Acoustique, ondes, vibrations/Acoustics, waves, vibrations
Parametric study of the influence of compression on the acoustical absorption coefficient of automotive felts
Bernard CASTAGNÈDEa, Julian TIZIANELb, Alexei MOUSSATOVa, Achour AKNINEa, Bruno BROUARDa
aLaboratoire d’acoustique, UMR CNRS 6613, Université du Maine, avenue Olivier-Messiaen, 72085 Le Mans cedex 9, France
bPlâtres Lafarge, 500 rue Marcel-Demonque, zone du pôle technologique Agroparc, 84915 Avignon cedex 9, France
(Reçu le 5 septembre 2000, accepté après révision le 30 novembre 2000)
Abstract. The present note describes some work related to the modelization of acoustical absorption properties of compressed porous materials. An ‘equivalent fluid’ model with five parameters is used. This prediction technique is validated with some industrial materials being 1-D compressed along their thickness, starting from a felt mat having a given basic weight. The obtained results tend to indicate that an adequate prediction of the decrease in the coefficient of absorption is possible with this method. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
felt materials / equivalent fluid theory / acoustical absorption coefficient / influence of compression
Étude paramétrique de l’influence de la compression sur le coefficient acoustique d’absorption des matériaux de type feutres
Résumé. La présente note décrit un travail relatif à la modélisation des propriétés d’absorption acoustique de matériaux poreux qui sont comprimés. Un modèle standard de « fluide équivalent » à cinq paramètres est utilisé. Cette technique de prédiction est validée pour des matériaux industriels ayant subi une compression 1-D le long de l’épaisseur, à partir d’un matériau type feutre de grammage initial donné. Les résultats obtenus indiquent qu’il est possible de prédire la diminution du coefficient d’absorption avec cette méthode.2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
matériaux type feutres / théorie de fluide équivalent / coefficient acoustique d’absorption / influence de la compression
Version française abrégée
Une étude précédente [1] s’était concentrée sur la comparaison de prédictions numériques avec des mesures réalisées pour un matériau comprimé mécaniquement le long de son épaisseur (compression 1-D). Le paramètre central est donc le taux de compression notén. Le modèle de fluide équivalent de Allard–Johnson est décrit succinctement dans la deuxième section, notamment à partir des expressions de la masse volumique effective du fluideρ(ω), ainsi que du module de compressibilité effectifK(ω), Note présentée par Évariste SANCHEZ-PALENCIA.
fournis dans les équations (1) et (2) tirées de l’ouvrage de référence [2]. Dans ces expressions on retrouve cinq paramètres physiques fondamentaux du modèle qui sont la porositéφ, la résistivité σ, la tortuosité α∞et deux longueurs caractéristiquesΛetΛ liées aux effets visqueux et thermiques [3,4]. On obtient le coefficient d’absorptionA(ω) = 1− |R(ω)|2, à partir de celui de réflexionR(ω) = [z(ω)−1]/[z(ω) + 1], lui-même défini à l’aide de l’impédance de surface Z(ω) = [K(ω)ρ(ω)]1/2, oùZ(ω) =z(ω)Z0, avec Z0 impédance de l’air libre. Les équations (3) définissent l’ensemble de ces cinq paramètres, alors que les équations (4) à (6) estiment leurs variations en fonction du taux de compressionn. Certaines de ces relations prédictives sont empiriques en accord avec des résultats d’expérience ultrasonores [5,6] (exemple de la tortuosité), d’autres sont issues de calculs théoriques de Tarnow [7–9] (exemple de la résistivité).
Des mesures de ces paramètres sont réalisées de manière systématique pour plusieurs plaques de feutre de l’industrie automobile ayant subi des compressions allant jusqu’à un taux maximal de 4,25. Des recalages numériques sont réalisés sur le coefficient d’absorption des diverses plaques mesurées au tube de Kundt par la méthode T.M.T.C. [10] ou bien les mesures brutes des paramètres sont fournies sans recalage [11]. Des calculs prédictifs sont aussi effectués à partir des relations simplifiées (4) à (6). L’ensemble de ces données numériques sont rassemblées dans le tableau. Les prédictions issues de ces formules sont acceptables pour plusieurs paramètres (porosité et longueurs caractéristiques), alors qu’elles sont plutôt moyennes pour la tortuosité, voire médiocres pour la résistivité. Des explications sont données pour indiquer la difficulté d’application de ce modèle mono-couche à des matériaux industriels ayant subi un traitement de surface (température et pression) venant modifier localement certains paramètres. Une modélisation plus fine de ces phénomènes de surface devra considérer les matériaux comme des tri-couches.
