Modelling of Porous Media and Systems

*Porous Fluids - Advances in Fluid Flow and Transport phenomena in Porous Media*

small cooling device with uni-directional porous copper, Proceedings of the 54th Japan Heat Transfer Symposium, C115

[52] Kazuhisa Yuki, Tomohiro Hara, Soichiro Ikezawa, Kentaro Anju, Koichi Suzuki, Tetsuro Ogushi, Takuya Ide, Masaaki Murakami, Immersion Cooling of Electronics utilizing Lotus-Type Porous Copper, Transactions of The Japan Institute of Electronics Packaging, 9 (2016), pp. E16-013-1-E16-013-7.

[53] K. Yuki, K. Takai, K. Anju, R. Kibushi, N. Unno, T. Ogushi, M.

of electronics by uni-directional porous heat sinks (CHF enhancement of saturated pool boiling toward immersion cooling of on-vehicle inverter), 2017 Annual Meeting of the Japan Society of Mechanical Engineers,

J0330103 (2017) (in Japanese).

204882 (2018).

no. 128, (2020).

[54] Yuki, K., Kibushi, R., Unno, N., Ide, T., Ogushi, T., Murakami, M., Patent publication number 2018-

[55] D. Negita, K. Yuki, N. Unno, R. Kibushi, T. Ide, T. Ogushi, M. Murakami, T. Numata, H. Nomura, Critical heat flux improvement of two-phase immersion cooling by controlling breathing phenomenon, Proceedings of the International Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Microsystems (InterPACK2020-Online), (2020).

[56] Daiki Suga, Kazuhisa Yuki, Risako Kibushi and Kio Takai, Visualization of two-phase flow in uni-directional porous copper, Proceedings of the the 31st International Symposium on Transport Phenomena (ISTP31), Paper

Murakami, T. Ide, Thermal management

(2017) (in Japanese).

Jun Li, Thermal contact resistance and thermal conductivity of a carbon nanofiber, Proceedings of 2005 ASME Summer Heat Transfer

[45] Hideo Nakajima, Fabrication, properties and application of porous metals with directional pores, Progress in Materials Science, 52, 7, pp. 1091-

[46] H. Chiba, T. Ogushi, H. Nakajima, Heat transfer capacity of lotus-type porous copper heat sink for air cooling, Journal of Thermal Science and

Technology (Journal of Thermal Science and Technology, 5, 2, pp. 222-237 (2010).

[47] K. Hokamoto, M. Vesenjak, Z. Ren, Fabrication of cylindrical unidirectional porous metal with explosive compaction, Materials Letters, 137, pp.

[48] K. Yuki, Y. Sato, R. Kibushi, N. Unno, K. Suzuki, T. Tomimura and K. Hokamoto, Heat transfer performance of porous copper pipe with uniformlydistributed holes fabricated by explosive welding technique, Proceedings of the 27th International Symposium on Transport Phenomena, Paper no.

[49] K. Takai, D. Suga, K. Yuki, R. Kibushi, N. Unno, K. Shimamoto, Development of Cooling Device utilizing, Uni-directional Porous Copper Manufactured by Metal 3D Printer Proceedings of the 56th Japan Heat Transfer Symposium, I114 (2019)

[50] Kio Takai, Kohei Yuki, Kazuhisa Yuki, Risako Kibushi, Noriyuki Unno, Teruya Tanaka, Heat transfer performance of an energy-saving heat removal device with uni-directional porous copper for divertor cooling, Fusion Engineering and Design, 136,

[51] K. Yuki, R. Tsuji, K. Takai, R. Kibushi, N. Unno, K. Suzuki, Development of a

Part A, pp. 518-521 (2018).

Conference (2005).

1173 (2007).

323-327 (2014).

126 (2016).

(in Japanese).

**104**

**Chapter 6**

Approach

*Mustapha Sadouki*

reflected wave, Darcy's regime

hospitals to minimize noise and reduce nuisance.

**1. Introduction**

**107**

**Abstract**

Inverse Measurement of the

Thickness and Flow Resistivity of

Low Frequency Waves-Frequency

A direct and inverse method is proposed for measuring the thickness and flow resistivity of a rigid air-saturated porous material using acoustic reflected waves at low frequency. The equivalent fluid model is considered. The interactions between the structure and the fluid are taken by the dynamic tortuosity of the medium introduced by Johnson et al. and the dynamic compressibility of the air introduced by Allard. A simplified expression of the reflection coefficient is obtained at very low frequencies domain (Darcy's regime). This expression depends only on the thickness and flow resistivity of the porous medium. The simulated reflected signal of the direct problem is obtained by the product of the experimental incident signal and the theoretical reflection coefficient. The inverse problem is solved numerically by minimizing between simulated and experimental reflected signals. The tests are carried out using two samples of polyurethane plastic foam with different thicknesses and resistivity. The inverted values of thickness and flow resistivity are compared with those obtained by conventional methods giving good results.

**Keywords:** acoustic characterization, porous materials, fluid equivalent model,

Porous materials are of great importance for a wide range of industrial and engineering applications, including transportation, construction, aerospace, biomedical and others. These materials, such as plastic foams, fibers and granular materials are frequently used for sound and heat insulation in buildings, schools and

The propagation of sound in a porous material is a phenomenon that governed by physical characteristics of a porous medium. Porous sound absorbers are materials in which sound propagation takes place in a network of interconnected pores such that the viscous and thermal interaction causes the dissipation of acoustic energy and converts it into heat. Knowledge of the acoustic and physical properties of these materials is of great importance in predicting their acoustic behavior and

Porous Materials via Reflected

### **Chapter 6**

Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected Low Frequency Waves-Frequency Approach

*Mustapha Sadouki*

### **Abstract**

A direct and inverse method is proposed for measuring the thickness and flow resistivity of a rigid air-saturated porous material using acoustic reflected waves at low frequency. The equivalent fluid model is considered. The interactions between the structure and the fluid are taken by the dynamic tortuosity of the medium introduced by Johnson et al. and the dynamic compressibility of the air introduced by Allard. A simplified expression of the reflection coefficient is obtained at very low frequencies domain (Darcy's regime). This expression depends only on the thickness and flow resistivity of the porous medium. The simulated reflected signal of the direct problem is obtained by the product of the experimental incident signal and the theoretical reflection coefficient. The inverse problem is solved numerically by minimizing between simulated and experimental reflected signals. The tests are carried out using two samples of polyurethane plastic foam with different thicknesses and resistivity. The inverted values of thickness and flow resistivity are compared with those obtained by conventional methods giving good results.

**Keywords:** acoustic characterization, porous materials, fluid equivalent model, reflected wave, Darcy's regime

#### **1. Introduction**

Porous materials are of great importance for a wide range of industrial and engineering applications, including transportation, construction, aerospace, biomedical and others. These materials, such as plastic foams, fibers and granular materials are frequently used for sound and heat insulation in buildings, schools and hospitals to minimize noise and reduce nuisance.

The propagation of sound in a porous material is a phenomenon that governed by physical characteristics of a porous medium. Porous sound absorbers are materials in which sound propagation takes place in a network of interconnected pores such that the viscous and thermal interaction causes the dissipation of acoustic energy and converts it into heat. Knowledge of the acoustic and physical properties of these materials is of great importance in predicting their acoustic behavior and

their insulate ability against noise and heat. For this reason, there are many works of research and studies in the literature [1–15] that are articulated in this line of inquiry where many mathematical and semi-phenomenological models have been developed to study the acoustic behavior of these materials. Among the most important of these models, we find the JCA model (Johnson-Champoux-Allard model) [1–4] used in the case of porous materials with a rigid structure saturated with air.

to the very low frequencies [11, 15], the air flow resistivity is the most important parameter describing the viscous losses caused by fluid/structure exchanges. In this case, the dynamic tortuosity α (ω) and the dynamic compressibility β (ω) are given

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected…*

*α ω*ð Þ¼ *σϕ jωρ*

Let us consider an acoustic wave arriving under normal incidence and striking a homogeneous porous material that occupies the region 0 ≤ x ≤ L (**Figure 1**). This wave generates an acoustic pressure field *p* and an acoustic velocity field *v* within the material that satisfies the following macroscopic equivalent fluid equations

*∂x* ,

The expression of a pressure field incident plane, unit amplitude, arriving at

In the medium (1) (x < 0), the movement's results from the superposition of

ð Þ¼ *x*,*ω e*

*β ω*ð Þ *Ka*

*<sup>j</sup>ω<sup>p</sup>* <sup>¼</sup> *<sup>∂</sup><sup>v</sup> ∂x*

, *k* and *c0* are, respectively, the wave number and the wave

*<sup>2</sup>* <sup>=</sup> �1, *<sup>ϕ</sup>* is the porosity, *<sup>σ</sup>* is the flow resistivity, *<sup>ρ</sup>* is

*β ω*ð Þ¼ *γ* (2)

�*j kx* ð Þ �*ω<sup>t</sup>* , (4)

(1)

(3)

by [2, 3, 13–15, 19]:

(along the x-axis):

where *<sup>k</sup>* <sup>¼</sup> *<sup>ω</sup>*

**Figure 1.** *Problem geometry.*

**109**

velocity of the free fluid.

incident and reflected waves,

*<sup>c</sup>*<sup>0</sup> ¼ *ω*

In these equations, *j*

*DOI: http://dx.doi.org/10.5772/intechopen.94860*

saturating fluid density and γ is the adiabatic constant.

