NUMERICAL AND EXPERIMENTAL CHARACTERIZATION OF ACOUSTIC POROUS …€¦ · This empirical model,...

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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 8, Issue 8, August 2017, pp. 919–930, Article ID: IJMET_08_08_100

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=8

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

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NUMERICAL AND EXPERIMENTAL

CHARACTERIZATION OF ACOUSTIC POROUS

MATERIAL - A REVIEW

Yuvaraj L, Jeyanthi S

School of Mechanical and Building Sciences,

VIT University, Chennai, India

ABSTRACT

The need of porous material is becoming increasing because of its high influence

in sector like aerospace and automotive due to their light weight, mechanical stability.

In sight of acoustic it is necessary to understand the comprehension zones of material

and acoustics to develop a good acoustic absorption material. This paper reviews the

various approaches and methods to determine the acoustic performance of porous

material. This paper serves as an effective source of literature for those interested in

conducting research in acoustic porous materials.

Key words: porous material, sound absorption co-efficient, porosity, flow resistivity,

tortuosity.

Cite this Article: Yuvaraj L, Jeyanthi S. Numerical and Experimental

Characterization of Acoustic Porous Material - A Review, International Journal of

Mechanical Engineering and Technology, 8(8), 2017, pp. 919–930.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=8

1. INTRODUCTION

The study of sound propagation in porous materials has started at the end of the 17th

century

with the work of Lord Rayleigh. In 1949, Zwikker and Kosten [1] stated fundamental to the

field of acoustic by offering a model of sound propagation in cylindrical pores of lemonade

straws and small glass tubes of different diameter and different angle to the wave front taking

into account the viscous and thermal interaction between air and the solid. In 1956, M. A.

Biot [2, 3] contributed the most elaborate model of the acoustic wave propagation of sound in

elastic porous materials. According to this theory, three different kinds of waves propagate

within the porous material. Two compression waves propagating in the fluid phase and solid

phase and one shear wave propagating in the solid phase. Biot studied the low frequency [2]

and the high frequency [3] behaviours. Early Biot theory was ignored until 1980. Later on

research community realized the importance of Biot theory and now many acoustic based

companies using Biot model. There are five different parameter were used in biot model to

predict sound propagation namely porosity, airflow resistivity, tortuosity, thermal

characteristic length, and viscous characteristic length. Other than Biot theory

Numerical and Experimental Characterization of Acoustic Porous Material - A Review


Figure 1 Types of pores a: closed pores, b,c,d,e,f: open pores b, f: blind pores (dead-end or saccate) e:

through pores

Several theoretical models have been proposed to predict the acoustic properties of porous

material. These model are classified in two: the time domain model and frequency domain

.only the frequency domain are common to describe the acoustic behaviour .some of the

simple and empirical models are there. For example in 1970 Delany and Bazley [4] proposed

an empirical which predict the sound wave propagation in fibrous material.

2. THEORETICAL APPROACH

2.1. Delany and Bazley’s method [4]

From a large number of measurements on fibrous materials with porosities close to 1.00,

Delany and Bazley have proposed empirical expressions for the values of the complex wave

number k (γ/jγ/j) and characteristic impedance Zc for such materials.

(

)

(

)

where ρ0 and c0 are the density of air and the sound speed in air, ω=2πf is the angular

frequency and σ is the static air flow resistivity in the wave direction of propagation

(expressed in N.m-4

.s).Boundaries, proposed by the authors, for the validity of these power

law expressions are:

This empirical model, which can provide reasonable estimations of k and Zc in the

approximate frequency range defined above, is still widely used for its simplicity: only one

parameter, σ is needed to describe the acoustic behavior of a material.



Qunli’s method [5]

Based on the numerous experimental data. A simple empirical relation was established

beween the characteristic impedance „Z0‟and „Γ‟ propagation constant of porous material

when flow resistivity value is 1.

⁄ ( )

Where „Z1‟ and „Z2‟ are the impedance with rigid backing and a ¼ wavelength air space

termination, and „l‟ is the thickness of the sample.

Miki [6]

Miki used Delany and Bazley‟s measurement data to create a new regression method.

Because they identified that the Delany-Bazley model produced an unphysical prediction at

low frequencies and amended the original equation regression coefficients. This Method

always resulted in positive absorption coefficients and valid for a slightly larger frequency

range than specified in following equation.

