Acoustical roerties of double orosity granular materials Venegas, R and Umnova, O htt://dx.doi.org/10.111/1.3644915 Title Authors Tye URL Acoustical roerties of double orosity granular materials Venegas, R and Umnova, O Article Published Date 011 This version is available at: htt://usir.salford.ac.uk/19008/ USIR is a digital collection of the research outut of the University of Salford. Where coyright ermits, full text material held in the reository is made freely available online and can be read, downloaded and coied for non commercial rivate study or research uroses. Please check the manuscrit for any further coyright restrictions. For more information, including our olicy and submission rocedure, lease contact the Reository Team at: usir@salford.ac.uk.
Acoustical roerties of double orosity granular materials Rodolfo Venegas a) and Olga Umnova Acoustics Research Centre, University of Salford, Salford M5 4WT, England (Received 14 Aril 011; revised 5 August 011; acceted 11 Setember 011) Granular materials have been conventionally used for acoustic treatment due to their sound absortive and sound insulating roerties. An emerging field is the study of the acoustical roerties of multiscale orous materials. An examle of these is a granular material in which the articles are orous. In this aer, analytical and hybrid analytical-numerical models describing the acoustical roerties of these materials are introduced. Image rocessing techniques have been emloyed to estimate characteristic dimensions of the materials. The model redictions are comared with measurements on exanded erlite and activated carbon showing satisfactory agreement. It is concluded that a double orosity granular material exhibits greater low-frequency sound absortion at reduced weight comared to a solid-grain granular material with similar mesoscoic characteristics. VC 011 Acoustical Society of America. [DOI: 10.111/1.3644915] PACS number(s): 43.55.Ev, 43.0.Bi, 43.0.G [KVH] Pages: 765 776 I. INTRODUCTION Proerties of granular materials are of great imortance in many areas of acoustics and noise control. This aer is focused on the acoustical roerties of double orosity granular materials, i.e., ackings of orous articles. The study of the long-wavelength sound roagation in orous media has mainly been focused on estimating two intrinsic quantities, i.e., the characteristic imedance and the wave number. 1 Biot investigated the roblem of elastic wave roagation in a statistically isotroic orous elastic solid saturated by a viscous fluid and found that two dilatational waves, a slow and a fast wave, and one rotational wave can roagate in the medium. In this aer, only the slow wave is considered as the solid frame is assumed to be motionless. A widely acceted semihenomenological model for the acoustical roerties of materials with a single ore size or narrow ore/grain size distribution, i.e., single orosity materials, has been roosed by Johnson et al. 3 and Lafarge et al., 4 and imroved by Pride et al. 5 and Lafarge. 6 These models make use of scaling functions that correctly match the low- and high frequency limits of the dynamic viscous 7 and thermal 4 ermeabilities, and rely on indeendently measurable macroscoic arameters; namely, orosity, static viscous and thermal ermeabilities, static viscous and thermal tortuosities, tortuosity, and viscous and thermal characteristic lengths. Allard et al. 8 have measured most of these macroscoic arameters for a random acking of beads and found good agreement between the model rediction and measured data. Asdrubali and Horoshenkov 9 have used a rational Padé aroximation to model sound roagation through exanded clay granulate. This modeling aroach assumes that the material ossesses caillary ores whose size distribution is log-normal. However, the ore size distribution of exanded clay may be better described as multimodal. The main drawback of these aroaches is the difficulty in measuring the model arameters. Moreover, they rovide a) Author to whom corresondence should be addressed. Electronic mail: r.g.venegascastillo@edu.salford.ac.uk little information about the microstructure influence on the acoustical roerties. In an attemt to overcome these difficulties and rovide ractical exressions to work with, emirical models have been roosed by Voronina and Horoshenkov. 10 They have measured flow resistivity, orosity, and tortuosity and fitted the so-called structural characteristic to redict sound roagation in granular materials such as erlite, vermiculite, and granulate nitrile. An emirical model has, however, limited redictive ower and does not rovide an understanding of the material morhology influence on the acoustical roerties. A microstructure-based aroach overcomes these roblems. It requires information about the material structure such as article shaes, sizes, and arrangement and relies on solving the oscillatory fluid flow and heat conduction roblems in a reresentative geometry. Chaman and Higdon 11 have calculated dynamic viscous ermeability of monodiserse arrays of sheres arranged in eriodic lattices. Umnova et al. 1 have extended the cell model aroach to calculating the drag arameters of acking of sheres considering a geometrically justified outer cell radius. These authors have also rovided exressions for dynamic bulk modulus based on a mathematical similarity between the oscillatory fluid flow and heat conduction roblems. 13 The homogenization of eriodic media theory 14,15 (HPM) has been alied by Gasser et al. 16 to numerically solve the boundary value roblems at different scales and calculate both the macroscoic material arameters and the acoustical roerties of a face-centered cubic (fcc) acking of sheres. A similar work, based on numerical HPM, has been ublished by Lee et al. 17 for simle cubic, bodycentered cubic (bcc), fcc, and hexagonal close-acked arrays. Boutin and Geindreau 18 have roosed analytical estimates of dynamic viscous ermeabilities for granular media using a combined HPM and self-consistent aroach. The descrition of the acoustical roerties of granular materials is comleted with an analytical exression for dynamic thermal ermeability also derived by these authors. 19 An emerging field is the study of the acoustical roerties of double orosity materials which are efficient at absorbing J. Acoust. Soc. Am. 130 (5), November 011 0001-4966/011/130(5)/765/1/$30.00 VC 011 Acoustical Society of America 765
