Measuring static viscous permeability of porous absorbing materials

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Measuring static viscous permeability of porous absorbing materials

M. Sadouki, Z.E.A Fellah, A. Berbiche, M. Fellah, F. G. Mitri, Erick Ogam, C. Depollier

To cite this version:

M. Sadouki, Z.E.A Fellah, A. Berbiche, M. Fellah, F. G. Mitri, et al.. Measuring static viscous

permeability of porous absorbing materials. Journal of the Acoustical Society of America, Acoustical

Society of America, 2014, 135 (6), pp.3163-3171. �10.1121/1.4874600�. �hal-01082367�

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bing materials.

M. Sadouki

Faulté des Sienes et Tehnique, Université deKhemis MilianaBP 44225 Ain Dea, Algérie.

Z.E.A. Fellah

LMA, CNRS,UPR7051, Aix-MarseilleUniv,Centrale Marseille, F-13402MarseilleCedex 20,

Frane

A. Berbihe and M.Fellah

Laboratoire de Physique Théorique,Faulté de Physique, USTHB, BP 32El Alia,Bab Ezzouar

16111, Algérie.

F.G. Mitri

Chevron,Area 52 -ETC, 5 Bisbee Ct., Santa Fe, New Mexio 87508, United States.

E. Ogam

LMA, CNRS,UPR7051, Aix-MarseilleUniv,Centrale Marseille, F-13402MarseilleCedex 20,

Frane

C. Depollier

LUNAM Universite du Maine. UMR CNRS 6613 Laboratoire d'Aoustique de l'Universite du

Maine UFR STSAvenue O. Messiaen 72085 Le MansCEDEX 09 Frane.

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rous material require a priori estimation of the porosity.In this work,an aoustial method is

presented,inwhihasimpliedexpression(independentofboth thefrequenyandporosity)for

thetransmitted wavesat theDary'sregime (lowfrequeny range) isderived, andusedfor the

inverse determinationof both the visousstatipermeability(orowresistivity)andthethik-

ness of air-saturated porous materials. The inverse problem issolved based on theleast-square

numerial method using experimental transmitted waves in time domain. Tests are performed

usingindustrialplastifoams.Experimental andnumerial validationresults ofthismethodare

presented, whih show the advantage of measuring the visous permeability and thikness of

a porous slab, without the required prior knowledge of the porosity, but by simply using the

transmitted waves.

(4)

The stati visous permeability

k

₀ îsône ôf ^the ^most împortant parameters,whih appears in the theories of sound propagation in porous media, in the low frequeny regime

1−5

. This

parameter is related to the spei ow resistivity

σ

^by ^the ^relation ^:

k

₀

= η/σ

^, ^where

η

^is

the uidvisosity. Several methods

6−23

have been developed in thepast to measure the stati

visous permeability or the ow resistivity.Among these methods, we distinguish between the

so-alled diret methods

6−11

whih do not use sound waves, and indiret methods

12−23

that

usesoundwavestransmitted orreeted bytheporousmaterial. Thepratial implementation

of the diret methods ould be both omplex and expensive. Most of the aousti (indiret)

methods

6−23

require a priori estimation of the porosity, or other non aousti parameters

1−3

(tortuosity, visous and thermal harateristi lengths, thermal permeability). The proposed

proedure is an indiret aoustial method for measuring the stati visous permeability (and

thereforeowresistivity),withoutknowinginadvanetheporosityorothernon-aoustisetting.

This method improves and simplies the approah developed in our previous work

18−20

. The

originalityofthis ontribution withrespetto Refs.18-20istheuseofasimpliedtransmission

oeient whih isindependent of the frequeny and porosity.Sometehniquesuseimpedane

tube, in whih standing waves are generated, and where two

12−16

or three

17

mirophones are

usedfor experimental measurements. Inthisase, a alibrationofthemirophones isneessary

for a good quality of the results. In our proposed method, a tube is used, in whih transient

soundwavespropagate. A singlemirophone

18−20

isusedfor themeasurement ofexperimental

signals,therefore, noalibration isrequired.