Mechel[7]

The acoustic characteristic variable, wave impedance and propagation constant can be

expressed by one non-dimensional variable for survey calculation

⁄

Density of air,N specific flow resistance, =wave impedance, =propagation

constant (For low frequency)

√ ⁄

⁄

With γ=1.403 the adiabatic exponent of air and h the porosity of the absorber. H=0.95 for

most materials

(For High frequency frequency)

Komatsu’s method [8]

This model were for predicting the acoustical properties of fibrous materials, the

characteristic impedance Zc and the propagation constant γ, only from their airflow

resistivity was constructed. It was found that the introduction of an expression involving the

common logarithm improved the conventional models. This new model is more effective than

the conventional models, particularly for the prediction for high-density fibrous materials



√

√

This new model is more effective than the conventional models, particularly for the

prediction for high-density fibrous materials .Above equations are temporarily called the basic

model in komatsu‟s paper. When this model is expanded, the real and imaginary parts to

predict the acoustic nature of the material

Allard and Champoux[9]

The Johnson-Champoux-Allard model is based on the work by Johnson, Koplik & Dashen to

describe visco-inertial dissipative effects inside the porous media. The work by Champoux &

Allard is used to describe the thermal dissipative effects. In 1987 Johnson Koplik and

Dashen[9] proposed a semi-phenomenological model to describe the complex density of an

acoustical porous material with a motionless skeleton having arbitrary pore shapes. This

expression is:

[

√

]

4 parameters are involved in the calculation of this dynamic density: the open porosity ϕ,

the static air flow resistivity σ, the high frequency limit of the tortuosityα∞ and the viscous

characteristic length Λ

In 1991, Champoux and Allard [10] introduced an expression for the dynamic bulk

modulus for the same kind of porous material based on the previous work by Johnson et al.

[

√

]

The open porosity ϕ and thermal characteristic length Λ′ are the two parameters involved

in the calculation of this dynamic bulk modulus

3. NUMERICAL APPROACH

3.1. Fem

Finite element method is most commonly used numerical method to solve the acoustic

engineering problems .many of the researchers (Atalla et al[12], 1998; Coyette and

Wynendale[13], 1995; Kang and Bolton, 1995[14]; Johansen et al., 1995[15]) developed the

fem model for modeling sound absorbing material using biot theory. There are certain

limitation finite element modeling of porous media in high computational cost this is because

of initial formulation proposed in parallel with the original Biot theory implied six degree of

freedom per node, involving the solid and fluid phase‟s displacement. The problem associated

with porous material is higher frequency propagation when compared to other material. The

http://apmr.matelys.com/Parameters/OpenPorosity.html

http://apmr.matelys.com/Parameters/StaticAirFlowResistivity.html

http://apmr.matelys.com/Parameters/HFLimitOfTortuosity.html

http://apmr.matelys.com/Parameters/ViscousCharacteristicLength.html

http://apmr.matelys.com/Parameters/ViscousCharacteristicLength.html

http://apmr.matelys.com/Parameters/OpenPorosity.html

http://apmr.matelys.com/Parameters/ThermalCharacteristicLength.html



sound absorption property decreases with increase of wavelength until the thickness of porous

material increases in same proportion.

Craggs[16] proposed equivalent fluid formulation to represent porous absorbing material

using eight noded isoparametric finite element. The theme is to explain porous material with

Helmholtz equation using complex and frequency dependent expression of the effective

density and effective bulk modulus. The main benefit of this formulation is that it involve

pressure fluctuation as a primary variable which is more efficient for one degree of freedom

per node. Based on biot-allard theory Kang and Bolton‟s published a full length paper of two

dimensional modelling of porous material.

In 1995 Goransson[17] proposed formulation for limp porous material in that he used

fluid pressure and frame displacement as degrees of freedom variables. In 1998 Goransson

[18] proposed five degrees of formulation to overcome the drawbacks associated kang and

Bolton six degrees of freedom model because it contain lot of matrices which are difficult to

solve. This models assumes that the fluid pressure which is rotational free but this is was

disagreed due to inertial and viscous coupling between solid and fluid phase (Allard and

Atalla[19]). Atalla developed a three dimensional formulation for modelling porous material

and this formulation is considered to be as accurate in terms of computational efforts

Nordgren[20]

3.2. Boundary Element Method

BEM is famous for its simplified mesh in order to improve the computational efficiency. It

consider entire problem to single boundary condition. Tanneau et al[21]. (2006) proposed a

new method solving insulation panel problem. The idea is to simplify problem in solving mid

–frequency range using BEM method. BEM reduces the mesh so it lead to computational

saving. Kinder and Hansen(2008) this method important in poro-elastic material deals large

matrices are frequency dependent.