low frequency sound. In these materials, two fluid networks with well-searated characteristic ore or inclusion sizes can be identified. The equations for sound roagation in double orosity materials have been derived by Auriault and Boutin 0 and Boutin et al. 1 using HPM. It has been ointed out in a later work that the material roerties strongly deend on the ratio e 0 between the characteristic size of the microscoic scale l m and that of mesoscoic scale l.atmoderateinterscale ratio values, e.g., e 0 ¼ l m =l 10 1, the two fluid networks strongly interact and influence both the macroscoic fluid flow and heat conduction. This case is usually referred to as low-ermeability contrast. Materials with this roerty have been exerimentally and theoretically investigated by Pisola et al. 3 For small interscale ratio values, e.g., e 0 10 3, the macroscoic flow is determined by the mesoscoic fluid network. The dynamic bulk modulus is modified due to ressure diffusion effects in the microdomain. This leads to an additional dissiation term which deends on the mesoscale geometry and the microdomain characteristics. This effect is secific to double orosity materials and has been exerimentally confirmed for erforated orous anels 4 and for orous materials with orous inclusions. 5 In this aer, the acoustical roerties of double orosity granular materials, i.e., ackings of orous articles, are investigated both theoretically and exerimentally. The aer is organized as follows. In Sec. I, the wave equation describing sound roagation in double orosity materials of arbitrary geometry is resented. The calculations of the dynamic density and bulk modulus are outlined. In Sec. II the general theory is alied to ackings of identical orous sheres. Exerimental validation of the models and discussion are resented in Sec. III. Measurements on both lowand high-ermeability contrast granular materials (exanded erlite and activated carbon, resectively) are reorted as well as the methods used to estimate their structural characteristics. The main findings are summarized in Sec. IV. FIG. 1. Three scales of the double orosity material (adated from Ref. ). be satisfied, e.g., e ¼ l =L 1 and e 0 ¼ l m =l 1. The existence of the reresentative elementary volumes REVm X m and REV X can then be ensured rovided that the conditions on the interscale ratios are fulfilled. The mesoscale and microscale orosities are / ¼ X f =X and / m ¼ X fm =X m, resectively, where X fi is the oen voids/ore volume and X i is the volume of the REVs. Here and in the following i ¼ ; m. The overall orosity of the material is given by / db ¼ / þ 1 / /m. The subscrit db denotes a double orosity quantity from now on. The wave equation for acoustic ressure in a rigidframe double orosity material is given by jx K db ðxþ r k db ðxþ r ¼ 0: (1) g II. THEORY A. Sound roagation in double orosity materials Overview Following Refs. 0, three scales can be identified and used to describe sound roagation in a rigid-frame double orosity granular material saturated by a Newtonian fluid, e.g., air. They are schematically shown in Fig. 1. The macroscoic characteristic size L is associated with the sound wavelength k in the material as L ¼ Oðjk=jÞ. The characteristic size l is determined by the size of the mesoheterogenities, which in the case under study corresonds to the article size. The characteristic size l m is determined by the size of the ores within the article. This characteristic size is assumed large enough so that the saturating fluid is continuously distributed throughout the sace it occuies. Rarefaction effects are therefore neglected. However, it has been shown that these effects do not change the form of the macroscoic isothermal acoustic descrition in single 6 and double 7 orosity materials but the way the effective quantities are calculated. To model the material as a homogenous equivalent fluid, the searation of scales assumtion should Time deendence in the form e jxt is assumed. Here, g is the dynamic viscosity and K db and k db are the dynamic bulk modulus and dynamic ermeability of the double orosity material. The latter becomes a scalar quantity for isotroic media, i.e., k db ¼ k db I, where I is the unitary second-rank tensor. The way of calculating K db and k db deends on the value of the interscale ratio e 0. When e 0 10 3 a highermeability contrast is achieved and the ores in the articles have negligible contribution to the macroscoic fluid flow. In this case, the dynamic ermeability k db coincides with that of a acking of solid articles, i.e., k db ðxþ ¼ k ðxþþ 1 / km ðxþ k ðxþ; () where k and k m are the dynamic ermeability tensors of the mesoscoic and the microscoic domains, resectively. The dynamic bulk modulus for a material with highermeability contrast is given by K db ðxþ ¼ 1 K ðxþ þ 1 /! 1 K m ðxþ F P 0 x d ; (3) / m K m ðxþ 766 J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media
where K and K m are the dynamic bulk moduli of the mesoscoic and microscoic domain, resectively. These can be exressed in terms of the dynamic thermal ermeabilities k 0 and k 0 m as4 K i ¼ cp 0 k 0 iðxþ 1 c jxq / 0 ðc 1ÞC ; (4) i j/ i where q 0 is the gas density, C is the secific heat at constant ressure, j is the thermal conductivity, and c is the adiabatic constant. In the case of low-ermeability contrast, e.g., e 0 10 1, the dynamic ermeability of the double orosity material is given, in a first aroximation, by a arallel flow model. Hence, the dynamic ermeability of the microscoic domain k m ðxþ in Eq. () cannot be neglected. This exression holds for mesoscoic geometries with straight ores or slits. The dynamic bulk modulus under low-ermeability contrast condition is obtained by relacing the function F d by 1 in Eq. (3). The way of calculating the dynamic viscous and thermal ermeabilities and the function F d is now detailed. The fluid flow at the microscoic and mesoscoic levels is described by an oscillatory Stokes forced roblem with no-sli boundary condition on C si. This roblem is linear and can be written in terms of the X i eriodic dynamic ermeability field k _ i and the zero mean value ressure _ i as 7 jx q 0 g k_ i þr _ i Dk _ i ¼ I in X fi ; (5) rk _ i ¼ 0 in X fi ; and k _ i ¼ 0 on C si : (6) The dynamic ermeability is then calculated by Daveraging E the solution over the REVi as k i ðxþ ¼ k _ i with i hi i ¼ ð1=x i Þ Ð X fi dx. For isotroic microscoic media it becomes k i ¼ k i I. The thermal exchanges between the saturating fluid and