Allard (JCA) model; a model that describes the viscous and thermal effects of the sound energy in porous materials with arbitrary shaped pores (Allard and Champoux 1992; Johnson et al. 1987). Two major physical parameters which are included in this model are the effective density and the effective bulk modulus which are then used to calculate the characteristic impedance and the wavenumber as follows:

$$\rho \left( \omega \right) = \alpha_{\infty } \rho_{0} \left[ {1 + \frac{\sigma \phi }{{j\omega \rho_{0} \alpha_{\infty } }}\left( {1 + \frac{{4{\text{i}}\alpha_{\infty }^{2} \eta \omega \rho_{0} }}{{\left( {\sigma\Lambda \phi } \right)^{2} }}} \right)^{1/2} } \right]$$
(8)
$$K\left( \omega \right) = kp_{0} \left( {k - \left( {k - 1} \right)\left[ {1 + \frac{{\sigma^{\prime}\phi }}{{i\omega \rho_{0} \alpha_{\infty } N_{pr} }}\left( {1 + \frac{{4{\text{i}}\alpha_{\infty }^{2} \eta N_{pr} \omega \rho_{0} }}{{\left( {\sigma^{\prime}\Lambda ^{\prime } \phi } \right)^{2} }}} \right)^{1/2} } \right]^{ - 1} } \right)^{ - 1}$$
(9)
$$\sigma^{\prime} = \frac{{8\eta \alpha_{\infty } }}{{\Lambda ^{'2} \phi }}$$
(10)

The physical parameters of the sample are airflow resistance \(\sigma\), porosity \(\phi\), tortuosity \(\alpha_{\infty }\), viscose characteristic length \(\Lambda\) and thermal characteristic length \(\Lambda ^{\prime }\). Among these equations, \(\rho_{0}\) is density of air, \(N_{pr}\) is Prandtl number for air, \(\eta\) is air viscosity, k is special heat for air and \(\omega\) is angular speed. It can be seen that air condition is an important factor in calculating the density and bulk module equivalent. The characteristic impedance and the wave number of the air in the materials \(Z_{\text{c}} \left( \omega \right)\) and \(K_{\text{c}} \left( \omega \right)\) are described by the effective density and the bulk modulus of the air in the materials \(\rho \left( \omega \right)\) and \(K\left( \omega \right)\), respectively [Eqs. (11) and (12)].

$$Z_{\text{c}} \left( \omega \right) = \sqrt {\rho_{\left( \omega \right)} .K_{\left( \omega \right)} }$$
(11)
$$K_{\text{c}} \left( \omega \right) = \omega \sqrt {\frac{{\rho_{\left( \omega \right)} }}{{K_{\left( \omega \right)} }}}$$
(12)

Finally, for obtaining the surface acoustical impedance \({\text{Z}}_{\text{s}}\) at normal incidence with the thickness of d, the following equation is adopted (13).

$${\text{Z}}_{\text{s}} = - i\frac{{Z_{\text{c}} \left( \omega \right)}}{\phi }.\cot \left( {K_{\text{c}} \left( \omega \right).d} \right)$$
(13)

The prediction error of the normal incidence absorption coefficient of the JCA model was also examined in the frequency range between 63 Hz and 6300 Hz by:

$$100 \times \left| {\frac{{\alpha_{\text{m}} - \alpha_{\text{p}} }}{{\alpha_{\text{m}} }}} \right|$$
(14)

where \(\alpha_{\text{m}}\) is the measured frequency-dependent narrow band absorption coefficient and \(\alpha_{\text{p}}\) is the predicted frequency-dependent narrow band absorption coefficient.

Correlation analysis

Pearson correlation test was used to examine the correlation between the sound absorption coefficients obtained from the experiments and the ones predicted by the existing models.

Results and discussion

Results

The results of experimental data for samples with two different thicknesses (20 and 40 mm), the effect of air gap on the values of sound absorption coefficient and the noise reduction coefficients (NRCs) which is defined as the arithmetic average of the absorption coefficients for a specific material at octave band center frequencies of 250, 500, 1000 and 2000 Hz are presented in Table 2.

