Modelling of the acoustical properties of hemp particles

Construction and Building Materials 37 (2012) 801–811

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Modelling of the acoustical properties of hemp particles Philippe Glé a,⇑, Emmanuel Gourdon a,b, Laurent Arnaud a a b

DGCB, Ecole Nationale des Travaux Publics de l’Etat, Membre de l’Université de Lyon, Rue Maurice Audin, 69518 Vaulx-en-Velin Cedex, France LTDS UMR CNRS 5513, ECL/ENISE/ENTPE, Rue Maurice Audin, 69518 Vaulx-en-Velin Cedex, France

h i g h l i g h t s " The acoustical dissipation in loose hemp particles is governed by the interparticle pores. " Double porosity approach is used to model their acoustical properties. " Shiv have higher tortuosities and smaller viscous lengths than spherical particles. " Acoustical parameters can be computed from basic parameters of shiv. " Acoustical measurements enable to characterize the key properties of shiv.

a r t i c l e

i n f o

Article history: Received 3 January 2012 Received in revised form 29 May 2012 Accepted 4 June 2012 Available online 15 September 2012 Keywords: Shiv Hemp particles Acoustical properties Characterization

a b s t r a c t This paper focuses on the acoustical properties of mixes of hemp particles. A physical analysis of the experimental data revealed that acoustical dissipation in this material is governed by inter-particle pores in the tested frequency range. The evolution of the acoustical parameters of mixes of hemp particles is studied as a function of density and it was found that these parameters are strongly affected by the parallelepipedal shape of the particles. The relationships existing between these acoustical parameters and basic parameters of the material, such as the apparent density of the mix, the particle apparent density, the characteristic dimension of the particles and their shape factor are investigated. It is shown first that these basic parameters can be directly used to model the acoustical properties of the mixes, then that acoustical data can be very useful to characterize the shape and microstructure of these hemp aggregates.  2012 Elsevier Ltd. All rights reserved.

1. Introduction Environmental protection has now become a major issue in today’s world. Saving energy and reducing carbon dioxide emissions range among the various means of protecting the environment and both can be achieved in buildings, which are responsible for 44% of power consumption in France [1]. Using thermally insulating ecological materials instead of traditional materials is one of the keys to success. Further, materials are now expected to be efficient from mechanical, thermal and acoustical points of view in order to reduce costs during building process, and most of such multifunctional materials are porous. Indeed, trapped air in materials endows them with interesting thermal and acoustical properties. When their structural properties are satisfactory, they can be directly used in building construction. The manufacturing parameters of materials allows to determine their porosity, which in turn allows an easy control of the thermal and acoustic dissipation processes: this is very helpful when designing new materials.

⇑ Corresponding author. E-mail address: [email protected] (P. Glé). 0950-0618/$ - see front matter  2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.06.008

To improve sound absorption, existing pores in the material must be opened to the outside. Cellular, fibrous and granular materials are usually distinguished as the three types of porous materials. Ecological materials can fit under any of these categories; they can be either natural or recycled [2] and in both cases feature very good performance in thermal insulation, thereby enabling a reduction of heat loss in buildings. Additionally, some of them can store carbon dioxide. For instance, materials of plant origin store carbon dioxide during their growth. A number of these porous materials are well-known, for example wood wool and hemp wool in the fibrous category, and are already used in building as insulation boards. The cellular type is rare, as for instance tannin foam. And the granular type is available under the form of cellulose, wood concrete or hemp concrete. In this paper, we study more closely the acoustical properties of hemp concrete. Hemp concrete is a building material that consists in plant particles (hemp particles) mixed with a binder (lime or cement). Developed over the past 20 years and initially used for the restoration, it is now used to build houses and multi-floor buildings. It can be applied in buildings in various ways (projection process, brick) as filling material for walls in timber frame structures and as floor and roof insulation. Life cycle analysis [3] shows that 1 m2 of hemp

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P. Glé et al. / Construction and Building Materials 37 (2012) 801–811

concrete with a width of 26 cm encasing a timber frame stores 35.5 kg carbon dioxide over a reference period of 100 years, which proves the ecological relevance of this material. Moreover, as shown in [4–7] it features very interesting properties from mechanical, thermal and acoustical points of view, so that it can be labelled as a multifunctional material. The acoustical behaviour of hemp concrete has already been the subject of several studies. The effect of the binder content in the formulation was observed in [5]. Furthermore, the effect of density, particles size distribution, type of binder, and water content was succinctly discussed in [7]. A first approach has been developed to model its acoustical properties in [7], however the model does not highlight the effect of multiple scales of porosity existing in the material. Indeed, the porous microstructure of hemp concrete is quite complex, because it is composite and has a natural origin. Therefore, pores have dimensions distributed over three scales, with the inter-particle pores size ranging from 1 mm to 10 mm (see Fig. 1a and b), intra-particle size varying between 10 and 60 lm (see Fig. 1c) [8,9], and with the intra-binder pores characteristic size of 1 lm. In acoustics, classical models suggest that the pores sizes are of the same order of magnitude throughout the material. In the case of multiple scales of porosity, other approaches have to be considered. In the case of two scales of porosity, an homogenization approach was developed to take into account both pore sizes [10] and several cases of permeability contrast were distinguished as a function of the ratio between the scales of porosity. Using this kind of approach may allow a more accurate modelling of the properties of hemp concretes. However, it requires knowledge of many acoustical parameters which depend, in our case, simultaneously on the physical properties of the components and the manufacturing process of the material. In a perspective of developing a user-friendly model, it seems more useful to use the ‘‘basic parameters’’ known at the manufacturing stage, such as the density of the material, the particles content, the binder content and the water content, rather than acoustical parameters which cannot be known a priori. The idea developed in this article is to use a model composed of two steps. The first step links the basic parameters to the acoustical parameters and the second step links the acoustical parameters to the acoustical properties (sound absorption and sound insulation). The first step has already been achieved in the case of granular material having spherical aggregates of the same size [11,12]. Unfortunately, in the case of hemp concrete, aggregates shape is parallelepidedic. Besides, the lengths, widths and thicknesses of the particles are not uniform but can be described by distributions. Existing models must be adapted to take into account both the shape of aggregates and their size distribution.

