Acoustics of sorptive porous materials

Accepted Manuscript Acoustics of sorptive porous materials Rodolfo Venegas, Claude Boutin PII: DOI: Reference:

S0165-2125(16)30126-3 http://dx.doi.org/10.1016/j.wavemoti.2016.09.010 WAMOT 2109

To appear in:

Wave Motion

Received date: 3 June 2016 Revised date: 13 September 2016 Accepted date: 19 September 2016 Please cite this article as: R. Venegas, C. Boutin, Acoustics of sorptive porous materials, Wave Motion (2016), http://dx.doi.org/10.1016/j.wavemoti.2016.09.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Acoustics of sorptive porous materials Rodolfo Venegas and Claude Boutin

Rodolfo Venegas

1

Université de Lyon - Ecole Nationale des Travaux Publics de l'Etat - LGCB/LTDS UMR-CNRS 5513, Rue Maurice Audin, 69518 Vaulx-en-Velin - France. e-mail: [email protected]

Claude Boutin Université de Lyon - Ecole Nationale des Travaux Publics de l'Etat - LGCB/LTDS UMR-CNRS 5513, Rue Maurice Audin, 69518 Vaulx-en-Velin - France. e-mail: [email protected]

1

Corresponding author

2

Abstract Sound propagation in conventional rigid-frame porous materials is aected by the viscosity of the saturating gas and the heat transfer between the gas and the solid frame.

This

paper investigates the acoustical properties of hierarchical porous materials that support mass transfer processes, such as sorption and dierent types of diusion, in addition to the previously mentioned eects. The two-scale asymptotic expansion method of homogenization for periodic media is used to derive the macroscopic acoustic description.

This description

allowed to conclude that, at the leading order, neither sorption nor diusion aect the macroscopic uid ow through the material. However, the eective compressibility is signicantly modied by these physical phenomena. Specically, the real part of the low-frequency bulk modulus can take values much smaller than those found in conventional materials while its imaginary part is increased around the characteristic frequency associated with diusion. As a consequence, the sound waves are slowed down and more attenuated at low frequencies. The results are exemplied for sorptive porous, brous, and granular materials and the inuence of their microstructural descriptors on their eective acoustical properties is discussed.

Keywords

:

sound propagation; porous materials; homogenization; sorption; adsorption;

diusion

1 Introduction Dissipation of sound energy in conventional rigid-frame porous materials is primarily determined by viscosity and heat conduction eects [1, 2, 3, 4, 5]. Viscous dissipation is caused by a local production of thermal energy through the generation of viscous stresses, while dissipation associated with heat conduction by heat exchanges between the uid and the solid frame of the material.

Under certain circumstances, to be detailed in this paper, another

mechanism of dissipation of energy associated to mass diusion and sorption can co-exist. Sorption is a general term used to refer to adsorption, desorption, and absorption (penetration of the uid into the solid phase). The former is a physical or chemical process in which the uid molecules are adhered on to a surface.

Adsorption can also be understood as an

increase of uid density in the vicinity of a uid-solid interface. Desorption is the opposite phenomenon, i.e. the uid molecules are released from a surface. The molecules adherence in physical adsorption is caused by weak van der Waals forces while chemical bonding is responsible for chemical adsorption. The release of the molecules is caused by either an increase of temperature or a decrease in pressure which leads to a break of the weak physical bond [6]. Adsorption/desorption is accompanied by diusion that governs the ux of molecules from a region of higher concentration to one of lower concentration [6]. The inuence of sorption on sound propagation in a single tube has been investigated in [7] and [8]. Herzfeld [7] assumed that the temporal variation of the molecular surface density caused by sorption gives rise to uid ow normal to the surface, whose rate compensates that of the mass transfer on the boundary.

He concluded that the eect of sorption at audible

frequencies is negligible compared to that of heat conduction. However, in his treatment the eect of such a ow on the eective compressibility was not considered.

Contrarily, it was

assumed in [8] that the compressibility of the uid is aected by the increase in density caused

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by sorption in the vicinity of the uid-solid interface. Specically, he postulated that there is an instantaneous conservation of the total number of molecules contained in a given volume (that extends one molecular mean free path from the tube wall) and the number of molecules adsorbed on the boundary intercepted by such a volume. Then, the resulting excess of density on the tube wall is communicated to the bulk of the tube via a mass diusion process. The main conclusion of his work is that the contribution to dissipation of sound energy of viscosity, heat conduction, and mass diusion are additive and of the same general type. However, this appeared as a consequence of the a priori decomposition of the total density variation produced by the sound eld into i) a density eld balanced by variations of volume, and ii) another eld resulting from mass diusion due to the excess of uid density on the tube wall caused by sorption. Since the models proposed in [7] and [8] were developed for a single tube, their applicability to bulk porous materials is limited. More recently, sorption processes have been considered as a potential explanation for the unusual acoustical properties of granular activated carbon (GAC). For example, it was experimentally shown in [9] that partially lling a loudspeaker cavity with GAC can increase its eective compliance. A similar behaviour was observed in Helmholtz resonators fully and partially lled with GAC in [10, 11]. In addition, it has been shown that GAC displays unusually large low frequency sound absorption [10, 11, 12]. The particularity of activated carbon is that the low-frequency eective compressibility of the saturating gas attains values larger than the isothermal one predicted by the current theory of acoustics of porous media [10, 12]. It was suggested in [12] that such a behaviour may be explained by adding an additional scale to the double porosity model introduced therein, as well as by accounting for sorption processes. This idea was developed further in [10, 13] where rarefaction and sorption eects were included into a model for sound propagation in granular activated carbon. However, the model proposed in those works assumed that the characteristic frequency of diusion is very high so that a quasi-static diusional regime is attained in the audible frequency range. The theoretical description introduced in the present work allows relaxing this assumption. On the other hand, the interaction between low frequency cyclic pressure variations and sorption and diusion has been studied in chemical engineering within the context of the socalled frequency response method [14, 15] (see also [16] for an extensive review). This method is a relaxation technique that aims at measuring the diusion and sorption parameters of porous materials. It is based on perturbing the equilibrium of a system with a periodic change in one of its properties. In a batch system, it is normally considered a sinusoidal change in volume of a container in which the sorptive material is placed. This change in volume leads to a change in pressure that is recorded (see [16] for more details on the type of instrumentation utilised) and further used to obtain the material parameters by tting a theoretical model to the measured response. Of those theoretical models, the ones published in [17, 18, 19] for bidispersed structured sorbents are relevant to the present work. The model in [17] features diusion and sorption in both pores larger than 50 nm and pores with size comparable to the size of the molecules (macropores and micropores after the IUPAC recommendation). In [18], the model includes diusion in absence of sorption in macropores, activated diusion in micropores, and sorption at the micropore mouth. This modelling approach was then extended in [19] to account for lm mass transfer and surface barrier resistance. The common features of these models are : i) the mass transport in both pore networks is modelled as a Fickian diusion process and ii) they are usually applied to describe diusion and sorption in granular materials made by agglomerating microparticles. In these materials the smaller pores are located in the microparticles, while the larger ones are formed in between them. It is

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pertinent to emphasise that the inter-granular physics is not considered in these models, while in acoustics the inter-granular voids, where viscosity and heat transfer eects take place, are responsible for signicant attenuation of sound [20, 21, 22, 12]. This issue has been somehow accounted for in [23] where sound propagation through two innite adsorbing microporous plates was theoretically investigated using the low reduced frequency approximation method. In the plates, the diusion of the adsorbate was modelled as a Fickian diusion process while sorption by expanding the concentration around its equilibrium value. The sound propagation through the slit pore formed by the two plates was modelled using the classical equations of conservation of momentum, mass, and energy, and the equation of state. The main assumptions in [23] were that i) the solid phase and the adsorbate are treated as a single homogeneous medium, and ii) at the solid surface the normal advective mass ux is balanced by the diusive mass ux, while the tangential velocity is null.

It can be inferred from the results of their

work that the eective compressibility of the single slit pore can be signicantly modied by sorption and diusion, and that the same type of frequency dependence of the attenuation coecient in [8] is obtained.

Since the model in [23] only considers a single slit pore, its

applicability to hierarchical bulk materials is limited. This paper investigates sound propagation through hierarchical porous materials in which viscosity and heat conduction eects at the pore scale and mass diusion and sorption eects at the micropore scale are accounted for. We focus on the case of a pure gas saturating a hierarchical sorptive porous material. A typical example may be the system nitrogen/activated carbon, which could approximate the acoustic behaviour of activated carbon saturated with air. The macroscopic acoustic description is established in Section 2 using the two-scale asymptotic expansion method of homogenisation for periodic media. The analysis of the eective parameters associated with this description is presented in Section 3. Models for hierarchical materials with dierent microstructure are introduced in Section 4. The results are illustrated and discussed in Section 5 and the main ndings are summarised in the conclusions.

2 Sound propagation in sorptive porous materials - Theory 2.1

Governing equations

The local equations governing sound propagation in hierarchical porous materials accounting for viscosity, heat conduction, mass diusion, and sorption eects are presented in this section. The two-scale asymptotic expansion method of homogenisation for periodic media [24, 4] is then used, in subsequent sections, to derive the wave equation in this type of materials. Consider a periodic rigid-frame hierarchical porous material saturated with a pure Newtonian gas. Figure 1 shows a diagram of the scales of the material and the relevant geometrical descriptors. through as

Ω.

The macroscopic characteristic length

L = |λ| /2π .

L

This is constituted by the volume of the pores

domain

Ωm .

is related to the sound wavelength

λ

The representative elementary volume (REV) of the material is denoted

The solid part of

Ωm

Ωf

and the volume of the microporous

is assumed perfectly impervious to gas transport.

volume of the micropores is represented by

Ωmf .