*ρα ω*ð Þ*jω<sup>v</sup>* <sup>¼</sup> *<sup>∂</sup><sup>p</sup>*

where *Ka* is the compressibility modulus of the fluid.

*pi*

normal incidence to the porous material is given by

ffiffiffiffi *ρ*0 *Ka* q

According to the JCA model [3, 4], The acoustic propagation in air saturated porous materials is described by the inertial, viscous, and thermal interactions between the fluid and the structure [1–5]. In the high frequency domain [1–4] the inertial, viscous and thermal interactions are taken into account, by the high limit of tortuosity for the inertial effects [3], and by the viscous and thermal characteristic length [1, 2, 4] for the viscous and thermal effects. In the low-frequency domain [1, 2, 11, 13], inertial, viscous and thermal interactions are described by the inertial and thermal tortuosity and by the viscous and thermal permeability. In very low frequency approximation, the viscous-inertial interactions [11, 14, 15] are only described by the flow resistivity. The determination of these parameters is crucial for the prediction of sound damping in these materials.

The objective of this work is to propose an acoustic method based on the resolution of the direct and inverse problem using reflected acoustic waves at low frequency to determine the thickness and flow resistivity describing the porous medium. The direct problem consists in constructing theoretically the reflected signal knowing the incident signal and the parameters of the medium; given the experimental incident signal denoted by *pi exp*ð Þ *ω* , and the reflection coefficient which plays the role of a transfer function of the medium denoted *R*ð Þ *σ*, *L*, *ω* as a function of the parameters to be found, we deduce the simulated reflected signal *pr sim*ð Þ *<sup>σ</sup>*, *<sup>L</sup>*, *<sup>ω</sup>* which must be compared to the experimental reflected signal *pr exp*ð Þ *ω* . The inverse problem therefore consists in minimizing the difference between the *pr exp*ð Þ *<sup>ω</sup>* and *pr sim*ð Þ *σ*, *L*, *ω* signals by varying the required parameters. The solution corresponds to the sets of parameters that give the minimum deviation between the simulated reflected signal and the experimental reflected signal.

#### **2. Acoustical model**

The porous material is a bi-phasic medium consisting of a solid part and a fluid part that saturates the pores. When the solid part is flexible, the two phases start moving simultaneously under excitation by an acoustic wave; in this case the dynamics of the movement is well described by Biot's theory [16–18]. In the case of a rigid material, the solid part remains immobile and the acoustic waves propagate only in the fluid. This case is described by the equivalent fluid theory [1–5]. In this theory the viscous and inertial interactions within the medium are described by the dynamic tortuosity introduced by Johnson et al. [2, 3] while the thermal effects are taken into account by the dynamic compressibility of the fluid given by Allard and Champoux [1, 4]. In the frequency domain, these factors are multiplied by the density and compressibility of the fluid.

To differentiate between high and low frequency regimes [1–3], the viscous and thermal layer thicknesses *<sup>δ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*η=ωρ* <sup>p</sup> and *<sup>δ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>δ</sup><sup>=</sup>* ffiffiffiffiffi *Pr* <sup>p</sup> are compared, at a given frequency, with the effective radius of the pores r (ρ is the density of the saturation fluid, ω the pulsation frequency, *Pr* the Prandtl number, and η the viscosity of the fluid). The low frequency range is defined when the viscous [3] and thermal [1] skin thicknesses are great relatively to the pore radius. Otherwise, it is the highfrequency range. In the Darcy regime (flow without inertial effect), corresponding

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected… DOI: http://dx.doi.org/10.5772/intechopen.94860*

to the very low frequencies [11, 15], the air flow resistivity is the most important parameter describing the viscous losses caused by fluid/structure exchanges. In this case, the dynamic tortuosity α (ω) and the dynamic compressibility β (ω) are given by [2, 3, 13–15, 19]:

$$a(a) = \frac{\sigma \phi}{j a \rho} \tag{1}$$

$$
\beta(\alpha) = \gamma \tag{2}
$$

In these equations, *j <sup>2</sup>* <sup>=</sup> �1, *<sup>ϕ</sup>* is the porosity, *<sup>σ</sup>* is the flow resistivity, *<sup>ρ</sup>* is saturating fluid density and γ is the adiabatic constant.

Let us consider an acoustic wave arriving under normal incidence and striking a homogeneous porous material that occupies the region 0 ≤ x ≤ L (**Figure 1**). This wave generates an acoustic pressure field *p* and an acoustic velocity field *v* within the material that satisfies the following macroscopic equivalent fluid equations (along the x-axis):

$$
\rho a(\alpha) j a v = \frac{\partial p}{\partial x}, \frac{\beta(\alpha)}{K\_a} j a p = \frac{\partial v}{\partial x} \tag{3}
$$

where *Ka* is the compressibility modulus of the fluid.

The expression of a pressure field incident plane, unit amplitude, arriving at normal incidence to the porous material is given by

$$p^i(\mathfrak{x}, \omega) = \mathfrak{e}^{-j(k\mathfrak{x} - \alpha \mathfrak{t})},\tag{4}$$

where *<sup>k</sup>* <sup>¼</sup> *<sup>ω</sup> <sup>c</sup>*<sup>0</sup> ¼ *ω* ffiffiffiffi *ρ*0 *Ka* q , *k* and *c0* are, respectively, the wave number and the wave velocity of the free fluid.

In the medium (1) (x < 0), the movement's results from the superposition of incident and reflected waves,

**Figure 1.** *Problem geometry.*

their insulate ability against noise and heat. For this reason, there are many works of research and studies in the literature [1–15] that are articulated in this line of inquiry where many mathematical and semi-phenomenological models have been developed to study the acoustic behavior of these materials. Among the most important of these models, we find the JCA model (Johnson-Champoux-Allard model) [1–4] used in the case of porous materials with a rigid structure saturated with air. According to the JCA model [3, 4], The acoustic propagation in air saturated porous materials is described by the inertial, viscous, and thermal interactions between the fluid and the structure [1–5]. In the high frequency domain [1–4] the inertial, viscous and thermal interactions are taken into account, by the high limit of tortuosity for the inertial effects [3], and by the viscous and thermal characteristic length [1, 2, 4] for the viscous and thermal effects. In the low-frequency domain [1, 2, 11, 13], inertial, viscous and thermal interactions are described by the inertial and thermal tortuosity and by the viscous and thermal permeability. In very low frequency approximation, the viscous-inertial interactions [11, 14, 15] are only described by the flow resistivity. The determination of these parameters is crucial

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

The objective of this work is to propose an acoustic method based on the resolution of the direct and inverse problem using reflected acoustic waves at low frequency to determine the thickness and flow resistivity describing the porous medium. The direct problem consists in constructing theoretically the reflected signal knowing the incident signal and the parameters of the medium; given the

which plays the role of a transfer function of the medium denoted *R*ð Þ *σ*, *L*, *ω* as a function of the parameters to be found, we deduce the simulated reflected signal

The inverse problem therefore consists in minimizing the difference between the

corresponds to the sets of parameters that give the minimum deviation between the

The porous material is a bi-phasic medium consisting of a solid part and a fluid part that saturates the pores. When the solid part is flexible, the two phases start moving simultaneously under excitation by an acoustic wave; in this case the dynamics of the movement is well described by Biot's theory [16–18]. In the case of a rigid material, the solid part remains immobile and the acoustic waves propagate only in the fluid. This case is described by the equivalent fluid theory [1–5]. In this theory the viscous and inertial interactions within the medium are described by the dynamic tortuosity introduced by Johnson et al. [2, 3] while the thermal effects are taken into account by the dynamic compressibility of the fluid given by Allard and Champoux [1, 4]. In the frequency domain, these factors are multiplied by the

To differentiate between high and low frequency regimes [1–3], the viscous and

*Pr*

<sup>p</sup> are compared, at a given

<sup>2</sup>*η=ωρ* <sup>p</sup> and *<sup>δ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>δ</sup><sup>=</sup>* ffiffiffiffiffi

frequency, with the effective radius of the pores r (ρ is the density of the saturation fluid, ω the pulsation frequency, *Pr* the Prandtl number, and η the viscosity of the fluid). The low frequency range is defined when the viscous [3] and thermal [1] skin thicknesses are great relatively to the pore radius. Otherwise, it is the highfrequency range. In the Darcy regime (flow without inertial effect), corresponding

*sim*ð Þ *σ*, *L*, *ω* signals by varying the required parameters. The solution

*sim*ð Þ *<sup>σ</sup>*, *<sup>L</sup>*, *<sup>ω</sup>* which must be compared to the experimental reflected signal *pr*

simulated reflected signal and the experimental reflected signal.

