Simultaneous determination of porosity, tortuosity, viscous and thermal characteristic lengths of rigid porous materials

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Simultaneous determination of porosity, tortuosity, viscous and thermal characteristic lengths of rigid

porous materials

Zine El Abiddine Fellah, Mustapha Sadouki, Mohamed Fellah, F. G. Mitri, Erick Ogam, Claude Dépollier

To cite this version:

Zine El Abiddine Fellah, Mustapha Sadouki, Mohamed Fellah, F. G. Mitri, Erick Ogam, et al.. Simul- taneous determination of porosity, tortuosity, viscous and thermal characteristic lengths of rigid porous materials. Journal of Applied Physics, American Institute of Physics, 2013, 114 (21), pp.204902.

�10.1063/1.4833546�. �hal-00903319�

ous and thermal harateristi lengths of rigid porous

materials.

Z.E.A. Fellah

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

Frane.

M. Sadouki

Faulté des Sienes et Tehnique, Université de Khemis Miliana, Route de Thénia, Khemis

MilianaBP 44225, Algérie.

M. Fellah

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

16111, Algérie.

F.G. Mitri

Los Alamos National Laboratory, MPA-11, Sensors Eletrohemial Devies, Aoustis Sensors

Tehnology Team,MS D429, Los Alamos,NM87545, USA.

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.

running title : Ultrasoniharaterization of porous materials.

Wepresentanimprovedmethodfortheharaterizationofair-saturatedporousmaterialsby

simultaneousmeasurementofporosity,tortuosity,visousandthermalharateristilengthsvia

ultrasoni transmission only.The proposedmethod is basedon a temporalmodelof thediret

and inverse sattering problem for the transient ultrasoni waves in a homogeneous isotropi

slabofrigidporousmaterial.Theadvantageoftheproposedmethodisthatthefourparameters

aredeterminedsimultaneouslyusingjusttransmittedexperimentalwavefromaporousmaterial

saturatedbyonegas(air).Inaddition,norelationshipisassumedbetweenthetwoharateristi

lengths.

Aoustidampinginair-saturatedporousmaterialsisdesribed bytheinertial,visous,and

thermal interations between the uid and the struture

1−3

. These materials are mainly used

to redue noise and vibration pollution. The physial parameters

1−4

desribing the ultrasoni

propagationinthesemedia are:porosity,tortuosity,visousandthermalharateristi lengths.

Theseparameters playanimportant roleintheattenuationand dispersionofaoustiwavesin

aporousmediumatthehighfrequenies

4

.Thehighfrequenydomain

4

orrespondstotherange

offrequeniessuhthatthevisousboundary layerthikness

δ = (2η/ωρ) ^1/2

^{issmallerthanthe}

radius

r

^{of thepores (}

η

^and

ρ

^arerespetively thevisosityand density of thesaturating uid and

ω

^{representsthe pulsation frequeny).}

The transmittedwaves

5−9

areoftenusedfor theultrasoni haraterization of air-saturated

porous materials in the frequeny

5−7

and time

8,9

domains. When the struture of the porous

materialsisrigid,twoindependentparametersaregenerallymeasuredintransmittedmodeusing

ultrasoniwaves;thetortuosityandthevisousharateristi length.Thethermalharateristi

length is dedued from a xed ratio with the visous harateristi length. When the porous

medium is subsequently saturated by two gases (air and helium), the determination of the

thermal harateristi length independently of the visous length is possible

5

.In thease of a

porous materialhaving a struture whih vibrates

10,11

the ultrasoni waves transmitted, allow

measurement of the porosity,andmehanial parameters.

The reeted waves by the rst interfae

12,13

of a slab of rigid porous material permit the

measurement of the tortuosity and porosity. When the reeted wave by the seond interfae

is deteted experimentally, the determination of the harateristi lengths beomes possible

14

.

Theuse ofboth transmitted andreeted wavessimultaneously

15,16

gives agoodestimation of

porosity, tortuosity,visous and thermal harateristi lengths. Other methods

17

−

21

,not using

ultrasoni waves, have been developed for the haraterization of rigid porous materials, mea-

In thiswork,we introdue an improvedmethodto measuretheporosity,tortuosity,thevis-

ous and thermal harateristi lengths simultaneously, by solving the inverse problem in the

time domain, without the use of reeted waves. Experimental data are used in the inversion

proess from waves transmitted by the porous material in the high frequeny range. No rela-

tionship is assumed between the harateristi lengths suh that both lengths are determined

independently.Thisworkshows thatitisnowpossibleto measuretheporosityusingultrasoni

transmitted waves only.In addition, the thermal harateristi length an be obtained regard-

less of the visous length, without saturating the porous medium by another gas, or the use

datareetedwave.Thus,themethodisreliable,rapidandpresentsadvantagesoverthelassi

tehniques used to date. These results open perspetives yet to be explored for the ultrasoni

haraterization tehniques ofair-saturated porous materials.

