MODELLING THE ACOUSTIC PROPERTIES OF EXTRUDED MATERIALS WITH COMPLEX PORE SIZE DISTRIBUTION Giulio Pispola1, Kirill V. Horoshenkov2 and Amir Khan3 1

2, 3

Department of Industrial Engineering, University of Perugia, Via Duranti 67, 06125, Perugia, Italy [email protected]

School of Engineering, Design and Technology, University of Bradford Bradford, England, UK

ABSTRACT This work presents the results of two distinct approaches for modelling the acoustic properties of highly heterogeneous porous media. Firstly, we show that the estimation of the Biot viscosity correction function through a direct numerical integration of the shear stress on the pore wall and the average seepage velocity over a representative range of pore sizes can provide an accurate alternative to analytical modelling techniques. The only requirement is the evaluation of the probability density function of the pore size, which was here achieved by an optical “pore-counting” technique, together with the non-acoustical properties of the material. Secondly, the observed pore size distributions could be distinctively split into two parts, a “micro-scale” and a “meso-scale”, suggesting the use of the double-porosity approach originally proposed by Olny and Boutin. Assuming a low permeability contrast and estimating the non-acoustical properties of the two scales, which may be related with the process parameters, allowed for a rough prediction of the performance. Numerical results of the two models were compared with impedance tube measurements showing good agreement.

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Euronoise 2006, Tampere, Finland

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Giulio Pispola, Kirill V. Horoshenkov and Amir Khan

INTRODUCTION

The progressive sustainability requirements, incorporating the environmental, economical and social points of view, represents a “driving force” for the development of new passive reducing devices, e.g. poro-elastic materials from domestic and industrial waste. Traditional solutions based on the use of virgin noise control elements can be replaced just if the improved sustainable ones show equivalent or even superior acoustic performance. A recently developed cold extrusion process is able to make use of recycled polymeric fibres and grains to produce highly heterogeneous porous structures, having a broad pore size distribution and exhibiting top-range acoustic performance together with relatively low density. This is achieved through an optimization of the chemical reaction, mechanical mixing and consolidation process controlling the parameters of the cold extrusion manufacturing method. Recent researches shown that it is not possible to assume a simple pore size distribution (e.g., log-normal) for modelling the acoustic behaviour of such materials [1]. In this work, two theoretical models are applied and compared to predict the intrinsic acoustic properties of extruded samples, aiming at identifying an engineering tool for correlating the process parameters and the performance of complex materials. 2

POROUS EXTRUDED MATERIALS

A controlled cold extrusion process was employed to manufacture the tested samples. A detailed description of process is provided in [1]. Such a technique, widely used in polymer production, was here tailored for mixing polymer grains and fibres, reclaimed from textile waste, with a binder and water in controlled proportions. A diphenylmethane diisocyanate (MDI) binder was chosen to create bonds between the granular and fibrous parts and, at the same time, react with water producing carbon dioxide. The structure of bonded grains and fibres may be considered responsible for the lower pore sizes, while the CO2 release and bubble creation for the bigger pores (see Fig. 1a). It was observed that an increase of the water/binder ratio gives rise to a wider pore size distribution.

probability density function

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -5

-3

-1

log2 (pore size, mm)

(a)

1

(b)

Fig. 1. (a) Photograph of one of the tested samples; (b) probability density function of the pore size.

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Euronoise 2006, Tampere, Finland

Giulio Pispola, Kirill V. Horoshenkov and Amir Khan

After an appropriate curing time, the sample was optically analysed through a microscope in order to estimate the porosity, φ, and probability density function (PDF), e(s), of pore sizes [1] (see Fig. 1b). Several samples, characterized by different water/binder ratios, were investigated. Their non-acoustical macroscopic parameters such as the flow resistivity, σ, and tortuosity, q, were measured with standard methods. Finally the acoustical characterisation of the samples was experimentally performed in the frequency range of 200-6400 Hz through a four-microphone impedance tube (diameter: 29 mm) employing the transfer-matrix approach detailed in [2]. 3

MODELLING MATERIALS WITH COMPLEX PORE SIZE DISTRIBUTION

The observed pore size distributions appear to be “multi-modal” (Fig. 1b). Two theoretical methods were chosen and used to predict the acoustic performance of the developed porous media. The first method is detailed in refs. [1,3] and here we denoted it as the “Arbitrary Pore Size Distribution” (APSD) method. This method is based on the numerical integration of the viscosity correction function and requires knowledge of the probability density function of the pore size. The second method is detailed in ref. [4] and here we denoted it as the “Double Porosity” (DP) method. The method assumes that the observed pore size distribution in a heterogeneous porous sample can be distinctively split into two scales, one around 10-3 m (the “meso-scale”), suggesting that the wave propagation is similar to a double porosity medium [4]. 3.1

