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This paper addresses the effect of high temperature on absorption performance of sandwich material coupled with microperforated panels (MPPs) in multiple configurations using a finite element model (FEM) over a frequency range from 10 to 3000 Hz. The structure is backed with a rigid wall which can either be Aluminium or Al-Alloy used in aeronautic or automobile. The wave propagation in porous media is addressed using Johnson Champoux Allard model (JCA). The FEM model developed using COMSOL Multiphysics software makes it possible to predict the acoustic absorption coefficient in multilayer microperforated panels (M-MPPs) and sandwich structure. It is shown that, when structures made by MPPs or sandwich materials are submitted to high temperature, the absorption performance of the structure is strongly modified in terms of amplitude and width of the bandgap. For application in sever environment (noise reduction in engines aircrafts), Temperature is one of the parameters that will most influence the absorption performance of the structure. However, for application in the temperature domain smaller than 50?C (automotive applications for example), the effect of temperature is not significant on absorption performance of the structure.

During the last decades, noise became one of the important issues of daily life. Indeed, noise is omnipresent, and particularly in built-up areas. It can come from road, rail or air transports for example [

A sandwich structure usually consists of three layers: two thin skin plates and a thick core. Solid foams are often used as core materials, and they are either closed-cell foams, in which the fluid is isolated in discrete pockets, or open-cell foams, in which the fluid is interconnected [

Sandwich and porous materials for sound transmission loss (STL) have attracted numerous interests and have many configurations. The simple sandwich panel is made of double walls with an air layer inside the structure. The STL was calculated for both infinite and finite sized double walls separated by air gaps using analytical modelling and statistical energy analysis [

To obtain lightweight structures with good absorption and insulation properties, combinations of MPPs and sandwich panels come into view of several researchers. An infinite MPP-solid plate coupling structure, both theoretically and experimentally was investigated [

Based on the MPP-plate coupling strategy, the goal of this present paper is to investigate the effect of high temperature on the SAC of structures that combines honeycomb sandwich panels with perforated faceplates and porous materials. Indeed, there are very few theoretical and experimental studies on the noise absorption performance of porous or sandwich materials under high temperature. Our interest is to analyse the effect of high temperatures on absorption coefficients of structures made by sandwich, MPPs or porous materials, therefore on noise reduction. Section 2 presents the mathematical models used to calculate SAC. Results and discussions are presented in section 4 and finally conclusions and prospects are given.

The numerical model is built on a commercial COMSOL Multiphysics software (version 5.5) using the built-in Pressure Acoustics and Plate modules [

According to simulations done on sandwich material [

thermal conductivity and viscosity, the Helmholtz equation governing the sound pressure in the Finite Element model (FEM) is given by:

∇ 2 p = 1 c 0 2 ∂ 2 p ∂ t 2 (1)

where p represents the sound pressure, t the time and c 0 the sound speed in air.

The Navier Stokes (Equation (2)), the mass continuity (Equation (3)) and the heat conduction equations (Equation (4)), assuming the fluid is a perfect gas, can be expressed by the following set of equations:

i ω ρ a v = ∇ ⋅ ( − P t I + η ( ∇ v + ( ∇ v ) T ) − 2 3 η ( ∇ ⋅ v ) I ) (2)

i ω ρ a ( P t P 0 − T T 0 ) + ρ a ∇ ⋅ v = 0 (3)

i ω ρ a c p T = − ∇ ⋅ ( − k T ∇ T ) + i ω P t (4)

where v is the fluid velocity, P t is the sound pressure of the thermal-acoustic field, I is the identity matrix, T is the temperature variation of the thermal-acoustic field, ρ a is the air density, ω is the angular frequency, P 0 is the ambient pressure of air, T 0 is the ambient temperature, c p is the specific heat of air at constant pressure, η is the dynamic viscosity of air and k T is the thermal conductivity of air.

The imposed pressure boundary representing the plane wave excitation incident to the MPP was employed. The sound hard representing the normal velocity vanishing at the side walls of the impedance tube and the end of the Back cavity were also employed. At the interface of the pressure acoustic field and MPP, the normal accelerations of the air and MPP are the same in the FE model, given as:

− n ⋅ ( − 1 ρ a ∇ p ) = − n ⋅ a n (5)

F p = p n (6)

where n is the surface normal direction, a n is the acceleration of the solid panel. F p is the total load of solid panel, which is decided by the normal sound pressure exerted on the panel.

While at the interface of the thermal acoustic field and pressure acoustic field, the continuous normal stress and acceleration and adiabatic conditions are applied in the FE model as:

( − P t I + η ( ∇ v + ( ∇ v ) T ) − 2 3 η ( ∇ ⋅ v ) I ) n = − p n (7)

− n ⋅ ( − 1 ρ a ∇ p ) = − n ⋅ i ω v (8)

− n ⋅ ( k T ∇ T ) = 0 (9)

As to the thermal acoustic field and solid panel coupling boundary, the velocity of the air is identical to that of the solid panel and the temperature variation is isothermal at the interface of the two fields in the FE model.