Sound propagation in air-saturated porous material is desribed by various physial pa-

rameters, whih aredierent aordingto the frequeny domain

1

. Thehigh and low frequeny

ranges

1−3

,aredenedbyomparingthevisousandthermalskinthiknesses

δ = (2η/ωρ)

^1/2 ^and

δ

^′

= (2η/ωρP

_r

)

^1/2 ^with^the^radius ^of^the^pores

r

⁽

ρ

îs^the^densityôf^the^saturating ûid^;

ω

^the

(5)

pulsationfrequeny;

P

_r ^the^Prandtl^number). ^In^the^asymptoti^domain^(high frequenies),the skinthiknessesbeomenarrowerandthevisouseetsareonentratedinasmallvolumenear

the frame

δ ≪ r

^and

δ

^′

≪ r

^. ^The uid-struture interations aredesribed by the tortuosity

1

,

visous

2

andthermal harateristi lengths

3

.In thevisous domain (lowfrequenies)

3

,theskin

thiknesses

δ

^and

δ

^′ âre^muh^larger^than ^the^radiusôf ^the^pores.^The^main împortant ^parame-

tersinthisfrequenydomainare;thestativisousandthermalpermeabilities

3

andvisousand

thermaltortuosities

3

.Inadditiontotheseparameters,theporosity

φ

^is^a^key^parameter^playing

animportantrole forallfrequenies. IntheDary'sregime

18−20,24,25

(verylowfrequenies),the

stativisous permeabilityis themostinuential parameter; thepropagationequationredues

to a diusion equation

7−9

. The aousti wave does not propagate, but is just attenuated. The

diretand inverse problems weresolved intimedomain

7−9

,using reeted

7

and transmitted

8,9

experimentaldata,thus obtainingagoodestimateofthestativisouspermeability(orspei

owresistivity), knowing inadvane,thevalueof the porosity.

Inthiswork,wepresent animprovedmethodtodeterminesimultaneouslythestativisous

permeability (orow resistivity)and thiknessof the porous material, without knowing inad-

vane,anyothernonaoustiparameter.Theinterestofsolvingtheinverseproblemwithrespet

to the thikness of the material is to verify the results of the inversion, sine the thikness of

thematerialiseasily measurable.Theinverseproblemissolved usingexperimentaltransmitted

waves. We derive asimplied expression for thetransmission oeient in theDary'sregime.

The obtained expression is independent of frequeny and porosity. The visous stati permea-

bility and thikness of the material are the only parameters involved. The transmitted waves

preditedbythesimpliedexpressionofthetransmissionoeientoinidewiththoseobtained

using the lassial expression

19,20

.Inversions on real experimental data are made, using waves

transmittedbysamplesofairsaturatedporousfoams.Theinversionresultsaresatisfatoryand

opennew perspetivesfor the haraterization of air-saturatedporousmaterials.

(6)

In the aoustis of porous materials, one distinguishes two situations aording to whether

the frame is moving or not. In the rst ase, the dynamis of the waves due to the oupling

between thesolidskeleton and the uidis welldesribedbytheBiot theory

24

.In air-saturated

porousmedia,thevibrationsofthesolidframeanoftenbenegletedinabseneofdiretontat

withthesoundsoure,sothatthewavesanbeonsideredtopropagate onlyinuid.Thisase

isdesribedbytheequivalent-uidmodel,whihisapartiularaseoftheBiotmodel,inwhih

uid-struture interations aretaken into aount in two frequeny response fators: dynami

tortuosityof the medium

α(ω)

^given ^by ^Johnson

et al

²^,^and ^the^dynamiompressibility ofthe air in the porous material

β(ω)

^given ^by ^Allard

et al

¹^. ^In ^the ^frequeny ^domain, ^these ^fators

multiply thedensityofthe uidandits ompressibilityrespetively andrepresent thedeviation

fromthebehaviorof theuidinfreespaeasthefrequenyinreases. Consider ahomogeneous

porous materialthat oupies theregion

0 ≤ x ≤ L

^.Â ^sound^pulse împinges ^normally ôn ^the

medium. It generates an aousti pressure eld

p

ând ân âousti ^veloity êld

v

^within ^the

material. Theaousti eldssatisfythefollowing equivalent-uid marosopi equations (along

the

x −

^axis)¹ ^:

ρα(ω)jωv = ∂p

∂x , β(ω)

K

_a

jωp = ∂v

∂x ,

⁽¹⁾

where,

j

²

= − 1

^,

ρ

îs^the ^saturating ûid^density ând

K

_a ^is ^the ompressibilitymodulus of the uid. In the low frequeny domain, the visous eets are important in all the pore volume,

andtheompressiondilatation yleintheporousmaterial isslowenough to favor thethermal

interations between uid and struture

3

. At the same time the temperature of the frame is

pratially unhanged bythepassage ofthe soundwavebeauseof thehighvalueofits spei

heat:theframeatsasathermostat

3

.Inaddition,thethermalondutivityofthesolidishigh,

andtheexessheatisimmediately evauatedbythesolid,whih thereforeremainsat thesame

temperature duringtheompression dilatation yle

3

.