The

The characteristic length associated with

the pores (or the period of the hierarchical medium) is denoted as l. Similarly, lm represents

2 and is assumed much smaller than the characteristic

the characteristic size of the micropores

2

From now on, the word micropore refers to the pores at the scale

lm

and does not determine their actual

size, i.e. the micropores are not necessarily pores with size smaller than 2 nm as in the IUPAC recommendation.

5

pore size, i.e.

lm << l.

Because of the separation of scales between the pore size and mi-

Ωm is modelled as an homogenised equivalent medium. Ωmf . These correspond to volumetric diusion of free gas

cropore size, the microporous domain Two diusion processes occur in

molecules in the bulk of the micropores and surface diusion on the walls of the micropores. These diusion processes are represented in Figure 1 by the horizontal dashed grey and black lines, respectively. Sorption occurs on the walls of the micropores. The mass exchange between the gas (hollow circles) and adsorbate (black circles) phases is depicted with vertical

Ωs = Γm N d, where Γm is molecules, and N is the number

lines with arrows in Figure 1. The adsorbate volume is given by the surface area of the micropores,

d

is the diameter of the

of adsorbed layers, which is assumed equal to

N = 1,

i.e. monolayer coverage. The size of

the molecules is assumed smaller than the micropore characteristic size, i.e. space available for the transport of free molecules is represented by of the material is denoted as

Φ = φ + (1 − φ)φm ,

φ = Ωf /Ω

where

Ωv .

d < lm .

The void

The overall porosity

and

φm = Ωmf /Ωm

are

the porosities associated with the pores and micropores, respectively. Finally, the disparity in length scales between the pore size and the macroscopic characteristic size associated with the acoustic phenomenon provides a small expansion parameter

=

l L

<< 1.

Ω n

Ωm

L

Γ

l

Ωf

lm

Ωs=Γ m×d

Ωv

Ωmf =Ωv +Ωs d

Γm

Figure 1: Diagram of the scales of a hierarchical sorptive porous material.

The linear harmonic response of a conventional single-porosity material to a smallamplitude pressure gradient is locally described by the following set of equations [1, 3, 4, 5]:

η∇2 u − ∇p = jωρ0 u jωρ + ρ0 ∇ · u = 0

in in

Ωf Ωf

jωCp ρ0 τ = κ∇ · ∇τ + jωp in Ωf p ρ τ = + in Ωf P0 ρ0 τ0

u=0 τ =0

on on

(1) (2) (3) (4)

Γ

(5)

Γ.

(6)

6

These equations correspond to the linearised equations of conservation of momentum (1), mass (2), and energy (3); and of state (4). The physical parameters involved are the dynamic viscosity

η,

ρ0 ,

density

specic heat capacity

Cp ,

thermal conductivity

κ,

and equilibrium pressure

P0 ,

τ0 . The oscillating velocity, pressure, density, and temperature τ , respectively. Note that harmonic dependence of the type ejωt

and temperature

are denoted as

u, p, ρ,

and

is assumed and, because of linearity, this term is omitted throughout the paper. The presence of the microporous domain modies the local description above. In particular, the no-slip boundary condition (5) is replaced, as it will be detailed below. The governing equations for diusion and sorption of a pure uid in the microporous domain are now posed. In doing so, it is assumed that i) sorption occurs on the walls of the micropores, ii) the adsorbed molecules (adsorbate) and the gas phase saturating the micropores are in dynamic instantaneous equilibrium, and iii) the concentrations of the gas phase and of the adsorbate are diused through the microporous domain via two diusion mechanisms in parallel, i.e. diusion in the bulk of the micropores and surface diusion on the micropore walls. Since the micropores are much smaller than the pores, the microporous domain can be considered as an homogenised equivalent medium governed by eective equations that are dened in the whole domain

Ωm

and reect the local physical processes. For simplicity, it is

assumed that the microporous domain is isotropic. The constitutive ux equation accounting for the two diusion mechanisms is given by [6]:

J = −φm (ϕDm ∇cm + (1 − ϕ)Ds ∇cs ), where

J

(7)

is the molar mass ux (relative to the volume of the microporous domain including

the micropores and its solid part),

cm

and

cs

are, respectively, the concentration of the gas

(in mole/volume of uid) and adsorbate phase (in mole/volume of adsorbed phase),

Dm

is

the micropore diusion coecient (which is not necessarily an activated process in this work), and

Ds

is the surface diusion coecient. Note that these parameters could be corrected by a

geometrical tortuosity factor. The transport void fraction

ϕ = Ωv /Ωmf

represents the fraction

of the microporous void space available for the transport of free molecules (see also Figure 1). Similarly,

(1 − ϕ) = Ωs /Ωmf

represents the fraction of space available for the transport of

adsorbed molecules.

Performing a mass balance in a volume element of the microporous domain leads to the following equation [6]:

jωφm (ϕcm + (1 − ϕ)cs ) = φm ∇ · (ϕDm ∇cm + (1 − ϕ)Ds ∇cs )

in

Ωm .

(8)

As mentioned above, the adsorbed phase is assumed to be in dynamic instantaneous equilibrium with the gas phase. This is valid when the local adsorption kinetics is much faster than the diusion processes. Such a situation is commonly found in microporous media and is justied by the fact that the average residence time of adsorption ranges from s for physical adsorption [6].

10−13

to

10−9

Then, the equilibrium relationship between the two phases is

given by :

cs = Hcm where

H

in

Ωm ,

(9)

(in units of volume of gas/volume of adsorbed phase) is the linearised sorption

equilibrium constant. It locally measures the concentration of the adsorbed phase with respect to the gaseous one and corresponds to the slope of the local isotherm at a given equilibrium point. This parameter depends on pressure, temperature, the microstructure of the material,

7

and the physical and chemical properties of the uid and the solid; and can be calculated from the linearised local isotherm, as shown in Appendix A for a Langmuir isotherm model. Moreover,

H

coincides at very low pressures with the Henry's constant (as dened in [6]) when

expressed in units of volume of gas/volume of adsorbent. Replacing Eq.

ρm = cm M ,

where

(9) into Eq.

M

(8) and writing the concentration in terms of density, i.e.

is the molar mass of the gas, lead to the following Fickian equation

[6, 25]:

jωρm He = De ∇ · ∇ρm where the eective diusion coecient constant

He

De

Ωm ,

in

(10)

and the eective linearised sorption equilibrium

are given by:

De = φm (ϕDm + (1 − ϕ)Ds H)

(11)

He = φm (ϕ + (1 − ϕ)H).

(12)

The local description in the pores and microporous domain is completed by the conditions of

Γ,

i.e.

on

Γ

continuity of mass ux and pressure on their boundary

ρ0 u · n = −De ∇ρm · n pm = p The

boundary

condition

u − (u · n)n = 0)

(13)

along

ρm =

i.e.

with

ρ0 p P0

setting

Γ.

on the

(13) (14)

tangential

velocity

to

zero

(i.e.

replace the no-slip boundary condition (5). It should also be noted that

in deriving (14) it was made use of the equation of state to relate the pressure to density. In addition, the thermal impedance of the solid is normally much smaller than that of the saturating uid [26] hence the boundary condition (6) holds. follows.

This is further justied as

The sorption phenomena, as a phase transition [27], is associated with generation

of heat by sorption.

However, for the reasons detailed in several steps below, this eect

can be disregarded and the condition of negligible temperature variation in the solid part of the microporous domain can be considered as consistent with the local physics.

i) The

heat associated with sorption is generated on the walls of the micropores. Since the thermal conductivity of the solid transferred to the solid. by

δms =

p

κms

is much higher than that of the gas, the thermal ux is mostly

ii) The thermal boundary layer thickness in the solid is given

κms /ρms Cpms ω .

δms ≈ 20 µm. Note κms = 1.7 W/mK , ρms = 2200 kg/m3 , and this estimation. iii) The typical (solid) distance tw

For acoustic frequencies, say 200 Hz,

that the properties of carbon black (i.e.

Cpms = 710 J/kgK )

has been used for

between two micropore walls in the microporous domain is in the order of the micropore size, i.e.

tw

is nanometric in size and is much smaller than

δms .

Therefore, the temperature in

the solid oscillates in time with a quasi-uniform distribution in space. iv) The temperature variation

δτms

in the solid is therefore mostly determined by the heat of adsorption

ρms Cpms δτms = (∆H)He δcm . Thus, δτms = δp(∆H)He ρ0 /ρms Cpms M P0 .

and the heat capacity of the solid through the pressure variation one deduces that

∆H

introducing At normal

pressure and temperature conditions and using typical values of the parameters for nitrogen (M

= 28 g/mole) and activated carbon (i.e. ∆H = 30 kJ/mole and He = 7.5,

which is an

average value deduced from measurements of the static bulk modulus of activated carbon in [13]), one obtains that

δτms ≈ 6.1 · 10−5 δp.

This value is much smaller than the temperature

variation in the gas, which is in the order of

δτ ≈ δpτ0 /P0 = 3.9 · 10−3 δp.

Consequently, the

temperature of the solid can be considered as constant and the boundary condition (6) as

8

further justied. This is in agreement with the results in [28] where it is concluded that the inhomogeneities of temperature in the microporous domain are of much less inuence when compared to those of concentration and pressure. In summary, the local description is given by Eqs. (1)-(4), and (10); and boundary conditions (6), (13), and (14). Next section deals with the physical analysis of this local description.