*exp*ð Þ *ω* , and the reflection coefficient

*exp*ð Þ *ω* .

for the prediction of sound damping in these materials.

experimental incident signal denoted by *pi*

density and compressibility of the fluid.

thermal layer thicknesses *<sup>δ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

*pr*

*pr*

**108**

*exp*ð Þ *<sup>ω</sup>* and *pr*

**2. Acoustical model**

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

$$p\_1(\mathbf{x}, \alpha) = e^{-j(k\mathbf{x} - \alpha t)} + \tilde{R}e^{-j(-k\mathbf{x} - \alpha t)} \tag{5}$$

*<sup>R</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup> � *<sup>C</sup>*<sup>2</sup>

ffiffiffiffiffi *jω* p *cosh LC*<sup>2</sup>

2*C*<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94860*

very low frequencies (see appendix):

and the thickness L of the material.

The incident *p<sup>i</sup>*

**3. Inverse problem**

Wherein *pr*

signal and *p<sup>r</sup>*

**111**

reflection coefficient *R*:

where

<sup>1</sup>*<sup>ω</sup>* � � *sinh LC*<sup>2</sup>

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected…*

ffiffiffiffiffi *<sup>j</sup><sup>ω</sup>* � � <sup>p</sup> <sup>þ</sup> <sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup>

> ffiffiffiffiffiffiffiffi *γρϕ σ*

et *C*<sup>1</sup> ¼

By doing the Taylor series expansion of the reflection coefficient (Eq. (15)), limited to the first approximation, the reflection coefficient expression is written at

> *<sup>R</sup>* <sup>¼</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup> *Lσ*

This simplified expression of the reflection coefficient is independent of the frequency and porosity of the material, and depends only on the flow resistivity σ

*sim*ð Þ¼ *<sup>x</sup>*, *<sup>ω</sup> R pi*

*sim*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>F</sup>�<sup>1</sup> *R p<sup>i</sup>*

The simplified expression of the reflection coefficient obtained at low frequency (Eq.(17)) depends only on the flow resistivity σ and thickness *L* of the medium. Our objective is to find this two parameters simultaneously, supposedly unknown, by minimizing between the simulated reflected signal given by the expression (18) and the experimental reflected signal. The inverse problem then consists in finding the flow resistivity σ and thickness *L* of porous samples that minimize the function:

*exp*ð Þ� *<sup>ω</sup> pr*

signal. The minimization is made in frequency domain. The experimental setup [15] is shown in **Figure 2**. The tube length is adaptable to avoid reflection, and to permit the propagation of transient signals, according to the desired frequency range. For measurements in the frequency range (20–100) Hz, a length of 50 m is sufficient. The tube diameter is 5 cm (the cut-off of the tube *fc* � 4 kHz). A sound source Driver unit "Brand" constituted by loudspeaker Realistic 40–9000 is used. Tone-bursts are provided by Standard Research Systems Model DS345–30 MHz synthesized function generator. The signals are amplified and filtered using model

*sim σi*, *Li* ð Þ , *ω*

� �<sup>2</sup>

*sim σi*, *Li* ð Þ ,*ω* are the discrete sets values of the simulated reflected

*exp*ð Þ *x*, *ω* are the discrete sets of values of the experimental reflected

r

*C*<sup>1</sup> ¼

*pr*

The time-domain simulated reflected signals P*<sup>r</sup>*

P*r*

*<sup>U</sup>*ð Þ¼ *<sup>σ</sup>*, *<sup>L</sup>* <sup>X</sup>

*i*¼*N*

*pr*

*i*¼1

by taking the inverse Fourier transform F�<sup>1</sup> of (18),

ffiffiffiffiffi *jω* � � p

<sup>1</sup>*<sup>ω</sup>* � � *sinh LC*<sup>2</sup>

ffiffiffiffiffiffiffiffi *γσϕ Ka*

r

ffiffiffiffiffiffiffiffi *ρKa*

and reflected *p<sup>r</sup>* fields are related in the frequency domain by the

ffiffiffiffiffi *<sup>j</sup><sup>ω</sup>* � � <sup>p</sup> (15)

p (17)

ð Þ *x*, *ω* (18)

ð Þ *<sup>x</sup>*, *<sup>ω</sup>* � � (19)

*sim*ð Þ *x*, *t* are obtained numerically

(16)

(20)

where *R*~ is the reflection coefficient.

According to Eq. (3), the expression of the velocity field in the medium (1) is written:

$$w\_1(\mathbf{x}, \alpha) = \frac{1}{Z\_0} \left( e^{-j(k\mathbf{x} - \alpha t)} - \tilde{R} e^{-j(-k\mathbf{x} - \alpha t)} \right) \tag{6}$$

where *<sup>Z</sup>*<sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi *ρ*0*Ka* p is the characteristic impedance of air.

In the medium (2) corresponding to the porous material, the expressions of the pressure and velocity field are:

$$p\_2(\mathbf{x}, \alpha) = \tilde{A}e^{-j\left(\tilde{k}\mathbf{x} - \alpha t\right)} + \tilde{B}e^{-j\left(-\tilde{k}\mathbf{x} - \alpha t\right)} \tag{7}$$

$$w\_2(\mathbf{x}, \alpha) = \frac{1}{\tilde{Z}\_\varepsilon} \left( \tilde{A}e^{-j\left(\tilde{k}\mathbf{x} - \alpha t\right)} - \tilde{B}e^{-j\left(-\tilde{k}\mathbf{x} - \alpha t\right)} \right) \tag{8}$$

In these expressions *A*~ and *B*~ are amplitude constants of the right-going and left-going waves, *Z*~*<sup>c</sup>* and ~ *k* are the characteristic impedance and wave number, respectively, of the acoustic wave in the porous medium. These are two complex quantities:

$$\check{\boldsymbol{k}} = \boldsymbol{a} \sqrt{\frac{\bar{\boldsymbol{\rho}}}{\bar{\mathbf{K}}}} = \boldsymbol{a} \sqrt{\frac{\rho\_0 \boldsymbol{a}(\boldsymbol{\omega}) \boldsymbol{\beta}(\boldsymbol{\omega})}{\boldsymbol{K}\_a}}, \text{and } \check{\boldsymbol{Z}}\_c = \sqrt{\boldsymbol{\rho} \check{\boldsymbol{K}}} = \sqrt{\frac{\rho\_0 \boldsymbol{K}\_a \boldsymbol{a}(\boldsymbol{\omega})}{\boldsymbol{\beta}(\boldsymbol{\omega})}} \tag{9}$$

Finally, in the medium (3), the expressions of the pressure and velocity fields of the wave transmitted through the porous material are,

$$p\_3(\mathbf{x}, \alpha) = \tilde{T}e^{-j(k(\mathbf{x}-L)-\alpha\mathbf{t})},\tag{10}$$

$$v\_{\mathfrak{J}}(\mathfrak{x}, \mathfrak{o}) = \frac{1}{Z\_0} \tilde{T} e^{-j(k(\mathfrak{x} - L) - \mathfrak{a}\mathfrak{t})} \tag{11}$$

In these Eqs. ((10) and (11)) *T*~ is the transmission coefficient.