1. Position of the problem
When modelling the acoustical properties of porous absorbing materials, one could wonder about the influence of compression of the porous network. It is well known that the absorption properties are decreasing with compression as it is occuring and readily observed for instance for thick snow layers.
In turn this drastic change is related to modifications of the porous network such as a decrease in porosity and similar trend for permeability. In a recent paper [1], some simple expressions have been proposed for these changes of the relevant parameters describing the absorption coefficient in the frame of the standard ‘equivalent fluid model’ [2], and numerical predictions were verified with fine experimental data obtained on controlled compressed mats of a fibrous material. In the present work, after briefly reviewing the theoretical tools of interest, we discuss and eventually modify some of the predictive formulae of our previous work on that topic. Some experiments are then described on some industrial materials studied during a recent research contract leaded by Renault company, and with other industrial partners. One of them, Rieter did provide some unique felt materials having controlled compression rate during their manufacturing. We deal here with 1-D compression effects across the thickness for unwoven fibrous materials having a constant basic weight (i.e., the mass per unit surface is kept unchanged), where the compression rate n is defined as being the ratio of the thickness before and after compression.
2. Predictions with the Johnson–Allard ‘equivalent fluid’ model
The general expressions for the effective fluid densityρ(ω)and the effective compressibility modulus of the fluidK(ω)are given by [2]:
ρ(ω) =α∞ρ0
1 + σφ jωρ0α∞
1 +4jα2∞ηρ0ω σ2Λ2φ2
(1)
K(ω) =
γP0/γ−(γ−1)
1 + σφ jB2ωρ0α∞
1 +4jα2∞ηρ0ωB2 σ2Λ2φ2
(2) In such expressionsσ= 8α∞η/φΛ2,γrepresents the adiabaticity constant (ratio of the specific heat at constant pressure onto the same quantity at fixed volume),P0the ambient pressure,B2the Prandtl number, φthe porosity,α∞the tortuosity,ω= 2πf the angular frequency,σthe resistivity,ρ0 the density of the fluid at rest,ηthe air viscosity,ΛandΛthe viscous and thermal characteristic lengths [3,4]. The complex
‘sound wavespeed’ inside the porous medium is then provided by the classical relationship which depends on frequency,c(ω) = [K(ω)/ρ(ω)]1/2. The characteristic acoustical impedance is defined by its usual form:Z(ω) = [K(ω)ρ(ω)]1/2. The reflection coefficient at normal incidenceR(ω)can be easily deducted from the non-dimensional normalized acoustical impedancez(ω) :R(ω) = [z(ω)−1]/[z(ω) + 1]. The absorption coefficientA(ω)is related to the reflection coefficientR(ω)as follows:A(ω) = 1− |R(ω)|2. This quantity is directly linked, through the various constitutive equations written above, to the 5 physical parameters (porosity, resistivity, tortuosity, and the two characteristic lengths) of the standard Johnson–
Allard ‘equivalent fluid’ model.
Then the prediction of the changes in terms of the 5 physical parameters occuring during a 1-D compression of a fibrous mat can be documented as follows. One starts from the definition of these parameters as given by the relationships:
φ= Vair
Vtotal
, σ=S∆p hQV
, α∞=
1 V
V u2dV
1 V
V udV2, Λ= 2
Vu2dV
Su2dS, Λ= 2
VdV
SdS = 2V S (3) In these expressions, the volume and surface integrals are calculated over an ‘average’ pore inside an homogeneization domain having a volumeV. The microscopic velocity of the air particles is noted byu, while ∆p represents a pressure difference between both sides of a porous layer having a thickness h, for which a QV air flow circulates through the surface S. The predictions for the changes occuring during compression of some of these parameters (tortuosity, resistivity, viscous characteristic length) is not trivial because they include in their definition the microscopic fluid velocities. Such dependance is eventually much easier to predict for the ‘geometrical’ parameters (porosity, thermal characteristic length).
