Most of the useful models for predicting acoustical properties of porous materials can be categorized in two groups: microstructural theoretical models and phenom­enological empirical models.14

Theoretical models generally deal with the behavior of sound waves in micro­structural elements of porous material. However, it is confoundedly complicated to explain the acoustical behavior of many sound absorbers based on theoretical mod­els due to structural and geometrical complexities of porous materials.13 As a conse­quence, in order to predict sound absorption behavior of fibrous materials, empirical models with a macroscopic point of view have been developed. Nevertheless, these empirical models, which do not deal with the microstructure, may only contribute with a limited guidance during the design phase of an absorber.14

Among these empirical models, one has encountered general acceptance for the last several decades: the empirical model, which was presented by Delany and Ba — zley.28 They investigated the sound absorption behavior of many fibrous materials with porosity close to unity for a large range of frequency. Based on their measure­ments, they found sound absorption indicators, the quantities of the wave number, k, and the characteristic impedance, Zc, to mainly depend on the frequency, f (or an­gular frequency, о), and the flow resistivity, r0, of the material, as explained below.

The flow of a fluid in a circular cylinder of diameter a flowing at a steady Poi — seuille flow can be given by

dp _ 128mu

Эх na4

where p represents pressure in Pa, p stands for dynamic viscosity of air in kg-m-1-s-1, u denominates the volume flow rate in m3-s-1, and a symbolizes the diameter of the channel in m.

Assuming that the number of parallel tubes is n per unit cross-sectional area in the porous material, the relation becomes,

dp _ _ 32mu (11)

Эх ha2

where u’ stands for the average volume flow rate over cross-sectional area, and h=(n/4)na2, where a represents the pore diameter, gives porosity. The ratio of the pore radius to the boundary layer, n, (rnp 0a2/4 ^.)m defines the acoustic character­istic of the sound absorber as explained by Fahy13. Here, can be replaced for 8 ц/rft in this ratio, n, using Eq. (11) to give

where rn, is the angular frequency, p0 is the density of air. The porosity, h, can be ig­nored, as h1/2~1 for most sound absorbers. This gives the nondimensional parameter (mp/r0) or:13

x _rf

r

f0

where X is the dimensionless sound absorber variable, which is denoted by E by some other researchers.1318

Delany and Bazley28 obtained sound absorption indicators, the values of the wave number, k, and the characteristic impedance, Zc, through the following expres­sions:

Zc _Poc0[l + 0.0571X4X754 _ І0.087X4X732 J, (14)

k _— [1 + 0.0978X-0700 _І0.189X“0595J, (l5)

c 0

where c0 is the sound propagation velocity in air. X became a universal descriptor of fibrous sound absorbers as it collapsed all the sound absorption data for the 70 different fibrous materials that Delany and Bazley28 measured.18 The boundary sug­gested for the validity of the model is

0.01<X<1.0. (16)

At low frequencies, Delany and Bazley’s28 formulas produce unfeasible results such as negative values for the real part of surface impedance of a hard-backed porous layer.17 Several researchers modify these formulas to give more accurate predictions, such as Mechel18. Mechel’s18 modified formulas are as follows:

Z= p °C0(1+0.08X-0-6")-i0.19X-0-556, (17)

^=(1+0.136X-0-641)-/0.322X-0-502 (18)

for X<0.025.

Zc= p °C0(1+0.0563X-0J25)-i0.127X-0-655, (19)

a

k=—(1+0.103X-0■76)-i0.322X-0■663 (20)

c 0

for X>0.025.

Yilmaz3 developed a model with the frequency, f, thickness, l, and air flow resis­tivity, ro, variables, presented in Eq. (21). The investigated materials included nu­merous fibrous composite materials from polypropylene, poly (lactic acid), hemp fi­bers and glassfibers in single, or multifiber, untreated, compressed, alkalized, heated and different layer-sequenced forms. The model estimates gave a goodness-of-fit, R2, value of 0.97 and the boundary suggested for the validity of the model is a basis weight of 1.13 to 2.36 kgm-2, fiber diameter of 16.3M0-6 to 33.3M0-6 m, a material thickness of 3.9M0-3 to 1.31×10-2 m, porosity of 0.62 to 0.92, and a frequency range of 500 Hz — 5 kHz.

an = sin(-5.27 x 10-1 + 2.54 x 10-4 f+1.72 x 10-6 r0 + 3.54 x 1041) (21)

The theoretical and empirical models provide guidance during the design of optimum sound absorbers in deciding which characteristics should be manipulated and how. Based on theoretical and empirical models, the factors, which directly or indirectly affect sound propagation through fibrous materials can be given as fiber properties (fiber size, fiber shape, etc.), material properties (flow resistance, density, thickness, and so on), process parameters and sound frequency. The following sec­tion investigates the factors that affect sound absorption.