II. MODEL

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

the frame is moving or not. In the initial ase, the dynamis of the waves due to theoupling

between thesolidskeleton and theuidis welldesribedbytheBiot theory

22

.In air-saturated

porousmedia,thevibrationsofthesolidframeanoftenbenegletedinabseneofdiretontat

withthesoundsoure,sothatthewavesanbeonsideredtopropagate onlyinuid.Thisase

isdesribedbytheequivalent-uidmodel,whihisapartiularaseoftheBiotmodel,inwhih

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

tortuosityof the medium

α(ω)

^{given by Johnson}

et al ⁴

^{,and thedynami}ompressibility 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 theuidinfreespaeasthe frequenyinreases. Consider ahomogeneous

porous materialthat oupies the region

0 ≤ x ≤ L

^{.A soundpulse impinges normally on the}

medium. It generates an aousti pressure eld

p

^{and an aousti veloity eld}

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

^{theuid density and}

K

_a ^isthe ompressibility modulus of theuid. In the high frequeny domain, the visous eets are onentrated in a small volume near the frame

and the ompression/dilatation yle is faster than the heat transfer between the air and the

struture, and it is a good approximation to onsider that the ompression is adiabati. The

high-frequeny approximation of the responses fators

α(ω)

^and

β (ω)

^when

ω → ∞

^{are given}

by therelations :

α(ω) = α

_∞

1 + 2 Λ

r η jωρ

, β(ω) = 1 + 2(γ − 1) Λ

^′

r η

jωP

r

ρ ,

⁽²⁾

where

j ² = − 1

^,

α

_∞ ^{is the tortuosity,}

Λ

^{the visous} harateristi length,

Λ

^{′ the thermal ha-}

rateristi length,

η

^{the uid visosity,}

γ

^{the adiabati onstant,}

P

_r ^{thePrandtl number. The}

expression,infrequenydomain,ofthetransmissionoeient

T (ω)

^{ofaslabofporousmaterial}

isgiven by :

T (ω) = 2Y (ω)

2Y (ω) coth (k(ω)L) + (1 + Y ² (ω)) sinh (k(ω)L) ,

⁽³⁾

where :

Y (ω) = φ s

β(ω)

α(ω) ,

^and

k(ω) = ω s

ρα(ω)β(ω) K

_a

,

φ

^{istheporosityofthe material.Inthe timedomain,}

α(ω)

^and

β(ω)

^{atasoperatorsandinthe}

highfrequeny approximationtheir expressions aregiven by

21

:

˜

α(t) = α

_∞

δ(t) + 2 Λ

η πρ

1/2

t

^−1/2

!

, β(t) = ˜ δ(t) + 2(γ − 1) Λ

^′

η πP rρ

1/2

t

^−1/2

!

,

⁽⁴⁾

inthese equations,

δ(t)

^{isthe Dirafuntion. In this modelthe timeonvolution of}

t

⁻

^1/2

^with

a funtion is interpreted asa semi derivative operator following thedenition of thefrational

derivative oforder

ν

^{given inSamko and oll}

²⁴

^,

D

^ν

[x(t)] = 1 Γ( − ν)

Z

t

0

(t − u)

^−ν−1

x(u)du,

⁽⁵⁾

where

Γ(x)

^{is the gamma funtion. Using equations (1) and (4)in the timedomain, it follows}

thefrational propagationequation :

∂ ² p

∂x ² − α

_∞

c ² ₀ ∂ ² p

∂t ² − B ∂ ^3/2 p

∂t ^3/2 − C ∂p

∂t = 0,

⁽⁶⁾

where theoeients

c ₀

^,

B

^and

C

^areonstantsrespetively givenby;

c ₀ = s K

_a

ρ , B = 2α

_∞

K

a

r ρη π

1

Λ + γ − 1

√ P rΛ

^′

C = 4α

_∞

(γ − 1)η K

_a

ΛΛ

^′

√

P r .

⁽⁷⁾

The inident

p

ⁱ

(t)

^and transmitted

p

^t

(t)

^{elds are related in time domain by the} transmission sattering operator

9 T

^:

p

^t

(x, t) = Z

t

0

T ˜ (τ )p

ⁱ

t − τ − (x − L) c ₀

dτ,

^{where :}

T ˜ (t) = 4φ √ α

_∞

(φ + √ α

_∞

) ² G

t + L

c , L c

.