The continuity conditions of the pressure field and of the velocity field at the boundary of the medium are given by:

$$p\_1(\mathbf{0}^-, \alpha) = p\_2(\mathbf{0}^+, \alpha) p\_2(L^-, \alpha) = p\_3(L^+, \alpha) \tag{12}$$

$$\upsilon\_1(\mathbf{0}^-,\alpha) = \phi \upsilon\_2(\mathbf{0}^+,\alpha)\phi \upsilon\_2(L^-,\alpha) = \upsilon\_3(L^+,\alpha) \tag{13}$$

the � superscript denotes the limit from right and left, respectively. Using boundary and initial condition (12)–(13), reflected coefficient can be derived:

$$\bar{R}(\boldsymbol{\omega}) = \frac{\left(\boldsymbol{\phi}^2 - \bar{\boldsymbol{Z}}^2\right)\sinh\left(\boldsymbol{j\bar{k}L}\right)}{2\boldsymbol{\phi}\bar{\boldsymbol{Z}}\cosh\left(\boldsymbol{j\bar{k}L}\right) + \left(\boldsymbol{\phi}^2 + \bar{\boldsymbol{Z}}^2\right)\sinh\left(\boldsymbol{j\bar{k}L}\right)}\tag{14}$$

where *<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>Z</sup>*~*<sup>c</sup> <sup>Z</sup>*<sup>0</sup> ¼ ffiffiffiffiffiffiffi *α ω*ð Þ *β ω*ð Þ <sup>q</sup> is the normalized characteristic impedance of the material.

Using the expressions of the dynamic tortuosity *α ω*ð Þ and the dynamic compressibility *β ω*ð Þ given by Eq.(1), the expression (14) of the reflection coefficient becomes:

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected… DOI: http://dx.doi.org/10.5772/intechopen.94860*

$$R(\omega) = \frac{\left(\mathbf{1} - \mathbf{C}\_1^2 \alpha\right) \sinh\left(LC\_2\sqrt{j\alpha}\right)}{2\mathbf{C}\_1\sqrt{j\alpha}\cosh\left(LC\_2\sqrt{j\alpha}\right) + \left(\mathbf{1} + \mathbf{C}\_1^2 \alpha\right)\sinh\left(LC\_2\sqrt{j\alpha}\right)}\tag{15}$$

where

*p*1ð Þ¼ *x*,*ω e*

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

*<sup>v</sup>*1ð Þ¼ *<sup>x</sup>*,*<sup>ω</sup>* <sup>1</sup>

*p*2ð Þ¼ *x*, *ω Ae*

*Z*~*c Ae*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ρ*0*α ω*ð Þ*β ω*ð Þ *Ka*

*p*3ð Þ¼ *x*, *ω Te*

*Z*0 *Te*

The continuity conditions of the pressure field and of the velocity field at the

the � superscript denotes the limit from right and left, respectively. Using boundary and initial condition (12)–(13), reflected coefficient can be derived:

*<sup>ϕ</sup>*<sup>2</sup> � *<sup>Z</sup>*~<sup>2</sup> � �

*kL* � �

Using the expressions of the dynamic tortuosity *α ω*ð Þ and the dynamic compressibility *β ω*ð Þ given by Eq.(1), the expression (14) of the reflection

2*ϕZ cosh j* ~ ~

*p*<sup>1</sup> 0� ð Þ¼ , *ω p*<sup>2</sup> 0<sup>þ</sup> ð Þ , *ω p*<sup>2</sup> *L*� ð Þ¼ ,*ω p*<sup>3</sup> *L*<sup>þ</sup> ð Þ , *ω* (12)

*v*<sup>1</sup> 0� ð Þ¼ , *ω ϕv*<sup>2</sup> 0<sup>þ</sup> ð Þ , *ω ϕv*<sup>2</sup> *L*� ð Þ¼ ,*ω v*<sup>3</sup> *L*<sup>þ</sup> ð Þ , *ω* (13)

*sinh j*~ *kL* � �

> *sinh j*~ *kL*

is the normalized characteristic impedance of the material.

� � (14)

<sup>þ</sup> *<sup>ϕ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Z</sup>*~<sup>2</sup> � �

*<sup>v</sup>*3ð Þ¼ *<sup>x</sup>*,*<sup>ω</sup>* <sup>1</sup>

In these Eqs. ((10) and (11)) *T*~ is the transmission coefficient.

*<sup>v</sup>*2ð Þ¼ *<sup>x</sup>*,*<sup>ω</sup>* <sup>1</sup>

*Z*0 *e*

p is the characteristic impedance of air.

where *R*~ is the reflection coefficient.

*ρ*0*Ka*

ffiffiffiffi ~*ρ K*~ r

boundary of the medium are given by:

*<sup>R</sup>*~ð Þ¼ *<sup>ω</sup>*

ffiffiffiffiffiffiffi *α ω*ð Þ *β ω*ð Þ q

*<sup>Z</sup>*<sup>0</sup> ¼

where *<sup>Z</sup>*<sup>~</sup> <sup>¼</sup> *<sup>Z</sup>*~*<sup>c</sup>*

coefficient becomes:

**110**

¼ *ω*

s

the wave transmitted through the porous material are,

written:

where *<sup>Z</sup>*<sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

left-going waves, *Z*~*<sup>c</sup>* and ~

~ *k* ¼ *ω*

pressure and velocity field are:

�*j kx* ð Þ �*ω<sup>t</sup>* <sup>þ</sup> *Re*

According to Eq. (3), the expression of the velocity field in the medium (1) is

�*j kx* ð Þ �*ω<sup>t</sup>* � *Re* ~ �*j*ð Þ �*kx*�*ω<sup>t</sup>* � �

In the medium (2) corresponding to the porous material, the expressions of the

<sup>~</sup> �*<sup>j</sup>* <sup>~</sup>ð Þ *kx*�*ω<sup>t</sup>* <sup>þ</sup> *Be*

In these expressions *A*~ and *B*~ are amplitude constants of the right-going and

tively, of the acoustic wave in the porous medium. These are two complex quantities:

<sup>~</sup> �*<sup>j</sup>* <sup>~</sup>ð Þ *kx*�*ω<sup>t</sup>* � *Be* <sup>~</sup> �*<sup>j</sup>* �<sup>~</sup> ð Þ *kx*�*ω<sup>t</sup>* � �

, and *<sup>Z</sup>*~*<sup>c</sup>* <sup>¼</sup>

Finally, in the medium (3), the expressions of the pressure and velocity fields of

*k* are the characteristic impedance and wave number, respec-

ffiffiffiffiffiffi ~*ρK*~ q

¼

s

~ �*jkx* ð Þ ð Þ� �*<sup>L</sup> <sup>ω</sup><sup>t</sup>* , (10)

~ �*jkx* ð Þ ð Þ� �*<sup>L</sup> <sup>ω</sup><sup>t</sup>* (11)

~ �*j*ð Þ �*kx*�*ω<sup>t</sup>* (5)

~ �*<sup>j</sup>* �<sup>~</sup> ð Þ *kx*�*ω<sup>t</sup>* (7)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ρ*0*Kaα ω*ð Þ *β ω*ð Þ

(6)

(8)

(9)

$$\mathbf{C}\_{1} = \sqrt{\frac{\chi \rho \phi}{\sigma}} \text{ et } \mathbf{C}\_{1} = \sqrt{\frac{\chi \sigma \phi}{K\_{a}}} \tag{16}$$

By doing the Taylor series expansion of the reflection coefficient (Eq. (15)), limited to the first approximation, the reflection coefficient expression is written at very low frequencies (see appendix):

$$R = \frac{1}{1 + \frac{2}{L\sigma}\sqrt{\rho K\_a}}\tag{17}$$

This simplified expression of the reflection coefficient is independent of the frequency and porosity of the material, and depends only on the flow resistivity σ and the thickness L of the material.

The incident *p<sup>i</sup>* and reflected *p<sup>r</sup>* fields are related in the frequency domain by the reflection coefficient *R*:

$$p\_{sim}^r(\mathbf{x}, \boldsymbol{\alpha}) = \boldsymbol{R} \, p^i(\mathbf{x}, \boldsymbol{\alpha}) \tag{18}$$

The time-domain simulated reflected signals P*<sup>r</sup> sim*ð Þ *x*, *t* are obtained numerically by taking the inverse Fourier transform F�<sup>1</sup> of (18),

$$\mathcal{P}^{r}\_{sim}(\mathbf{x},t) = \mathcal{F}^{-1}(\mathbb{R}\,p^{i}(\mathbf{x},o))\tag{19}$$

#### **3. Inverse problem**

The simplified expression of the reflection coefficient obtained at low frequency (Eq.(17)) depends only on the flow resistivity σ and thickness *L* of the medium. Our objective is to find this two parameters simultaneously, supposedly unknown, by minimizing between the simulated reflected signal given by the expression (18) and the experimental reflected signal. The inverse problem then consists in finding the flow resistivity σ and thickness *L* of porous samples that minimize the function:

$$U(\sigma, L) = \sum\_{i=1}^{i=N} \left( p\_{exp}^r(\boldsymbol{\alpha}) - p\_{sim}^r(\sigma\_i, L\_i, \boldsymbol{\alpha}) \right)^2 \tag{20}$$

Wherein *pr sim σi*, *Li* ð Þ ,*ω* are the discrete sets values of the simulated reflected signal and *p<sup>r</sup> exp*ð Þ *x*, *ω* are the discrete sets of values of the experimental reflected signal. The minimization is made in frequency domain. The experimental setup [15] is shown in **Figure 2**. The tube length is adaptable to avoid reflection, and to permit the propagation of transient signals, according to the desired frequency range. For measurements in the frequency range (20–100) Hz, a length of 50 m is sufficient. The tube diameter is 5 cm (the cut-off of the tube *fc* � 4 kHz). A sound source Driver unit "Brand" constituted by loudspeaker Realistic 40–9000 is used. Tone-bursts are provided by Standard Research Systems Model DS345–30 MHz synthesized function generator. The signals are amplified and filtered using model

#### *Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

SR 650-Dual channel filter, Standford Research Systems. The signals (incident and reflected) are measured using the same microphone. The incident signal is measured by putting a total reflector [15] in the same position as the porous sample. **Figures 3**, **4** show the incident and reflected signals and their spectrum of the two samples in frequency bandwidth of 50 Hz.