4. EXPERIMENTAL ANALYSIS

4.1. Porosity

Porosity is one of the most important physical parameter in term of prediction and modeling

of porous acoustic media. By definition volume proportion of fluid that occupied in the pores

or other words total volume of the fluid in the pores shared by the volume of sample .Beranek

[23] suggested a simple method for the determination of porosity without making the sample

saturation with water. The apparatus working on the concept of great difference in

compressibility between a solid and a gas. In this method, a porous sample is introduced in an

air-tight chamber that is connected to a U-tube manometer as shown in Fig 2

Figure 2 Beranek u-tube setup for porosity



4.1.1. Pressure difference method

Champoux, Stinson and Daigle [23] compared the atmospheric pressure and the pressure

inside the closed enclosure by pressing the piston against the sample .For better result the

sample size should be larger. Increase of pressure depends on the volume of the enclosure and

the volume accessible to the air inside the porous medium. Schematic diagram shown in fig 3

Figure 3 Champoux pressure difference setup for porosity

4.1.2. Fluid saturation method (Archimedes principle)

Pores in sample is saturate by means of fluid by knowing the volume of gas that needed for

saturate porous material then it is easy to deduce its open porosity some of researchers

applied Archimedes principle to acoustic porous material are listed as follows. Panneton &

Gros [24] weight the porous sample before and after by removing saturated fluid in the pores

using air pump. Salissou & Panneton[25] used the perfect gas law to deduce the open porosity

and mass density of a porous material by accounting four masses at four static pressure.

4.2. Air Flow Resistivity

The air flow resistance is the resistance offered by a porous sample when air passes in it. The

sound absorption property of a material is directly related to this property. Bies and Hansen

[26] stated an empirical formula to determine the airflow resistance of a fibrous material by

using bulk density and fiber diameter. In order to maintain linear flow it is necessary to keep

low flow velocity in the range between 5 × 10–4 and 5 × 10–2 ms–1.the airflow resistance can

be found by various method by like direct flow method (ASTM C522) and alternate flow

method (Ignard) [28].

4.2.1. Direct flow method

The measurement of flow resistivity has been in ASTM standard c522 where you can found

the detail procedure.in direct method the steady state air supply rushes towards the sample,

the differential pressure sensor measure pressure drop across the sample and air flow meter

calculate the velocity of air and hence the air flow resistance is obtained [27].this is shown in

figure 4



Figure 4 Direct flow setup for flow resistivity

4.2.2. Alternate flow method

In this method, the piston is moved at the frequency rate of 2 Hz which produce the alternate

air flow towards the specimen. A condenser microphone is used to measure the pressure

difference. The main drawback of this method is repeatability and reproducibility. The setup

is shown in Figure 5. . Garai and Pompoli[29] found that 2.5 percent of error for five samples

of same material within the laboratory in case of repeatability, whereas Reproducibility

between laboratories were around 15 percent of error. Ignard [28] construct a measurement

system which is not actually need of any air supply, flow and pressure sensors. A piston falls

under gravity within the tube and the air pushes toward the porous material. Air flow

resistivity is given by mass of piston, time taken to terminal velocity. Cross section area of

tube and calibration factor

Figure 5 (a) Alternate flow setup for flow resistivity,5 (b) Falling piston method

4.3. Tortuosity

Tortuosity is the deviation from a straight line that is used to signify the complex path of

electrical conduction and fluid diffusion. It is the property of a porous material geometric

having many curves. The actual definition for tortuosity was given by Biot [30].



Predominantly the tortuosity is needed because the actual particle velocity in microscopic

pores deviate from macroscopic scale and hence there is a difference in kinetic energy. The

ratio of these energy is multiply with density as a correction factor.

Brown[31] shown that tortuosity of non -conducting porous material can be calculated by

electrical conductivity method. The sample porous material is saturated by electrolyte solution

then the resistivity of saturated sample and resisitivity of electrolyte solution can be compared

.the tortuosity is given by ratio of those resistivity. same method was optimized by champoux

Figure 6 Electric resistivity setup for tortuosity

4.3.2. Ultrasonic method

Transmission method of ultrasonic tortuosimeter was proposed by allard [33] and leclaire

[34]. This method is based on phase velocity of high frequencies to the sound velocity of free

air. The ultrasonic frequency is emitted from the transmitter end and signal captured in

received end on both condition like with sample and without sample. There is a delay in the

phase because of placing sample between two transducers when compared to no sample and it

is taken as reference signal.by doing fast Fourier transform of those signal to calculate phase

velocity.