Arbitrary Pore Size Distribution (APSD) approach

It has been shown already that the direct numerical integration of the terms in the Biot viscosity correction function over a given pore size PDF can provide an accurate alternative to other existing analytical modelling techniques [1,3]: ∞

1 F (ω ) = ∞ 4

∫ κ T (κ ) e ( s ) dr 0

∫ 1 − 2κ T (κ ) e ( s ) dr

, T (κ ) =

−1

I1 (κ ) ωρ 0 , κ =s I 0 (κ ) η

(1)

0

being I1 and I0 the modified Bessel functions (of the first kind) of the first and zero order respectively, ρ0 the static fluid density and η the dynamic viscosity of air. The only requirement here is the knowledge of the probability density function , e(s), of the pore size together with data on the porosity, flow resistivity and tortuosity – all are routinely measurable characteristics of the porous medium. 3.2

Double Porosity (DP) approach

The double porosity theory [4,5] can be employed to model the acoustic behaviour of multi-scale materials with complex pore size distribution. In the following, the subscripts p, m, db denote quantities respectively related to meso-pores, micro-pores and to the multi-scale medium. The following assumptions were adopted in the present case:

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Euronoise 2006, Tampere, Finland

Giulio Pispola, Kirill V. Horoshenkov and Amir Khan

a. Low permeability contrast [4], i.e. a low ratio between the characteristic dimensions of the meso and micro-porous parts of the medium: looking at the PDFs (Fig. 1b), it can be argued that this ratio should be of the order of 10. b. Only macroscopic, non-acoustical quantities, estimated from the optical analysis and from the measured properties of the multi-scale medium, were used to characterize the micro and meso-porous parts (see Table 1). Porosities could be computed from the results of the “optical-counting” technique, discriminating the overall areas to be attributed to the micro and the meso-pores (respectively, Afm and Afp) and employing Eq. (14) of ref. [4]:

φdb = φ p + φm (1 − φ p ) =

 A  Afm + 1 − fp  A  A  ( A − Afp )

Afp

(2)

where A denotes the sample area. Regarding the flow resistivity σ, the micro and meso-pores were considered as isolated tubes acting in parallel [4]: 1

σ db

 (1 − φ p ) 1   = + σ p   σ m

(3)

A rough estimation of the flow resistivity to be attributed to the micro-porous media could be obtained from the following [5]:

σm =

8η qm 2 sm

2

Ωm

b

,

sm = ∫ sm ⋅ e ( sm )ds

(4)

a

being the mean micro-pore size calculated from the PDF of the micro-pores within a given pore size range. The tortuosity q was, as a first approximation, considered equal for the two single porosity media: such assumption could be justified thinking that both the micro and meso-pores are consequence of a common process of gas release and blowout of bubbles. Table 1. Macroscopic non-acoustical characteristics of the sample (see also Fig. 1b for the PDF). “Overall” “Micro” “Micro” “Meso” “Overall” Flow Flow Tortuosity Thickness Porosity Porosity Porosity resistivity resistivity q [-] d [m] φdb [%] φm [%] φp [%] σdb [N m-4 s] σm [N m-4 s]

4000

20000

81

79

11

1.73

0.038

The semi-phenomenological models of Johnson et al. [6] and Lafarge et al. [7] were adopted to estimate the viscous and thermal dynamic permeabilities, Π and Θ, of the single porosity media: −1

1/ 2   Mi ω  Π i (ω ) = Π i ( 0 )  1 − j ϖ ν ,i  − jϖ ν ,i  , ϖ ν ,i =   2 ων ,i   

4

( i = m, p )

(5)

Euronoise 2006, Tampere, Finland

Giulio Pispola, Kirill V. Horoshenkov and Amir Khan

−1

1/ 2   M′ ω  Θi (ω ) = Θi ( 0 )  1 − j i ϖ t ,i  − jϖ t ,i  , ϖ t ,i =   2 ωt ,i   

( i = m, p )