The inverse problem is solved for two cylindrical polyurethane (PU) foams named (M1) and (M2) with a rigid frame and an open cell structure. Polyurethane foam is a leading member of the large and very diverse family of polymers or plastics and has many uses in the automotive sector and for the thermal insulation of buildings. The flow resistivity and thicknesses of the two samples M1 and M2 are measured by conventional methods [20, 21] and given in **Table 1**.

The inverse problem is to find the parametric vector *V* ¼ f g *σ*, *L* which satisfies

s) 27,500 � 500 7500 � 500

*The incident and reflected signals and their spectrum of the sample (M2) in frequency bandwidth of 50 Hz.*

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected…*

**Samples M1 M2** Thickness (cm) 2.6 � 0.5 5.0 � 0.5

> *U*ð Þ! *σ*, *L* 0 *LV* <sup>≤</sup>*<sup>V</sup>* <sup>≤</sup> *UV* (

*σ* ≥3000 *Nm*�<sup>4</sup>*s* <sup>0</sup>≤*L*≤<sup>10</sup> *cm* �

The inverse problem is solved by the last-square method. For its iterative solution, we used the simplex search method [22–26] which does not require numerical or analytic gradient. The flow resistivity and the thickness are inverted using experimental reflected signals by two PU porous material samples (M1 and M2). The variations in the cost function present one clear minimum corresponding to the solution of the inverse problem. **Figures 5**, **6** show the variation of the cost function U when varying the flow resistivity and the thickness in different

frequency bandwidths for the samples (M1, M2). The results of the inverse problem

where *LV* and *UV* are the lower and upper bounds that limit the research domain on the adjustable parametric vector V. For plastic foam samples, the value

of the flow resistivity is greater than 3000 Nm�<sup>4</sup>

*Flow resistivity and thickness of the sample M1 and M2.*

*DOI: http://dx.doi.org/10.5772/intechopen.94860*

Eq. (20) can be built from the following constraints:

(21)

(22)

s. The lower and upper limits in

the conditions:

Resistivity (Nm�<sup>4</sup>

**Figure 4.**

**Table 1.**

**113**

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected… DOI: http://dx.doi.org/10.5772/intechopen.94860*

**Figure 4.** *The incident and reflected signals and their spectrum of the sample (M2) in frequency bandwidth of 50 Hz.*


#### **Table 1.**

SR 650-Dual channel filter, Standford Research Systems. The signals (incident and

measured by putting a total reflector [15] in the same position as the porous sample. **Figures 3**, **4** show the incident and reflected signals and their spectrum of the two

The inverse problem is solved for two cylindrical polyurethane (PU) foams named (M1) and (M2) with a rigid frame and an open cell structure. Polyurethane foam is a leading member of the large and very diverse family of polymers or plastics and has many uses in the automotive sector and for the thermal insulation of buildings. The flow resistivity and thicknesses of the two samples M1 and M2 are

*The incident and reflected signals and their spectrum of the sample (M1) in frequency bandwidth of 50 Hz.*

reflected) are measured using the same microphone. The incident signal is

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

measured by conventional methods [20, 21] and given in **Table 1**.

samples in frequency bandwidth of 50 Hz.

**Figure 2.**

**Figure 3.**

**112**

*The experimental set up.*

*Flow resistivity and thickness of the sample M1 and M2.*

The inverse problem is to find the parametric vector *V* ¼ f g *σ*, *L* which satisfies the conditions:

$$\begin{cases} \ U(\sigma, L) \to 0 \\ LV \le V \le UV \end{cases} \tag{21}$$

where *LV* and *UV* are the lower and upper bounds that limit the research domain on the adjustable parametric vector V. For plastic foam samples, the value of the flow resistivity is greater than 3000 Nm�<sup>4</sup> s. The lower and upper limits in Eq. (20) can be built from the following constraints:

$$\begin{cases} \sigma \ge 3000 \text{ Nm}^{-4} \text{s} \\ 0 \le L \le 10 \text{ cm} \end{cases} \tag{22}$$

The inverse problem is solved by the last-square method. For its iterative solution, we used the simplex search method [22–26] which does not require numerical or analytic gradient. The flow resistivity and the thickness are inverted using experimental reflected signals by two PU porous material samples (M1 and M2). The variations in the cost function present one clear minimum corresponding to the solution of the inverse problem. **Figures 5**, **6** show the variation of the cost function U when varying the flow resistivity and the thickness in different frequency bandwidths for the samples (M1, M2). The results of the inverse problem

*Samples Frequency (Hz) Thickness (cm) Resistivity (Nm<sup>4</sup>*

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected…*

60 2.56 33,125 70 2.39 27,500

60 5.55 8500 70 4.88 8250

M1 50 2.52 28,750

*DOI: http://dx.doi.org/10.5772/intechopen.94860*

M2 50 4.75 7500

*Inverted parameters obtained of the flow resistivity and the thickness of the two samples M1 and M2.*

*Comparison between an experimental reflected signal and simulated reflected signal of the sample M1.*

*Comparison between an experimental reflected signal and simulated reflected signal of the sample M2.*

**Table 2.**

**Figure 7.**

**Figure 8.**

**115**

*s)*

*Variation of the cost function U when varying the flow resistivity and the thickness in different frequency bandwidths for the samples M1.*

#### **Figure 6.**

*Variation of the cost function U when varying the flow resistivity and the thickness in different frequency bandwidths for the samples M2.*

are summarized in **Table 2**, in which inverted values of flow resistivity and thickness are given for different frequency bandwidths. A comparison between an experimental reflected signal and simulated reflected signal is given in **Figures 7**, **8** for the optimized values of the inverted flow resistivity and thickness of the porous samples (M1, M2), respectively. The frequency bandwidth of the incident signals is (40–60) Hz. It can be seen that the agreement between experiment and theory is good for the two samples and the inverted values are close to those given by conventional methods.

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected… DOI: http://dx.doi.org/10.5772/intechopen.94860*


**Table 2.**

*Inverted parameters obtained of the flow resistivity and the thickness of the two samples M1 and M2.*

**Figure 7.**

*Comparison between an experimental reflected signal and simulated reflected signal of the sample M1.*

**Figure 8.** *Comparison between an experimental reflected signal and simulated reflected signal of the sample M2.*

are summarized in **Table 2**, in which inverted values of flow resistivity and thickness are given for different frequency bandwidths. A comparison between an experimental reflected signal and simulated reflected signal is given in **Figures 7**, **8** for the optimized values of the inverted flow resistivity and thickness of the porous samples (M1, M2), respectively. The frequency bandwidth of the incident signals is (40–60) Hz. It can be seen that the agreement between experiment and theory is good for the two samples and the inverted values are close to those given by

*Variation of the cost function U when varying the flow resistivity and the thickness in different frequency*

*Variation of the cost function U when varying the flow resistivity and the thickness in different frequency*

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

conventional methods.