⁽⁸⁾

Asinmost porous materialssaturated air, themultiplereetions arenegligible beause ofthe

high attenuation of the sound wave, the expression of the transmission operator

T ˜

^{takes into}

aount only the reetions at interfaes

x = 0

^and

x = L

^.

G

^{is the Green funtion of the}

mediumgiven by

25

:

G(t, k) =



 

 

 



 

 

 



0

^if

0 ≤ t ≤ k,

b^′

4

√ π

k

(t−k)

^3/2

exp

− _16(t

^b′2₋^k2_k)

+ ∆ R

t−k

0 h(t, ξ)dξ

^if

t ≥ k,

(9)

where :

h(ξ, τ ) = − _4π ¹

³/2

√ 1

(τ−ξ)

²−k²

1

ξ^3/2

R 1

−1

exp

−

^χ(µ,τ,ξ)

₂

(χ(µ, τ, ξ) − 1) √

^µdµ

1−µ

²

,and :

χ(µ, τ, ξ) =

∆µ p

(τ − ξ) ² − k ² + b

^′

(τ − ξ) 2

/8ξ

^,

b

^′

= Bc ² ₀ √

π

^,

c

^′

= C.c ² ₀

^,

∆ ² = b

^′2

− 4c

^′.

The inverseproblem isto ndtheparameters

α

_∞^,

φ

^,

Λ

^and

Λ

^{′ whih minimize numerially}

thedisrepanyfuntion

U (α

_∞

, φ, Λ, Λ

^′

) = P

i=N

i=1

(p

^t_exp

(x, t

i

) − p

^t

(x, t

i

)) ² ,

^wherein

p

^t_exp

(x, t

i

)

i=1,2,...n

is thedisrete set of values of the experimental transmitted signal and

p

^t

(x, t

_i

)

_i=1,2,...n ^thedis-

rete set of values of the simulated transmitted signal predited from Eq. (8). The inverse

problem is solved numerially by the least-square method. For its iterative solution, we used

the simplexsearh method (Nedler Mead)

26

whih doesnot require numerial or analyti gra-

dients. Experiments are performed in air using a broadband Ultran NCT202 transduer with

a entral frequeny of 190 kHz in airand a bandwith of 6 dBextending from 150 to 230 kHz.

Pulsesof400Vareprovided bya5058PR Panametris pulser/reeiver.The reeived signalsare

lteredabove1MHz toavoidhigh-frequenynoise.Eletroniinterfereneiseliminatedby1000

aquisition averages. The experimental setup is shown in Fig. 1. Consider a sample of plasti

polyurethanefoam M1,ofthiknesses

0.8 ± 0.01cm

^{.Sample M1 was}haraterized using lassi methods

3,6,9,12,13,27

andgavethefollowingphysialparameters

φ = 0.85 ± 0.05

^,

α

_∞

= 1.45 ± 0.05

^,

Λ = (30 ± 1)µm

^,

Λ

^′

= (60 ± 3)µ.m

^{. The porosity is measured by the diret}

²⁷

^{and ultrasoni}

methods

12−14

, giving the same results. Tortuosity is related to the formation fator

28

used to

desribetheeletrialondutivityofaporoussolidsaturatedwithondutinguid.Thediret

methodforthemeasurementoftortuosity,isbasedonthemeasurementofformationfator.For

air-saturated plasti foams

1

−

3

,itis not possible to saturatethematerialbya onduting uid.

Theultrasoni methods

3,5,6,9,12−14

arebest suited forthemeasurement of tortuosity.The stan-

dard methods usedfor measuring the visous and thermal harateristi lengths of theporous

samplearethe ultrasoni method

4−6

.

Fig. 2 givesa sanning EletronMirosope image of a polyurethane foam sampleshowing

the mirostruture of the pores. Fig. 3 shows the experimental inident signal (dashed line)

generatedbythetransduer,andtheexperimentaltransmittedsignal(solidline).Theamplitude

Aftersolving the inverseproblem simultaneously forthe porosity

φ

^,tortuosity

α

_∞^{,visous and}

thermalharateristi lengths

Λ

^and

Λ

^{′,wendthefollowingoptimizedvalues:}

φ = 0.87 ± 0.01

^,

α

_∞

= 1.45 ± 0.01

^,

Λ = (32.6 ± 0.5)µm

^and

Λ

^′

= (60 ± 0.5)µm

^{. The values of the inverted}

parameters arelose tothose obtainedbyonventional methods

3,6,9,12,13,27

.We present inFigs.