the solid matrix are described by an oscillatory heat conduction roblem with isothermal condition alied on C si. This linear roblem can be written in terms of the X i eriodic dynamic thermal ermeability distribution _ k i as follows: 4 jx q _ 0C j k0 i Dk 0 i ¼ I in Xfi ; and ki 0 ¼ 0onCsi : (7) The dynamic thermal ermeabilities of the microdomains and mesodomains are then calculated by averaging the solution of Eq. (8) over the REVi as ki 0 _ ðxþ ¼hki 0 ii. The function F d in Eq. (3) is defined as the ratio of the average ressure in the microscoic domain to that in the mesoscoic domain. It is related to the dynamic ressure diffusion function DðxÞ as F d ðxþ ¼ 1 jxg / m k0m 1 P 0 DðxÞ ; (8) 1 / where k 0m is static viscous ermeability of the microdomain and is calculated from the solution of Eqs. (5), (6) for x ¼ 0. In this exression, the material has been assumed isotroic or just the referential flow direction is taken into account. In addition, the sound roagation in the microscoic domain has been assumed isothermal [this also imlies relacing K m ðxþ by P 0 =/ m in Eq. (3)], and in a viscous regime. The ressure in the microscoic domain is not uniform and governed by a diffusion equation with boundary condition imosed by ressure in the mesoscoic domain. This roblem can be also formulated in terms of the X eriodic dynamic ressure diffusion distribution D _ as DD _ jxg / m P 0 k 1 0m D_ ¼ 1in X s and D _ ¼ 0onC s : (9) Pressure diffusion function DðxÞ is then calculated by averaging the solution over the solid hase of the REV as D E DðxÞ ¼ D _ ¼ 1 ð D _ dx (10) X X s This comletes the direct aroach. All the quantities can be calculated from their definitions for every frequency of interest. This could, however, require a significant amount of comutation time when dealing with realistic geometries. To overcome this roblem, one can use scaling functions that correctly match the low- and high-frequency asymtotics of the dynamic ermeabilities, 5 bulk moduli, 6 and ressure diffusion function. 8 In this way, the acoustic descrition is reduced to solving three static roblems er scale and one for the ressure diffusion function. The form of the scaling functions is the same for all of the abovementioned quantities and is the following: H i ðxþ¼h 0i s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jx þ1 P!i þp!i 1þ jx! 1 M!i x!i x!i P :!i (11) The symbol H is used to reresent viscous, thermal or ressure diffusion quantities and the subscrit! indexes the characteristic frequencies x!i and shae factors M!i and P!i accordingly. For viscosity related quantities H i ¼ k i and! ¼ v with P vi ¼ M vi =4ða 0i =a 1i 1Þ, M vi ¼ 8k 0i a 1i =/ i K i, and the viscous characteristic frequency defined as x vi ¼ g/ i =k 0i a 1i q 0. For temerature related quantities H i ¼ ki 0 and! ¼ t with P ti ¼ M ti =4 a 0 0i 1, M ti ¼ 8k0i 0 =/ ik 0 i, and the thermal characteristic frequency defined as x ti ¼ j/ i =C q 0 k0i 0. In these exressions, k 0i and k0i 0 are the static viscous and thermal ermeabilities. The static viscous and thermal tortuosities are calculated as a 0i ¼ / i hk _ i i 0ii 0i hk_ i and a 0 0i ¼ / _ ihk 0i _ i 0i hk0 0ii i. High frequency tortuosities and viscous characteristic lengths for isotroic materials are obtained from 3,9 / i a 1 1i ¼ h E i ei i and K i ¼ Ð X fi E i E i dx= Ð C si E i E i dc, where the scaled electric field (local electrical field divided by the alied macroscoic otential gradient) is given by E i ¼ e r# i and # i is the X i eriodic deviatoric art of an electric otential. This can be calculated from the solution of 9 J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media 767
D# i ¼ 0 in X fi and n r# i ¼ n e on C si : (1) The thermal characteristic length is a geometrical arameterdefined as twice the volume-to-ore-surface ratio, 30 i.e., K 0 i ¼ X fi=c si. For ressure diffusion related quantities H i ¼ Dand! i ¼ d with P d ¼ M d =4ða d 1Þ, M d ¼ 8D 0 = 1 / K d, and the ressure diffusion characteristic frequency defined as x d ¼ 1 / P0 k 0m =g/ m D 0. The static ressure diffusion arameter D 0 is calculated from the solution of Eq. (9) for x ¼ 0. The static ressure diffusion tortuosity is defined as 8 _ a d ¼ 1 / hd 0i and the 0 ihd_ ressure diffusion characteristic length as K d ¼ X s =C s. The characteristic imedance and the wave number are related to the dynamic density, q db ðxþ ¼ gkdb 1ðxÞ=jx, and dynamic bulk modulus as 1 Zdb c ðxþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi q db K db and q c db ðxþ ¼ x q db Kdb 1 : Surface imedance of a rigidly-backed material layer of thickness d is then calculated as 1 Zdb w ðxþ ¼ jzc db cot qc db d : Pressure reflection coefficient R db and normal incidence sound absortion coefficient a db are given by 1 R db ðxþ ¼ Zw db Z 0 Zdb w þ Z 0 and a db ðxþ ¼ 1 jr db j ; (13) where Z 0 is the characteristic imedance of air. In summary, dynamic viscous and thermal ermeabilities of the microdomain and mesodomain are needed to model the acoustical roerties of a double orosity lowermeability contrast material. To model materials with high-ermeability contrast dynamic viscous and thermal ermeabilities for the mesodomain, the static values of the bulk modulus and viscous ermeability of the microdomain and the ressure diffusion function are required. The analytical exressions used for modeling double orosity granular materials are resented in the next section. B. Analytical model for double orosity granular materials In a first aroximation, an unconsolidated double orosity granular material can be modeled as an array of identical orous sheres. The ores in the sheres can be assumed cylindrical. The mesoscoic domain is characterized by the article radius r and the intergranular orosity / ¼ X f =X, while the ore radius r m and the microorosity / m ¼ X fm =X m characterize the microscoic domain. First, dynamic viscous and thermal ermeabilities of the mesodomain are calculated. Different estimates for dynamic viscous ermeability of single orosity granular materials have been deduced using HPM combined with a selfconsistent aroach. 