Prediction of the acoustic behavior of a parallel assembly of hollow cylinders
Introduction
The acoustic behavior (i.e. sound absorption and sound transmission) of rigid porous materials can be predicted following, mainly, three model types: empirical model [1], phenomenological or semi-phenomenological model [2] and microstructural model [3], [4], [5]. Unlike the first two model types, microstructural models have the advantage of linking the microstructure to the acoustic properties. Those models are based on the observation of the microstructure of porous materials to define an elementary unit cell. Then, finite element method [3] or semi-empirical approach [4] are used to obtain the parameters of an acoustic model to predict absorption coefficient and transmission loss. In particular, for cylindrical pores and slits, the link is well modeled [5]. However, the microstructural modeling of cylindrical pores is generally applicable to perforated solid and not a stack of cylinder. In this second case, there is not only one type of pore, but several types of pore in parallel, see Fig. 1. The main objective of this paper is to develop a model that can predict the acoustic behavior of a stack of hollow cylinders containing, in parallel, different pores. To do this, the approach of the PTMM is used to model this parallel assembly [6]. In the literature, Oldham worked on bamboo reeds stacking [7] (i.e. hollow cylinders) oriented along the sound propagation direction, as in Fig. 1. However, only sound absorption was measured and not simulated (i.e. it is not a prediction). Liu et al. [8] worked on fibrous material made of a parallel tubes array oriented normal to the sound propagation by presenting a micro–macro model and by offering a tool to maximize absorption coefficient for a given thickness. This paper distinguished in that the tubes are hollow, mounted compactly and oriented in the wave propagation direction. Kulpe et al. [9] worked on a numerical prediction of acoustic absorption on smooth and rough packed microtubes. They modeled the packing of non-uniform microtubes by a multi-scale homogenization technique, where the packing is seen as a homogeneous volume. In the present paper, the construction is rather seen as a parallel assembly of different types of pores. Additionally, it examines the effects of the geometrical parameters of the assembly, such as the type of stacking, and the dimensions of the straws. Finally, the model is applied to a real material made of a stack of switchgrass straw.
Section snippets
Parallel transfer matrix method
Parallel Transfer Matrix Method (PTMM) is a method to evaluate absorption coefficient and transmission loss of a parallel assembly expressed as a 2 by 2 transfer matrix [6]. It is based on pressure and flow balances upstream and downstream of the parallel assembly. Five key assumptions have to be respected: (1) only plane waves propagate upstream and downstream the assembly; (2) only normal incidence plane waves propagate in the assembly; (3) no-exchange exists between adjacent parallel
Materials and methods
In this work, two types of straw stackings are studied: (1) the stacking of uniform plastic straws see Fig. 1; (2) the stacking of non-uniform natural straws see Fig. 6. The plastic straws are simple drinking straws used to validate the proposed approach. They are regarded as perfect circular straws (i.e. constant and uniform geometry). Their characteristic dimensions are given in Table 3, and their length is 110 mm. The natural straws are made of switchgrass. To determine their external
Validation of the model on the stacking of plastic straws
This section deals with the circular plastic straws stacked into the acoustic tube as shown in Fig. 1. The stack is composed of 64 straws and generates the four types of cylindrical tubes discussed in Section 2: circular tube; concave triangular tube; concave square tube; concave–convex triangular tube. Their surface ratios, as determined by image analysis, are 72.46% (i.e. 64 straws), 4.71%, 10.78%, and 4.64% (i.e. 22 contacts with the boundary), respectively. The remaining 7.41% is the solid
Conclusion
In this paper, it was demonstrated that in a free assembly of straws, four types of cylindrical tubes in parallel exist: (a) circular; (b) concave square; (c) concave triangular and (d) concave–convex triangular. To predict the acoustic behaviors of this parallel assembly, PTMM was used. In that case, for each type of tube, an effective fluid was defined by the means of the JCA model which is well adapted for this type of cylindrical pore.
The approach was experimentally validated with
Acknowledgements
This work was supported by Natural Sciences and Engineering Research Council of Canada (N.S.E.R.C.).
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