4(a)-4(b), the variation of the minimization funtion

U

^{with the porosity, tortuosity, visous}

harateristi length,andtheratiobetween

Λ

^{′ and}

Λ

^{.InFig.5,we showaomparison between}

anexperimentaltransmitted signalandsimulatedtransmittedsignalfortheoptimizedvaluesof

φ

^,

α

_∞^,

Λ

^and

Λ

^{′.Thedierenebetweenthetwourvesissmall,whihleadsustoonludethat}

theoptimizedvaluesofthephysialparametersareorret.Westudiedtheimpatof(

Λ

^′

/Λ 6 = 2

⁾

onthedeterminationofotherparameters,and wearrived atthefollowing result:Byinreasing

theratio

(Λ

^′

/Λ)

^{of2 to 3,theporosityinreases from 13}

%

^{ofits initial value,while thevisous}

harateristi length dereases by 9

%

^{of its initial value. The value of the tortuosity remains}

unhanged. We an therefore onlude that the porosity and visous harateristi length, are

theparameters thatweremost aetedby thehange intheratio between

Λ

^{′ and}

Λ

^.

The use of the expression of thetransmission operator inthe timedomain (Eq. 8), or that

thetransmission oeient inthe frequeny domain(Eq. 3),thentaking theinverse transform

numerially, gives exatly the same result for the inversion proess. The time and frequeny

approahes are omplementary. For transient signals in very short time, the temporal model

is best suited. However, for signals with a wide temporal ontent (narrow frequeny ontent),

the frequeny approah is preferred. For the experimental signals used in this work, the two

approahes areequivalent and give the same results.

Given thelow sensitivityof theporosityandthethermal length,relative to thatof thetor-

tuosity and the visous length of the transmitted waves, we thought

9,12,13

it was not possible

to solve the inverse problem with respet to porosity and thermal length.However, this study

ame toontradit thisearlier onlusion.The measurement of thesetwo quantities (

φ

^and

Λ

^′)

zonsyetto be exploredfortheultrasoniharaterization ofporousmaterials, thuslimitingthe

experimentaldata. We reall that until now, themeasurement of porosityand thermal hara-

teristi length required theombination of data intransmission and reetion. The alternative

ultrasoni method

5

formeasuring

Λ

^{′ usingonlythe}transmittedwaves,istosaturatetheporous material byanother gas, whih is not always easy to do experimentally, without damaging the

internal struture of the material porous. The proposed method has the advantage of being

simple,without anyintervention on the uid saturating theporous material. Inreasinglylimi-

ting theexperimental data to the transmitted signals,the inversion is faster to perform. Solve

theinverse problem inthe timedomain for transient signals hasthe advantage of treating the

full information of the experimental signal, ating simultaneously on speed, attenuation and

dispersionofthe ultrasoni wave.

IV. CONCLUSION

An inversesattering estimateof theporosity,tortuosity,visous andthermalharateristi

lengthswasgivenbysolvingtheinverseproblemintimedomainforwavestransmittedbyaslab

of air-saturated porous material. Theinverse problem issolved numerially bytheleast-square

method. The reonstruted values of these parameters are in agreement with those obtained

usinglassial methods.The proposedexperimentalmethodhastheadvantageof beingsimple,

rapid,andeient forestimatingthoseparameters andfurtherharaterizingporousmaterials.

This study shows that it is not neessary to use the reeted waves for a omplete ultrasoni

haraterization ofair-saturated porousmaterial.

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Fig.1.Experimentalset-up of the ultrasoni measurements.

Fig.2. Asanning EletronMirosopeimage of a polyurethane foamshowing themirostru-

tureof thepores

Fig. 3. Experimental inident signal (solid line) and experimental transmitted signal (dashed

line).

Fig.4-a. Variationof the minimizationfuntion

U

^{withporosityand tortuosity}

Fig. 4-b. Variation of the ost funtion

U

^{with the visous} harateristi length

Λ

^{and the}

ratio

Λ

^′

/Λ

Fig. 5. Comparison between the experimental transmitted signal (blak dashed line) and the

simulated transmittedsignals ( redline)usingthe reonstrutedvaluesof

φ

^,

α

_∞^,

Λ

^and

Λ

^′.

pulse generator

Computer Digital oscilloscope High frequency filtering

Pre-amplifier

Triggering Sample Transducers

3.8 3.9 4 4.1

x 10

⁻⁴

−1 0 1 2

Time (s)

Amplitude (a.u.)

0.75 0.8 0.85 0.9 0.95 1 0.035

0.04

Porosity

U

1.45 1.5 1.55

0 0.5 1 1.5

Tortuosity

U

2.5 3 3.5 4 4.5 x 10

⁻⁵

0

5 10 15

Viscous characteristic length Λ

U

1 1.5 2 2.5 3

0 10 20

Λ’/Λ

U

3.8 3.9 4 4.1 4.2 x 10

⁻⁴

−0.1

−0.05 0 0.05

Time (s)

Amplitude (a.u.)

Theory

Experience