Developing such a model would facilitate the prediction of the acoustical properties of the finished material at the manufacturing stage. Furthermore, it would enable to reach expected performances related both to acoustical absorption and sound insulation. The model could also be used to deduce the characteristics of constituents, such as the size of hemp particles, from acoustic measurements, and to perform a quality investigation of the material. This is particularly interesting with hemp concrete since hemp particles can have very different properties, depending on their variety, cultivation and harvesting method. The article focuses on shiv, particles in bulk obtained from the central wood of straw of hemp after extraction of the surrounding fibres. Shiv are hemp concretes without binder, which simplifies the modelling significantly. This paper is organized in three sections. Firstly, the materials studied and the characterization methods are described. Then the semi-phenomenological approach used to model this material is presented and illustrated in several cases. In the last section, the relationships between the basic parameters and the acoustical parameters are investigated. 2. Materials and methods 2.1. Presentation of the hemp mixes In this study, the acoustical properties of shiv were investigated. In order to discuss the effect of the origin of the plant and of the density on its acoustical properties, a wide range of shiv was tested as described in Table 1. There are five shiv with different origins (from SA to SE). For shiv SB, particles were sieved (from SB1 to SB5) in order to test the incidence of the particles size distribution on the acoustical properties. Moreover, these shiv were tested using several densities ranging between 80 kg m3 and 160 kg m3 as given in Table 1. These densities were adapted to the shiv so that particles were not compacted. 2.2. Characterization methods The first step of the characterization consisted in determining the particles size distribution and the open porosity of the shiv. Particles size distributions were investigated using a two-dimensional image analysis protocol developed in [13]. A representative mass of particles (5 g) is spread on coloured paper sheets (from 1 to 4 sheets of size 210  297 mm2 depending on the shiv) in such a way that the particles do not touch each other. These particles have usually a parallelepipedical shape, they are spread as to rest on the sheets on their largest section (length times width). Then, photographs of the sheets are analyzed in order to evaluate area A, length L, width W and slenderness L/W of each particle (please see the list of symbol in Table 9). The distributions of these parameters A, L, W, and L/W are presented in Fig. 2. Finally, the mass surface qs, defined as the ratio of the total mass of the particles on the cumulated area A of the particles, is calculated and is given in Table 1. These results show that shiv are very different from each other in size, except for shiv SA and SD which appear to be very similar. Moreover, in a same shiv, it appears that the particle size is distributed over a wide scale, this is due to the natural variety of size existing in the plant and to the scutching process (separation between the woody part and the fibres). However, slenderness L/W seems to have the same distribution regardless of the origin of the shiv.

Fig. 1. Photographs showing mixes of hemp particles SA ((a) shows the characteristic dimension of the particles, (b) shows the configuration of the particles in a cylindrical tube and (c) shows the intra-particle porosity).

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P. Glé et al. / Construction and Building Materials 37 (2012) 801–811 Table 1 Characteristics of the shiv.

3

SB

SC

SD

SE

SB1

SB2

SB3

SB4

SB5

100 150 0.181 1050

100 140 0.193 1350

100 160 0.232 1200

100 140 0.192 1140

100 140 0.140 870

80 120 0.328 1330

90 130 0.222 1270

90 130 0.183 1320

100 140 0.148 1340

110 150 0.117 1300

Cumulative percent

qmin (kg m ) qmax (kg m3) qs (kg m2) qframe (kg m3)

SA

1

Cumulative percent

Shiv

0.8 0.6 0.4 0.2 0

0

0.6 0.4 0.2 0

2

10

1 0.8

10

L/W

1

Cumulative percent

Cumulative percent

A (mm2 )

0.8 0.6 0.4 0.2 0

0

1 0.8 0.6 0.4 0.2 0

1

10

0

10

10

0

10

L (mm)

W (mm)

Fig. 2. Particles size distribution of the shiv SA ( ), SB ( ), SC ( ), SD ( ), SE ( ), SB1 ( ), SB2 ( ), SB3 ( ), SB4 ( ), SB5 ( ).

For such granular materials, porosity is a direct function of the density knowing the frame density of the particles qframe [7]. The measurement of these frame densities was performed for all shiv on three different samples having a mass Mshiv of 4 g, which ensures the representativity of the sample. The measurement consists in evaluating the volume of the frame Vframe of the shiv using air porosimetry [14]. The average frame densities are given in Table 1. The porosity can be calculated using Eq. (1). Among the tested shiv, the frame density ranges from 870 to 1350 kg m3. These values for frame density are of the same order as in [7]. One notices that for shiv SB, porosity does not depend on the particle size distribution of the shiv, but seems to depend only on its origin. The differences of frame densities can be attributed to the harvesting methods of hemp, and particularly to the retting process that consists in leaving the harvested plants on the ground so that sun, dew and rain help to separate the fibrous part from the woody part of hemp. During this process, pectins are washed out of the plant and porosity increases (pores closed by pectins are opened), so that the frame volume decreases and the frame density increases. This phenomenon is clearly visible in Table 1, where retted shiv SB, SBi and SC present greater frame densities than non-retted shiv SA, SD and SE.