2.2

Physical analysis

The physical analysis of the local description is presented in this section. Such analysis for Eqs. (1)-(4) is a well-established result. The arguments and procedure can be found in more detail in [1, 4] and are now recalled. Describing long-wavelength sound propagation in porous media leads to consider that the pressure uctuates at the macroscopic scale, and, while the velocity and its rate of deviatoric deformation vary at the microscopic scale, the microscopic divergence itself is of the order of the macroscopic divergence, i.e.

p ∇p = O( ) L

η∇2 u = O(

;

ηu ) l2

and

u ∇ · u = O( ). L

(15)

These estimates, meaning that the velocity eld is asymptotically divergence-free at the pore scale, are valid as long as the geometry remains suciently simple, excluding in particular the presence of cavities leading to local resonances [29, 30].

Using later on the two-scale

asymptotic expansion homogenisation process, the absence of such cavities is tacitly assumed. The more general ow regime occurs when the viscous and inertial terms balance the pressure gradient. This means that the three terms in the oscillatory Stokes equation (1) are of the same order of magnitude, i.e.

O(

ηu p ) = O(ωρ0 u) = O( ). l2 L

(16)

The three terms in the equation of state (4) are of the same order of magnitude and the temporal relative variations of the density are balanced out by the rate of volume variations, i.e.

O(

p ρ τ ) = O( ) = O( ) P0 ρ0 τ0

and

O(ω

ρ u ) = O( ). ρ0 L

(17)

The temperature varies at the pore scale while the three terms in the equation of conservation of energy (3) are of the same order of magnitude, i.e.

κ∇2 τ = O(

κτ ) l2

and

O(

κτ ) = O(ωρ0 Cp τ ) = O(ωp). l2

The physical analysis of mass diusion/sorption is now presented. The density

(18)

ρm

varies

at the pore scale and the richest regime of mass diusion/sorption occurs when both terms in Eq. (10) are of the same order of magnitude, i.e.

De ∇2 ρm = O(De

ρm ) l2

and

O(De

ρm ) = O(ωρm He ). l2

(19)

It remains to assess the ratio between the diusive and the advective mass uxes on the pore boundary

Γ.

The long-wavelength condition imposes that the diusive mass ux is of

one order smaller than the advective one, i.e.

J =

|De ∇ρm | = O(). |ρ0 u|

(20)

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This estimate is justied by the following argument. the ingoing mass ux on one face (of surface S) as the opposite face as

Sρ0 u2 ,

Consider a cell

Sρ0 u1 ,

Ω

and denote

the outgoing mass ux on

and the mass ux pulsed by the microporous domain

Ωm

as

|De ∇ρm · n| Γ. By hypothesis, a regime of long wavelength L >> l is considered, thus (Sρ0 u2 −Sρ0 u1 )/Sρ0 u1 ≈ l/L. Since by conservation of mass Sρ0 u2 ≈ Sρ0 u1 +|De ∇ρm · n| Γ, it follows that |De ∇ρm · n| Γ/Sρ0 u1 ≈ l/L = . 2 It should be noted that the estimate J = O( ) leads to the classical description of sound propagation in single porosity non-sorptive materials (see appendix A), while J = O(1) is not consistent with the hypothesis of separation of scales. Consequently, the latter case cannot be homogenised.

The estimate (20) can be rewritten in terms of physical parameters. From Eq. (16) one

O(ηu/l2 ) = O(p/L), while from Eq.(14) that O(P0 ρm /ρ0 p) = O(1). expressions with the fact that ρm uctuates at the local scale leads to:

has that two

J =

|De ∇ρm | ηDe L ρm De ) = O( = O( ) = O() |ρ0 u| u lρ0 P0 l 3

Combining these

ηDe = O(2 ). l2 P0

i.e.

As an example, the eective diusion coecient may be estimated as where

vT

is the mean thermal speed, while the term

is the molecular mean free path and both speeds are of the same order, i.e.

η/P0

(21)

De = O(lm vT ) `/c0 , where `

as in the order of

c0 is the speed of sound c0 = O(vT ), one obtains

in the saturating gas. that

ηDe /P0 = O(lm `).

Since This

indicates that the macroscopic description to be derived corresponds to situations where the combination of physical lengths satises

lm `/l2 = O(2 ).

This condition is satised, for

lm = O(l), = O(`).

example, when the pore and micropore scales are well separated, i.e. micropore size is in the order of the molecular mean free path, i.e.

2.3

lm

and the

Homogenisation procedure

The scale separation between the wavelength and the size of the period allows using the two-scale asymptotic expansion method of homogenisation for periodic media to derive the equivalent macroscopic description. To represent the evolution at the two spatial scales, one introduces the following dimensionless space variables x/L

= x∗

and x/l

= y∗,

where x stands

for the usual space variable. The introduced space variables are associated with the variations at the wavelength and the period scales, respectively. length, two space variables gradient operator

∇

x=

y= ∇(xy) = ∇x +

Lx∗ and

is given by

Equivalently, taking

L

as reference

= xL/l = −1 x will be used. Then, the usual −1  ∇y (and ∇2(xy) = ∇2x + 2−1 ∇xy + −2 ∇2y ).

Ly ∗

To reect the physics of the phenomena, the use of two space variables should be combined

with a rescaling of the usual equations based upon a single space variable. The reason for the rescaling lies in the fact that when expressed with the two space variables physical gradient of a quantity

Q

that varies at the large scale, i.e.

∇x Q,

(x, y),

∇(xy) Q. ∇y Q reads

becomes

Similarly, if the quantity varies at the micro scale, the actual physical gradient

∇(xy) Q.

the actual

Therefore, the gradient of variables oscillating at the local scale should be rescaled.

For instance,

∇2 u

should be rewritten as

2 ∇(xy) u

to express that the velocity actually varies

at the pore scale. This procedure based upon the physical analysis ensures that the limiting description obtained when

→0

keeps a physics of the same nature as in the real situation

where the scale separation is nite. Using the estimates identied in the physical analysis, equations (1)-(4), (6), (10), (13),

10

and (14) are rewritten in the two-space-variable formulation in rescaled form. Note that we adopt the usual homogenisation convention that consists in keeping the same notation as for the single-space-variable formulation for both the variables and the gradient operator. example,

∇

and

u stand for ∇(xy)

and

u(x, y), respectively.

2 η∇2 u − ∇p = jωρ0 u jωρ + ρ0 ∇ · u = 0

Ωf

in

(22)

Ωf

in

(23)

2 κ∇ · ∇τ = jωCp ρ0 τ − jωp in Ωf p ρ τ = + in Ωf P0 ρ0 τ0 2 De ∇ · ∇ρm = jωρm He

in

ρ0 u · n = −2 De ∇ρm · n ρ0 ρm = p on Γ P0 τ =0

Note that the

2 -rescaling

(24) (25)

Ωm

(26)

Γ

(27)

on

(28)

Γ.

on

For

(29)

in the boundary condition (27) results from (i) the physical

estimate (20) stating that the diusive ux is of one order smaller than the advective one, and (ii) the fact that

ρm

varies at the local scale.

The physical variables are looked for in the form of asymptotic expansions in powers of the small parameter



as

Q(x, y) =

P∞

i (i) i=0  Q (x, y) where

Q = p, u, τ, ρ, ρm

.

These are

then substituted into Eqs. (22)-(29) and the terms of the same order are identied. At follows from the equation of conservation of momentum that the pressure is a macroscopic variable, i.e.

p(0) = p(0) (x).

∇y

p(0)

= 0,

−1

it

which means that

Further identication provides the

following leading-order cell problems. Fluid ow :

η∇2y u(0) − ∇y p(1) = jωρ0 u(0) + ∇x p(0) ∇y · u(0) = 0

u(0) = 0

in on

in

Ωf

Ωf

(30) (31)

Γ.

(32)

Heat conduction :

κ∇y · ∇y τ (0) = jωCp ρ0 τ (0) − jωp(0) τ (0) = 0

on

in

Γ.

Ωf

(33) (34)

Mass diusion :

(0) De ∇y · ∇y ρ(0) m = jωρm He in Ωm ρ0 (0) ρ(0) p on Γ. m = P0

(35) (36)

Other equations to be used in the upscaling process are those of state and mass balance at

0 ,

and the boundary condition (27) at

1 .

These are given by:

p(0) ρ(0) τ (0) = + P0 ρ0 τ0

in

Ωf

(37)

11

  jωρ(0) + ρ0 ∇x · u(0) + ∇y · u(1) = 0 ρ0 u(1) · n = −De ∇y ρ(0) m ·n

on

in

Ωf

(38)

Γ.

(39)

It is worth highlighting that the three problems above are decoupled.

This means that

diusion and sorption do not aect the uid ow and heat conduction in the pores. The uid ow (i.e. Eqs. (30)-(32)) and heat conduction (i.e. Eqs. (33)-(34)) problems correspond to those of the classical case of sound propagation in single-porosity non-sorptive materials. The solution of these problems is given by [1, 31, 3]:

u(0) = −

k¯ (y, ω) η

· ∇x p(0)

(40)

p(1) = −¯ π (y, ω) · ∇x p(0) + p¯(1) (x) τ (0) = where

k¯ (y, ω)

and

¯ ω) θ(y,

represent the

(41)

¯ ω) θ(y, jωp(0) , κ

Ω − periodic

(42)

local elds of velocity and temperature

respectively. The pressure eld has been expressed in terms of its zero mean part an integration constant

π ¯ (y, ω)

and

p¯(1) (x).

By using the change of variable and (36)) can be rewritten as:

(0)

ρˆ(0) = ρm −

ρ0 (0) P0 p , the diusion problem (i.e. Eqs.(35)

ρ0 De ∇y · ∇y ρˆ(0) = jω ρˆ(0) − (−jω p(0) ) He P0 ρˆ(0) = 0

on

in

Ωm

Γ.