To calculate the absorption coefficient α, the two microphones method by Bodén et al. [_{1} and x_{2} are chosen and the pressure values are averaged by section at these two positions as p_{1} and p_{2}. Therefore, the absorption coefficient is calculated as:

α = 1 − | R | 2 (10)

With the reflection coefficient given by:

R = p 2 p 1 (11)

The tube is cylindrical with a circular cross-section and has a diameter of 100 mm, the first measuring point is placed at x_{1} = 273 mm from the stressed end by an acoustic pressure of 1 Pa, the other end, x_{2} delimits the depth of the air cavity after the sample and is placed at 50 mm from the first point of measure and from 100 mm from the first perforated panel. The walls of the tube are supposed to be rigid. In order to take into account, the dissipative phenomena within the porous domain, the poroacoustic domain is used for the JCA approach. The determination of the two-point pressures representing the positions of the microphones allows us to evaluate the reflection coefficient [

The mesh is built-in meshing tool by COMSOL [

The surface acoustic impedance of the n^{th} layer of the perforated plates in low-sound pressure level without mean flow is given as [

Γ p n = ρ a ε n 8 ω η ( 1 + t p n 2 a n ) + i ω ρ a ε n [ 8 η ω ( 1 + t p n 2 a n ) + t p n + δ n ] (12)

where ρ a is the air density, η is the dynamic viscosity of air, ω is the angular frequency and i = − 1 , t p n , a n , ε n = π a n 2 / b n 2 , δ n = 0.85 ( 2 a n ) ϕ n ( ε n ) are the thickness, hole radius, porosity and viscous boundary layer thickness of the n^{th} layer of the perforated plates respectively. b_{n} is the hole pitch of the n^{th} layer of the perforated plates and ϕ n ( ε n ) = 1 − 1.47 ε n + 0.47 ε n 3 .

The complex wave propagation constant γ_{an} and characteristic impedance z_{an} of airspaces can be found as:

γ a n = i k a (13)

z a n = ρ a c 0 (14)

where k a = ω / c 0 is the wave number of air and c_{a} is the sound speed of air.

If the n^{th} layer of airspaces is backed with a rigid wall, the surface acoustic impedance Γ a n of the n^{th} layer of airspaces with the thickness t p n can be expressed as [

Γ a n = − ρ a c 0 cot ( k a t p n ) (15)

For homogeneous and isotropic porous materials, the acoustic impedance can be predicted by the useful empirical relations [^{th} layer of porous materials, the empirical relations for the complex wave propagation constant γ m n and characteristic impedance z_{mn} can be expressed by the flow resistivity σ n as:

γ m n = k a { c 5 n ( f ρ a / σ n ) c 6 n + i [ 1 + c 7 n ( f ρ a / σ n ) c 8 n ] } (16)

z m n = ρ a c 0 { [ 1 + c 1 n ( f ρ a / σ n ) c 2 n ] − i [ c 3 n ( f ρ a / σ n ) c 4 n ] } (17)

where f is the sound frequency, c 1 n , c 2 n , ⋯ , c 8 n are the material constants for the n^{th} layer of porous materials. If the n^{th} layer of porous materials is backed with a rigid wall, the surface acoustic impedance Γ m n of the n^{th} layer of porous materials with the thickness t_{mn} can be expressed as [

Γ m n = z m n coth ( γ m n t m n ) (18)

The JCA model gives the expressions for the dynamic effective densities and bulk modulus of a porous material saturated by a fluid of density ρ a and bulk modulus K 0 considering a rigid frame. The poroelastic material is characterized by its porosity ε n , its tortuosity α ∞ , its flow resistivity η , static resistivity of air σ and the thermal and viscous characteristic lengths Λ and Λ 0 , respectively.

ρ p = α ∞ ρ a [ 1 − i σ ε n ω ρ a α ∞ 1 + i 4 α ∞ 2 η ρ a ω σ 2 Λ 2 ε n 2 ] (19)

K P = γ P 0 γ − ( γ − 1 ) [ 1 − i 8 η Λ ′ 2 P r ω ρ a 1 + i ρ a ω P r Λ ′ 2 16 η ] − 1 (20)

Material | Poroelastic core | Aluminium | Al-alloys | MPP1 | Air |
---|---|---|---|---|---|

Young’s modulus (E), [Pa] | 414 × 10^{3} | 70 × 10^{9} | 73.1 × 10^{9} | ||

Density ( ρ ) [kg/m^{3}] | 57 | 2700 | 2780 | - | - |

Thermal conductivity ( k t ) [W/(m.K)] | - | - | - | - | 0.026 |

Prandt Number (Pr) | - | - | - | - | 0.71 |

Air Density ( ρ a ) [kg/m^{3}] | - | - | - | - | 1.21 |

Ambient Pressure ( P 0 ) [Pa] | - | - | - | - | 101,325 |

Dynamic viscosity ( η ) [Pa.s] | - | - | - | - | 1.81 × 10^{−}^{5 } |

Speed of sound ( c 0 ) [m/s] | - | - | - | - | 343 |

Specific heat ( c p ) [J/(kg.K)] | - | - | - | - | 1004 |

Poisson ratios ( υ c ) [-] | 0.24 | 0.33 | - | - | - |

Frame loss factor ( η c ) [-] | 0.19 | 0.01 | - | - | - |

Statistic flow resistivity ( σ ) [N.s/m^{4}] | 55 × 10^{3} | 4.43 × 10^{3} | - | - | - |