These measurements done with ultrasonic techniques [5,6] did show that the tortuosity slightly increases, following a simple law of variation, as:
α(n)∞ = 1−n 1−α(0)∞
(4) An analogous equation exists for the porosity, that could easily be derived by some simple calculations, in the case of an unidirectional compression:
φ(n)= 1−n 1−φ(0)
(5) Moreover, one could also derive with some simplified assumptions the following relationships for the characteristic lengths and the resistivity:
Λ(n)=Λ(0)
n , Λ(n)=Λ(0)
n , σ(n)=n2σ(0) (6) The result for the thermal characteristic length Λ can be approximately derived (see [1] for some simplified geometrical configurations and distributions of the fibres). The formula dealing with the resistivityσis obtained from the result of the 2-D compression for which there exist a few theoretical calculations made by Tarnow [7–9]. In the case of a 1-D compression, the above results can then be generalized.
3. Experimental validation, parametric study and conclusions
A complete experimental validation has been done for a serie of felt materials, by measuring the coefficient of absorption with a research Kundt tube working with the T.M.T.C. (Two Microphones, Three Calibrations) method [10]. The measurements were performed on various materials, having different compression rates from 1 to 4, and having physical parameters that were independently measured with and without any fitting [11]. The comparison between the experimental data on the absorption coefficient and the corresponding predictions based on the Johnson–Allard ‘equivalent fluid’ model (see section 2) are presented on the figure. For further modelling of the compression effect, we have used the transformation equations (4) to (6) corresponding to a 1-D compression in order to make further numerical predictions.
The results exhibit an acceptable qualitative agreement between theoretical predictions and experimental data on some parameters.
The table collects the numerical data and the caption explains further the origin of each individual number. The last row results in applying the above transformation equations (4) to (6). The agreement is generally fair on some parameters (porosity and characteristic lengths). It is simply approximate for tortuosity and it is really poor for resistivity. For this last parameter the law of variation, i.e., equation (6a), with the influence of the square on the compression rate is certainely not rigorous as the simple prediction from Tarnow work tends to indicate it should instead be proportional ton. In fact this last assumption is not either fitting the data. These results prove that the crude modelling that is used here is not sufficient.
A close look to the tested materials show that the outer layers are slightly (and in some cases significantly) different from the inner core layer. This difference is due to some peculiar thermal treatment which is done
Figure. Experimental data compared to numerical simulations of the absorption coefficient for a felt material having different compression rate. On each curve the same notation applies that is: experimental data(), numerical predictions (—). The physical parameters are the following: (a)h0= 4.8mm;φ= 0.73;σ= 404000Nm−4s;
α∞= 1.333;Λ= 13.7µm;Λ= 41.1µm. (b)h0= 9.6mm;φ= 0.86;σ= 170000Nm−4s;α∞= 1.15;
Λ= 26.4µm;Λ= 79.2µm. (c)h0= 14.8mm;φ= 0.90;σ= 41600Nm−4s;α∞= 1.048;Λ= 42.9µm;
Λ= 128.7µm. (d)h0= 20.4mm;φ= 0.92;σ= 30000Nm−4s;α∞= 1.025;Λ= 52.2µm;Λ= 156.6µm.
Figure. Données expérimentales comparées aux prédictions numériques du coefficient d’absorption pour un matériau type feutre ayant différents taux de compression. Pour chaque courbe, les mêmes notations s’appliquent, à savoir:
données expérimentales(), prédictions numériques (—). Les paramètres physiques sont les suivants:
(a)h0= 4,8mm;φ= 0,73;σ= 404000Nm−4s;α∞= 1,333;Λ= 13,7µm;Λ= 41,1µm. (b)h0= 9,6mm;
φ= 0,86;σ= 170000Nm−4s;α∞= 1,15;Λ= 26,4µm;Λ= 79,2µm. (c)h0= 14,8mm;φ= 0,90;
σ= 41600Nm−4s;α∞= 1,048;Λ= 42,9µm;Λ= 128,7µm. (d)h0= 20,4mm;φ= 0,92;σ= 30000Nm−4s;
α∞= 1,025;Λ= 52,2µm;Λ= 156,6µm.