18 The inclusion comosed of a concentrically arranged solid article in a fluid shell is assumed to be surrounded by a Darcy medium. The oscillatory flow in the fluid shell is described by a Stokes forced roblem. A boundary condition of zero velocity is alied on the article surface. At the inclusion boundary, the normal velocity comonents in the fluid and in the equivalent Darcy medium are assumed equal. Following energy consistency arguments, two ossible conditions can be imosed on the stress on the inclusion boundary. For ressure or static aroach (P-estimate) the normal stress comonent in the fluid matches the ressure in the equivalent Darcy medium. In this case, the tangential velocity comonent does not match that in the Darcy medium, as is the case for the velocity or kinematic aroach (V-estimate). The P-estimate is considered here because it is energy consistent, correct u to the first order (in terms of the exansion arameter e) comared to the macroscoic descrition obtained using HPM, and rovides closer agreement with numerical results for regular acking of sheres. 18 It results in the following exression for dynamic viscous ermeability: k ðxþ ¼ d v ; 1 3C (14) x where Ax þ B tanh x b 1 C ¼ ax þ b tanhðxðb 1ÞÞ (15) A ¼ 3 þ ðbxþ 1 þ x 3b 1 þ x ; 6 (16) B ¼ 3 þ ðbxþ 1 þ x 3bx 1 þ x ; (17) 6 a ¼ 1 3þ bx 3b 3 ð 4 Þ b 1þx þ 6 coshðxðb 1ÞÞ ; (18) b ¼ 3 þ bb ð 1Þx b 1 þ x ; (19) rffiffiffiffiffiffiffiffiffi g 1=3; r d v ¼ ; b ¼ 1 / jq 0 x and x ¼ : (0) bd v The static viscous ermeability is recovered from Eq. (14) assuming that frequency tends to zero: 18 k 0 ¼ r 3b! þ 3b 5 b 3 þ b 5 1! : (1) The corresonding dynamic thermal ermeability is calculated from the solution of an oscillatory heat conduction roblem with isothermal boundary condition on the article surface and zero temerature gradient on the inclusion boundary. 19 It is given by k 0 ðxþ¼d t 1 b 3 þ 3b 1þx t tanhðx t ðb 1ÞÞ x bx t t x t þtanhðx t ðb 1ÞÞ 1 ; () where d t ¼ ffiffiffiffiffiffiffi ffiffiffiffiffiffiffi N rdv and x t ¼ x= N r. The Prandlt number is defined as N r ¼ C g=j. The exressions for dynamic viscous and thermal ermeabilities of a cylindrical ore network for the microdomain 768 J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media
are calculated from the solution of Eqs. (5) (7). These are given by 1 where k m ðxþ ¼/ m d v ð 1 Gs ðþ Þ; and km 0 ¼ / md ffiffiffiffiffiffiffi t 1 G N rs : (3) Gs ðþ¼ J 1 ðþ s s J 0 ðþ s and s ¼ ffiffiffiffiffi r m j jd v j : (4) Here J 0;1 are Bessel functions of the first kind. The static viscous ermeability is 1 k 0m ¼ / mrm 8 : The ressure diffusion function DðxÞ has been analytically calculated for a acking of identical sherical orous articles as follows. Allowing only for radial diffusion, Eq. (9) is written in sherical coordinates as @D r @r þ @ D @r jd d ¼ 1and D _ r ¼ r D_ ¼ 0; (5) where the ressure diffusion skin deth is defined as qffiffiffiffiffiffiffiffi P d d ¼ 0 k 0m / m gx: Multilying Eq. (5) by r and making the change of variable f ¼ ffiffiffiffiffi j d 1 d r, one can obtain a nonhomogenous sherical Bessel equation f @ ff D _ þ f@ f D _ þ f D _ ¼ jd d f ; with D _ f ¼ ffiffiffiffiffi j r d 1 ¼ 0: The general solution of this equation is 31 D _ h ¼ Aj 0 ðfþþby 0 ðfþ jd d ; where j 0 ; y 0 are sherical Bessel functions of first and second kind of order 0. The constant B is set to zero as the solution cannot be infinite at the article center. The coefficient A is calculated from the boundary condition as A ¼ jd z d sinðþ z ; where z ¼ ffiffiffiffiffi j r d 1 d. Here, the identity31 j 0 ðfþ ¼sinðfÞ=f was used to derive this exression. The ressure diffusion function is then obtained by integrating D _ over the volume of the orous shere as DðxÞ ¼ 1 / r ð3 3z cotðþ z z Þ=z 4 : The function F d is related to DðxÞ through Eq. (8) as F d ðxþ ¼ 3 z ð1 z cot ðþ z Þ: (6) Sound roagation in the microdomain has been assumed isothermal and in a viscous regime. In a more general case of non-isothermal and visco-inertial sound roagation, z should be relaced by z ¼ r x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð/ m =P 0 Þðg=jxk 0m Þ ¼ r x ffiffiffiffiffiffiffiffiffiffiffiffiffi q m Km 1 ¼ r q c mð x Þ. However, this generalization does not seem to be necessary for common acoustic materials in the audible range of frequencies. III. MODEL VALIDATION AND DISCUSSIONS To validate the models, measurements of sound absortion coefficient of double orosity materials have been conducted in a vertically installed B&K 406 imedance tube. The two-microhone method described in the standard 3 ISO 10534-:001 has been used. The analytical model for single orosity granular materials is validated first by conducting measurements on a acking of lead shots. Validations for both low and high ermeability contrast double orosity materials are erformed after that. Exanded erlite serves to comare redictions given by a hybrid model for low-ermeability contrast granular materials. The term hybrid model is used in this aer if at least one of the quantities is numerically calculated using scaling functions [Eq. (11)]. Two activated carbon samles are used to validate the model for double orosity high-ermeability contrast granular materials. For every case, a detailed descrition of the methods used to estimate the characteristic sizes and orosities is resented. A. Single orosity granular material 1. Characterization The article radius distribution of lead shots has been measured using otical granulometry. The image rocessing consists of adjusting the contrast, converting to black and white, and erforming morhological oerations to fill the holes. 33 The rocessed image is then analyzed to obtain the equivalent radius of each article. This has been done by fitting a circle with the same area as the ixel region. Figure shows the comlementary cumulative distribution function of the equivalent article radius obtained using 108 articles. The article radius distribution is close to normal, with a mean value of r ¼ 0:5507 mm and a standard deviation of 0.009 mm. Considering the non-erfect nature of the lead shots and that the nominal radius is 0.5 mm the agreement between this value and the exected value of the fitted distribution can be considered satisfactory. The bulk density of the acking has been measured by weighting a small cubic container of a known volume filled with the shots. Using the tabulated value of lead density (q ls ¼ 11:34 g/cm 3 ) the orosity of the acking has been estimated as / ¼ 1 q b =q ls ¼ 0:3905, where q b ¼ 6:9117 g/cm 3 is the bulk density of the acking.. Comarison with data The sound absortion coefficient of a lead shot layer is calculated using Eq. (13) with dynamic viscous and thermal ermeabilities given by Eqs. (14) and (). The subscrit db in Eq. (13) should be relaced by to reflect the single orosity nature of the acking. Measured and redicted values of ressure reflection coefficient are shown in Figs. 3(a) and 3(b) for frequencies between 150 to 6 400 Hz. The inset lot [Fig. 3(c)] shows the sound absortion coefficient. The slight disagreement between the data and the redictions could be due to limited alicability of the analytical model for low orosity materials (/ < 0:6) as suggested in Ref. 19. The inaccuracy, due to finite article size, in defining the layer thickness could also contribute to the disagreement. If the layer thickness is set to d ¼ 3:165 cm, a better agreement is obtained as is also shown in Figs. 3(a) 3(c). The model for the acoustical roerties of single orosity granular materials has been therefore validated and will be further used in modeling double orosity granular materials. J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media 769