/¼1

q qframe

with qframe ¼

M shiv V frame

ð1Þ

The different shiv were tested in a B&K type 4106 impedance tube having a diameter of 10 cm in the frequency range [150; 2000 Hz]. During measurement, the tube was held vertically since shiv were in bulk. The loose particles were poured in the tube and shaken to get the desired density with a thickness of 5 cm. The diameter of the tube and the thickness of the samples were chosen so as to ensure a low ratio between the characteristic sizes of the particles and the tested sample, so that the properties of the granular mixes are not affected by the shape of the containing tube. It has been shown that significant dispersion can exist in the measured acoustical properties of low permeability media [15]. For this reason, measurements were repeated three times in each configuration in order to check the reproducibility of the experiment. The shiv were characterized using the three microphone method [16] to enable the measurement of the intrinsic properties of the material: the equivalent dynamic ~ eq , which accounts for visco-inertial effects and the equivalent bulk density q e eq , which accounts for thermal effects (tilde is used in this paper for modulus K frequency-dependent properties). These two frequency-dependent properties fully describe the material, and enable to compute sound absorption and sound

insulation as a function of the thickness. Furthermore, viscous and thermal parameters can be determined from these intrinsic properties, using the indirect characterization relationships developed by Olny and Panneton [17,18].

3. Modelling of the acoustical properties of shiv: from acoustical properties to acoustical parameters In [7], a model was proposed to evaluate the acoustical properties of shiv and hemp concretes. Here, a new model gives more accurate predictions since it takes into account the multiple scales of porosity existing in the material. 3.1. Physical observations of the acoustical properties and interpretations Throughout the different measurements performed on shiv and hemp concretes, the same particularity can be observed. As shown in Fig. 3 in the case of shiv SA (q = 130 kg m3), the real part of the normalized bulk modulus is larger than 1// and c// (c being the ratio of the specific heats of the air and / the open porosity of the material) above 300 Hz. In porous material, the bounds of the real part of the bulk modulus are usually governed by the asymptotic behaviour of the thermal exchanges [19], which are isothere =P0 Þ ¼ 1=/ for x  xt) and adiabatic mal at low frequency (Reð K e =P0 Þ ¼ c=/ for x  xt), and delimited at higher frequency (Reð K by the thermal characteristic frequency xt defined in [20]. This can also be observed in Figs. 19, 20 and 21 of Ref. [7] where the measurement of the real part of the bulk modulus is underestimated by the model. This difference corresponds to an overestimate of the open porosity of more than 10%, although it was performed with a

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P. Glé et al. / Construction and Building Materials 37 (2012) 801–811

as the microsopic scale. The double porosity model developed by Olny and Boutin [10] has already been applied to porous granular media [21]. However, this is the first time to our knowledge that this model is used in the third acoustical behaviour described by Olny and Boutin [10] for a material with ‘‘natural’’ double porosity.

1.8

0

Re (K/P )

1.6 1.4

3.2. The adapted double porosity model 1.2 1

200

300

400

500

600

700

800

900

1000

f (Hz) Fig. 3. Measurement of the real part of the normalized bulk modulus ( )) compared to asymptotic limits 1// (. –) and c// (- –), for shiv SA (q = 130 kg m3).

precision of about 1%. This leads to the conclusion that classical porous models fails to predict the acoustical properties of such materials. However, these observations can be explained using the double porosity approach developed by Olny and Boutin [10]. For a porous medium described by two characteristic sizes (one microscopic size lm and one mesoscopic size lp), the acoustical e eq can be ex~ eq and K intrinsic properties at the macroscopic scale q pressed as a function of the intrinsic properties at the microscopic and mesoscopic scales. When an important contrast of permeability exists between the pores and the micropores, for llmp  103 , the model is described by Eqs. (2) and (3). In these equations, subscripts p and m refer respectively to the mesoscopic scale (the pores) and to the microscopic scale (the micropores).

q~ eq ðxÞ ¼

e eq ðxÞ ¼ K



 1  /p 1 1 þ q~ p ðxÞ q~ m ðxÞ !1 ð1  /p Þ e F d ðx; xd Þ þ e m ð xÞ e p ð xÞ K K 1

ð2Þ

xd 

P0 2

/m rm lp

ð3Þ

ris the resistivity (N m4 s), and P0 is the static pressure of the air (Pa). e F d is a pressure diffusion function that enables to separate three acoustical behaviours:  At low frequencies when x  xd ; e F d  1, which means that all pores take part into the acoustical dissipation process, with a uniform pressure in pores and micropores,  At frequencies such as x  xd, there is a coupling between the pressure in pores and the pressure in micropores that increases the acoustical dissipation in the material,  At higher frequencies when x  xd ; e F d  0 and the micropores do not participate anymore in the acoustical dissipation. When xd is below the tested frequency range, only the third e eq ðxÞ  K e p ðxÞ behaviour can be experimentally observed, so that K and the real part of the bulk modulus is between 1//p and c//p, higher than usual acoustical behaviour, as observed in Fig. 3. Shiv are characterized by several scales of porosity [7] with an inter-particle pores dimension ranging from 1 mm to 10 mm depending on the density and on the particles size distribution, and intra-particle pores dimensions which vary between 10 and 60 lm [8,9]. Therefore, the scale separation ratio meets the condition llmp 2 ½102 ; 103 , and suggests a high permeability contrast between inter-particle and intra-particle pores. Besides, the mesoheterogeneities (at the inter-particle scale) are small compared to the wavelength in the tested frequency range since it ranges approximately from 17 cm to 2.26 m, whereas particles length does not exceed few centimetres and inter-particle pores are less than 1 cm. Consequently, the conditions needed to apply the high contrast double porosity model are satisfied by considering the interparticle pores as the mesoscopic scale and the intra-particle pores