*bandwidths for the samples M2.*

**Figure 5.**

**Figure 6.**

**114**

*bandwidths for the samples M1.*

#### **4. Conclusion**

Simultaneous determination of the flow resistivity and the thickness of a rigid porous medium are obtained by solving the inverse problem using experimental signals at very low frequencies. The model is based on a simplified expression of the reflection coefficient which is independent on frequency and porosity and depends only on the flow resistivity and thickness of the medium. Two plastic foam samples having different values of flow resistivity and different thickness are tested using this proposed method. The results are satisfactory and the inverted values of flow resistivity and thickness are close to those given by conventional methods. The advantage of the proposed method is that the two parameters, resistivity and thickness of the porous medium, were determined simultaneously without knowing previously any other parameter describing the porous medium, including its porosity. The suggested method opens new perspectives for the acoustic characterization of porous materials.

with,

and,

coth *LC*<sup>2</sup>

1 þ coth *LC*<sup>2</sup>

ffiffiffiffiffi *jω* � � <sup>p</sup> <sup>2</sup>*C*<sup>1</sup>

1

*<sup>R</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup>

is given by the first term

**Author details**

Mustapha Sadouki

Algeria

**117**

ffiffiffiffiffi *jω* � � p <sup>2</sup>*C*<sup>1</sup>

> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*C*<sup>1</sup> *LC*<sup>2</sup>

!

ffiffiffiffiffi *jω* p

ffiffiffi *<sup>j</sup><sup>ω</sup>* p <sup>1</sup>þ*C*<sup>2</sup> <sup>1</sup> *jω*

Using Eqs. (A.1), (A.2) and (A.6), one obtains

1 � 2

@

<sup>1</sup>*j<sup>ω</sup>* <sup>¼</sup> <sup>2</sup>*C*<sup>1</sup> *LC*<sup>2</sup> þ 2 3

> <sup>¼</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*C*<sup>1</sup> *LC*<sup>2</sup> � 2 3 *L*3 *C*3

<sup>3</sup> *LC*1*C*<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>3</sup>*C*<sup>1</sup>

*<sup>R</sup>* <sup>¼</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*C*<sup>1</sup> *LC*<sup>2</sup>

*<sup>C</sup>*1*C*2*<sup>L</sup>* <sup>1</sup> � <sup>3</sup>*C*<sup>2</sup>

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected…*

1 *L*2 *C*2 2

1 *L*2*C*<sup>2</sup> 2 � �

� �<sup>2</sup> *<sup>j</sup><sup>ω</sup>* <sup>þ</sup> <sup>O</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>2</sup> � �, (A.6)

*<sup>j</sup><sup>ω</sup>* <sup>þ</sup> <sup>O</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>2</sup> � �, (A.5)

1

p (A.8)

A (A.7)

!

<sup>2</sup>*C*<sup>1</sup> <sup>1</sup> � <sup>3</sup>*C*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>2</sup>*C*<sup>1</sup> *LC*<sup>2</sup>

*LC*<sup>2</sup> <sup>þ</sup> <sup>3</sup> <sup>3</sup>*C*<sup>2</sup>

� �

� � *<sup>j</sup><sup>ω</sup>* <sup>þ</sup> <sup>O</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>2</sup> � � <sup>0</sup>

<sup>1</sup> <sup>þ</sup> <sup>2</sup>*C*<sup>1</sup> *LC*<sup>2</sup>

Acoustics and Civil Engineering Laboratory, Matter Sciences Department, Faculty of Sciences and Technology, Khemis-Miliana University, BP. 44225 Ain Defla,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: mustapha.sadouki@univ-dbkm.dz

provided the original work is properly cited.

As a first approximation, at very low frequencies, the reflection coefficient (A.7)

<sup>¼</sup> <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup> *Lσ*

1 *L*2*C*<sup>2</sup> 2

> ffiffiffiffiffiffiffiffi *ρKa*

<sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94860*

#### **Acknowledgements**

This work is funded by the university training research project (PRFU) under number: B00L02UN440120200001 and by the General Direction of Scientific Research and Technological Development (DGRSDT).

#### **Conflict of interest**

The authors declare that they have no conflict of interest.

#### **Appendix: Taylor series expansion of the reflection coefficient**

The reflection coefficient given by Eq. (15) can be rewritten as [14, 15]:

$$R(\boldsymbol{\alpha}) = \left(\frac{1 - C\_{\text{J}}^{2}i\nu}{1 + C\_{\text{J}}^{2}i\nu}\right) \left(\frac{1}{1 + \coth\left(L C\_{2} \sqrt{j\boldsymbol{\alpha}}\right) \frac{2C\_{1} \sqrt{j\boldsymbol{\alpha}}}{1 + C\_{\text{J}}^{2}i\nu}}\right) \tag{A.1}$$

where *C*<sup>1</sup> and *c*<sup>2</sup> are given by Eq. (16). Taylor's limited serial expansion in the vicinity of zero of the expressions <sup>1</sup>�*C*<sup>2</sup> <sup>1</sup> *jω* <sup>1</sup>þ*C*<sup>2</sup> <sup>1</sup> *jω* � �, 2*C*<sup>1</sup> ffiffiffi *<sup>j</sup><sup>ω</sup>* p <sup>1</sup>þ*C*<sup>2</sup> <sup>1</sup> *<sup>j</sup><sup>ω</sup>* and coth *LC*<sup>2</sup> ffiffiffiffiffi *jω* � � p is given by:

$$\left(\frac{\mathbf{1} - \mathbf{C}\_{\mathbf{j}}^{2} j o}{\mathbf{1} + \mathbf{C}\_{\mathbf{j}}^{2} j o}\right) = \mathbf{1} - 2\mathbf{C}\_{\mathbf{j}}^{2} j o + \mathcal{O}\left((j o)^{2}\right),\tag{A.2}$$

$$\frac{2\mathcal{C}\_1\sqrt{j\rho}}{1+\mathcal{C}\_1^2 j\rho} = 2\mathcal{C}\_1\sqrt{j\rho} - 2\mathcal{C}\_1^3 (j\rho)^{3/2} + \mathcal{O}\left((j\rho)^{5/2}\right),\tag{A.3}$$

and,

$$\coth\left(LC\_2\sqrt{jao}\right) = \frac{1}{LC\_2\sqrt{jao}} + \frac{1}{3}LCjao - \frac{1}{45}L^3C\_2^3(jao)^{3/2} + \mathcal{O}\left(\left(jao\right)^{5/2}\right),\tag{A.4}$$

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected… DOI: http://dx.doi.org/10.5772/intechopen.94860*

with,

**4. Conclusion**

of porous materials.

**Acknowledgements**

**Conflict of interest**

Simultaneous determination of the flow resistivity and the thickness of a rigid porous medium are obtained by solving the inverse problem using experimental signals at very low frequencies. The model is based on a simplified expression of the reflection coefficient which is independent on frequency and porosity and depends only on the flow resistivity and thickness of the medium. Two plastic foam samples having different values of flow resistivity and different thickness are tested using this proposed method. The results are satisfactory and the inverted values of flow resistivity and thickness are close to those given by conventional methods. The advantage of the proposed method is that the two parameters, resistivity and thickness of the porous medium, were determined simultaneously without knowing previously any other parameter describing the porous medium, including its porosity. The suggested method opens new perspectives for the acoustic characterization

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

This work is funded by the university training research project (PRFU) under number: B00L02UN440120200001 and by the General Direction of Scientific

Research and Technological Development (DGRSDT).

*<sup>R</sup>*ð Þ¼ *<sup>ω</sup>* <sup>1</sup> � *<sup>C</sup>*<sup>2</sup>

vicinity of zero of the expressions <sup>1</sup>�*C*<sup>2</sup>

2*C*<sup>1</sup>

ffiffiffiffiffi *jω* � � <sup>p</sup> <sup>¼</sup> <sup>1</sup>

and,

**116**

coth *LC*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup>

The authors declare that they have no conflict of interest.

**Appendix: Taylor series expansion of the reflection coefficient**

<sup>1</sup>*jω*

<sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup> <sup>1</sup>*jω*

<sup>1</sup> � *<sup>C</sup>*<sup>2</sup> <sup>1</sup>*jω*

!

<sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup> <sup>1</sup>*jω*

<sup>1</sup>*j<sup>ω</sup>* <sup>¼</sup> <sup>2</sup>*C*<sup>1</sup>

ffiffiffiffiffi *<sup>j</sup><sup>ω</sup>* <sup>p</sup> <sup>þ</sup> 1 3

ffiffiffiffiffi *jω* p

*LC*<sup>2</sup>

!