Table 2 Acoustic absorption measurements with different air gaps in octave bands
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Using the presented empirical and phenomenological models, the sound absorption coefficients for the frequency range 63–6300 Hz were determined via coding in MATLAB software (Figs. 2, 3). As can be seen in Table 3, the estimated values for physical parameters of coir and date palm fibers were calculated by differential evolution algorithm in MATLAB software using data such as thickness, density, air flow resistance along with absorption coefficients provided by the experimental data.

Fig. 3
figure 3

Comparison of the experimental versus empirical and phenomenological models for the sound absorption coefficients of date palm fiber samples (a 20 mm and b 40 mm thickness)

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Table 3 Basic properties and physical parameters for the acoustic model
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Table 4 presents results from the Pearson correlation test between the sound absorption coefficients obtained from the experiments and predicted by the existing models. Moreover, prediction error rates calculated by JCA model were compared with the values obtained from the experimental (impedance tube) tests (Figs. 2, 3).

Table 4 Relation between correlation analysis experimental data and mathematical models
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Discussion

From Figs. 3 and 4, it is evident that as frequency increases, acoustic absorption coefficient for the samples of coir and date palm fibers with 20 and 40 mm thicknesses significantly increases as well. In other words, while the values of absorption coefficient showed an increase at higher frequencies, for lower-frequency ranges, a decreasing trend was observed.

Fig. 4
figure 4

Comparison of the experimental versus empirical and phenomenological models for the sound absorption coefficients of coir fiber samples (a 20 mm and b 40 mm thickness)

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At lower frequencies (< 1000 Hz), the average sound absorption coefficients for the samples with thicknesses of 20 and 40 mm were 0.06 and 0.20 for coir fibers and 0.09 and 0.31 for date palm fibers, respectively.

Also, for samples with the same thicknesses, the absorption coefficients at the middle frequencies (1000–3000 Hz) were 0.43 and 0.74 for the coir fibers and 0.53 and 0.77 for date palm fibers, respectively. As mentioned earlier, for the frequencies above 3000 Hz, the 20- and 40-mm-thick samples showed higher values of acoustic absorption coefficients for coir fibers (0.7 and 0.85) and date palm fibers (0.68 and 0.87), respectively.

The thickness of the absorber is therefore assumed to play a major role in dissipating the sound energy and enhancing the values of sound absorption, particularly at lower-frequency ranges.

Acoustic tests performed on coir and date palm fibers also indicated that at 1000 Hz, the values of sound absorption coefficient for the samples with 40 mm thickness and a constant density (100 kg/m3) were 0.55 and 0.58, respectively, which were comparatively greater than the same sample with 20 mm thickness (0.19 and 0.14). Moreover, it is worth mentioning that the peak for absorption coefficient also shifted to the lower-frequency region as well. Therefore, increase in the thickness of the sound absorbing material would similarly impact the levels of sound absorption, irrespective of absorber type.

In this regard, Putra et al. (Asdrubali et al.) showed that with increasing the thickness of the samples (from 2 to 5 cm), the absorption coefficient at frequencies below 2000 Hz would reach 0.9. The increase in the values of sound absorption is assumed to be related to the longer depreciation process in the thermal and viscous conductivity between the air and absorbing materials in the composite, which in turn will increase the levels of sound absorption by increasing the thickness of samples.

In order to make an easy visual comparison among the different fibers, the NRCs are summarized in Table 1. It was found that NRC values of coir and date palm were slightly high (~ 0.44), in 40 mm thickness.

Currently, the trend for the development and application of modeling approaches is growing increasingly and this brings about the need for employing applicable, easy to use and accurate empirical or phenomenological models. This is also the case for the field of noise control and acoustic absorption; in a way, the models employed in this field must be able to predict the sound absorption coefficient of various types of sound absorbing materials including fibrous absorbers. Accordingly, in the present study, prediction of absorption coefficients of the samples of coir and date palm fibers was investigated by three of the most important empirical models, namely Delany–Bazley, Miki and Johnson–Champoux–Allard. The outputs from these models were then compared with the experimental data obtained from the tests conducted in the impedance tube. As shown in Figs. 3 and 4, the increase in the thickness of the samples enhanced the values of sound absorption coefficients approximated by the models and made them comparable to the data from experimental tests. Referring to Figs. 3 and 4 again, one will see that the DB and Miki models fairly reliably predict the values of sound absorption coefficients at higher and lower-frequency ranges for the thicker samples (40 mm). JCA model, however, enjoys higher accuracy in predicting the sound absorption coefficients for different thicknesses compared to the two former models. In fact, the results from Pearson correlation test revealed that there was a significant and strong correlation between the outputs from the existing models and the experiments. Although not equally significant, the correlations between experimentally obtained sound absorption coefficients and the JCA model are slightly higher than the two other models.

