In recent years, building regulations and the growing social demand for sustainability, have led to the emergence of new absorbent materials. For these new materials to be usable in the design stage, it becomes necessary to establish simplified models to easily incorporate them into simulation programs. This is one reason why, for a while now, various strategies have been proposed based on empirical models of material behavior or other linear equations. The materials studied in this respect have been the rock wool and glass wool, which are the most widely used in engineering practice. One of the most known and used models in this line is that of Delany and Bazley (1), which has established itself as a reference model in UNE_EN 12354:2004 standard (Part 6) (2) for fiber-based materials. These standards also cite Miki's later works (3, 4) models for foams, as well as Dunn and Davern (5) reference model in (2) for porous fibers, as well as models for textiles like those proposed by Garai and Pompoli (6). There are other empirical works, such as Wang et al. (7), based on a regression of different variables, or Shoshani and Yakubov (8) for finding maximum absorption. One should not forget the different empirical models proposed by Voronina (9–12), which make use of parameters such as the tortuosity factor or the structure associated with the distribution of the fibers or pores which characterize these materials.

In this work, a specific natural material is studied, namely coconut fiber. Coconuts, originated from the cocos nucifera variety of palms, can be harvested either from the ground (once they have fallen) or directly from the tree. After harvesting, the external fibrous layer of the fruit is separated from the hard shell in a so-called de-husking process, which can either be performed manually or mechanically. If green coconuts are harvested (after about six to twelve months on the tree), pliable white fibers may be extracted; a brown fiber may be obtained by harvesting fully mature coconuts. Each of these types of fibers is processed differently, and is used for different purposes. We here focus on the brown fibers, extracted from mature coconuts, and which are of interest to many technological and industrial applications. After extraction, the fibrous husks are soaked in pits or in nets in a slow moving body of water to swell and soften the fibers. Fibers are separated in two groups, namely longer (bristle) and shorter (mattress) fibers. After a cleaning and drying process, longer bristle fibers are then tied into bundles, while mattress fibers are compressed and packed into large bales.

These coconut fibers (known as coir) are abundant in many Asian countries such as India, Sri-Lanka or Malaysia, and have been sometimes been regarded as a natural waste product. In recent years, sustainability concerns lead to an increasing interest in its application in construction, industrial and technological solutions, and raised the attention of both the industry and of the scientific community. Traditionally, these fibers were used in floor mats, doormats, brushes or mattresses; recently, technological industries, such as the automotive industry in Europe, have been increasingly using brown coir pads sprayed with rubber latex (bonding the fibers together) as upholstery padding. In the construction industry, its application has been studied in many forms, such as a reinforcing fiber in cement boards (see for example (13)), or as an insulation material (see, for example, (14)). In fact, physically, these fibers have appealing properties in what concerns their application in the construction industry. The high porosity and low density of those elements allows them to exhibit low thermal conductivity and good sound absorption coefficients (as illustrated by Fouladi et al. (15)), and thus makes them an interesting product.

Due to their natural unprocessed origin, coir fibers are typically heterogeneous both in color, thickness and length, and, as natural untreated materials, they also exhibit performance limitations in some respects. For example in what concerns fire reaction, these fibers are a combustible material, and so their application without protective measures becomes limited. Due to this behavior, commercial products incorporating a fireproof treatment became available for use when the mattresses are to be used with some possibility of being exposed to fire. Two forms of industrially produced coconut fiber mattresses can thus be commonly found in western countries, corresponding to semi-rigid mattresses (usually higher density panels) with fireproof treatment and to flexible mattresses (usually rolls with lower density), without any synthetic treatment.

Studies regarding its applicability from the acoustic viewpoint jobs include Fouladi et al. (15, 16), Nor et al. (17, 18) and Zulkifli et al. (19).

Figure 1 shows an image of absorbent material made from coconut fiber. As can be appreciated, this is a porous and fibrous. Thus, by their nature, are expected to play a similar role to other porous and fibrous materials such as rockwool.

This work has developed and implemented a method for obtaining an empirical model for the case of absorbent materials obtained from natural fibers, in particular coconut fiber, following a procedure similar to that described in (20, 21), inspired in (6).