Table. Physical parameters versus the compression raten. The thermal characteristic length is taken as three times the viscous length, i.e.,Λ= 3Λ.
Tableau. Paramètres physiques en fonction du taux de compressionn. La longueur caractéristique thermique est prise égale à trois la longueur visqueuse, c.a.d.Λ= 3Λ.
Parameter Sample 1 Sample 2 Sample 3 Sample 4
Thickness (mm) 4.8±4% 9.6±4% 14.8±4% 20.4±4%
4.8 9.7 14.5 20.0
Porosityφ 0.73±4% 0.86±4% 0.90±5% 0.92±4%
0.77 0.86 0.90 0.92
0.66/0.73 0.83/0.83 0.89/0.89 0.92/0.92
Resistivityσ 404±6% 170±6% 41.6±7% 30.0±7%
(Nsm−4)×103 320 160 42.0 29.0
542/404 135/101 57.0/42.5 30.0/22.4 Tortuosityα∞ 1.33±14% 1.15±2% 1.05±1.4% 1.04±0.6%
1.30 1.15 1.06 1.04
1.17/1.33 1.09/1.17 1.06/1.11 1.04/1.08 Viscous lengthΛ(µm) 13.7±10.9% 26.4±8% 42.9±6.5% 52.2±6.5%
14 26 43 52
12/13.7 25/27 38/42 52.2/58
In each box, the numbers represent: For the first row, the experimental data including an estimated figure related to the precision of the measurements (there is an exception for the thermal characteristic length which is simply taken as three times the value of the viscous length). For the second row, the recovered parameter is numerically obtained in order to fit the experimental data with the help of the numerical predictions (see figure). For the third row, estimated values for the same parameters through the predictive equations (4) to (6) as described in the text. The first value refers to sample 4 taken as a reference, while the second value is related to sample 1. Further explanations and additional comments are included along the text.
Dans chaque case, les nombres représentent: Pour la première ligne, la valeur expérimentale incluant une estimation de la précision de la mesure (il y a une exception pour la longueur caractéristique thermique qui est simplement prise égale à trois fois la longueur visqueuse).
Pour la deuxième ligne, il s’agit du paramètre obtenu numériquement pour recaler les données expérimentales à l’aide des prédictions numériques (voir figure). Pour la troisième ligne, les valeurs des mêmes paramètres sont estimées à l’aide des équations de prédiction (4) à (6), telles que décrites dans le texte. La première valeur se réfère à l’échantillon 4 pris comme référence, alors que la seconde valeur correspond à l’échantillon 1. Des explications supplémentaires et des commentaires additionnels sont inclus dans le texte.
during the compression with the aim of providing smooth and neat surfaces. Accordingly, the local density of the surfaces might be higher, with different physical/acoustical parameters and even in some cases with chemical additives. The material is then finally no more homogeneous and should be modelled as a three layers system. Further work will then be needed (and already undertaken) in that area in order to model such tri-layer system.
In this short report, some simple heuristic formulae have been described in order to take into account some changes occuring due to the compression of the fibrous network on the basic physical parameters
describing the acoustical properties in terms of the standard ‘equivalent fluid’ model. A 1-D unidirectional compression is quite useful to describe the cycles of loading and unloading of fibrous materials occuring during their utilization. Some extra clues could also be deduced from a systematical use of simulation computer programs in order to model the influence of the microgeometry of the materials. These trends could also be efficiently used in order to optimize some preliminary design in terms of some peculiar applications.
Acknowledgments. The French Ministery of Research and Technology is gratefully acknowledged for authorizing the publication of the present work. We also want to thank Engineers at RENAULT and at RIETER for warm support during the course of this work, especially Eng. Franck Le Brazidec at RENAULT and Eng. Sylvain Simonnin at RIETER.
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