(43)

(44)

This problem is formally identical to that of pressure diusion in double porosity materials with highly contrasted permeabilities [26, 32]. Therefore, its solution is given by:

ρˆ(0) = − which can be rewritten in terms of

ρ(0) m

(0)

ρm

g¯(y, ω) ρ0 (0) jω p , Dapp P0

as:

  jω¯ g (y, ω) ρ0 (0) p 1− , = P0 Dapp

g¯(y, ω) represents the Ω − periodic diusivity Dapp is dened as:

where

Dapp =

(45)

(46)

microscopic diusive density eld and the apparent

De ϕDm + (1 − ϕ)Ds H = . He ϕ + (1 − ϕ)H

(47)

Equations (40), (42), and (46) provide the solution of the local problems arising from homogenisation. description.

These will be used in the next section to derive the macroscopic acoustic

12

2.4

Macroscopic acoustic description

Integrating the mass balance equation (38) over the pore space

Ωf ,

dividing by the volume

Ω,

and using the equation of state (37) lead to:

jωh

p(0) τ (0) − i + h∇x · u(0) i + h∇y · u(1) i = 0, P0 τ0

(48)

where the averaging operator is dened as:

Z

1 h·i = Ω

Ωf

· dΩ.

(49)

h∇y · u(1) i

The attention is now paid on the term

in Eq.

(48).

This is calculated by

making successive use of the divergence theorem, noting that the surface integral vanishes on the periodic boundaries, and using Eqs. (39) and (35), i.e.

h∇y · u

(1)

1 i= Ω

Z

u

(1)

Γ

1 · n dΓ = Ω

Z

De 1 − ∇y ρ(0) m · (−n) dΓ = ρ Ω 0 Γ

Z

De ∇y · ∇y ρ(0) m dΩ ρ Ωm 0 Z (0) jωρm He 1 = dΩ. Ω Ωm ρ0

Substituting this expression into Eq. (48) yields:

p(0) τ (0) 1 jωh − i + jωHe P0 τ0 Ω

Z

(0)

Ωm

ρm dΩ + ∇x · hu(0) i = 0. ρ0

(50)

(0)

τ (0) and ρm (Eqs. (42) and (46) respectively) and using the thermodynamic identity P0 /τ0 = ρ0 Cp (γ − 1)/γ , where γ is the adiabatic exponent; one nally obtains the macroscopic Replacing

acoustic description:

∇x · h u

(0)

i + jωp

(0)

   φ Θ(ω) jωG(ω) γ − jωρ0 Cp (γ − 1) + γMh 1 − =0 γP0 φκ (1 − φ)Dapp hu(0) i = −

Here the dynamic viscous

k(ω) η

· ∇x p(0) .

(51)

(52)

k(ω) and thermal Θ(ω) permeabilities, the diusion function G(ω),

and the ratio of the adsorbate to gas phase hold-up

k(ω) = hk¯ (y, ω)i G(ω) = Mh =

1 Ω

;

Z

Mh

are given by:

¯ ω)i Θ(ω) = hθ(y,

(53)

g¯(y, ω) dΩ

(54)

Ωm

(1 − φ)He (1 − φ) = φm (ϕ + (1 − ϕ)H). φ φ

(55)

The wave equation is then obtained by eliminating the velocity in eq. (51), i.e.

∇x ·



k(ω) η

· ∇x p

(0)



=

jωp(0) , E(ω)

(56)

13

where the dynamic bulk modulus

E(ω)

is given by the geometric mean between the classical

bulk modulus accounting for thermal exchanges in the pores,

Et (ω),

modulus that results from the mechanisms of diusion and sorption,

and an additional bulk

Eds (ω),

i.e.

−1 E −1 (ω) = Et−1 (ω) + Eds (ω)

Et (ω) =

(57)

γP0





(58)

 .

(59)

φ γ − jωρ0 Cp (γ − 1) Θ(ω) φκ

Eds (ω) =





γP0

φ γMh 1 −

jωG(ω) (1−φ)Dapp

Further assuming that the material is isotropic, the dynamic viscous permeability becomes

k = KI,

where

wave number

I

kc ,

is the unitary second-rank tensor. Then, the characteristic impedance and speed of sound

Zc (ω) =

r

η E jωK

;

C

Zc ,

through the material are given by:

kc (ω) = ω

r

η 1 jωK E

ω C(ω) = = kc (ω)

;

s

jωK E. η

In summary, the derived macroscopic acoustic description, given by Eqs. (52), allows concluding that the macroscopic constitutive uid ow law, i.e. Darcy's law, and its associated eective parameter, i.e.

(60)

(51) and

the dynamic

the dynamic viscous permeability,

are not modied by the presence of diusion and sorption in the microporous domain. This comes from the fact that, in the long-wavelength regime, the macroscopic uid ow at the leading order is not aected by these physical processes (see Eqs.

(30)(32), (40), and

(53)). Conversely, the dynamic bulk modulus becomes signicantly modied by diusion and sorption. This modication comes from the appearance of a source term in the macroscopic balance equation (i.e.

the second term in Eq.

of diusion and sorption.

(50)) that accounts for the contribution

Moreover, since the eective parameters in Eq.

on the bulk modulus, these are also aected by diusion and sorption.

(60) depend

The eective pa-

rameters associated with the macroscopic acoustic description are analysed in the next section.

3 Analysis of the eective parameters The frequency behaviour of The characteristic frequency

Et (ω) is characterised by that of the thermal permeability Θ(ω). ωt determining the transition from isothermal to adiabatic sound

propagation and the limiting values of the thermal permeability are given by [3]:

ωt = Here

Θ0

φκ ρ0 Cp Θ0

;

Θ(ω << ωt ) = Θ0

;

represents the static thermal permeability and

ary layer thickness. The bulk modulus

Et (ω << ωt ) =

Et (ω)

δt =

p

κ/ρ0 Cp ω

(61)

is the thermal bound-

therefore varies as:

P0 φ(1 −

Θ(ω >> ωt ) = −jφδt2 .

γ−1 jω γ ωt )

;

Et (ω >> ωt ) =

γP0 . φ

(62)

14

Similarly, the behaviour of

Eds (ω)

is determined by that of

terised by the diusion characteristic frequency

ωd = Here

G0

(1 − φ)Dapp G0

;

ωd ,

G(ω << ωd ) = G0

The latter is charac-

G(ω >> ωd ) = −j(1 − φ)δd2 .

;

(63)

p δd = Dapp /ω is the diusion values of G(ω), one can conclude that

is the static value of the diusion function and

boundary layer thickness. Considering the asymptotic

Eds (ω)

G(ω).

i.e.

varies as:

Eds (ω << ωd ) =

P0 φMh (1 −

jω ωd )

;

Eds (ω >> ωt ) → ∞.

(64)

On the other hand, the ratio between the diusion and thermal characteristic frequencies is given by:

Since

G0

and

Θ0

κ G0 ωt φ . = ωd 1 − φ ρ0 Cp Dapp Θ0

(65)

are of the same order of magnitude, this frequency ratio is approximately

given by the ratio between the eective thermal and apparent mass diusivities. For the case of interest (i.e. materials with relatively low porosity

φ) this frequency ratio is generally larger

than one and increases as the apparent diusivity decreases. This means that the dissipation of sound energy induced by diusion and sorption could be maximised at frequencies much smaller than those where losses due to heat conduction at the pore scale are observed. From the analysis above, it is concluded that the dynamic bulk modulus

E(ω)

has the

following asymptotic behaviour:

E(ω << ωd ) =

P0 φ(1 + Mh −

jω ωtd )

;

E(ω >> ωt ) = Et (ω >> ωt ) =

where the thermo-diusive characteristic frequency

ωtd

γP0 , φ

(66)

is dened through:

1 γ−1 1 1 = + Mh . ωtd γ ωt ωd

(67)

From Eq. (66) it can be deduced that the static bulk modulus is given by:

E(ω → 0) =

P0 . φ(1 + Mh )

(68)

This equation is a key result of this paper and shows that, as a consequence of sorption, the normalised (to equilibrium pressure) low-frequency bulk modulus can attain a value substantially smaller than that of conventional porous materials. Such atypical behaviour has been experimentally observed in activated carbon [12, 10, 11, 13].

Et = 0. Hence its static value is given by E(ω → 0) = Et (ω → 0) = P0 /φ.

In general, the bulk modulus reduces to that of conventional single porosity materials when

Mh → 0, i.e. φm

The case of diusion without sorption is obtained when i) the concentration of the adsorbed and gaseous phase are identical (H (N

= 0),

= 1),

ii) the number of adsorbed layers is equal to zero

or iii) the characteristic size of the micropores is much larger than the size of the

molecules, i.e.

lm >> d.

In all these cases, the bulk modulus tends to that of a mate-

rial with eective diusion coecient

De = φm Dm

and

Mh = (1 − φ)φm /φ.

Therefore, at

15

low frequencies the bulk modulus tends to

E(ω → 0) = P0 /Φ = P0 /(φ + (1 − φ)φm ).

In

addition to this, it is worth noting that when sorption is negligible, the macroscopic description (56) is similar to that of double porosity materials with highly contrasted micro- and pore-scale viscous permeabilities (i.e.

2 /l2 ) = O(2 )) Km /Kp = O(lm p

[26, 32]. This similarity

lies in the fact that in both descriptions the mass ux pulsed by the microporous domain is of one order smaller than the mass ux in the pores (cf. Eq. 20 and Eqs. 20 and 40 in [32]). This is justied as follows. Considering that in double porosity materials with highly contrasted permeabilities the pressure at the micropore-scale

pm

(resp. pore-scale

pp )

varies

at the pore (resp. as

macroscopic) scale, the ratio between the mass uxes can be estimated |ρ0 um | / |ρ0 up | = Km η −1 pm /lp / Kp η −1 pp /L . Using the continuity of pressures on the

pore boundaries and reminding that the pore- and macroscopic scales are well separated, i.e.

lp /L = O(),

lead to

to the same estimate

2 L/l2 l ) = O(2 −1 ) = O(). This corresponds |ρ0 um | / |ρ0 up | = O(lm p p of J in Eq. (20). On the other hand, eective diusion parameters

in double porosity materials with highly contrasted permeabilities can be identied as follows.