The model presented in this paper aims to predict the acoustic behavior of the fibrous material under study using the fewest number of non-intrinsic physical parameters, thus mitigating some propagation errors people make when using parameters obtained experimentally. Basically, it comes to finding the coefficients C _i (i = 1…8) that best fit the following equations for the case of the tested material (21). α = ( 2 × π × f c 0 ) × [ c 5 × ( ρ 0 × f r ) - c 6 ] β = ( 2 × π × f c 0 ) × [ 1 + c 7 × ( ρ 0 × f r ) - c 8 ] R = ρ 0 × c 0 [ 1 + c 1 ( ρ 0 × f r ) - c 2 ] X = - ρ 0 × c 0 [ c 3 × ( ρ 0 × f r ) - c 4 ]

where α and β are the real and imaginary parts of the propagation constant Γ of the material, R and X are the real and imaginary parts of its characteristic impedance Z, σ the air flow resistivity in Ns/m⁴, ρ₀ is the air density, ƒ the frequency in Hz and c ₀ the sound propagation velocity in air in m/s. The flow resistance r of a sample of material can be obtained through multiplying the flow resistivity value by its thickness e.

To determine the specific acoustic impedance of a material, the commonly used transfer function method is used, which considers a plane-wave tube with a speaker mounted in one end. The sample of the material to be characterized is placed at the other end of the tube, at a distance x ₁ from the microphone placed further away from the sample, and spaced a distance s from the second microphone, as shown in Figure 2.

In the standards ASTM E1050 (22) and ISO 10534-2:1998 (23) or UNE EN ISO 10534-2:2002 (24), the measurement procedure is explained in detail, as well as its precision, limitations and sources of error or uncertainty.

Driving the speaker with a random noise signal (allowing an assessment of the full frequency range) and recording the sound pressures in the two microphone positions, these pressures can be written as the sum of the incident and reflected sound waves. The complex transfer function H ₁₂ is then given by the relation between both pressures, and the reflexion coefficient can be obtained as: R = H 12 - e - j k s e j k s - H 12 e j 2 k x 1

where and k is the wave number in rad/m.

Thus, it is possible to obtain the specific acoustic impedance (real and imaginary) using the equation Z s = Z 0 1 + R 1 - R = Z R + j Z I

being Z ₀ =ρ₀ c ₀ the characteristic impedance of air and Z _R and Z ₁ the real and imaginary parts of specific acoustic impedance, that can also be obtained from the characteristic impedance of the material, its propagation constant and the sample thickness through: Z s = Z Z 0 c o t h ( Γ e )

From this reflexion coefficient, it becomes possible to compute an acoustic parameter of significant practical relevance, namely the sound absorption coefficient for normal incidence α _n as: α n = 1 - | R | 2

This technique has the disadvantage of requiring the calculation of the transfer functions using successive measurements with exchanged measurement channels, in order to allow for phase correcting mismatches between microphones. Otherwise large errors can occur in the impedance calculation.

The relation between this absorption coefficient for normal incidence and the acoustic impedance can be defined by the equation: α = 4 Z R Z 0 | Z s | 2 + 2 Z 0 Z R + Z 0 2

where Z ₀ =ρ₀ c ₀.

To allow fitting the material model, it becomes necessary to obtain experimental data concerning the airflow resistance and absorption coefficient under normal incidence.

The following quadratic error function for the iterative method used here can be defined as: ε = ∑ i = 1 n ( α n , i - α ^ n , i ) 2

where α _n,i is the absorption coefficient for normal incidence, measured for the absorbing material under study, for the i-th frequency, and is the estimation of the that value obtained from application of equations [1] to [4]. To minimize the error function, the following expression must be set to zero: ∂ ε ∂ A i = 2 ∑ i - 1 N ( α n , i - α ^ n , i ) ∂ α ^ n , i ∂ A i = 0 i = 1 , … , 8

To obtain the coefficients that best describe the measured acoustic behavior of the samples, an iterative method of decrease of quadratic error function was used. As initial values (input values), different values proposed by different authors were tried, and all converged to the coefficients values as obtained and presented in this paper.

The experimental basis taken for adjusting the model consisted of:

Coconut fiber material without fireproof treatment with 2, 3 and 4 cm nominal thickness.

As for the distribution of fiber diameter, its analysis was conducted on a sample including 200 random fibers, leading to an average fiber diameter of 0.25 mm, with a size distribution between 1.17 and 0.48 mm.

This paper described the process for obtaining empirical equations for the acoustic behavior of an absorbent material made natural fibers, particularly coconut fibers. The procedure, which has been successfully tested in other materials, involves performing impedance and airflow resistance measurements for samples of the material under study. The results are useful since the availability of empirically derived analytical equations, simplifies the incorporation of these materials into simulation programs in order to make predictions of their acoustic behavior when used as part of noise control devices. Note that there are different types of coconut fiber and, obviously, not all have the same degree of absorption.