The Olny and Boutin [10] porous model is a fluid equivalent model that is based on the rigid frame hypothesis. As shown by Zwikker and Kosten [22], this assumption can be made above a characteristic frequency, when the vibrations of the frame and of the fluid in the porous material are decoupled. This decoupling frequency is given in Eq. (4) and it is evaluated through several examples in Table 2.

xdec ¼

r/2 q

ð4Þ

The frequencies calculated in the examples given in Table 2, and for all other shiv, being below the experimental frequency range [150; 2000 Hz], we can make the rigid frame assumption. e eq , we can use Eqs. ~ eq and K To compute the intrinsic properties q (2) and (3) in the case of high permeability contrast, which leads to the simplified Eqs. (5) and (6)

q~ eq ðxÞ  q~ inter ðxÞ e inter ðxÞ e eq ðxÞ  K K

ð5Þ ð6Þ

Since we assume a high permeability contrast between intraparticle and inter-particle pores, we have x  xd, which leads to rintra  / Px0 l2 > 80; 000 N m4 s. intra inter Hence, the viscous characteristic frequency of the intra-particle intra medium xv defined in [23] by Eq. (7) is above the tested frequency range. This shows that the flow in the material is only due to the flow in the inter-particle pores [10] so that the dynamic density can be approximated by the dynamic density of the interparticle medium.

xintra ¼ v

rintra /intra ¼ Oðrintra Þ  80; 000 rad s1 q0 aintra 1

ð7Þ

The intrinsic properties of the inter-particle porous medium e inter ðxÞ could be computed using different semi~ inter ðxÞ and K q phenomenological models. As shown in [7] on shiv and hemp concrete, four-parameter models are necessary to describe this kind of material. For this reason, the model by Johnson et al. [23] ~ inter and the visco-inertial effects, and has been used to predict q e inter and the the model by Zwikker et al. [24] has been used for K thermal effects. Their expressions are recalled in Eqs. (8) and (9). A different approach taking the pore size distribution existing in the material directly into account [25] could also be developed, since particularly suitable in the case of granular materials; this will be presented by the authors in a future paper.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 u inter 2 jrinter /inter u t1 þ j 4a1 gq0 x 5 q~ inter ðxÞ ¼ 1 inter 2 /inter xq0 a1 rinter K2inter /2inter pffiffiffiffiffiffiffi pffiffiffiffiffiffi11 0 Te NPr ~k j c P 0 e inter ðxÞ ¼ @1 þ 2ðc  1Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffi A K /inter NPr k j 2

ainter 1 q0 4

ð8Þ

ð9Þ

Table 2 Decoupling frequencies of shiv SA. Density q (kg m3)

100

110

120

130

140

150

xdec (rad s1)

11 1.8

13 2.0

14 2.3

19 3.1

22 3.5

27 4.2

fdec (Hz)

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P. Glé et al. / Construction and Building Materials 37 (2012) 801–811

e is the ratio between the Bessel functions of first and zero where T order, and ~ k is defined by Eq. (10).

~k ¼ sinter

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8ainter 1 q0 x rinter /inter

Firstly, the viscous parameters r, a1 and K are indirectly charac~ eq Þ and Iðq ~ eq Þ of terized from the real and imaginary parts Rðq the dynamic density measurement using the analytical method presented in [18]. The resistivity is estimated from the low frequency range of the imaginary part of the dynamic density while the tortuosity and viscous length are analytically derived from the model by Johnson et al. [23]. Then, as in [7], the shape factor s is kept constant at unity. Finally, porosity /inter is adjusted so that the model by Zwikker and Kosten agrees with the measurement of the real part of the bulk modulus. The expressions needed are given in Eqs. (11)–(14). In a first step, tortuosity was sought under the form /a1 since porosity /inter inter is determined at the end of the characterization process.

ð10Þ

where q0 is the density of the air (kg m3), g its viscosity (Pa s) and NPr its Prandtl number. Five acoustical parameters describe the inter-particle porous medium: the porosity /inter, the resistivity rinter, the high frequency limit of the tortuosity ainter 1 , the viscous characteristic length Kinter (m) and the shape factor sinter. In the following, the subscript inter will be removed of the notations except for the   inter-particle porosity /inter. e 2 ~ can be calculated as a ~ ¼ 1   Z Z0  , The sound absorption a eZ þZ0 e the surface impedance of the sample, and Z0 the impedance with Z of the air. For a hard backing behind the sample in the impedance ~ with Z e can be calculated as Z e ¼ j Z e c cotðkeÞ, e c the charactertube, Z  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ the wave number e eq ; k ~ eq K istic impedance of the media f Zc ¼ q ! rffiffiffiffiffiffi ~ ¼ x q~ eq and e the thickness of the material (m). in the media k e K eq

r ¼ lim  xIðq~ eq Þ

ð11Þ

x!0

a1 /inter

¼

1

"

q0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ~ eq Þ  Rðq

~ eq Þ2  Iðq

r2 x2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q0 g K¼ ~ eq Þðq0 /a1  Rðq ~ eq ÞÞ /inter xIðq inter

ð13Þ

s¼1

ð14Þ

a1

3.3. Characterization of the acoustical parameters In order to model the dynamic density and the bulk modulus (Eqs. (8) and (9)), five acoustical parameters must be determined.

ð12Þ

(a) α

1

0.5

0

200

400

600

800

1000

1200

1400

1600

1800

2000

f (Hz)

(c) 0

0

Im (ρ/ρ )

Re (ρ/ρ0)

(b) 5 0

200

400

600

800

−5 −10

1000

200

400

f (Hz)

0

1.5 1

600

1000

800

1000

(e) Im (K/P )

0

Re (K/P )

(d)

400

800

f (Hz)

2

200

600

800

1000

0.5

0 200

f (Hz)

400

600

f (Hz)

(f) TL (dB)

10

5

0

200

300

400

500

600

700

800

900

1000

f (Hz) Fig. 4. Comparison of the measured and modelled sound absorption (a), real and imaginary part of normalized dynamic density ((b) and (c)), real and imaginary part of normalized bulk modulus ((d) and (e)), and transmission loss (f). Three configurations of shiv are presented, SC ( ), SA ( ), and SB5 ( ). Markers correspond to data obtained through measurements and lines correspond to the model. Their acoustical parameters are given in Table 3.