FIG.. Comlementary cumulative distribution function for lead shot article radius. Circles data. Continuous line fitted normal distribution. Dashed lines 95% confidence interval. The inset lots show the original (a) and the rocessed (b) images. B. Double orosity granular material with low-ermeability contrast 1. Characterization Exanded erlite is a chemically-inert industrial mineral made out of naturally occurring siliceous volcanic rock. This material is commonly used in ceiling tiles and roof insulation roducts. The samle used in this work is a commercial roduct called Exanded Perlite P3 (P3 from now on). The estimated density of the frame is.774 g/cm 3. The bulk density has been measured using the rocedure described in the revious section and is equal to 0.06 g/cm 3. The overall orosity is therefore / db ¼ 0:9776. This value does not necessarily correlate with the oen orosity commonly used in acoustics but includes closed ores as well. The fact that this value is used in calculations inevitably affects the accuracy of the redictions. The P3 grains are fairly ellisoidal with a nominal 80%-band grain size of 0.75 1. mm. This means that 10% of the grain sizes fall below or above this range. It has not been ossible to aly otical granulometry to this samle due to its extremely light-weight nature. In the absence of the actual article size distribution, the equivalent article radius has been calculated as half of the average between the uer and lower limit of the 80%-band grain size, i.e., r ¼ 0:4875 mm. It is known that identical nonsherical articles can ack better than sherical articles. 34 Therefore, the mesoorosity can be smaller than that for a random close acking of identical sheres, which is given by 35 / 0:36. On the other hand, the effect of olydisersity is manifested through a decrease in orosity as small articles can fit in between the voids formed by larger articles. An arrangement that reflects this henomenon is the bcc acking. The mesoorosity is therefore set equal to that of the bcc close acking, i.e., / ¼ 0:3. It is, however, recognized this value is somewhat arbitrary. FIG. 3. Real (a) and imaginary (b) arts of the reflection coefficient of a 3-cm hard-backed layer of lead shots. Circles data. Continuous line redictions. Dashed line model redictions for a layer thickness of 3.165 cm. The inset lot (c) shows the sound absortion coefficient. Scanning electron microscoe (SEM) images, shown in Fig. 4, have been taken in order to obtain information about the inner structure of P3. Figure 4(b) shows the wall junctions, which resemble Plateau borders 36 (three and four junctions meet at angles close to 10 and 109.47, resectively). From this image, the average wall thickness has been estimated as h w ¼ 0:47460:088 mm. The oly-diserse semiclosed foam-like microstructure of P3 can be identified in Fig. 4(c). A cell size C s is defined as the largest distance between oosite walls in each cell. The mean cell size has been calculated using manual image segmentation and is C s ¼ 47:5869:659 mm. The interscale ratio is therefore given by e 0 ¼ C s =r ¼ 0:0976. This confirms the lowermeability contrast assumtion.. Microstructure modeling of exanded erlite One can notice from Fig. 4(c) that each cell can be aroximated by a olyhedron. The inner geometry of P3 must therefore obey the rules of sace-filling acking of olyhedra. 37 A geometry that satisfies this condition, and 770 J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media
that related to the Plateau borders law, is the Kelvin foam. 38 Other ossible geometries for modelling foams are the Weaire Phelan foam, 38 array of cylinders arranged in a hexagonal lattice, 39 and array of sherical voids connected by cylindrical ores. 40 The Weaire Phelan foam has eight cells in the REV. This low symmetry limits its use. Although an array of cylinders can reresent oen-cell foams with thin struts, it does not aear to aroriately reresent semi-closed foams. Meanwhile, an array of sherical voids connected by cylindrical ores does not obey sace-filling rules. The inner geometry of the P3 granules is therefore modeled as a monodiserse array of erforated truncated octahedra. The erforations have been included in an attemt to reresent the holes observed in the P3 granules [see Fig. 4(a)]. This imlicitly assumes that the fluid hase is connected. The truncated octahedron is an Archimedian solid that has eight regular hexagons and six squares faces (N ¼ 14). It corresonds to a flat-faced version of the Kelvin foam. It is generated by subtracting six ffiffi square yramids of side a and height a= from a regular octahedron of side 3a and half height 3 ffiffi a=. A cell size is defined as the distance between the square faces and is given by C s ¼ ffiffi a. The volume and the area of the truncated octahedron are V ¼ C 3 s = and A ¼ ð 3=4 Þ 1 þ ffiffi 3 C s. The elementary fluid cell has been built as follows. A truncated octahedron with cell size C s is generated. Then, fourteen cylinders of radius r er ¼ nc s =4 ffiffi and height hw = are located at the center of each face [see Fig. 5(d)]. The arameter n ½0; 1Š controls the erforation size with resect to half of the smallest face (square of side a). The microorosity and thermal characteristic length can be calculated as / m ¼ C 3 s þ Nr er h w = ðc s þ h w =Þ 3 and 3K 0 m =4C s ¼ ð1þ n Nh w =3C s Þ=w, with w ¼ 1 þ ffiffiffi ffiffiffi 3 þ Nn=3 ð hw =C s n=4 ffiffi Þ. The viscous and thermal ermeabilities of the microscoic domain have been calculated using the semi-henomenological models given by Eq. (11). To obtain the values of the model arameters the numerical solution of the static fluid flow [Eq. (5),(6) for x ¼ 0], the heat transfer [Eq. (7) for x ¼ 0], and the high-frequency oscillatory fluid flow (electrical conduction) [Eq. (1)] roblems have been erformed using the finite element method software COMSOL MULTIPHYSICS. Second-order Lagrangian elements have been used to model the velocity comonents, whereas the linear elements aroximated the ressure field (P-P1 velocityressure formulation), as suggested in Ref. 41. An arbitrary reference ressure has been set in one of the vertices of the fluid cell to ensure uniqueness of the solution for the fluid flow roblem. 