The reflection coefficient given by Eq. (15) can be rewritten as [14, 15]:

0

B@

1

ffiffiffiffiffi *jω* � � p <sup>2</sup>*C*<sup>1</sup>

<sup>1</sup> *<sup>j</sup><sup>ω</sup>* and coth *LC*<sup>2</sup>

<sup>1</sup>*j<sup>ω</sup>* <sup>þ</sup> <sup>O</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>2</sup> � �

<sup>1</sup>ð Þ *<sup>j</sup><sup>ω</sup>* <sup>3</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>O</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>5</sup>*=*<sup>2</sup> � �

<sup>2</sup>ð Þ *<sup>j</sup><sup>ω</sup>* <sup>3</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>O</sup> ð Þ *<sup>j</sup><sup>ω</sup>* <sup>5</sup>*=*<sup>2</sup> � �

ffiffiffi *<sup>j</sup><sup>ω</sup>* p <sup>1</sup>þ*C*<sup>2</sup> <sup>1</sup> *jω*

1

ffiffiffiffiffi *jω* � � p is given by:

, (A.2)

, (A.3)

, (A.4)

CA (A.1)

1 þ coth *LC*<sup>2</sup>

where *C*<sup>1</sup> and *c*<sup>2</sup> are given by Eq. (16). Taylor's limited serial expansion in the

<sup>1</sup> *jω* <sup>1</sup>þ*C*<sup>2</sup> <sup>1</sup> *jω* � � , 2*C*<sup>1</sup> ffiffiffi *<sup>j</sup><sup>ω</sup>* p <sup>1</sup>þ*C*<sup>2</sup>

<sup>¼</sup> <sup>1</sup> � <sup>2</sup>*C*<sup>2</sup>

*LC*2*j<sup>ω</sup>* � <sup>1</sup>

45 *L*3 *C*3

ffiffiffiffiffi *<sup>j</sup><sup>ω</sup>* <sup>p</sup> � <sup>2</sup>*C*<sup>3</sup>

$$\coth\left(LC\_2\sqrt{j\omega}\right)\frac{2C\_1\sqrt{j\omega}}{1+C\_1^2j\omega} = \frac{2C\_1}{LC\_2} + \frac{2}{3}C\_1C\_2L\left(1-\frac{3C\_1^2}{L^2C\_2^2}\right)j\omega + \mathcal{O}\left(\left(j\omega\right)^2\right),\tag{A.5}$$

and,

$$\frac{1}{\mathbf{1} + \coth\left(LC\_2\sqrt{j\boldsymbol{\omega}}\right) \frac{2C\_1\sqrt{j\boldsymbol{\omega}}}{\mathbf{1} + \coth\left(LC\_2\sqrt{j\boldsymbol{\omega}}\right)}} = \frac{\mathbf{1}}{\mathbf{1} + \frac{2C\_1}{LC\_2}} - \frac{\frac{2}{3}L^3C\_2^3C\_1\left(\mathbf{1} - \frac{3C\_1^2}{L^2C\_2^2}\right)}{\left(\mathbf{1} + \frac{2C\_1}{LC\_2}\right)^2}j\boldsymbol{\omega} + \mathcal{O}\left(\left(j\boldsymbol{\omega}\right)^2\right), \quad \text{(A.6)}$$

Using Eqs. (A.1), (A.2) and (A.6), one obtains

$$R(\boldsymbol{\omega}) = \left(\frac{1}{1 + \frac{2C\_1}{LC\_2}}\right) \left(1 - \frac{\frac{2}{3}LC\_1C\_2\left(1 + \frac{3C\_1}{LC\_2} + 3\frac{3C\_1^2}{L^2C\_2^2}\right)}{\left(1 + \frac{2C\_1}{LC\_2}\right)} j\boldsymbol{\omega} + \mathcal{O}\left((j\boldsymbol{\omega})^2\right)\right) \tag{A.7}$$

As a first approximation, at very low frequencies, the reflection coefficient (A.7) is given by the first term

$$R = \frac{1}{1 + \frac{2C\_1}{LC\_2}} = \frac{1}{1 + \frac{2}{L\sigma}\sqrt{\rho K\_a}}\tag{A.8}$$

#### **Author details**

Mustapha Sadouki Acoustics and Civil Engineering Laboratory, Matter Sciences Department, Faculty of Sciences and Technology, Khemis-Miliana University, BP. 44225 Ain Defla, Algeria

\*Address all correspondence to: mustapha.sadouki@univ-dbkm.dz

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Allard. J. F, Propagation of Sound in Porous Media Modelling. Sound Absorbing Materials. Elsevier, London, UK, 1993, p. 1–284; DOI 10.1007/ 978-94-011-1866-8

[2] Lafarge D, Materials and Acoustics Handbook, edited by M. Bruneau and C. Potel. ISTE-Wiley, London, 2009, p. 149–202.

[3] Johnson D. L, Koplik J, and Dashen R, "Theory of dynamic permeability and tortuosity in fluidsaturated porous media," J. Fluids Mech, 1987; 176–379; https://doi.org/10.1017/ S0022112087000727

[4] Champoux Y, and Allard J. F, Dynamic tortuosity and bulk modulus in air-saturated porous media. J. Appl. Phys.1991; 70: 1975–1979; https://doi. org/10.1063/1.349482

[5] Zwikker C, and Kosten C. W, Sound Absorbing Materials. Elsevier, 1949, New York.

[6] Henry M, Lemarinier P, Allard J. F, Bonardet J. L, and Gedeon A, Evaluation of the characteristic dimension for porous sound absorbing materials. J. Appl. Phys, 1995;77: 17–20.

[7] Tizianel J, Allard J. F, Castagnéde B, Ayrault C, Henry M and Moussatov A, Transport parameters and sound propagation in an air-saturated sand. J. Appl. Phy. Am.1999; 86: 5829; https:// doi.org/10.1063/1.371599

[8] Attenborough K, On the acoustic slow wave in air-filled granular media. J. Acoust. Soc. Am. 1987; 81, 93; https:// doi.org/10.1121/1.394938

[9] Nagy P. B, Adler L, and Bonner B. P, Slow wave propagation in air-filled porous materials and natural rocks. Appl. Phys. Lett. 1990;56: 2504; https:// doi.org/10.1063/1.102872

[10] Delaney M, and Bazley E, Acoustical properties of fibrous absorbent materials, Appl. Acoust. 1970;3: 105– 116; https://doi.org/10.1016/0003-682X (70)90031-9

[17] Sadouki M, Fellah M, Fellah Z.E.A, and Ogam E, Depollier C. Ultrasonic propagation of reflected waves in cancellous bone: Application of Biot theory. ESUCB 2015, 6th European

*DOI: http://dx.doi.org/10.5772/intechopen.94860*

2018; **35:** 045005; https://doi.org/

Fellah Z E. A and Depollier C. Characterization of rigid porous medium via ultrasonic reflected waves at oblique incidence, Proc. Mtgs. Acoust. 2015; **25:** 045005; https://doi.

org/10.1121/2.0000201

[25] Sadouki M, Berbiche A, Fellah M,

[26] Sadouki M, Berbiche A, Fellah M,

Measurement of tortuosity and viscous characteristic length of double-layered porous absorbing materials with rigidframes via transmitted ultrasonic-wave, Proc. Mtgs. Acoust. 2015; **25**: 045001; https://doi.org/10.1121/2.0000184.

Fellah Z E. A and Depollier C,

10.1121/2.0000991.

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected…*

Characterization of Bone, 10–12 June 2015; Corfu, (Greece). p.1–4; DOI: 10.1109/ESUCB.2015.7169900

characterization of human cancellous bone via the first ultrasonic reflected wave – Application of Biot's theory, J. Ap.Acoust, 2020; 107237: 163; https:// doi.org/10.1016/j.apacoust.2020.107237

[19] Sadouki M, Direct problem for reflected wave at the first interface of a rigid porous medium in Darcy's regime. Proc. Mtgs. Acoust. 2015; **25**: 045004; https://doi.org/10.1121/2.0000200

[20] Leonard R. W, Simplified flow resistance measurements. J. Acoust. Soc. Am.,1946; 17: 240; https://doi.org/

[21] Stinson M. R and Daigle G. A, Electronic system for the measurement of flow resistance, J. Acoust. Soc. Am. 1988;83: 2422. https://doi.org/10.1121/

[22] Lagarias JC, Reeds JA, Wright MH, et al. Convergence properties of the Nelder-Mead Simplex method in low dimensions. SIAM J Optim 1998; 9: 112–147; https://doi.org/10.1137/

[23] Sadouk M, Thickness measurement of rigid porous material through reflected acoustic waves at Darcy's regime, Proc. Mtgs. Acoust. 2018;**35:** 045003; https://doi.org/10.1121/

measurement of tortuosity, viscous and thermal characteristic lengths of rigid

transmitted waves, Proc. Mtgs. Acoust.

[24] Sadouki M., Experimental

porous material via ultrasonic

10.1063/1.2099510

S1052623496303470

2.0000973.