Considering that the mass ux pulsed by the microporous domain can be expressed

ρ0 um = ρ0 (−Km η −1 ∇y pm ) = −(Km P0 /η)∇y ρm , an eective diusion coecient can be identied, i.e. De = Km P0 /η , that leads to Dapp = De /He = Km P0 /φm η . This apparent as

diusivity signicantly determines the pressure diusion phenomenon in double porosity materials with highly contrasted permeabilities (cf. Eqs. 102, B12, and B15 in [32]). Despite the similarities between the macroscopic descriptions and the identication of equivalent diusion parameters, it should be emphasised that the physics investigated in this paper is dierent. As discussed previously, the uid ow at the leading order remains unaected by diusion and sorption under the conditions established through homogenisation (i.e.

J = O()).

Hence

the properties of the dynamic viscous permeability are the same as those of this parameter for single porosity non-sorptive materials. These have been analysed in [2] and are as follows. Considering leading-order terms only, the dynamic viscous permeability tends at low frequencies to its static value, i.e. where

p δv = η/ρ0 ω

K(ω → 0) = K0 ,

while at high frequencies to

is the viscous boundary layer thickness and

Using the asymptotic values for

K(ω → ∞) = −jφδv2 /α∞ ,

α∞

is the tortuosity.

E(ω) and K(ω), the following limiting values for the speed

of sound and wave number are obtained:

s

C0φ jωK0 P0 C0Φ =q =√ η φ(1 + Mh ) φ 1 + Mh Φ (1 + Mh ) c0 C(ω → ∞) = C∞φ = C∞Φ = √ α∞ s r p η φ φ kc (ω → 0) = ω (1 + Mh ) = kc0φ 1 + Mh = kc0Φ (1 + Mh ) jωK0 P0 Φ ω ω kc (ω → ∞) = = . C∞φ C∞Φ C(ω → 0) =

The subscripts

φ

and

Φ

(69)

(70)

denote the limiting values for a non-sorptive single and double poros-

ity material respectively. These expressions show that, at low frequencies, the sound waves are both slowed down and more attenuated by a factor of

√

1 + Mh

or

p (φ/Φ)(1 + Mh )

in

comparison with single and double porosity materials, respectively. At high frequencies, the inuence of sorption and diusion vanishes.

16

4 Models for materials with dierent microstructure Analytical models for materials with dierent microstructure are introduced in this section. The micropore domain is modelled as an array of cylindrical micropores with radius porosity

φm .

rm

and

Taking into account that the ratio between the surface area and the volume is

inversely proportional to the characteristic size of the micropores, the transport void fraction is approximated as

ϕ = 1 − N d/rm

with

N = 1.

It is further considered that the diusion

Dm = Dk = 2rm vT /3, Ds = (1/4)vT ζ exp (−Ea /Rg τ0 ). Here

mechanism in the bulk of the micropores is Knudsen diusion [6], i.e. while the surface diusion coecient is calculated as

Rg

ζ is the distance between adjacent sites (which is approximated by the ζ ≈ d), and Ea is the energy of activation needed for a jump, which is in

is the gas constant,

molecular size, i.e.

the order of a third of the heat of adsorption [6]. It must however be reminded that these are

approximations that allow illustrating, in a simple manner, the physical phenomenon investigated in this work. Further improvement in the modelling may consider some of the models for the surface diusion coecient discussed in section 7.9.5 in [6]. Alternatively, one may use measured values of the diusion coecients or the apparent diusivity as inputs to the model. The remaining parameter to be specied is the linearised sorption equilibrium constant

H.

To the authors' knowledge, there is no analytical expression to estimate this parameter

from the material microstructure and the physical and chemical properties of the uid and solid. However, it can be obtained from isotherm measurements [6] (see Appendix A where

H

is related to the parameters of a Langmuir isotherm model).

be mentioned that at very low pressures (or Henry's region)

H

In addition to this, it can varies [6] from 10 to 10000

[volume of gas/volume of adsorbent]. Note that these values should be multiplied by a factor of

dφm /lm (1 − φm )

to express them in units of [volume of gas/volume of adsorbate].

In the remaining part of this section the modelling of the acoustical properties of hierar-

chical sorptive porous materials is completed by introducing the expressions for the dynamic viscous

K(ω)

and thermal

Θ(ω)

permeabilities and diusion function

G(ω)

for materials with

dierent microstructure. These include materials with slit and cylindrical pores, and brous and granular materials. Semi-phenomenological models for materials with complex microstructure are presented in Appendix B. The parameters (or inclusion) characteristic length, porosity

δd

φ,

K(ω), Θ(ω),

and the viscous

and

δv ,

G(ω) depend on the pore thermal δt , and diusion

boundary layer thickness, respectively.

4.1

Materials with slit pores

The frequency-dependent eective parameters for a material with slit pores of half-width and porosity

φ

are calculated as [5, 32]:

K(ω) = χ(h, φ, δv ) where

h

;

Θ(ω) = χ(h, φ, δt )

;

G(ω) = χ(h(1 − φ)/φ, (1 − φ), δd ),

√   tanh ( jxδ −1 ) √ χ(x, ϑ, δ) = −jϑδ 2 1 − . jxδ −1

(71)

(72)

The static values of these parameters are given by:

K0 = Θ0 = φ

h2 3

;

G0 = (1 − φ)

h2 (1 − φ)2 . 3 φ2

(73)

17

4.2

Materials with cylindrical pores

The frequency-dependent eective parameters for a material with cylindrical pores of radius

rp

and porosity

φ

are given by [5] :

K(ω) = χcyl (rp , φ, δv ) where

χcyl (x, ϑ, δ) = −jϑδ and

J0,1

2

Θ(ω) = χcyl (rp , φ, δt ),

;

J1 (j 3/2 xδ −1 ) 2 1 − 3/2 −1 j xδ J0 (j 3/2 xδ −1 )

!

(74)

,

(75)

are Bessel functions of the rst kind of order 0 and 1.

The diusion function

G(ω)

is obtained by solving Eqs. (43)-(44) using a cell approach.

The resolution is formally identical to the calculation of the thermal permeability of a gassaturated array of solid cylinders arranged in a square lattice [33]. Specically, the diusion problem is solved in an annulus geometry with inner radius is chosen to match the porosity

φ,

i.e.

√

ro = rp / φ.

rp

and outer radius

ro .

The latter

In addition, the periodicity condition

used to solve the diusion problem in a periodic cell is replaced by an energetically consistent boundary condition set on the outer boundary of the annulus. It is shown in [22] that such a condition results in null normal diusive ux on the outer boundary of the annulus. Then, averaging the solution of the cell problem one can obtain the following expression for the diusion function :

where

G(ω) = χf ib (rp , (1 − φ), δd ),

(76)

  1−ϑ χf ib (x, ϑ, δ) = −jϑδ 1 − Ψ(x, ϑ, δ) ϑ

(77)

2

Ψ(x, ϑ, δ) = 2 Ri (x, ϑ, δ) = Here



√

1 jxδ −1

i



R1 (x, ϑ, δ) R0 (x, ϑ, δ)

(78)

 √ −1 K1 ( √jxδ ) p p 1−ϑ Ki ( jxδ −1 ) + (−1)i √ Ii ( jxδ −1 ) −1 jxδ I1 ( √1−ϑ )

;

i = 0, 1.

Ii and Ki are modied Bessel functions of the rst and second kind of order i, respectively.

The static values of the parameters are given by :

rp2 K0 = Θ0 = φ 8 4.3

;

rp2 G0 = 4φ

  1 3 φ2 ln ( ) − + 2φ − φ 2 2

(79)

Fibrous materials

The frequency-dependent eective parameters of a brous material modelled as a square lattice of cylinders with radius

a

and porosity

φ

have been investigated in [33]. The dynamic

viscous permeability for sound propagation perpendicular (subscript

||)

⊥) and parallel (subscript

to the bre axes, and dynamic thermal permeability (which is the same for both sound

propagation directions) are given by :

K⊥ (ω) =

−jφδv2



1 − φ Ψ(a, φ, δv )(1 + φ) + φ 1− φ Ψ(a, φ, δv )(1 − φ) + (2 − φ)



(80)

18

K|| (ω) = χf ib (a, φ, δv )

Θ(ω) = χf ib (a, φ, δt ).

;

(81)

The diusion function for brous material has been calculated in [10]. Its expression is given by:

G(ω) = χcyl (a, (1 − φ), δd ). It is worth highlighting that the diusion function using the function

χcyl

G(ω)

(82) for brous materials is calculated

for a material with cylindrical pores. This comes from the fact that i)

the heat conduction and diusion problems are formally identical, and ii) the domain where these problems are solved are equivalent, i.e.

the heat conduction problem is solved in a

cylindrical pore while the diusion problem in a cylinder. The static values of the parameters above are given by :

K⊥0 = a2 4.4

−2 ln (1 − φ) − 2φ − φ2 16(1 − φ)

;

K||0 = Θ0 = 2K⊥0

;

G0 = (1 − φ)

a2 . 8

(83)

Granular materials

The frequency-dependent eective parameters of a granular material with particle radius and porosity

φ

have been introduced in [21, 22].

rg

The expression for the dynamic viscous

permeability reads :



K(ω) = −j(1 − β 3 )δv2 1 −

where

χg =

3

β3

χg 1−β +1 β3

1 − β3

χg − 1



,

(84)

3 Ag z + Bg tanh (z(β − 1)) z 2 ag z + bg tanh (z(β − 1))

z2 z2 ) − 3β(1 + ) 6 2 2 1 2 z 4 ag = (3 + (βz)2 ) − 3β − (1 + ) + 3 β 6 cosh (z(β − 1)) Ag = (3 + (βz)2 )(1 +

z2 z2 ) − 3βz 2 (1 + ) 2 6 2 2 z bg = 3 + β(β − 1)z 2 − (1 + ) β 2 p p rg β = 3 1−φ ; z = j . βδv √ rg permeability is given by (with zt = j βδt ) : Bg = (3 + (βz)2 )(1 +

The dynamic thermal

Θ(ω) = −j(1 − β

3

)δt2



β 3 1− 3 1 − β zt2



1 + zt tanh (zt (β − 1)) 1 − βzt zt + tanh (zt (β − 1))



.