4 Second-order Lagrangian elements have been used for the temerature and the electric otential for the heat and electrical conduction roblems, resectively. In all these roblems, eriodic boundary conditions were rescribed on the boundaries corresonding to the erforation faces. The convergence of the numerical method has been tested by a careful mesh refining analysis. The microorosity has been set to / m ¼ 0:967 in order to match the measured overall orosity. The diameter of the erforations has been set to half of the smallest face of the truncated octahedron FIG. 4. Scanning electron microscoe images of exanded erlite P3: P3 grain (a), wall junctions (b), and the inner structure (c). (q ¼ 0:5) while the wall thickness to h w ¼ 1:3883 mm. This value is about three times larger than the one estimated by image rocessing and has been chosen to match the overall orosity value. The cell size has been, however, set to the measured value. Table I shows the arameters utilized to model the acoustical roerties of the array of erforated truncated octahedra. Only results for which the ressure gradient has been alied in the negative z-direction are resented here and in the following. These do not differ more than.534% from those obtained when the ressure gradient was alied in the negative x- or y-direction. This reflects the quasi-isotroic nature of the microscoic domain. 3. Comarison with data Figure 5 shows the real and imaginary arts of ressure reflection coefficient for a 3-cm hard backed layer of P3. The single orosity model is not caable of reroducing the J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media 771
TABLE I. Parameters for the microdomain of exanded erlite P3. 10 3 k 0m =C s 10 3 k 0 0m =C s a 0m a 0 0m a 1m K m =C s K 0 m =C s 0.7051 14.844.3519 1.45.0431 0.1066 0.339 erforated truncated octahedra. On the other hand, the arallel fluid flow model used to calculate the viscous ermeability might not be a realistic assumtion as this exression only holds for mesoscoic geometry with straight arallel cylindrical or slit ores. A more general aroach may be based on solving the set of equations A1 in Ref.. A better agreement can be obtained by fitting the analytical double orosity model to the data. This rocedure has been imlemented by using the differential evolution algorithm. 43 The fitted values for the microore and article radii and the orosities are r m ¼ 8:055 mm, / m ¼ 0:999, r ¼ 0:599 mm, and / ¼ 0:871. This results in an overall orosity of / db ¼ 0:997, which is.6% larger than the measured value. The fitted microore diameter aears to be comarable to the measured average cell size lus its standard deviation while the fitted article diameter corresonds to the uer limit of the nominal 80%-band grain size. Although a much closer agreement is obtained, it is recognized that a staggered array of circular ores does not corresond to the actual microdomain geometry of P3. The disagreement between the data and the redictions is not very ronounced for sound absortion coefficient as is shown in Fig. 5(d). The general trend is that a double orosity low-ermeability contrast granular material resents larger sound absortion at low frequencies comared to a solid-grain material with the same mesoscoic characteristics at reduced weight. C. Double orosity granular materials with high-ermeability contrast 1. Characterization Two samles of granular activated carbon made of coal have been used as examles of double orosity granular FIG. 5. Real (a) and imaginary (b) arts of the reflection coefficient and sound absortion coefficient (c) of a 3-cm hard-backed layer of exanded erlite P3. Circles data. Continuous black line analytical single orosity model. Dashed black line hybrid double orosity model. Dashed gray line fitted analytical double orosity model. (d) geometry and mesh of the elementary fluid cell. data over the whole range of frequency. A reasonably good agreement is found between the data and the hybrid model. The following simlifications used in the modeling could exlain the observed differences between data and the redictions. The analytical model for the mesodomain assumes that the articles are sherical and monodiserse while P3 is clearly a olydiserse granular material with nonsherical grains. The same argument alies to the microscoic domain as it has been modeled as a monodiserse array of FIG. 6. Comlementary cumulative distributions of the equivalent article radius for the activated carbon samles. Gray circles: data for SRD71. Black circles: data for SRD75. Dashed dotted lines fitted distributions. The inset lots show the rocessed images. (a) SRD75 and (b) SRD71. 77 J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media
TABLE II. Proerties of activated carbon samles. Samle Bulk density q b (g/cm 3 ) N surface area (m /g) CO surface Area (m /g) Nanoore volume V n (0 nm) (cm 3 /g) Overall orosity / db Equivalent article radius (mm) SRD71 0.566 665 371 0.83 0.747 0.7536 (0.1609) SRD75 0.335 174 847 0.774 0.8477 0.7364 (0.056) materials with high ermeability contrast. Activated carbon is normally manufactured by carbonizing raw material followed by an activation rocess by either oxidization of CO or steam. This rocess creates a hierarchical orosity ranging from nanometer to micrometer size ores within the granules. Activated carbon is commonly used in filtration and urification rocesses due to its remarkably large surface area and sortion caacity. The samles will be referred to as SRD71 and SRD75 in the following. Otical granulometry has been alied to these two samles. The image rocessing is the same as the one imlemented for the lead shot samle. Figure 6 shows the comlementary cumulative distribution of the equivalent article radius for SRD71 and SRD75. The inset lots show the rocessed