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In the Dary's regime

25,26

(very low-frequeny approximation), the expressions of the res-

ponsesfators

α(ω)

^and

β(ω)

^when

ω → 0

^are^given ^by^the^relations²⁵ ^:

α(ω) = − ηφ

ρk

₀

jω , β(ω) = γ.

⁽²⁾

where

k

₀ ^is^the ^stati permeability,

φ

^the^porosity^and

γ

^the^adiabati ^onstant.

The inident

p

ⁱ

(t)

^and transmitted

p

^t

(t)

^elds ^are ^related ⁱⁿ ^time ^domain ^by ^the transmission

sattering operator

19,20

T

^:

p

^t

(x, t) = Z

_t

0

T (τ )p

ⁱ

t − τ − (x − L) c

₀

dτ.

⁽³⁾

The temporal operator kernel

T

îs âlulated ^by ^taking ^the înverse ^Fôurier ^transform ôf ^the

transmissionoeient of aslab ofporousmaterial given by (Appendix):

T ˜ (ω) = 2Y (ω)

2Y (ω) cosh (jk(ω)L) + (1 + Y

²

(ω)) sinh (jk(ω)L) ,

⁽⁴⁾

where :

Y (ω) = φ s

β(ω)

α(ω) ,

^and

k(ω) = ω s

ρα(ω)β(ω) K

_a

.

Usingtheexpressions(2)ofthedynamitortuosityandompressibility,weobtainthefollowing

expressionfor the transmission oeient :

T ˜ (ω) = 2C

₁

√

jω 2C

₁

√

jω cosh LC

₂

√ jω

+ 1 + C

₁²

jω

sinh LC

₂

√

jω ,

⁽⁵⁾

where

C

₁

= s

γρk

₀

φ

η , C

₂

=

r γηφ

K

_a

k

₀ ⁽⁶⁾

By doing theTaylorseries expansion ofthe transmission oeient, when the frequeny tends

to zero(

ω → 0

^),^we ^obtain ^:

T ˜ (ω) = 1 1 +

^LC_2C₁²

!





 1 −

LC

1

C

2

1 +

^LC_C₁²

+

¹₆

LC2

C¹

2

2 1 +

^LC_2C₁²

jω + ...







(7)

(8)

rstterm :

T ˜ = 1

1 +

^LC_2C₁²

= 1 1 +

^L₂ ^η

k⁰√ ρKa

(8)

Thissimpliedexpressionof theoeient of transmissionisindependent ofthefrequeny and

the porosity of the material, and depends only on the stati permeability and thikness of the

material. In the next paragraph,we ompare the expression (5)of thetransmission oeient,

withits simpliedexpression(8), usingnumerial simulationsof transmitted signals(Eq. 3) by

aslab of air-saturated porousmaterial.

Consider two samplesM1 andM2 ofair-saturated porous foams,having thesame thikness

L = 0.05m

^, ând ^two ^dierent ^vâlues ôf ^their ^stati permeability. M1 is more permeable (less

resistive)thanM2.Theinternational systemofunitsforpermeabilityism

2

.Apratialunitfor

permeability is the Dary (

D

^),⁽¹ ^Dary

=0.97 × 10

⁻¹²

m

²^). ^The permeability value ofM1 is:

k

₀

= 3092.8D

^;^(owresistivity:

σ = 6000

^N^m⁻⁴^s),^and^of^M2^;

k

₀

= 185.56D

^(owresistivity:

σ = 10

⁵ ^N^m⁻⁴^s)^.

Theinidentsignalanditsspetrumaregiveninthegures(1-a)and(1-b),respetively.The

frequeny bandwidth of theinident signal is(450-550)Hz. Fig.2 shows a omparison between

twosimulated transmittedsignalsomputed withdierentexpressions ofthetransmissionoef-

ientsfor thesampleM1.Therstsignal(solidline)orrespondsto thesimulated transmitted

signalusing the expression(5)of thetransmissionoeient, andtheseond one (dashedline)

using the relation (8). The amplitude is represented by an arbitrary unit and the point num-

berrepresented in the absissa is proportional to time. We note that for this frequeny range

(450-550)Hz, the transmitted waves predited bythe two terms of thetransmission oeient

areslightly dierent,a small shift isobserved between thetwo signals;10

%

^for ^the^amplitude,

and 0.2

%

^for ^the ^phase. ^By ^making ^the ^same ^omparison ^with ^the ^sample ^M2, ^whih ^is ^less

permeable than M1, theresults given in gure 3 show a signiant dierene between the two

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simulatedsignals (shiftof49

%

^for ^theâmplitude,ând ôf^0.5

%

^for ^phase).^We^an ^onlude^that

the approximation (8) of the transmission oeient is muh more aurate when the porous

mediumis morepermeable (lessresistive).