(6)

being ω the angular frequency. It was approximately assumed that the thermal characteristic frequencies ωt,i were close to their viscous counterparts ων,i and that the dimensionless shape factors M and M’ were equal to 1 for both single porosity media. The complex dynamic density ρdb and complex dynamic compressibility Cdb of the double porosity medium were then computed through the expression suggested by Olny et al. [4]:

ρ db (ω ) = j

η (1 − φ p ) Π m (ω ) + Π p (ω ) ω

(

)

−1

 (1 − φ p ) Θ (ω )  γP  1   , K i (ω ) = 0  γ + j ( γ − 1) i 2  + Cdb =   K m (ω ) K p (ω )  φi  δ t ,i φi   

(7) −1

( i = m, p )

(8)

being γ the specific heat ratio, P0 the ambient pressure, Ki the bulk moduli and δt,i the thermal skin depths. Finally, the dynamic density and the complex compressibility of the double porosity medium were used to compute the normalised characteristic impedance zc, the complex wave number kc and the normalised surface impedance zs for a sample of thickness d (ρ0c is the characteristic impedance of air): 1 ρ db ρ 0c Cdb

(9)

kc = ω ρ db Cdb

(10)

zs = zc coth ( − jkc d )

(11)

zc =

7

Normalised surface impedance [-]

6 5 4

APSD model - real APSD model - imaginary DP model - real DP model - imaginary exp data - real exp data - imaginary

3 2 1 0 -1 -2

3

10 frequency [Hz]

Fig. 2. Normalised surface impedance - comparison between experimental data and theoretical predictions for sample in Fig. 1.

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Euronoise 2006, Tampere, Finland

Giulio Pispola, Kirill V. Horoshenkov and Amir Khan

In Fig. 2 a comparison between experimental data and theoretical predictions, in terms of normalised impedance, is reported. The employed approaches show equivalent predictive capabilities with an accuracy level suitable for engineering purpose. Some discrepancies can be noticed only at the higher frequencies. Slightly superior results are obtained in the case of the APSD method. This is to be attributed to the somewhat arbitrary estimates of the nonacoustical properties of the micro and meso-porous scales deduced from the observed PDFs. 4

CONCLUSIONS

Two distinct methodologies were applied for modelling materials with wide, complex pore size distribution, manufactured through a recently developed mixing and binding process (cold extrusion) of polymer grains and fibres. While both the methods allow for accurate predictions of the acoustic behaviour, they look appropriate for different purposes. The Arbitrary Pore Size Distribution technique requires the knowledge of the pore size probability density function, here achieved by an optical “pore-counting” technique. Thus, it is suitable for a detailed material optimisation. On the other hand, a semi-phenomenological approach requiring only macroscopic quantities could be more easily correlated to the manufacturing process parameters. The scale separation observed for the tested sample allows assuming wave propagation analogous to a double porosity medium. The double porosity method, apart from the non-acoustical properties of the multi-scale medium, which can be measured, requires an estimation of the properties of the two porous scales (micro and meso): further research is needed to manage this complication. REFERENCES

[1]

[2]

[3]

[4] [5]

[6] [7]

A. Khan, K. V. Horoshenkov and H. Benkreira, “Controlled extrusion of porous media for acoustic applications”, Proceedings of the 1st Symposium on the Acoustics of PoroElastic Media (SAPEM), Lyon, France, 2005. B. H. Song and J. Stuart Bolton, “A transfer-matrix approach for estimating the characteristic impedance and wave numbers of limp and rigid porous materials”, J. Acoust. Soc. Am. 107 (3), 1131-1152, 2000. K. V. Horoshenkov, I. Rushforth and M. J. Swift, “Acoustic Properties of Granular Materials with Complex Pore Size Distribution, Proceedings of 18th Int. Congress on Acoustics, Kyoto, Japan, 2004. X. Olny and C. Boutin, “Acoustic wave propagation in double porosity media”, J. Acoust. Soc. Am. 114 (1), 73-89, 2003. K. V. Horoshenkov and M. J. Swift, “The acoustic properties of granular materials with pore size distribution close to log-normal”, J. Acoust. Soc. Am. 110 (5), Pt. 1, 23712378, 2001. D. L. Johnson, J. Koplik, and R. Dashen, “Theory of dynamic permeability and tortuosity in fluid-saturated porous media”, J. Fluid Mech. 176, 379–402, 1987. D. Lafarge, P. Lemarinier, J. F. Allard and V. Tarnow, “Dynamic compressibility of air in porous structures at audible frequencies”, J. Acoust. Soc. Am. 102 (4), 1995–2006, 1997.

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