(85)

Considering once again the similarity between the heat conduction and diusion problems, it is worth mentioning that the expression for thermal permeability (85), which is for granular materials, can be used to calculate the function

G(ω)

for a sorptive cellular material (e.g.

foam) modelled as a microporous material with large embedded spherical pores. It would only

19

be required to replace

δt → δd , φ → (1 − φ),

and

spherical pores or half of the cell size. The diusion function

G(ω)

where

rsp

is the radius of the

for granular materials, which has been calculated in [12], is

given by:

G(ω) = −j(1 −

rg → rsp ,

φ)δd2

  3 1 − 2 (1 − ξ cot (ξ)) ξ

with

ξ = j 3/2

rg . δd

(86)

The static values of the parameters are given by :

rg2 K0 = 3β 2



2 + 3β 5 −1 β(3 + 2β 5 )



;

rg2 Θ0 = 15



5 − 9β + 5β 3 − β 6 β3



;

G0 = (1−φ)

rg2 . 15

(87)

5 Illustrating examples and discussion The eective properties of hierarchical sorptive porous materials are illustrated in this section. Figure 2 shows the normalised bulk modulus for a material with slits pores of half-width

µm

and porosity

φ = 0.4.

h = 10

The normalisation was made to the isothermal bulk modulus of

Dapp = Dk and Mh = (1 − φ)φm /φ). The rm = 2.5 nm and φm = 0.1, while the H = 75. The heat of adsorption is, for example,

a non-sorptive double porosity material (with

micropore radius and microporosity were set to linearised sorption equilibrium constant to in the order of 20 to 40

kJ/mole

for activated carbons [6]. Hence, the energy of activation

needed for a jump, which is about a third of the heat of adsorption, was set to

kJ/mole.

Ea = 10

Although the developed theory applies to hierarchical sorptive porous materials

saturated with a pure uid, for simplicity, the parameters of the saturating uid are set equal to those of air (d (P0

≈ 0.38 nm),

= 101325 P a)

which are close to those of nitrogen at the considered pressure

and temperature (τ0

= 293.15 K )

conditions. It is observed that sorption

induces three eects: i) it reduces the real part of the bulk modulus at low frequencies (cf. Eq. (68)), ii) it decreases the apparent diusivity, and as a result, the characteristic frequency of diusion (cf. Eq. (63)), and iii) it increases the imaginary part of

Mh G(ω)

E(ω)

(cf. the product

in Eq. (59)), and thereby the attenuation caused by diusion and sorption. These

eects, as predicted by Eqs.

(69)-(70), are traduced onto a decrease in the speed of sound

and an increase of the overall sound attenuation. This is shown in the main and inset plots in Figure 3, respectively.

The same general trends are found for materials with cylindrical

pores, and granular and brous materials in comparison with the respective double and single porosity non-sorptive materials.

20

1.6 1.4

Φ E(ω)/P0

1.2 1 0.8 0.6 0.4 0.2 0 0 10

1

10

Figure 2: Normalised bulk modulus

2

10

3

4

5

6

10 10 10 Frequency [Hz]

ΦE(ω)/P0

7

10

10

as a function of frequency for a material with

slit pores. Continuous lines: real part. Dashed lines: imaginary part. Black lines: Sorptive material. Gray lines: Double porosity non-sorptive material. Light gray lines: Single porosity non-sorptive material

0

(a) (b) (c)

−1

10

3

10

−Im(kc) c0/ω

Normalised speed of sound

10

−2

10

2

10

1

10

−3

0

10

0

10

1

10

2

10

10 0 10

10

3

4

1

10 10 Frequency [Hz]

Figure 3: Normalised real part of the speed of sound

2

10

5

10

C(ω)/c0

3

4

10

10 6

10

7

10

as a function of frequency for

sorptive (a), non-sorptive double (b) and single (c) porosity materials with slit pores. The inset plot shows the normalised attenuation coecient The material parameters are as in Figure 2.

−Im(kc (ω))c0 /ω

as a function of frequency.

21

The inuence of the material microstructure on the bulk modulus is now illustrated. The eective bulk modulus of sorptive materials with slit and cylindrical pores is compared with that of a sorptive granular material in Figure 4. The porosity is set to a value commonly found

φ = 0.4, while the cylindrical pore radius to rp = h = 10 rg is calculated so that the inter-granular void size is comparable and rp , i.e. rg = hβ/(1 − β) = 53.87 µm. Figure 5 shows the same comparison but with respect to a sorptive brous material with inter-bre half-distance comparable to h. p Specically, the bre radius was calculated for φ = 0.8 as a = h/( π/4(1 − φ) − 1) = 10.18 µm. Note that in brous materials the porosity normally ranges from 0.6 up to values close in granular materials [20, 12], i.e.

µm. to h

The particle radius

to 1. Here an intermediate value has been considered. The parameters of the microporous

domain are as in Figure 2. As predicted by Eq. (68), the low-frequency bulk modulus does not vary with the characteristic length of the pore scale (either depends on the equilibrium pressure, porosity, and

Mh .

terial geometry becomes stronger as the frequency increases. behaviour in frequency of

ωt

E(ω)

h, rp , rg ,

or

a)

as it only

Contrarily, the inuence of the maAs shown in Section 3, the

is determined by the characteristic frequencies

ωtd , ωd ,

and

through their dependence on physical parameters and geometrical descriptors. The results

shown in Figure 4 are discussed rst. The low-frequency peak of Im(E(ω)) is associated with

the thermo-diusive characteristic frequency for relatively dense materials,

ωtd

ωtd

(Eq. (67)). Since

can be approximated by

ωd

ωd /Mh .

is much smaller than

ωt

Keeping the micropore-

scale parameters and porosity constant, the ratio between the thermo-diusive characteristic frequencies for materials with cylindrical (superscript c) and slit (superscript s) pores is given

c /ω s = (4/3)(h/r )2 (1 − φ)3 (φ(− ln φ − 3/2 + 2φ − φ2 /2))−1 . ωtd p td quency ratio is a decreasing function of φ and varies in between 9.74

by

the range 0.1 to 0.5.

For

h = rp , this freφ values in

to 4.89 for

The frequency ratio for granular materials (superscript g) and those

g s = 5(h/r )2 ((1 − φ)/φ)2 . Replacing r = hβ/(1 − β) yields /ωtd ωtd g g g s ωtd /ωtd = 5((1 − β)(1 − φ)/φβ)2 . This is also a decreasing function of φ but its range of variability is more limited (from 0.51 down to 0.33 for φ in between 0.1 and 0.5). The highfrequency peak of Im(E(ω)) associated with ωt is discussed by following the same approach. c s 2 The frequency ratio ωt /ωt = (8/3)(h/rg ) for h = rp gives a value of 8/3, while that for grang 2 3 6 −1 s ular materials and materials with slit pores reads ωt /ωt = 5φβ(1 − β) (5 − 9β + 5β − β ) for rg as previously chosen. This ratio is also a decreasing function of φ and takes values in between 0.96 and 0.79 for φ within the range 0.1 to 0.5. The estimations of the frequency ratios allow explaining the variability of the location in frequency of the peaks of Im(E(ω)) observed

with slit pores reads

in Figure 4. In general, a granular material provides characteristic frequencies smaller than those of materials with cylindrical or slit pores for the same porosity and inter-granular voids size comparable to

h

or

rp .

The results shown in Figure 5 exhibit similar trends. However,

as the porosity increases the inuence of diusion and sorption becomes smaller. This is evidenced by both the real part of the normalised bulk modulus approaching to the classical value of unity and the smaller amplitude of the peaks of Im(E(ω)) associated with

ωtd .

In fact, for

highly porous materials, and depending on their geometry, the amplitude of the peaks associated with to

ωt ;

ωtd

may either become negligible with respect to or be merged with that associated

as it can be seen from the curve for the material with cylindrical pores. On the other

hand, for materials with higher porosity the approximation able as the contribution of

ωt

to

ωtd

ωtd ≈ ωd /Mh

becomes question-

may not be negligible. However, to simplify the analysis,

the ratio between the diusion characteristic frequencies is still considered. This is given by

ωda /ωds = (8/3)(h/a)2 ((1 − φ)/φ)2 and ωda /ωdc = 2(rp /a)2 (− ln φ − 3/2 + 2φ − φ2 /2)/φ(1 − φ). By replacing the chosen value of a one obtains that these ratios are, respectively, smaller than

22

1.6 1.4

Φ E(ω)/P0

1.2 1 0.8 0.6 0.4 0.2 0 0 10

1

10

Figure 4: Normalised bulk modulus

2

10

3

4

10 10 Frequency [Hz]

ΦE(ω)/P0

5

10

6

10

7

10

of sorptive materials as a function of frequency.

Dark grey lines: granular material. Black lines: material with cylindrical pores. Light grey lines: material with slit pores. Continuous lines: real part. Dashed lines: imaginary part.