**119**

1.396321

Symposium on Ultrasonic

[18] Sadouki M, Experimental

[11] Sadouki M, Fellah Z. E. A, Berbiche A, Fellah M, Mitri F. G, Ogam E, and Depollier C, Measuring static viscous permeability of porous absorbing materials, J. Acoust. Soc. Am, 2014; 135: 3163; https://doi.org/10.1121/1.4874600

[12] Sadouki M, Experimental characterization of rigid porous material via the first ultrasonic reflected waves at oblique incidence. J.ApAcoust.2018;133: 64–72; https://doi.org/10.1016/j.apac oust.2017.12.010

[13] Sadouki M, Experimental Measurement of the porosity and the viscous tortuosity of rigid porous material in low frequency. Journal of Low Frequency Noise Vibration and Active Control, 2018;37(2): 385–393; https://doi.org/10.1177% 2F1461348418756016

[14] Sadouki M, Theoretical modeling of acoustic propagation in an inhomogeneous porous medium. (in French),( Ph. D. Thesis), Université des sciences et de la technologie ' Houari Boumediène', Algiers, Algeria; 2014,

[15] Berbiche A, Sadouki M, and Fellah Z.E.A, Fellah M, Mitri F.G, Ogam E, Depollier C, Experimental determination of the viscous flow permeability of porous materials by measuring reflected low frequency acoustic waves. J. Appl. Phys. 2016; 119: 014906; https://doi.org/10.1063/ 1.4939073

[16] Biot M. A, The theory of propagation of elastic waves in a fluidsaturated porous solid, low frequency range. J. Acoust. Soc. Am, 1956; 28:168; https://doi.org/10.1121/1.1908241

*Inverse Measurement of the Thickness and Flow Resistivity of Porous Materials via Reflected… DOI: http://dx.doi.org/10.5772/intechopen.94860*

[17] Sadouki M, Fellah M, Fellah Z.E.A, and Ogam E, Depollier C. Ultrasonic propagation of reflected waves in cancellous bone: Application of Biot theory. ESUCB 2015, 6th European Symposium on Ultrasonic Characterization of Bone, 10–12 June 2015; Corfu, (Greece). p.1–4; DOI: 10.1109/ESUCB.2015.7169900

**References**

978-94-011-1866-8

S0022112087000727

org/10.1063/1.349482

New York.

p. 149–202.

[1] Allard. J. F, Propagation of Sound in Porous Media Modelling. Sound Absorbing Materials. Elsevier, London, UK, 1993, p. 1–284; DOI 10.1007/

[10] Delaney M, and Bazley E, Acoustical

[11] Sadouki M, Fellah Z. E. A, Berbiche A,

characterization of rigid porous material via the first ultrasonic reflected waves at oblique incidence. J.ApAcoust.2018;133: 64–72; https://doi.org/10.1016/j.apac

properties of fibrous absorbent materials, Appl. Acoust. 1970;3: 105– 116; https://doi.org/10.1016/0003-682X

Fellah M, Mitri F. G, Ogam E, and Depollier C, Measuring static viscous permeability of porous absorbing materials, J. Acoust. Soc. Am, 2014; 135: 3163; https://doi.org/10.1121/1.4874600

[12] Sadouki M, Experimental

[13] Sadouki M, Experimental

https://doi.org/10.1177% 2F1461348418756016

acoustic propagation in an

Measurement of the porosity and the viscous tortuosity of rigid porous material in low frequency. Journal of Low Frequency Noise Vibration and Active Control, 2018;37(2): 385–393;

[14] Sadouki M, Theoretical modeling of

inhomogeneous porous medium. (in French),( Ph. D. Thesis), Université des sciences et de la technologie ' Houari Boumediène', Algiers, Algeria; 2014,

[15] Berbiche A, Sadouki M, and Fellah Z.E.A, Fellah M, Mitri F.G, Ogam E, Depollier C, Experimental determination of the viscous flow permeability of porous materials by measuring reflected low frequency acoustic waves. J. Appl. Phys. 2016; 119: 014906; https://doi.org/10.1063/

[16] Biot M. A, The theory of

propagation of elastic waves in a fluidsaturated porous solid, low frequency range. J. Acoust. Soc. Am, 1956; 28:168; https://doi.org/10.1121/1.1908241

1.4939073

oust.2017.12.010

(70)90031-9

*Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media*

[2] Lafarge D, Materials and Acoustics Handbook, edited by M. Bruneau and C. Potel. ISTE-Wiley, London, 2009,

[3] Johnson D. L, Koplik J, and Dashen R, "Theory of dynamic permeability and tortuosity in fluidsaturated porous media," J. Fluids Mech, 1987; 176–379; https://doi.org/10.1017/

[4] Champoux Y, and Allard J. F,

Dynamic tortuosity and bulk modulus in air-saturated porous media. J. Appl. Phys.1991; 70: 1975–1979; https://doi.

[5] Zwikker C, and Kosten C. W, Sound Absorbing Materials. Elsevier, 1949,

[6] Henry M, Lemarinier P, Allard J. F, Bonardet J. L, and Gedeon A, Evaluation of the characteristic dimension for porous sound absorbing materials. J.

[7] Tizianel J, Allard J. F, Castagnéde B, Ayrault C, Henry M and Moussatov A, Transport parameters and sound propagation in an air-saturated sand. J. Appl. Phy. Am.1999; 86: 5829; https://

[8] Attenborough K, On the acoustic slow wave in air-filled granular media. J. Acoust. Soc. Am. 1987; 81, 93; https://

[9] Nagy P. B, Adler L, and Bonner B. P, Slow wave propagation in air-filled porous materials and natural rocks. Appl. Phys. Lett. 1990;56: 2504; https://

Appl. Phys, 1995;77: 17–20.

doi.org/10.1063/1.371599

doi.org/10.1121/1.394938

doi.org/10.1063/1.102872

**118**

[18] Sadouki M, Experimental characterization of human cancellous bone via the first ultrasonic reflected wave – Application of Biot's theory, J. Ap.Acoust, 2020; 107237: 163; https:// doi.org/10.1016/j.apacoust.2020.107237

[19] Sadouki M, Direct problem for reflected wave at the first interface of a rigid porous medium in Darcy's regime. Proc. Mtgs. Acoust. 2015; **25**: 045004; https://doi.org/10.1121/2.0000200

[20] Leonard R. W, Simplified flow resistance measurements. J. Acoust. Soc. Am.,1946; 17: 240; https://doi.org/ 10.1063/1.2099510

[21] Stinson M. R and Daigle G. A, Electronic system for the measurement of flow resistance, J. Acoust. Soc. Am. 1988;83: 2422. https://doi.org/10.1121/ 1.396321

[22] Lagarias JC, Reeds JA, Wright MH, et al. Convergence properties of the Nelder-Mead Simplex method in low dimensions. SIAM J Optim 1998; 9: 112–147; https://doi.org/10.1137/ S1052623496303470

[23] Sadouk M, Thickness measurement of rigid porous material through reflected acoustic waves at Darcy's regime, Proc. Mtgs. Acoust. 2018;**35:** 045003; https://doi.org/10.1121/ 2.0000973.

[24] Sadouki M., Experimental measurement of tortuosity, viscous and thermal characteristic lengths of rigid porous material via ultrasonic transmitted waves, Proc. Mtgs. Acoust.

2018; **35:** 045005; https://doi.org/ 10.1121/2.0000991.

[25] Sadouki M, Berbiche A, Fellah M, Fellah Z E. A and Depollier C. Characterization of rigid porous medium via ultrasonic reflected waves at oblique incidence, Proc. Mtgs. Acoust. 2015; **25:** 045005; https://doi. org/10.1121/2.0000201

[26] Sadouki M, Berbiche A, Fellah M, Fellah Z E. A and Depollier C, Measurement of tortuosity and viscous characteristic length of double-layered porous absorbing materials with rigidframes via transmitted ultrasonic-wave, Proc. Mtgs. Acoust. 2015; **25**: 045001; https://doi.org/10.1121/2.0000184.

### *Edited by Vallampati Ramachandra Prasad*

Written by authoritative experts in the field, this book discusses fluid flow and transport phenomena in porous media. Portions of the book are devoted to interpretations of experimental results in this area and directions for future research. It is a useful reference for applied mathematicians and engineers, especially those working in the area of porous media.

Published in London, UK © 2021 IntechOpen © Fernando Rico Mateu / iStock

Porous Fluids - Advances in Fluid Flow and Transport Phenomena in Porous Media

Porous Fluids

Advances in Fluid Flow and Transport

Phenomena in Porous Media

*Edited by Vallampati Ramachandra Prasad*