Specific ratio ( γ ) | - | - | - | - | 1.4 |

Tortuosity ( α ∞ ) [-] | 1.05 | - | - | - | - |

Porosity ε n [-] | 0.95 | - | 0.022 | - | |

Viscous characteristic length ( Λ ) [μm] | 37 | - | - | - | - |

Thermal characteristic length ( Λ ′ ) [μm] | 120 | - | - | - | - |

Perforations radius a n [mm] | - | - | - | 0.5 | - |

MPP thickness t p n [mm] | - | - | - | 1 | - |

Skin thickness layer h [mm] | - | - | - | 0.2 | - |

Core/Air cavity H [mm] | - | - | - | 27 | - |

To make our sandwich panel with layers made of face sheets of MPPs, we needed one configuration of poroelastic core (PC) used as core material. The thickness of the skin of sandwich material (or sheet) is 0.2 mm and the rigid wall which can be Aluminum or Al-alloy backing the whole structure has thickness of 1 mm. Alloy materials in aircraft industries are special metal material. They are subjected to stress and strain excitation because their energy are consumed as a result of a magnetoelastic effect, grain boundary effect, and so on. And the loss factor of damping alloys is large for good damping properties [

First, a structure made by one, two, three and four MPPs with air cavities or porous materials were considered and the effect of temperature was analysed. Then four types of configuration made by sandwich materials are considered. They are described in

The parameters associated with the panel system are taken as those in previous studies of double wall sandwich panels lined with poroelastic material [

In this section, the effect of temperature is analysed on structures made by one, two, three and four MPPs with air cavities or porous materials and backed with rigid wall. Then, the cases of structures made by sandwich materials are considered.

The structure made by one MPP and one air cavity filled with porous material (poroelastic core) and backed with a rigid wall is now analyse in

With the above sound absorption coefficient, it can be noticed that the fluid velocity is affected by the temperature. When the temperature increases, the air viscosity increases too, which increases the flow resistance.

In this section, the acoustic absorption of structure made by two MPPs and two air cavities or porous materials and backed with a rigid wall are considered.

In

Now we consider the case of a structure made by three MPPs and three air cavities or porous materials and backed with a rigid wall.

In

the performances in terms of amplitudes and width of the spectrum of the absorption coefficient. The acoustic absorptions of each curve are similar with the nearly same amplitude in the range of the frequency < 1500 Hz. For frequencies > 1500 Hz, the amplitude of the absorption coefficient resulting from high temperature increases and those of the small temperature (between 20˚C to 45˚C) decreases. The amplitude of absorption coefficient goes from 0.72 for 20˚C to 0.94 for 960˚C, either a relative variation of +23.40%. In this case, the acoustic absorption performance is enhanced by increasing temperature. The increases of temperature and presence of the poroelastic core within the structure is therefore interesting for high frequencies (>1500 Hz).

The case of a structure made by four MPPs and four air cavities or porous materials and backed with a rigid wall is considered. As for the previous case of

In this section, structures made by sandwich materials (listed in

The structures in case C and D are firstly considered and the effect of temperature is analysed. In these two cases sandwich materials have MPP as upper and lower face filled with air.

We consider now the cases A and B. In

This study reveals the effect of temperature on the absorption coefficient. Two different approaches were used with MPPs and sandwich structures filled with air and poroelastic core using FE model. Increasing temperatures from 20˚C to 960˚C lead to the shift of the absorption coefficient to higher frequency and a decrease or an increase (depending on the configuration for MPPs or sandwich with air or poroelastic core) in amplitude of the absorption coefficient. Therefore, when structures made by MPPs or sandwich materials are submitted to high temperatures, the absorption performance of the structure is strongly modified in terms of amplitude, position of the peaks and width of the bandgap. For applications submitted in environment (greater than 500˚C), the temperature is one of the parameters that will most influence the absorption performance of the structure. However, for application in the temperature domain lesser than 50˚C, the effect of temperature is not significant on absorption performance of the structure. The highest absorption coefficient achieved by the structure is equal to 1; a level researched by constructors of automotive and aircraft engines. During the manufacturing of the structure, the varying characteristics of acoustic parameters with temperature and frequency must be considered.

The authors declare no conflicts of interest regarding the publication of this paper.

Bainamndi, D.J., Syriabe, E., Doka, S.Y. and Ntamack, G.E. (2020) High Temperature Effect on Absorption Coefficient of M-MPPs and Sandwich Structures Coupled with MPPs. Open Journal of Acoustics, 10, 1-18. https://doi.org/10.4236/oja.2020.101001