1.6 1.4

Φ E(ω)/P0

1.2 1 0.8 0.6 0.4 0.2 0 0 10

1

10

Figure 5: Normalised bulk modulus

2

10

3

4

10 10 Frequency [Hz]

ΦE(ω)/P0

5

10

6

10

7

10

of sorptive materials as a function of frequency.

Dark grey lines: brous material. Black lines: material with cylindrical pores. Light grey lines: material with slit pores. Continuous lines: real part. Dashed lines: imaginary part.

23

0.2 and 0.045 for

φ > 0.7.

A brous material provides much lower diusion characteristic

frequency than materials with cylindrical or slit pores when

h

or

rp

is comparable to the

inter-bre half-distance for a given porosity. However, sorptive materials with large values of porosity

φ

are of less interest as the relative inuence of diusion and sorption is smaller in

comparison to denser materials.

c0 ) = 5 µm) and granular (rg = 50 µm) sorptive materials is shown.

The inuence of porosity is further highlighted in Figure 6 where the normalised (to speed of sound for brous (a

Sound propagation perpendicular to the bre axes is considered (note that the same trends are found for sound propagation parallel to the bre axes). The inset plots show the speed of sound normalised to that of single porosity non-sorptive materials, i.e.

C/Cφ .

The micropore

scale parameters are as in Figure 2. It is clear that when porosity decreases the speed of sound does and the eect of sorption becomes stronger.

ωtd for granular materials (rg = 5 µm and rg = 80 µm) and materials with cylindrical pores (rp = 0.92 µm and rp = 14.85 µm) as a function of the eective diusion coecient De for several values of He . It should be noted that, as previously, the intergranular void size is comparable to rp and the porosity is set to φ = 0.4. For materials with slow kinetics (i.e. small values of De ), ωtd can Finally, Figure 7 shows the thermo-diusive characteristic frequency

take very small values. Furthermore, strong sorption makes this characteristic frequency to

De increases and/or the particle (or pore) radius decreases, ωtd can fall in the audible frequency range. As an example, for an apparent diusivity of Dapp = 10−9 m2 /s, He = 2 (i.e. Mh = 3), and rp = 0.92 µm, the thermo-diusive characteristic frequency is ftd = 434.2 Hz. Considering the same parameters but for a granular material with inter-granular void size comparable to rp , the thermo-diusive characteristic frequency ftd becomes 31.86 Hz. As exemplied in these gures, the inuence of the microstructure can become even smaller. However, as

be signicant.

24

1

0.8

c(ω)/cφ(ω)

Normalised speed of sound c(ω)/c0

1 0.9

0.6

0.8 0.7 0.6 0.5 0.4 0.3 −1 10

1

10

3

5

10

10

7

10

0.4

φ=0.7 φ=0.8 φ=0.9

0.2

0 1 10

2

3

10

10

4

5

10 10 Frequency [Hz]

6

7

10

10

1

0.8

c(ω)/cφ(ω)

Normalised speed of sound c(ω)/c0

1 0.9

0.6

0.8 0.7 0.6 0.5 0.4 0.3 −1 10

1

10

3

5

10

10

7

10

0.4

φ=0.2 φ=0.3 φ=0.4

0.2

0 1 10

2

10

Figure 6: Inuence of porosity

φ

3

10

4

5

10 10 Frequency [Hz]

6

7

10

on normalised speed of sound

10

C(ω)/c0

for sorptive brous

(top) and granular (bottom) materials. The inset plots show the speed of sound normalised to that of single porosity non-sorptive materials, i.e.

C(ω)/Cφ (ω).

low-frequency asymptotic value of this ratio (cf. Eq. (69)).

The circles represent the

25

6

10

5

10

2

10

0

4

10

10

−2

10

3

10

−4

10

−6

2

10

1

10

10 ωtd [rad/s]

−8

10

−14

10

−13

10

−12

10

−11

10

−10

10

−9

10

−8

10

−7

10

0

10

−1

10

−2

10

−3

He=1

10

He=2

−4

10

He=4 He=8

−5

10

He=16

−6

10 −14 10

−13

10

−12

10

−11

10

−10

−9

10 De [m /s]

−8

10

−7

10

10

2

6

10

5

10

4

10

3

10

He=1 He=2 He=4 He=8 He=16

2

10 ωtd [rad/s]

1

10

0

10

−1

3

10

10

1

10

−2

10

−1

10

−3

10

−3

10

−4

−5

10

10

−7

−5

10

10

−14

10

−13

10

−12

10

−11

10

−10

10

−9

10

−8

10

−7

10

−6

10 −14 10

−13

10

−12

10

−11

10

−10

10 De [m /s]

−9

10

−8

10

−7

10

2

Figure 7: Thermo-diusive characteristic frequency

De

ωtd

as a function of the eective diusion

He . The porosity is φ = 0.4. Top: granular material (main plot: rg = 5 µm, inset plot: rg = 80 µm). Bottom: material with cylindrical pores (main plot: rp = 0.92 µm, inset plot: rp = 14.85 µm).

coecient

for several values of the eective linearised sorption equilibrium constant

26

6 Conclusions Sound propagation in hierarchical rigid-frame porous materials accounting for the eects of viscosity and heat conduction at the pore scale and sorption and diusion at the micopore scale has been investigated in this paper. The two-scale asymptotic expansion method of homogenisation for periodic media has been used to derive the macroscopic acoustic description. This description shows that, at the leading order, sorption and diusion do not modify the macroscopic uid ow, provided that the diusive mass ux pulsed by the microporous domain on the pore boundaries is of one order smaller than the advective mass ux (i.e.

J = O()).

Therefore, both the macroscopic constitutive uid ow law, i.e. the dynamic Darcy's law, and its associated eective parameter, i.e. the dynamic viscous permeability, correspond to those of single porosity non-sorptive materials. Contrarily, the eective bulk modulus is signicantly altered by sorption and diusion. At the microscopic level, sorption increases the local density and slows down the diusion. Macroscopically, these eects are manifested through i) a signicant reduction of the real part of the bulk modulus at low frequencies, ii) a decrease of the diusion characteristic frequency

ωd , and iii) an increase of the imaginary part of the bulk modulus around the thermo-diusive characteristic frequency ωtd which results in larger attenuation caused by the combined eect of sorption and diusion. In turn, these eects lead to a slower sound speed and larger sound attenuation at low frequencies. Favourable conditions for increasing the eects of sorption and diusion are those that maximise the ratio of the adsorbate to gas phase hold-up to have i) a large value of microporosity length

l,

φm ,

Mh .

In particular, it is appropriate

ii) small values of porosity

φ

and characteristic

iii) the occurrence of relatively strong sorption, i.e. a large value of

large such that the thermo-diusive characteristic frequency

ωtd

He

but not so

lies in the audible frequency

range, and iv) a carefully selected pore-scale geometry due to the strong dependence of the diusion characteristic frequency

ωd

on the morphology.

Potential applications of the results presented in this paper may concern materials for room and building acoustics and inverse methods in chemical engineering and geophysics.

Acknowledgments This work has been supported by the project METAUDIBLE, co-funded by the Agence Nationale de la Recherche ANR (ANR 13-BS09-0003-03) and Fondation de Recherche pour l'Aéronautique et l'Espace FRAE, and was conducted within the framework of CeLyA of Université de Lyon operated by ANR (ANR 10 Labex 0060- ANR 11 IDEX - 0007).

The

authors are grateful to anonymous reviewers for their valuable remarks and suggestions.

Appendix A a. Case of

J = O(2 )

Estimating the diusive mass ux pulsed on mass ux, i.e.

J = O(2 ),

Γ

to be two orders smaller than the advective

the application of the homogenisation procedure leads to the

boundary condition (13) to become

ρ0 u(0) · n = 0 at 0

and

ρ0 u(1) · n = 0 at 1 .

Therefore, the

right-most term in Eq. (48) vanishes and the macroscopic acoustic description corresponds to that of single porosity non-sorptive materials.

27

b. Dynamic sorption model The Langmuir kinetic model [34] is used to relate the rate of adsorption and desorption via Eq. (A.1). This model assumes that [34, 6] : i) the surface of the solid is homogeneous, i.e. the adsorption energy is constant over all sites of the surface; ii) adsorption is localised, i.e. the molecules are adsorbed at denite localised sites; iii) each site can accommodate only one molecule, i.e. monolayer coverage; and iv) the adsorbed molecules do not interact with each other.

Considering that obtain :

dces = k¯a cf m (cN − ces ) − kd ces . dt jωt , this ces = cs0 + cs ejωt and cf m = cm0 + cm e cs =

(A.1) equation can be linearised to

k¯a (cN − cs0 ) cm jω + k¯a cm0 + kd

(A.2)

In deriving this equation, the following relation between the equilibrium values of the concentrations has been used.

Noting that

ωa

k¯a cm0 (cN − cs0 ) − kd cs0 = 0

k¯a cm0 = ka Rg τ0 cm0 = ka P0

and dening

(A.3)

ωa = ka P0 + kd = kd (1 + bP0 ),

where

ka is the adsorption constant (in 1/Pa/s), kd is b = ka /kd is the Langmuir constant (in 1/Pa); Eq. (A.3) can be rewritten as cs0 /cN = ka P0 /ωa = bP0 /(1 + bP0 ). Replacing these expressions into Eq. (A.2) and further considering that i) 1 − cs0 /cN = kd /ωa = 1/(1 + bP0 ), and ii) k¯a cN = ka (Rg τ0 /M )ρN = ka (P0 /ρ0 )ρN , where ρN = cN M is the maximum density increment is the sorption characteristic frequency,

the desorption constant (in 1/s), and

due to sorption, lead to :

cs = H(ω)cm ,

(A.4)

where the linearised sorption dynamic equilibrium constant

H(ω) = Eq. (A.4) replaces Eq. (9) and

H

H(ω)

is given by :

ρN bP0 1 . 2 ρ0 (1 + bP0 ) (1 + ωjω )

(A.5)

a

in Eqs. (11) and (12) is consequently given by Eq. (A.5).