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P. Glé et al. / Construction and Building Materials 37 (2012) 801–811

Table 3 Acoustical parameters used for the modelling.

1

Shiv

/inter

r (N m4 s)

a1

K (lm)

SC q = 100 kg m3 SA q = 130 kg m3 SB5 q = 150 kg m3

0.79 0.75 0.77

1530 3090 9290

1.72 2.07 2.15

389 215 101

0.98 0.96 0.94 0.92

φ

3.4. Experimental validation of the model The model was applied successfully to more than 50 different configurations of shiv. Three of them are presented in Fig. 4, with three different densities and particle size distributions. In Fig. 4, the measured sound absorption coefficient, normalized dynamic density, normalized bulk modulus and transmission loss are compared to the modelling. The predictions are very accurate, allowing to validate the hypothesis of high permeability contrast for the use of the double porosity model. However, the validity of this hypothesis can be confirmed only for frequencies above fmin = 150 Hz where it is checked that e F d  0, and below fmax = c0/(2Plp)  5400 Hz (for lp = 1 cm) so that frequency range is homogenizable [10]. This figure also enables to discuss the performance of shiv in sound absorption and sound insulation. It appears that the sound absorption is high given that thickness is 5 cm, especially for small particles and high densities (SB5q = 150 kg m3). This has already observed in [7] with a thickness of 10 cm. Concerning sound insulation, 5 cm of shiv provides a transmission loss lower than 10 dB for all configurations in the tested frequency range. However, this material is commonly used in buildings with a thickness ranging between 20 and 30 cm and with a binder, which increases significantly the transmission loss. For instance, hemp concrete bricks having a thickness of 31 cm can provide a weighted transmission loss of 43 dB [26]. 4. Modelling of the acoustical parameters of shiv: from acoustical parameters to basic parameters Several studies were performed to predict acoustical or transport parameters of porous materials as a function of manufacturing parameters or compression parameters. For fibrous materials, Castagnede et al. [27] have studied the evolution of the acoustical parameters as a function of a compression ratio knowing the parameters of a reference configuration. For shiv, the aim is to link the acoustical parameters to the characteristics of the particles – such as the particles size distribution and the intra-particle porosity – and to the configuration of the bed – such as its orientation and its density. 4.1. Porosity As explained in Section 2.2, the open porosity of the shiv can be determined from their density and their frame density using the linear relationship Eq. (1). The open porosity is represented in Fig. 5. In this figure, we can see that porosity ranges globally from 85% to 95% depending on the density. In the present study, another useful parameter is the inter-particle porosity since acoustical properties depend mainly on it. From Eq. (1), one can derive Eq. (15) to describe the inter-particle porosity as a function of both the density of the shiv and the apparent density of the particles qparticle.

/inter ¼ 1 

q qparticle

ð15Þ

This relationship is true as long as the particles are not compressed, for densities lower than a maximum qmax, which is the

0.9 0.88 0.86 0.84 0.82 0.8 80

90

100

110

120

130

140

150

160

−3

ρ (kg.m ) Fig. 5. Open porosity / of the shiv SA ( ), SB ( ), SC ( ), SD ( ), SE ( ), SB1 ( ), SB2 ( ), SB3 ( ), SB4 ( ), SB5 ( ) as a function of density.

case when particles are only shaken. And it has been observed during the experimental investigations that maximum density of shiv can range between 120 and 160 kg m3 (see Table 1) depending on its characteristics (particles size distribution and apparent density of particles). The particle density was determined using a least squares method to minimize the error between Eq. (15) and the inter-particle porosity estimates. This error  was evaluated using Eq. (16), where yexp(q) and ycal(q) are respectively the mean experimental parameter and the calculated parameter for the density q.

!2 X yexp ðqÞ  ycal ðqÞ ¼ yexp ðqÞ q

ð16Þ

The estimated apparent densities are provided in Table 4 with their coefficient of determination while inter-particle porosity is presented and compared with the theoretical evolution in Fig. 6. On this figure, there are little differences between porosity estimations and regressions. However, the results confirm the validity of Eq. (15) since almost all the coefficients of determination R2 are above 0.9. Inter-particle porosity varies between 65% and 85% and is from 10% to 20% less than the open porosity of the samples, which enables to validate the choice of the double porosity approach. Apparent particle densities range from 360 to 630 kg m3 and are very low compared to the frame densities. This is attributed to the important intra-particle porosity of the particles. This parameter /particle is evaluated from Eq. (17) and is also given in Table 4 for the different shiv. The results show that intra-particle porosity varies between 43% and 73% and highly depends on the particle size distribution. This can be due to the fact that small particles size distribution occurs when particles come from the top of hemp stem where the plant is less porous and more dense. This effect will be checked and further investigated in another study employing microscopy.

/particle ¼ 1  qparticle =qframe

ð17Þ

Through the mass surface qs deduced from image analysis and the apparent particle density qparticle, it is also possible to evaluate the mean thickness E of the particles of hemp. Indeed, by using the definition of these parameters, one can derive Eq. (18). The values were calculated for the shiv and are presented in Table 4. Mean thickness ranges from 0.2 to 0.9 mm, which corresponds to a visual estimate.