Anothertestisperformedbytakinganinidentsignal(Figs.4-a,4-b)withlowerfrequenies

(30-70)Hz. The transmitted signals alulated from equations (5) and (8), are ompared in

gures 5 and 6, for the samples M1 and M2,respetively, in thefrequeny domain (30-70)Hz.

Theseomparisonsshowavery goodagreement, sineitis pratiallyimpossibleto distinguish

between thetwo urves for both M1 and M2 samples. Indeed, the simplied expression of the

transmission oeient given by equation (8) is developed in very low frequenies. This study

showed that the simplied expression (8) gives the same results as the expression (5) for the

lowerfrequenies, espeiallyfor themost permeable(less resistive) materials. Itwould be more

advantageous to usethe simpliedexpression (Eq. 8) ofthe transmissionoeient, sine it is

fast, and doesnot depend onthe frequeny or porosity,andis simpler.

Thetransmission oeient

T(ω) ˜

^given ^by Êq.⁷ ân^be ^written âs^:

T ˜ (ω) = 1 1 +

^LC_2C₁²

! 1 − j ω

ω

c

,

⁽⁹⁾

where

ω

_c

=

²

1+^LC_2C²

1

LC1C2

1+^LC_C²

1 +¹₆_LC

2 C1

2^. ^The ^modulus ^of ^the transmission oeient is given by:

| T ˜ (ω) | =

1 1+^LC_2C²

1

s

1 +

ω ωc

2

.

Table (I)shows values of

ω/ω

_c ^and

| T ˜ (ω) |

^for ^various ^values ôf ^frequenyând ôwresistivity. It an be seen that the values of (

ω/ω

_c⁾ âre ^small ômpared ^to ^1, ând ^those ôf

| T ˜ (ω) |

^are

almostonstant for thesamevalueof theowresistivity,espeiallyfor lowfrequeniesand low

resistivities. These results onrm those obtained in the previous paragraph, and again shows

thepossibilityof using the simpliedexpression(8).

(10)

The experimental setupisskethed inFig.7.ItspitureisgiveninFig.8.Thetube length

isadaptableto avoidreetion, andtopermitthepropagationoftransientsignals,aordingto

thefrequeny rangedesired. Thetube material isPVC, itswall thiknessisof 3mm.For mea-

surementsinthefrequenyrange(20-100)Hz, alengthof50missuient.Itisnotimportantto

keep thepipe straight; itan be rolled inorderto save spae withoutperturbationson experi-

mentalsignals.Thetubediameteris5m(theut-ooftube

f

_c

∼ 4

^kHz).^A^sound^soure^Driver

unit"Brand"onstituted byloudspeakerRealisti40-9000isused. Tone-burstsareprovided by

synthesized funtion generator Standford Researh Systems model DS345-30MHz. The signals

areampliedand lteredusing modelSR650-Dualhannellter,StandfordResearh Systems.

Thesignals(inident and transmitted)aremeasuredusingthesamemirophone(Bruel

&

^Kjaer,

4190) inthe same position in the tube, avoiding theneed of a alibration. The inident signal

is measured without any porous sample, however, thetransmitted signal is measured withthe

poroussample.

ConsideraylindrialsampleofplastifoamF1ofdiameterof5m.SampleF1washarate-

rizedusingdiret

6,9

andindiretmethods

18,19

,giventhevaluesofthestativisouspermeability

and thiknessmarked by

∗

ⁱⁿ^Table^II.^Fig.^9-a ^shows^theexperimental inident signal (dashed line) generated bythe loudspeakerin thefrequeny bandwidth (20-40)Hz, and theexperimen-

tal transmitted signal (solid line). Fig.9-b shows their spetra. The inverse problem is to nd

the visous stati permeability

k

0 ^and ^the ^thikness

L

^of ^the ^porous ^sample, ^whih ^minimize

numerially the ost funtion

U (k

₀

, L) = P

_i=N

i=1

(p

^t_exp

(x, t

_i

) − p

^t

(x, t

_i

))

²

,

^wherein

p

^t_exp

(x, t

_i

)

^is

the disrete set of values of the experimental transmitted signal and

p

^t

(x, t

_i

)

^the ^disrete ^set

of values of the simulated transmitted signal. This inverse problem is solved numerially by

the least-square method. For its iterative solution, we used the simplex searh method (Ned-

ler Mead)

27

whih does not require numerial or analyti gradients. A large variation range is

(11)

applied for eah estimated parameter in solving the inverse problem :

k

₀

∈ [1, 9] × 10

³

D

^and

L ∈ [3, 7]

^m.^The^vâriationsⁱⁿ^theôst^funtion^presentône^lear^minimumorrespondingtothe

following solution of the inverse problem:

k

₀

= 560D

^, ^or

σ = 33137

^N ^m⁻⁴^s, ^and

L = 2.49

^m.