In fact, for the samples with 40 mm thickness, the JCA model yields the prediction accuracy values of 59%, 63%, 51%, 23%, 10% and 17% at six central frequencies (125 Hz to 4000 Hz) for date palm fiber samples and 64%, 58%, 39%, 30%, 5% and 22% for coir fiber samples, respectively.

While JCA model takes into account more physical parameters of material such as thickness, mass density, resistance to airflow, tortuosity and viscose and thermal characteristic length, DB and Miki models are only dependent upon the samples’ thickness and air flow resistance and therefore they might not be able to comprehensively demonstrate the fluctuations caused by the maximum absorption and critical frequency of the samples.

Given that unlike synthetic fibers, the natural fibers diameters are not usually uniform, the sound waves at certain frequencies cannot easily pass through the sample matrix; hence, we see the drop at the frequency range of 3000 to 4000 Hz. In other words, it is assumed that such nonuniformity affects the two major parameters of tortuosity and porosity of the samples which in turn minimize the values of sound absorption coefficients predicted by JCA model at certain frequencies. The thicker samples would therefore seem to be more affected in terms of decline in sound absorption coefficient levels.

Placing the samples with a distance from their rigid backing in the impedance tube and hence introducing an air cavity (gap) behind them shifted the maximum sound absorption values from higher-frequency ranges to the lower frequencies. Findings also indicated that increasing the air cavity distance behind the sample (to 3 cm) in frequency ranges lower than 1000 Hz would raise the sound absorption coefficients as well. In a study conducted by Fouladi et al. on coir fiber, increasing the spacing between the sample and its rigid backing enhanced the levels of sound absorption coefficient at lower-frequency ranges. They also found that as maximum values of absorption move toward lower-frequency ranges, absorption coefficients decrease at higher- and medium-frequency ranges (Mamtaz et al. 2016). This behavior may be the result of increased impedance in the absorbing materials. Acoustic resonance value moves toward lower-frequency ranges and hence improves the level of absorption in those ranges. It can therefore be assumed that using sound absorbing materials with an air cavity gap behind them plays a significant part in decreasing expenses, since a better absorption coefficient can be achieved with thinner samples. The potential of natural fibers as an alternative to common synthetic fibers in acoustic absorbers is approved in this study. Additionally, date palm and coir natural fibers demonstrate a favorable potential of dissipating the sound waves energy.

Conclusion

Empirical and phenomenological models which were studied in this research showed almost acceptable results in certain ranges of thickness. This has led to the elimination of many types of natural fibers from research processes in this field, since these fibers are irregularly shaped and have larger diameters compared to synthetic fibers which in turn decreases their accuracy of prediction. This issue must be taken into account in developing the future empirical models for analyzing the acoustic properties of natural fibers. The present study makes this noteworthy contribution that it is possible to apply more accurate models to predict the sound absorption coefficient of samples fabricated from natural fibers. As it was stated, there exist several models with superior performance in lower-frequency ranges, while some are more accurate in approximating absorption coefficients mostly in higher-frequency ranges. Therefore, models and combined approaches can be employed to predict the acoustic absorption values in all frequency ranges. Moreover, referring to the comparisons and correlation analysis, it is evident that JCA model remains the best option to predict the sound absorption coefficient for the sound absorbers made of natural fibers and their composites for the time being. In other words, the outputs from JCA model are likely to be more realistic and relatively follow the results from the experimental measurements performed by the impedance tube system.

As a result, developing empirical models with capability of accurately predicting the sound absorption coefficients based on the acoustic performance of natural fibers is very useful since these models enjoy advantages such as being cost-efficient, time saving and simplifying the process of determining acoustic absorption coefficients.