images. The number of articles considered for SRD71 is 439 and the average mass er article is 0.8319 mg. In the case of SRD75, these corresond to 08 and 0.479 mg, resectively. For both samles, the equivalent article radius follows a lognormal distribution, fðtjl; hþ¼ 1=th ffiffiffiffiffi ex ðlnðþ l t Þ =h. The arameters of the fitted distribution for SRD71 are l ¼ 7:19 and h ¼ 0:11 while those for SRD75 are l ¼ 7:513 and h ¼ 0:741. These two samles are very similar in terms of mesoscoic characteristics. The difference in their inner structure is significant as the samle SRD75 is more orous than SRD71. This can be deduced from the average mass er article and more exlicitly from Table II where the bulk density, ore surface area, nanoore volume, and the estimated overall orosity are resented along with the exected value and standard deviations (in arenthesis) of the equivalent article radius for SRD71 and SRD75. The tabulated density of carbon black q c ¼ :g/ cm 3 has been used to estimate the overall orosity. In double orosity materials with high-ermeability contrast the fluid flow is not affected by the microores. To justify the alicability of this aroximation, flow resistivities of the two activated carbon samles have been measured. The rocedure is detailed in the standard 44 BS EN 9053:1993. Table III resents the flow resistivity for SRD71 and SR75 for a layer of 4 cm and two samles with layer thicknesses of cm. The other columns show a theoretical estimation of mesoorosity that matches the measured flow resistivity assuming identical sheres with their radii given by the exected values of the equivalent article radius distribution (last column of Table II). The flow resistivity was calculated as r 0 ¼ g=k 0, where the static viscous ermeability of the sheres acking is given by Eq. (1). The observed variability in flow resistivity is 6.85% for SRD71 and 6.14% for SRD75. This can be exlained with an average variability in mesoorosity of 1.77 and 1.6%, resectively. The average flow resistivity of SRD75 is slightly larger than that of SRD71 due to its higher dust content. Consequently, its mesoorosity is smaller. The flow resistivity and the mesoorosity can be considered thickness indeendent and not significantly affected by the activated carbon acking conditions. The fact that the two samles with very similar mesoscoic characteristics and different microscoic characteristics have close flow resistivity values justifies the high-ermeability contrast assumtion. These results also allow estimating the microorosity as / m ¼ / db / = 1 /.. Comarison with data The data of ressure reflection coefficient for a rigidlybacked -cm layer of SRD71 along with the model redictions is resented in Figs. 7(a) and 7(b). The article radius and the mesoorosity values are given in Tables II and III. The microorosity / m ¼ 0:68 has been calculated using mesoorosity and overall orosity values. The microore radius has been set to r m ¼ 0:715 mm. This has been calculated through a best fitting routine using the differential evolution algorithm. 43 This value correlates well with the mean size of the macroores (or transort ores) commonly found in activated carbon 45 and rovides an interscale ratio of e 0 ¼ r m =r ¼ 0:9455 10 3, which is in line with the high-ermeability contrast assumtion. On the other hand, the static viscous ermeability including rarefaction effects is given by 46 k 0m ðk n Þ ¼ / m rm ð 1 þ 4K nþ=8, where the Knudsen number is defined as 46 K n ¼ l mean =r m. Considering that the molecular mean free ath l mean is aroximately 60 nm at normal conditions, 47 the calculation of the static viscous TABLE III. Flow resistivity data and mesoorosity estimation for activated carbon. Flow resistivity (kpa s/m ) Mesoorosity / SRD71 SRD75 SRD71 SRD75 A(d¼4cm) 0.041 3.0667 0.31 0.305 B(d¼cm) 0.4436 4.63 0.30 0.99 C(d¼cm).7196 6.0870 0.311 0.95 Average 1.068 6 1.444 4.593 6 1.5104 0.3083 6 0.0055 0.997 6 0.005 J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media 773
ermeability would differ by 33.6%. This seemingly large difference does not significantly modify the redictions of the acoustical quantities in the audible frequency range [see, for examle, Eq. (64) in Ref. 47]. The double orosity model rovides much closer agreement with the data than the single orosity model. However, its accuracy is affected by the fact that only a limited amount of information about the material microstructure was available. Figures 7(c) and 7(d) resent the same information as Figs. 7(a) and 7(b) but for the samle SRD75. The calculated microorosity value is / m ¼ 0:785 while the fitted microore radius is r m ¼ 0:9881 mm. This value gives an interscale ratio of e 0 ¼ 1:3419 10 3. The agreement between the double orosity model and the data is better than that for the single orosity model. This is emhasized in Fig. 8 where absortion coefficient data for both samles is resented along with redictions of the single orosity and double orosity models. The single orosity model redictions are close due to similar mesoscoic characteristics of the two samles. The double orosity models redictions of absortion coefficient are also close at low frequencies. This is exected since the static flow resistivity values of the samles are close. To rovide an insight into the low-frequency behavior, the dynamic bulk moduli for both activated carbon samles and the lead shots have been deduced from the low-frequency characteristic imedance and wave number measured using the two-thickness method 48 as K db ¼ xzdb c =qc db. It is shown in Fig. 9 that the normalized static value of the dynamic bulk modulus of the lead shots is accurately redicted and given by 1=/ 1/0.3905 ¼.5608. For SRD71 the redicted value is 1/0.747 ¼ 1.3464 while for SRD75 is 1/0.8477 ¼ 1.1797. By extraolating the real art of the dynamic bulk modulus to zero frequency one can conclude that the static limit is 0.5777 for SRD71 and 0.474 for SRD75. These values cannot be exlained by the roosed model and suggest that the theory may need to be extended to account for hysical rocesses that modify the static bulk modulus. This quantity is modified when the finite heat caacity of the solid is taken into account. 