Using these values,we present inFigs. 10(a)-10(b), thevariations of theost funtion

U

^when

varying only one of the parameters around the minimum. This result is onsistent with what

has been found using lassial methods

6,9,18,19

(marked by

∗

ⁱⁿ ^Tâble ÎI). Â ômparison ^bet-

ween an experimental transmitted signal and simulated transmitted signal is given in Fig. 11

for the optimized values of the inverted parameters. The agreement between theoretial and

experimentaldata isgood,whih leadsus to onlude thatthe optimized values ofthe visous

permeabilityandthe thiknessofthesampleareaurate within10

%

^.^This^study^has^been ^also

arried,outinotherfrequenybandwidthsummarizedinTableII.Theinverseproblemhasalso

been solved if the material thikness is held onstant, the results of the inversion are markek

by

♯

^. Ît ân ^be ^seen ^that ^for ^the ^dierent ^frequeny ^bandwidths ôf ^the experimental inident signals,theoptimized valuesobtainedusing thismethodarelose tothoseprodued usinglas-

sial methods

6,9,18,19

. The results of the inversion for the permeability when the thikness is

known or unknown are slightly dierent but lose enough in general. Two other plasti foams

samples F2 and F3, having a very dierent values of their permeability arealso studied. Their

harateristis

6,9,18,19

aremarkedby

∗

ⁱⁿ^Tâbles ÎIIândÎV.Âfter^solving^theînverse^problemⁱⁿ

dierent frequenyregimes,theresultsarepresentedinthesameTables (IIIand IV).Notethat

forthesampleF3,whihishighlypermeable(notresistive),itwaspossibletouseexperimental

dataatrelatively highfrequeny(upto1kHz).Inthisase, theapproximation(8)remainsvalid

forhigherfrequenies,relativetosamplesF1andF2,whiharemuhlesspermeable.Here,again

theinverted values of the permeability ofthesamples F2and F3, arevery lose to those given

bythelassialmethods

6,9,18,19

(markedby

∗

ⁱⁿ^Tâbles ÎIIând ÎV).^The^dierene ^between ^the

optimized valuesand those givenbyother methods

6,9,18,19

doesnot exeed 10

%

^,^exept^for ^the

inverted valueofthe thikness(sample F2)inthefrequeny band (20-40)Hz.

(12)

ases)theresultsoftheinversionforthe permeabilitiesarebetterwhenthematerialthiknessis

assumedunknown. The advantageof solving theinverseproblem withrespetto thethikness,

isto hekthe results ofthe inversion,sine thethiknessofthematerial iseasily measurable.

Thissimplemethodseemstobeeetive formeasuring thepermeabilityoftheporousmaterial

saturatedwithairandoersanotherfasterandsimpleralternativetoonventionalmethods

6−23

.

Notethatthisexperimentannotbeusedforveryhighlyresistiveporousmaterials(permeability

<

9 × 10

⁻¹¹^m²^). În ^fat, ^the âousti êxitation ^generates ^Biot's ^vibrations ôf ^the ^struture²⁷^,

whih indues a strutural disturbane resulting from the elastiity. These vibrations are not

taken into aount intheequivalent uidmodelusedinthis work.

The great advantage of this method ompared to other indiret methods using aousti

waves

12−23

,is thatitisnotneessary,to knowinadvane,thevalueoftheporosity,inorderto

measurethepermeabilityoftheporousmaterial.

*

IV. CONCLUSION

A simple and eetive method isdeveloped for theexperimental measurement of thestati

visouspermeability(orowresistivity)andthiknessofanair-saturated porous material.The

development of the transmission oeient in the Dary's regime (low frequeny), was used

to extrat a simplied expression. This study shows that this new expression gives the same

results as the general one dependent on the frequeny

19,20

, but has the advantage of being

more reliable,simpler andfaster.The inverseproblemis solvedusing experimental transmitted

data. The reonstruted values of permeability and thikness are lose to those using lassial

methods

7−9,11

. The most important result in this study is that it is now possible to measure

thevisouspermeabilityand thikness, withoutknowingtheporosityofthematerials, andjust

(13)

methods usingaousti methods

7−9,17−21

,or non-aousti methods

10−16

.