E ¼ qs =qparticle

ð18Þ

807

P. Glé et al. / Construction and Building Materials 37 (2012) 801–811 Table 4 Characterization parameters of the shiv. Shiv 3

qparticle (kg m ) /particle E (mm) R2(qparticle) Rparticle(r) (mm) R2(r) n R2(a1) Rparticle(K) (mm) R2(K)

SA

SB

SC

SD

SE

SB1

SB2

SB3

SB4

SB5

523 0.502 0.346 0.95 0.279 0.97 2.36 0.83 0.101 0.73

460 0.659 0.419 0.92 0.282 0.98 2.66 0.87 0.093 0.64

486 0.595 0.477 0.99 0.305 0.97 2.4 0.92 0.109 0.59

605 0.469 0.317 0.83 0.280 0.97 2.01 0.98 0.113 0.82

499 0.426 0.280 0.98 0.195 0.95 2.24 0.42 0.050 0.44

362 0.728 0.905 0.88 0.498 0.99 2.45 1.00 0.257 0.93

425 0.665 0.523 0.91 0.364 0.91 2.87 0.88 0.145 0.72

463 0.649 0.395 0.83 0.294 0.93 3.05 0.66 0.106 0.62

531 0.604 0.279 0.93 0.220 0.98 3.16 0.95 0.068 0.68

633 0.513 0.184 0.98 0.151 0.97 2.24 0.30 0.040 0.56

φ

inter

0.8

0.75

0.7

0.65 80

90

100

110

120

130

140

150

160

−3

ρ (kg.m ) Fig. 6. Inter-particle porosity /inter of the shiv SA ( ), SB ( ), SC ( ), SD ( ), SE ( ), SB1 ( ), SB2 ( ), SB3 ( ), SB4 ( ), SB5 ( ) as a function of density. Markers correspond to averaged measurements and lines correspond to Eq. (15).

As for the estimate of the inter-particle porosity from the real part of the bulk modulus measurement, it seems that the method used here is very interesting. It enables an adequate estimate of key parameters such as particle density or intra-particle porosity that otherwise may only be found using tomography or mercury density methods that are applied to only a few particles, and therefore less representative. 4.2. Resistivity The static air flow resistivity r, usually reduced to resistivity, characterizes viscous effects in porous materials at low frequencies. It is related to the static permeability k0 by Eq. (19).

r¼

g k0

ð19Þ

The resistivity measurements were studied and compared to several models of the literature. These models are either empirical or physical models, but they all describe the air permeability of granular materials from the inter-particle porosity /inter and a size Rparticle characterizing the equivalent radius of the aggregates. The models have the same dependence in 1=R2particle but various dependences in /inter. Their expressions are given in Table 5. Attenborough extensively studied the acoustical properties of granular materials. In [28], he establishes a relationship to describe the resistivity of spherical particles. Voronina and Horoshenkov have developed an empirical model for the acoustical properties of granular materials [29]. The low frequency limit of their model leads to another equation.

Using self-consistent methods, Umnova et al. [11], and Boutin and Geindreau [12] have derived analytical relationships for the resistivity, the tortuosity and the viscous characteristic length of a bed of spherical particles having the same radius. For the Boutin - Geindreau model, three equations are given for the resistivity, depending on the boundary condition chosen for the elementary volume. They are specified by subscripts p for the pressure approach, v for the flow approach and c for the zero vorticity approach. Finally, a different model was tested. This model is a theoretical validation of the empirical Ergun equation widely used for chemical applications and to predict pressure drop in wood particles beds. Prieur du Plessis and Woudberg generalized it to the entire porosity range in [30] using a self-consistent approach based on a rectangular geometry. These models were tested for shiv. The inter-particle porosity was computed from Eq. (15) while the particle dimension Rparticle(r) was adjusted to the measurement using the least squares method described in Eq. (16). Most analytical models are in good agreement with the experimental data and yield very similar optimal radius. However, Prieur du Plessis–Woudberg model was chosen since it best fits the experimental results. The evolution of the resistivity of the different shiv is given in Fig. 7 and is compared to the model. The estimated radii Rparticle(r) and the coefficients of determination R2(r) were evaluated and are given in Table 4. From Fig. 7, we can see that resistivities range from 500 to 10,000 N m4 s and that the agreement between this model and experimental data is very good (all coefficients of determination are above 0.9). Moreover, the estimated particle radii are consistent since they follow the size of the sieve for shiv SB1–SB5. However, the interpretation of these radii is tricky, because of the parallelepipedical shape of hemp particles. When comparing these radii to the results of the particle size distribution analysis, it appears that the estimated radii and the mean thickness of the particles have the same order of magnitude. According to the photograph presented in Fig. 1b the acoustic flow is perpendicular to the greatest section of the particles, so the characteristic dimension in this direction represents the thickness of the particles. A linear regression with a good fit (R2 = 0.94) has been performed between these two parameters and is given in Eq. (20).

Rparticle ðrÞ ¼ 0:462E þ 9:66  105

ð20Þ

Thus, this modelling of resistivity is very useful, since it is demonstrated here that, knowing the inter-particle porosity and the equivalent radius of the particles, it is possible to accurately evaluate the resistivity of the shiv. An illustration of this result is given in Fig. 8 where the normalized static permeabilities

k0 R2particle

are plotted as a function of /inter and describe the same evolution. In other words, this result means that, in terms of resistivity, shiv is completely equivalent to a mix of mono-sized spherical particles.

808

P. Glé et al. / Construction and Building Materials 37 (2012) 801–811 Table 5 Resistivity models for granular media.

r (N m4 s)

Authors

27g ð1  /inter Þ2

Attenborough

R2particle

/3:5 inter

100gð1  /inter Þ2 ð1 þ /inter Þ5

Voronina–Horoshenkov

/inter R2particle 9g

Umnova et al.