Figure 7 Leclaire ultrasonic setup for tortuosity



4.4. Characteristic Length

Basically viscous characteristic length is the effect of viscosity at higher frequencies or the

viscous force between the solid and fluid phase at higher frequency. Viscous force are

responsible for shear force production in porous material. Ultrasonic attenuation has been

introduced by Leclaire et al to measure viscous characteristic length. The viscous forced

exists only at certain frequency which is beyond inertia force. The same tortuosity has been

used for measuring viscous characteristic length using helium and air filled saturated porous

material. Saturating porous material using helium was a precise method to estimate both

viscous and thermal characteristic length.

The thermal interaction between solid and fluid phase at higher frequency defining the

thermal characteristic length. After exceeding certain frequency, the thermal effects are seen

near the skeletal wall of the porous material.at lower frequency isothermal compressibility

and higher frequency adiabatic modulus will takes place. Ultrasonic setup which is discussed

in previous section as same here also. Mostly the tortuosity, viscous characteristic length and

thermal characteristic length of porous material are measured simultaneously with single

technique.

4.6. Sound Absorption Co-efficient

Measurement techniques used to characterize the sound absorptive properties of a material

are:

Reverberant Field Method

Impedance Tube Method

Steady State Method

4.6.1. Reverberant Field Method

This method is detailed in ASTM C 423 – 72 which is used for measuring sound absorption

coefficient is calculated by exposing material to a randomly incident sound wave, which

technically called as diffusive field. The main drawback of this techniques is creation of a

diffusive sound field which requires a large and costly reverberation chambers [35].A

completely diffuse sound field can be achieved only rarely. Moreover, an accurate value of

complex impedance cannot be derived from the absorption coefficient alone. Since sound is

allowed to strike the material from all directions, the absorption coefficient determined is

called random incidence sound absorption coefficient,

4.6.2. Impedance Tube Method

Impedance tube method is also called as standing wave tube in this method plane sound

waves generated and allow to travel down the pipe to strike sample material. Based on the

upper frequency the tube diameter has been taken .for higher frequency analysis lower

diameter pipe and vice versa for lower frequency. Usually the pipe ranges from 29mm to 100

mm. and thus size of the sample is large enough to fill the cross section for accurate reading

.the main advantage of this method is to avoid fabricating of larger size sample and very easy

to find sound absorption co-efficient .the impedance tube method employs two technique to

sac

Movable microphone which is 1/3 octave frequencies technique (ASTM C 384) is based on

the standing wave ratio principle and uses an audio frequency and spectrometer is to measure

the absorption coefficients at various center frequencies of 1/3 octave bands[36].



Two fixed microphone impedance tube or transfer function method (ASTM E 1050), which is

relatively advanced one. In this technique, a broad band random signal is used as a sound

source. The normal incidence absorption coefficients and the impedance ratios of the test

materials can be measured much faster and easier compared with the first technique.

4.6.3. Steady State Method

This method is applicable when all other fails the ASTM E336-71 standard describe to

measure transmission co-efficient of the material Using three microphone and four

microphone can be used a pair of microphone which placed behind the test sample in a second

impedance tube.

5. CONCLUSIONS

This paper has reviewed and discussed various model that characterize and estimate the

various acoustic properties as follows, Different empirical model can predict the approximate

estimate values of acoustic property in that Delaney and Bazley model was accurate when

compare to other model fibrous porous material. The five physical parameter of Biot are

required to predict the acoustic properties of sound absorption at various frequency range. The

tortuosity, viscous and thermal characteristic length where exhibit the property at higher

frequency level whereas flow resistivity at low frequency. Porosity play important role at all

range of frequency .This

Biot parameters is mostly used in all porous material characterization, at present this is

more accurate. Direct impedance method can be used to measure sound absorption coefficient

using transfer function method which is simple and quicker method compared other

technique. The back work or reverse characterization can estimate other parameter like

tortuosity, viscous, thermal characteristic length, flow resistivity and porosity of obtained

sound absorption values from impedance tube technique.

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