τa = 1/ωa , ranges −9 10 s for physical adsorption [6]. This means that the sorption characteristic

As mentioned in Section 2.1, the average residence time of adsorption, i.e.

−13 to from 10

frequency takes very large values and

H(ω)

can be approximated at low frequencies by :

H = H(ω << ωa ) =

bP0 ρN . ρ0 (1 + bP0 )2

(A.6)

Further considering that the micropore size is in the order of size of the molecules leads to

He = φm H

and

Mh = (1 − φ)φm H/φ,

which corresponds to the

Mh

value of the nano-micro

scale model in [13].

Appendix B a. Models for materials with complex microstructure Since the diusion function in this work and the pressure diusion function in double porosity materials with highly contrasted permeabilities are formally identical one can calculate

G(ω)

28

using a semi-phenomenological model as proposed in [32] (see also [2] and [3]), i.e.

G(ω) = G0 where

G0

jω + ωd

r

jω Md 1+ ωd 2

!−1

,

(B.1)

is calculated from the solution of the problem (43)-(44) for

characteristic frequency

ωd

ω = 0,

the diusion

(63), and the diusion shape factor is dened

Λd is related to the thermal R R 0 . The latter is dened as Λ0 = 2 Λd = 1−φ Λ φ Ωf dΩ/ Γ dΓ and 0 0 is given by Λ = h and Λ = r for materials with slit and cylindrical pores, respectively; and 0 0 0 by Λ = Λ⊥ = aφ/(1 − φ) and Λ = 2rg φ/3(1 − φ) for brous and granular materials, respec|| as

Md = 8G0 /(1 − φ)Λ2d .

is given by Eq.

The diusion characteristic length

characteristic length through

tively.

Analogously, a semi-phenomenological model as in Eq.

(B.1) can be used to calculate

the dynamic viscous and thermal permeabilities [2, 3]. For the dynamic viscous permeabil-

G0 , Md , and ωd are respectively replaced by K0 , Mv = 8K0 α∞ /φΛ2 , ωv = ηφ/K0 ρ0 α∞ , where Λ is the viscous characteristic length. This parameter and the

ity, the terms

and tor-

tuosity can be calculated from the solution of an inviscid ow problem (or, analogously, an electrical conduction one) as shown in [2] [5] [10]. The semi-phenomenological model for the dynamic thermal permeability is obtained by replacing the terms

Mt = 8Θ0 /φΛ02 ,

and

ωt ,

G0 , M d ,

and

ωd

by

Θ0 ,

respectively.

These semi-phenomenological models allow calculating the frequency-dependent eective parameters (i.e. dynamic permeabilities and diusion function) of materials with complex microstructure using parameters calculated from the solution of four limiting problems, namely the inviscid ow problem and the static uid ow, heat conduction, and mass diusion problems.

This may be preferred over calculating the eective parameters from the solution of

the oscillatory uid ow, heat conduction, and mass diusion problems for a discrete set of frequencies.

References [1] J.L. Auriault, L. Borne, R. Chambon, Dynamics of porous saturated media, checking of the generalized law of Darcy, J. Acoust. Soc. Am. 77 (5) (1985) 1641-1650. [2] D. L. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in uid-saturated porous media, J. Fluid Mech. 176 (1987) 379-402. [3] D. Lafarge, P. Lemarinier, J.F. Allard, V. Tarnow, Dynamic compressibility of air in porous structures at audible frequencies, J. Acoust. Soc. Am. 102 (4) (1997) 1995-2006. [4] J.L. Auriault, C. Boutin, C. Geindreau, Homogenization of Coupled Phenomena in Heterogeneous Media, ISTE Ltd and John Wiley & Sons, London and Hoboken, 2009. [5] J. F. Allard, N. Atalla, Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, John Wiley & Sons, West Sussex, 2009. [6] D. D. Do, Adsorption analysis: Equilibria and Kinetics, Imperial College Press, London, 1998. [7] K. Herzfeld, Reection of sound, Phys. Rev. 53 (1938) 899-906.

29

[8] J.G. Parker, Eect of adsorption on acoustic boundary layer losses, J. Chem. Phys. 36 (1962) 1547-1554. [9] J.R. Wright, The virtual loudspeaker cabinet, J. Audio Eng. Soc. 51 (4) (2003) 244-247. [10] R. Venegas, Microstructure inuence on acoustical properties of multiscale porous materials, Ph.D. dissertation, University of Salford, UK, 2011. [11] F. Bechwati, M. Avis, D. Bull, T.J. Cox, J. Hargreaves, D. Moser, D. Ross, O. Umnova, R. Venegas, Low frequency sound propagation in activated carbon, J. Acoust. Soc. Am. 132 (1) (2012) 239-248. [12] R. Venegas, O. Umnova, Acoustical properties of double porosity granular materials, J. Acoust. Soc. Am. 130 (5), (2011) 2765-2776. [13] R. Venegas, O. Umnova, Inuence of sorption on sound propagation in granular activated carbon, J. Acoust. Soc. Am. 140 (2), (2016) 755-766. [14] L. M. Naphtali, L. M. Polinski, A novel technique for characterization of adsorption rates on heterogeneous surfaces, J. Phys. Chem., 67 (2) (1963) 369-375. [15] Y. Yasuda, Frequency response method for study of the kinetic behavior of a gas-surface system. 1. Theoretical treatment, J. Phys. Chem. 80 (17) (1976) 1867-1869. [16] R. Song, L.V.C. Rees, Frequency response measurements of diusion in microporous materials, in:

Karge H.G., Weitkamp, J. (Eds.), Adsorption and Diusion, Molecular

Sieves Science and Technology vol. 7, Springer, Berlin, 2008, pp. 235276. [17] E. Ruckenstein, A. S. Vaidyanathan, G. R. Youagquist, Sorption by solids with bidisperse pore structures, Chem. Eng. Sci. 26 (1971) 1305-1318. [18] K. Kawazoe, M. Suzuki, K. Chihara, Chromatographic study of diusion in molecular sieve carbon, J. Chem. Eng. Japan 7 (3) (1974) 151-157. [19] R. Jordi, D.D. Do, Analysis of the frequency response method for sorption kinetics in bidispersed structured sorbents, Chem. Eng. Sci. 48 (6) (1993) 1103-1130 . [20] O. Umnova, K. Attenborough, K. M. Li, Cell model calculations of dynamic drag parameters in packings of spheres, J. Acoust. Soc. Am. 107 (6) (2000) 3313-3119. [21] C. Boutin, C. Geindreau, Estimates and bounds of dynamic permeability of granular media, J. Acoust. Soc. Am. 124 (6) (2008) 3576-3593. [22] C. Boutin, C. Geindreau, Periodic homogenization and consistent estimates of transport parameters through sphere and polyhedron packings in the whole porosity range, Phys. Rev. E 82 (2010) 036313. [23] M. Nori, S. Brandani, A model for sound propagation between two adsorbing microporous plates, J. Acoust. Soc. Am. 135 (5) (2014) 2634-2645. [24] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, Berlin, 1980.

30

[25] J. Lewandowska, J.L. Auriault, S. Empereur, P. Royer, Solute diusion in fractured porous media with memory eects due to adsorption, C. R. Mécanique 330 (12) (2002) 879-884. [26] C. Boutin, P. Royer, J.L. Auriault, Acoustic absorption of porous surfacing with dual porosity, Int. J. Solids Struct. 35 (1998) 4709-4737. [27] J. L. Auriault, C. Boutin, Waves in bubbly liquids with phase change, Int. J. Eng Sci. 39 (2001) 503-527. [28] L. M. Sun, F. Meunier, B. Mischler, Etude analytique des distributions de température et de concentration à l'intérieur d'un grain sphérique d'adsorbant solide soumis à un échelon de pression de vapeur adsorbable, Int. J. Heat Mass Transfer 29 (9) (1986) 1393-1406. (Calculation of temperature and concentration distributions inside a spherical solid adsorbent grain submitted to a pressure step of adsorbable vapor) [29] D. Lafarge, N. Nemati. Nonlocal Maxwellian theory of sound propagation in uidsaturated rigid-framed porous media, Wave Motion 50 (6) (2013) 1016-1035. [30] C. Boutin, Acoustics of porous media with inner resonators, J. Acoust. Soc. Am. 134 (6) (2013) 4717-4729. [31] T. Levy, E. Sanchez-Palencia, Equations and interface conditions for acoustic phenomena in porous media, J. Math. Anal. Appl. 61 (1977) 813-834. [32] X. Olny, C. Boutin, Acoustic wave propagation in double porosity media, J. Acoust. Soc. Am. 113 (6) (2003) 73-89. [33] O. Umnova, D. Tsiklauri, R. Venegas, Eect of boundary slip on the acoustical properties of microbrous materials, J. Acoust. Soc. Am. 126 (4) (2009) 1850-1861. [34] I. Langmuir, The constitution and fundamental properties of solids and liquids, J. Am. Chem. Soc. 38 (11) (1916) 2221-2295.

1 Research highlights

Acoustics of sorptive porous materials Rodolfo Venegas and Claude Boutin

• Upscaling of wave equation in hierarchical sorptive porous materials. • Eective bulk modulus is signicantly altered by diusion and sorption. • Sound waves are slowed down and more attenuated at low frequencies. • Analytical models for sorptive porous, brous, and granular materials are presented.