Empirical Model for Predicting the Sound Absorption of Polyfelt Fibrous Materials - Article Example

?A New Empirical Model for Predicting the Sound Absorption of Polyfelt Fibrous Materials for Acoustical Applications I. Introduction Accurate modeling of the acoustical properties of porous materials is of considerable interest to acoustic and noise engineers and material manufacturers. Hence, there are many theoretical and empirical models which have been developed to predict the characteristic acoustic impedance and propagation coefficient of these materials. These models may be divided into three categories: empirical, phenomenological, and microstructural models. Empirical models do not require detailed knowledge of the internal structure of the material nor are they derived from theoretical considerations. Delany and Bazley [1] showed that the values of the characteristic acoustic impedance and propagation coefficient for a range of fibrous materials, normalized as a function of frequency divided by flow resistivity could be presented as simple power law functions. Model for Impedance The model is based on numerous impedance tube measurements and is good for determining the bulk acoustic properties at frequencies higher than 250 Hz, but not at low frequencies [2,3]. The validity of this model for lower and higher frequencies was further extended by Bies and Hansen [4].Dunn and Davern [5] calculated new regression coefficients between characteristic acoustic impedance and propagation coefficient for low airflow resistivity values of polyurethane foams and multilayer absorbers. To that effect, engineers can obtain the absorption coefficient of sound at normal incidence by using the equation below: ZR = P0 * C0 (1 + C1 ((P0f)/r)-c2) The final model which comes as a derivative of the first model is Zt = (ZR + iZl)[coth(a + iB) * l] Zt = ZIR + iZIl Qunli [6] later extended this work to cover a wider range of flow resistivity values by considering porous plastic open-cell foams.Miki [7, 8] generalized the empirical models developed by Delany and Bazley for the characteristics acoustic impedance and propagation coefficient of porous materials with respect to the porosity, tortuosity, and the pore shape factor ratio. Moreover, he showed that the real part of surface impedance computed by the Delany’s model converges to negative values at low frequencies. Therefore, he modified the model to give it real positive values even in wider frequency ranges. Other empirical models include those of Allard and Champoux [9]. These models are based on the assumption that the thermal effects are dependent on frequency. The models work well for low frequencies. The Voronina model [10] is another simple model that is based on the porosity of a material. This model uses the average pore diameter, frequency and porosity of the material for defining the acoustical characteristics of the material. Voronina [11] further extended the empirical model developed for porous materials with rigid frame and high porosity, and compared it with that of Attenborough's theory. A significant agreement was found between their empirical model and Attenborough's theoretical model. Recently, Gardner et al. [12] implemented a specific empirical model using neural networks for polyurethane foams with easily measured airflow resistivity. The algorithm embedded in the neural networks substitutes the usual power-law relations. The phenomenological models are based on the essential physics of acoustic propagation in a porous medium such as their universal features and how these can be captured in a model [13]. Biot [14] established the theoretical explanation of saturated porous materials as equivalent homogeneous materials. His model is believed to be the most accurate and detailed description till now. Among the significant refinement made to Biot theory, Johnson et al. [15] gave an interpolation formula for “Dynamic tortuosity” of the medium based on limiting behavior at zero and infinite frequency. The dynamic tortuosity employed by Johnson et al. is equivalent to the structure factor introduced by Zwikker and Kosten [16] and therefore represent a surrogate form of the complex effective fluid density. Johnson et al. interpolate between zero and infinite frequencies by the use of characteristic length, which is unambiguously related to the geometry, though difficult to calculate or measure exactly for practical types of materials. Champoux and Allard [17] extended this work and showed that an additional characteristic length was required in order to obtain an expression for the dynamic bulk modulus of the fluid in the pores. Allard and Champoux [9] later applied this approach to bulk fibrous materials and compared their prediction to the empirical power law of Delany and Bazley. They found that for random fibrous materials, the characteristic lengths could not easily be measured and some degree of approximation was inevitable. Work by Pride et al. [18] showed that at low frequencies the effective density of the fluid given by Johnson et al. should be modified. This was because Johnson’s model does not give the correct low-frequency behavior as w tends to zero. Therefore, they suggested five possible functions that can connect the low and high frequency limits, but they are also complicated expressions and difficult to interpret. Acoustical Model Sample of acoustical Model is shown below: r = APB * m In this model, A = K2 (d)-2, A and B being independent parameters. Model for airflow Resistivity This is a simple presentation of a model which enables engineers to calculate the airflow. Resistivity beginning with the values of the bulk density for fibrous material as well as the fiber diameter: The model for air flow Resistivity is shown below: Rd2Pm-K1 = K2 In this model r is known as the airflow resistivity, Pm is known as the bulk density while d is the average value of fiber diameter. Integrated Model Integrated model is a collection of an entire set of equations which are used to describe the acoustical properties of the fiber material, having the full knowledge of its bulk density and its thickness. Integrated model is essential since the manufacture of polyester fiber blankets combines many combinations of thicknesses and density as well as the measured values of coefficient of sound absorption or the values of airflow resistivity. The model is as shown below: ZR = P0 * C0 (1 + C1 * ((P0 * F)/r)-C2) B = (2 * ? * f)/C0 * (1 + C7 * ((P0 * f)/r) -Cg ) ZR and is the actual part of the characteristic acoustic impedance Z. The micro structural models are developed by calculating the exact solution for propagation in pores of constant, usually circular, cross-section, and then tuning the equations to accommodate more complicated geometries using shape factors [13]. Attenborough [19, 20] derived rigid frame models that require five parameters, including the static and dynamic shape factors for more complicated pore microstructures. He also showed that these models can be effectively used both to granular and fibrous materials. Champoux and Stinson [21] later came up with another five parameters model, which included two varying shape aspects accounting for thermal and viscous effects. They verified their model on various porous materials having an exactly know geometry. Wilson [13] developed a general three parameter model by matching relaxation characteristics of viscous and thermal properties and compared it to previous models. Although microstructural models are advantageous over other models owing to their adequacy for various high and low range frequencies and types of materials, they suffer from the disadvantage of being complicated and the need for determining about three to five parameters. This task’s main aim is to come up with an empirical model of polyfelt materials in order to effectively support acoustic engineering in noise control. The empirical models are highly advantageous as it only needs a single input, flow resistivity, which is easily measurable. References [1] Delany, M.E., and Bazley, E.N. (1970), ‘Acoustical properties of fibrous absorbent materials’, Applied Acoustics, Vol. 3, No. 1, pp. 105–116. [2] Fouladi, MH, Ayub, M, Nor, MJM (2011), ‘Analysis of coir ?ber acoustical characteristics’, Applied Acoustics, Vol. 72, pp. 35-42 [3] Rey, R., Alba, J., Arenas, J. P., and Sanchis, V. J. (2012), “An empirical modeling of porous sound absorbing materials made of recycled foam”, Applied Acoustics, 73 (6-7), 604-609. [4] Bies D.A., and Hansen, C.H. (1980a) ‘Flow resistance information for acoustical design’, Applied Acoustics, Vol.13, pp.357–91. [5] Dunn I.P., Davern W.A., (1986) ‘Calculation of acoustic impedance of multi-layer absorbers’, Applied Acoustics, Vol.19, pp.321–34 [6] Qunli, W., (1988) ‘Empirical relations between acoustical properties and flow resistivity of porous plastic open cell foam’, Applied Acoustics, Vol.25, pp.141–148. [7] Miki, Y. 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