X¼

/2inter R2particle inter Þð1HÞX 2

2 ð1/ Boutin–Geindreaup



5

3

5  9H1=3 þ 5H  H2

H ¼ pffiffiffi ð1  /inter Þ 2P

3b g  2þ3b5 1 þ bð3þ2b R2particle 5 Þ b = (1  /inter)1/3

18b2 g

Boutin–Geindreauv

2 2

ð1b Þ 2 ð4 1b b  5 1b5 ÞRparticle

Boutin - Geindreauc

45b2 g 5

2 59bþ5b b Du Plessis–Woudberg

b6

R2particle

Agð1  /inter Þ2 2=3 2 Rparticle /3inter 3P

A¼

4

10000

25:4/3inter ð1  /inter Þ

2=3

ð1  ð1  /inter Þ1=3 Þð1  ð1  /inter Þ2=3 Þ2

Table 6 Tortuosity models for granular media.

9000 8000

Authors

a1

Boutin–Geindreau

3  /inter 2 1  /inter 1þ 2/inter /n inter (1  pln(/inter))2

−4

σ (N.m .s)

7000 Umnova et al.

6000 Attenborough Comiti–Renaud

5000

e

p ¼ e0:55þ0:18a

4000 3000 2000

3

1000

2.8

0 80

2.6 90

100

110

120

130

140

150

160 2.4

−3

ρ (kg.m ) ∞

2.2 2

α

Fig. 7. Resistivity r of the shiv SA ( ), SB ( ), SC ( ), SD ( ), SE ( ), SB1 ( ), SB2 ( ), SB3 ( ), SB4 ( ), SB5 ( ) as a function of density. Markers correspond to averaged measurements and lines correspond to the modelling.

1.8 1.6

0.35

1.4 1.2

0.3

1 80

0.25

90

100

110

120

130

140

150

160

−3

ρ (kg.m ) Fig. 9. Tortuosity a1 of the shiv SA ( ), SB ( ), SC ( ), SD ( ), SE ( ), SB1 ( ), SB2 ( ), SB3 ( ), SB4 ( ), SB5 ( ) as a function of density. Markers correspond to averaged measurements and lines correspond to the modelling.

0

k /R

2

0.2 0.15 0.1

4.3. Tortuosity 0.05 0 0.65

0.7

0.75

φ

0.8

0.85

inter

Fig. 8. Normalized permeability Rko2 of the shiv SA ( ), SB ( ), SC ( ), SD ( ), SE ( ), SB1 ( ), SB2 ( ), SB3 ( ), SB4 ( ), SB5 ( ) as a function of inter-particle porosity /inter. Markers correspond to averaged measurements and lines correspond to the modelling.

The high frequency limit of the dynamic tortuosity defined in [23], reduced to tortuosity, is also a key parameter for porous materials and characterizes the sinuosity of the porous network. Its definition is recalled in Eq. (21), where V is the volume of the pores and ~ v the velocity of the fluid in the pores. 1 V

R

v 2 dV ~ v dV 2 V

a1 ¼ 1 RV V

ð21Þ

809

P. Glé et al. / Construction and Building Materials 37 (2012) 801–811 Table 7 Viscous characteristic length models for granular media. Authors

K(m)

Umnova et al.

4ð1  HÞ/inter a1 Rparticle 9ð1  /inter Þ 4/inter a1 R 9ð1  /inter Þ particle 4/inter a1 1 R 9ð1  /inter Þ 1 þ b4 particle

Boutin–Geindreaupc Boutin–Geindreauv

Table 8 Range of validity of the models. Model /inter

r a1

6 5

Λ (m)

Range of validity

q < qmax "q qmin < q < qmax qmin < q < qmax 150 < f < 5400 Hz

Homogenization f < 5400 Hz Homogenization f < 5400 Hz

particles. Indeed, in [32], it was shown by testing several beds having a given porosity but different shapes, that the tortuosity is more important in the case of cubic and parallelepipedical particles. An empirical relationship was developed to predict parameter p as a function of the ratio height to side ae of the particles. Finally, Attenborough’s expression was chosen to model the tortuosity of the shiv. In Fig. 9, experimental data for the tortuosities of the shiv are compared to this model. For all shiv, parameter n was adjusted using the least squares method and is given in Table 4. From this graph, it is apparent that the modelled tortuosity is consistent with the experimental data. The shape factor n is included between 2.0 and 3.2 and is affected by the differences existing between shiv. However, at this stage of the study, no clear link has been found between n and the particle size distribution parameters derived from the image analysis. Further investigations will be led to understand this evolution of the shape factor.

x 10

4 3 2 1 0 80

Eq. (15) Physical du Plessis and Woudberg [30] Homogenization Attenborough [31] Empirical Umnova et al. [11] Semi-empirical Olny and Boutin [10] Homogenization

K e eq ~ eq ; K q ~ inter Johnson et al. [23] q e inter Zwikker and Kosten [22] K

−4

7

Description

90

100

110

120

130

140

150

160

−3

ρ (kg.m )

4.4. Viscous characteristic length

Fig. 10. Viscous characteristic length K of the shiv SA ( ), SB ( ), SC ( ), SD ( ), SE ( ), SB1 ( ), SB2 ( ), SB3 ( ), SB4 ( ), SB5 ( ) as a function of density. Markers correspond to averaged measurements and lines correspond to the modelling.

For granular materials, expressions were established by Boutin and Geindreau [12] or Umnova et al. [11]. For these models adapted to identical spherical particles, tortuosity only depends on the porosity of the bed. The expressions of these models are presented in Table 6. In the case of particles having an arbitrary shape, empirical models have also been developed and are widely used, for example the Attenborough model [31], or the Comiti and Renaud model [32]. They have an adjustable parameter to account for the shape of the aggregates. The models of Boutin–Geindreau and of Umnova et al. were tested first but they significantly underestimated the experimental tortuosity. The values of tortuosity range from 1 to 3, which is very high; this can be attributed to the parallelepipedical shape of the

Finally, the viscous characteristic length of the materials was investigated. This parameter is defined by Johnson et al. in [23] using the expression given in Eq. (22), where S is the surface in interface between the fluid and the frame of the material. It gives an order of magnitude of the size of interconnection between pores.