4 According to Eq. (B10) in Ref. 4, the static bulk modulus is given by /Kðx! 0Þ=P 0 ¼ ð1 þ hþ= ð1 þ h=cþ,whereh is the ratio of the heat caacity of air to that of the solid. For the activated carbon samles h ¼ Oð10 3 Þ and the exerimental trend cannot therefore be exlained. Thermal sli effects modify the dynamic bulk modulus but not its static value 46,47 Mass transfer rocesses also alter the dynamic bulk modulus. For examle, Raset et al. 49 have studied the sound attenuation in rigid cylindrical ores filled with air and saturated water vaor accounting for the mass transfer of vaor from the wet tube wall. According to Eq. (47) in Ref. 49 and equation of state, the static bulk modulus for an array of cylindrical ores with wet walls can be written in terms of the ratio between the artial ressure of air 1 and that of water vaor as /Kðx! 0Þ=P 0 ¼ 1= ð1 þ = 1 Þ. However, this ratio is small at normal conditions 49 and might not be able to exlain the measured activated carbon static bulk modulus value. The resence of adsorbed films of water at the contact oint of the granules has been identified as the main cause of both damening structure-borne sound and shifting the resonant frequency of a bar filled with tungsten articles FIG. 7. Real (a) and imaginary (b) arts of the reflection coefficient of a -cm hard-backed layer of SRD71. Circles data. Continuous gray line double orosity model. Continuous black line single orosity model for a acking of solid articles [Eqs. (14) and ()]. Dashed black line single orosity model for the microscoic domain alone [Eqs. (3)]. The same information but for SRD75 is shown in (c) and (d). towards lower frequencies. 50 A similar shift has been documented in loudseakers 51 artially filled and resonators 5 fully filled with activated carbon. This effect has been however attributed to sortion rocesses that occur in the small ores within the grains. The effect of sortion on sound 774 J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media
might be necessary. It should also be noted that the friction caused by the article vibration has been suggested to increase the low-frequency sound absortion in granular materials. 54 Particle motion, mass transfer and sortion rocesses are likely to be occurring during sound roagation in activated carbon and need to be included into a more general theory. This theory could also include an additional orosity scale at the nanometre level, which seems to be the case that would better describe the acoustical roerties of activated carbon. This idea is motivated by the larger nanoorosity of SRD75 (/ n ¼ q b V n ¼ 0:593) comared to that of SRD71 (/ n ¼ 0:160) (see Table II). FIG. 8. Sound absortion coefficients of -cm hard-backed layers of SRD71 (gray circles) and SRD75 (black circles). Continuous lines double orosity model redictions. Dashed lines single orosity model redictions for acking of solid articles. attenuation in straight cylindrical ores has been studied by Mellow et al., 53 who aroximated the sortion dynamics by a Langmuir isotherm model. From Eq. (39) in Ref. 53, the static bulk modulus can be deduced as /Kðx! 0Þ=P 0 ¼ 1= 1 þ P 0 k a k d q N = ðk a P 0 þ k d Þ q 0, where k a and k d are the adsortion and desortion constants and q N ¼ r s =r n is the maximum adsorbed density. Here, the surface mass density that can be accommodated is denoted as r s while the ore radius as r n. This exression might be able to exlain the general trends in the behaviour of the bulk modulus of activated carbon. However, the Langmuir model is a oor aroximation to the sortion characteristics of activated carbon. A more comrehensive aroach based for instance on the Freundlich adsortion isotherm model 45 IV. CONCLUSIONS Double orosity materials have been intensively studied in the ast decade. It has been roven that they can achieve larger low frequency sound absortion comared to single orosity materials. However, granular materials with orous articles, which are commonly used in building and chemical industries, have received relatively little attention. These materials come with a large variety of article and ore sizes which allows low as well as high ermeability contrast. In this aer, the acoustical roerties of double orosity granular materials have been studied both theoretically and exerimentally. The existing self-consistent model for a acking of identical solid sheres has been extended to allow for article orosity. Exanded erlite has been used as an examle of a granular material with low ermeability contrast. High ermeability contrast has been achieved in samles of activated carbon. It has been demonstrated that ackings of orous articles rovide much imroved low frequency sound absortion comared to that of solid articles with the same mesoscoic characteristics at reduced weight. This makes these materials otentially attractive for acoustic alications. It has also been found that the low frequency roerties of activated carbon cannot be comletely exlained by their double orosity structure as the measured static values of the bulk moduli are lower than those redicted by the theory of sound roagation in double orosity materials. This might be an indication of mass transfer and sortion rocesses haening in smaller ores. The investigation of these effects and their use in designing new acoustic materials are interesting toics for future research. ACKNOWLEDGMENTS R.V. gratefully acknowledges the ORSAS award and the University of Salford Research Studentshi. The authors are grateful to Chemviron and William Sinclair Holdings PLC for kindly sulying the samles investigated in this aer. FIG. 9. Normalized measured real art of the low-frequency dynamic bulk modulus for SRD75 (black diamonds), SRD71 (gray circles), and lead shots (gray asterisks). The lines reresent the redicted static bulk moduli 1=/ ;db. 1 J. F. Allard and N. Atalla, Proagation of Sound in Porous Media: Modeling Sound Absorbing Materials (Wiley, New York, 009), Chas. 5. M.A, Biot, Theory of roagation of elastic waves in a fluid saturated orous solid. I. Low-frequency range, J. Acoust. Soc. Am. 8, 168 178 (1956); M. A. Biot, Theory of roagation of elastic waves in a fluid saturated solid. II. High-frequency range, ibid. 8, 179 191 (1956). 3 D. L. Johnson, J. Kolik, R. Dashen, Theory of dynamic ermeability and tortuosity in fluid-saturated orous media, J. Fluid Mech. 176, 379 40 (1987). 4 D. Lafarge, P. Lemarinier, J. F. Allard, and V. Tarnow, Dynamic comressibility of air in orous structures at audible frequencies, J. Acoust. Soc. Am. 10(4), 1995 006 (1997). J. Acoust. Soc. Am., Vol. 130, No. 5, November 011 R. Venegas and O. Umnova: Acoustics of double orosity granular media 775