R 2 ~ v dS 2 R ¼ S 2 ~ K v dV V

ð22Þ

Few models exist to predict the characteristic viscous length in a granular material. Relationships are developed in the self-consistent approaches of Boutin and Geindreau, and of Umnova et al. for mixes of spherical particles. As shown in Table 7, the models are very similar, and depend on the radius of the particles, on the tortuosity and on the porosity of the bed. Then, these models were applied to shiv. Since they yield to similar results, only Umnova et al. model was finally used. In a first

Fig. 11. Diagram describing the two steps of the modelling.

810

P. Glé et al. / Construction and Building Materials 37 (2012) 801–811

4.5. Synthesis of the proposed method

Table 9 Appendix – list of symbols. j c0 P0 Z0

g c NPr

q0 q qs qframe qparticle /particle Rparticle n A L W L/W E e /

r k0

a1 K s

x xdec xv xt xd f Fd ~ q e K ec Z

pffiffiffiffiffiffiffi 1 Sound speed in air (m s1) Static pressure of air (Pa) Characteristic impedance of air (Pa m1 s) Viscosity of air (Pa s) Ratio of the specific heats of air Prandtl number Density of air (kg m3) Apparent density of shiv (kg m3) Surface density of particles (kg m2) Frame density of particles (kg m3) Apparent density of particles (kg m3) Porosity of particles Equivalent radius of particles (m) Shape factor of particles Area of particle (m2) Length of particle (m) Width of particle (m) Ratio length/width of particle Mean thickness of particles (m) Thickness of sample (m) Porosity Airflow resistivity (N m4 s) Static permeability (m2) High frequency limit of the dynamic tortuosity Characteristic viscous length (m) Shape factor of particles Frequency (rad s1) Decoupling frequency (rad s1) Visco-inertial characteristic frequency (rad s1) Thermal characteristic frequency (rad s1) Pressure diffusion characteristic frequency (rad s1) Pressure diffusion function Dynamic density (kg m3) Bulk modulus (Pa) Characteristic impedance (Pa m1 s)

~ k e Z

Wave number (rad m1)

a~

Sound absorption Transmission loss (dB)

f TL R2

Surface impedance (Pa m1 s)

Coefficient of determination

step, the equations were computed using the equivalent particle radii determined from resistivity measurements. Yet with these radii the model overestimates the experimental viscous characteristic length (see Fig. 10). This shows that spherical granular models are not suitable for the modelling of K for parallelepipedicalshaped particles. However, according to the authors knowledge, there is no other model adapted to non-spherical granular media. So, optimal radii for the viscous lengths Rparticle(K) have been determined using the least squares method. The radii Rparticle(K) and the coefficient of determination are given in Table 4. It appears that this second estimate of the radii of the particles is significantly lower than Rparticle(r). On the basis of the ten tested shiv, a linear regression was performed between Rparticle(r) and the mean thickness of the particles E. Another good fit (R2 = 0.93) was found using Eq. (23).

Rparticle ðKÞ ¼ 0:294E  1:32  105

ð23Þ

This regression, based on very different mixes of particles, suggests that characteristic viscous length can be calculated for all shiv regardless of their shape, apparent particle density and characteristic dimension. The idea here is to combine Eq. (23) and the relationship of Umnova et al. model to predict this parameter. However, the validity of this modelling cannot be extended out of the investigated densities, as for the modelling of tortuosity.

From the above, we can model the acoustical parameters of any shiv knowing four basic parameters. These basic parameters are the apparent density q of the shiv, the mean apparent density of the particles qparticle, the equivalent dimension of the particles Rparticle (directly deduced from the mean thickness of the particles E), and the shape factor n. q is easily found by measuring the volume and weight of the bed of particles, qparticle and n can be deduced from acoustical data using the models presented in this section, and the thickness can be determined from the image analysis. Then, the acoustical properties can be predicted from the four acoustical parameters (inter-particle porosity /inter, resistivity r, tortuosity a1 and viscous length K) using Johnson et al. and Zwikker and Kosten models. These two steps are illustrated in Fig. 11. The conditions of application of this modelling have been discussed throughout the paper and are recalled in Table 8.

5. Conclusion In this study, the acoustical properties of various shiv were investigated. Shiv is a granular material having porous aggregates that can be described by two characteristic pore networks, the intra-particle and inter-particle pores. Here, it is proved that the contrast of permeability existing between these two networks is sufficient to assume that the acoustical flow is governed by the bigger pores, the inter-particle pores. The double porosity model developed by Olny and Boutin was used and validated to model shiv acoustical properties. For these materials, four acoustical parameters have to be determined: inter-particle porosity, resistivity, tortuosity and viscous characteristic length. The experimental data of these parameters were studied as a function of the density of the shiv. Compared to spherical granular materials, the experimental results suggest higher tortuosities and smaller characteristic viscous lengths for a given inter-particle porosity. Then, these acoustical parameters were successfully compared to empirical and physical models describing them as a function of the characteristics of the bed of particles. It was thus proven that the acoustical parameters can be directly determined knowing the following basic parameters: the density of the shiv, the apparent particle density, the mean thickness of the particle and the shape factor. This allows to predict the acoustical properties of shiv by knowing the characteristics of the particles and their configuration. Further investigation is currently in progress to understand the meaning of the shape factor and to take into account how the binder affects the acoustical properties of hemp concrete.

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