Finite element based acoustic analysis of dissipative silencers with high temperature and thermal-induced heterogeneity
Introduction
The presence of high temperature and heterogeneous properties in dissipative silencers modifies their acoustic attenuation performance. These spatial variations can arise, for example, from uneven filling processes [1], [2], [3], [4], non-uniform mean flow fields [5] and thermal gradients [6], [7]. In the latter case, the difficulties associated with experimental measurements at high temperatures make it necessary to find computational approaches useful to evaluate the acoustic behaviour of the silencer.
Several theoretical models were developed to incorporate the influence of high temperature and thermal variations in ducts. Dong and Liu [8] presented a finite element approach for rectangular ducts including flow and temperature gradient. Prasad and Crocker [9] considered a wave equation with uniform mean flow in terms of velocity potential and obtained analytically the four-pole parameters for a straight pipe in the presence of a linear temperature gradient. Munjal and Prasad [10] noted that a temperature gradient would introduce a corresponding variation in the mean density and a gradient in the mean flow velocity and Mach number, and incorporated this influence in a plane wave propagation model for a uniform pipe. Sujith [11] presented a formulation for uniform ducts with arbitrarily large temperature gradients in the absence of flow, while Dokumaci [12] and Karthik et al. [13] extended the one-dimensional approach to include the presence of mean flow.
Significant temperature variations can be found along the exhaust system of internal combustion engines [14], [15], [16], [17], [18]. Concerning the acoustic performance of silencers, it is straightforward to account for the effects of uniform high temperature fields if only reactive elements are present [19], since attenuation curves at different temperatures overlap when a normalized frequency f/c is considered, c being the speed of sound at working temperature T [20]. Therefore, a single computation at a known temperature allows the calculation of the acoustic attenuation performance for a wide temperature range. The previous comments cannot be applied, in general, to dissipative silencers, since the acoustic properties of the absorbent material cannot be scaled in the same way [6], [7]. This indicates that the approach considered for reactive configurations cannot be used to describe sound propagation in the presence of a dissipative medium. In this case, a specific calculation is required for each temperature.
For reactive configurations, a number of works can be found where the influence of temperature and the associated gradients has been modelled and analysed. Kim et al. [21] presented a multidimensional analytical approach for the acoustic modelling of expansion chambers with mean flow and a temperature gradient. A segmentation technique was applied dividing the silencer into segments with constant temperature and mean flow, and matching the acoustic fields through the corresponding continuity conditions. The approach was extended by Kim and Choi [22] to circular reversing chambers with temperature variations and a stationary medium. Wang et al. [23] combined the segmentation procedure and the boundary element method (BEM) to compute the transmission loss of expansion chambers with uniform mean flow and a linear temperature gradient. Siano [24] presented some three-dimensional BEM results for perforated silencers with different uniform temperatures. It is worth noting that, for a continuously varying temperature field, a suitable version of the wave equation is required to account for the heterogeneous properties of the propagation medium [25], [26]. For reactive configurations, this wave equation provides a model for computing the sound attenuation, once the coordinate-dependent density and speed of sound have been evaluated from the ideal gas law [7], [27].
Few articles in the literature consider the effect of temperature on the performance of dissipative silencers. In principle, the equivalent bulk acoustic properties of fibres can be estimated by including the effect of temperature on the material resistivity [20]. This approach was experimentally validated by Christie [28], who predicted with reasonable agreement the characteristic impedance and propagation constant of mineral wool fibrous material at different temperatures from the combination of the flow resistivity measurements at those temperatures and a modified version of Delany and Bazley’s formulae [29], [30]. Williams et al. [31] have recently provided further experimental validation at high temperatures for additional fibrous materials such as basalt wool and E glass. From an acoustical point of view, a suitable material model can be obtained by using the results obtained at room conditions while updating the material resistivity to the actual working temperature. Concerning dissipative silencers, Ref. [6] presented a mode matching–based multidimensional analytical approach to assess thermal effects on the acoustic performance of circular dissipative reversing chamber silencers. A significant temperature influence was found on the sound attenuation characteristics. To account for the temperature-induced heterogeneity within the absorbent material, a segmentation procedure was considered with a number of dissipative regions with different but axially uniform temperature. The regression formulas (similar to Delany and Bazley expressions) for texturized fibre glass at room conditions [32] were extended to high temperature applications by including the thermal influence on the material resistivity while keeping the rest of coefficients and exponents constant. The same procedure was applied in Ref. [7], where a numerical approach based on the finite element method (FEM) was presented to analyze the effect of a continuously varying temperature field on the transmission loss of perforated dissipative silencers. A suitable version of the wave equation was required to account for heterogeneous density and speed of sound [3], [4], [25], [26]. In the previous approaches [6], [7], [31], the effect of the temperature on the acoustic properties of the absorbent material were accounted for by modifying the steady air-flow resistivity in the initial model of the material obtained at room temperature.
More general problems involving the simultaneous consideration of variable temperature and mean flow effects for perforated dissipative silencers containing an absorbent material require further research. The objective of the present work is to model and analyse the sound propagation in dissipative configurations including: (1) A central perforated passage carrying a non-uniform mean flow, and (2) high temperature and thermal variations in the central duct and the outer chamber. A mixed finite element approach is developed in the current investigation that couples a convective wave equation for inhomogeneous moving medium (in terms of an acoustic velocity potential) associated with the central duct and a pressure-based wave equation for heterogeneous stationary medium corresponding to the dissipative region. The influence of a number of parameters on the acoustic attenuation performance is investigated, including the effect of temperature, axial and radial thermal gradients and mean flow Mach number.
Section snippets
Mathematical approach
Fig. 1 shows the sketch of a dissipative silencer, which consists of a perforated central duct (subdomain Ωa) carrying a mean flow and an outer chamber (subdomain Ωm) with absorbent material. The corresponding boundary surfaces are denoted by Γa and Γm, respectively, the inlet and outlet sections are represented by Γi and Γo, and the perforated surface is Γp. The temperature field is assumed one-dimensional in the central passage, reaching its maximum value at the inlet while decreasing
Absorbent material. Spatial variations of the equivalent acoustic properties
Absorbent materials can be modelled as equivalent fluids [30] by using complex and frequency dependent values of speed of sound cm and density ρm (or, equivalently, the characteristic impedance Zm and wavenumber km). Empirical models such as the one proposed by Delany and Bazley [29] for rigid fibrous materials are commonly used to calculate cm and ρm in terms of the steady airflow resistivity R. Once the resistivity is known, the equivalent acoustic properties can be expressed in terms of a
Results
The problem under study consists of an axisymmetric configuration whose relevant dimensions are Lm=0.3 m, R1=0.0268 m and R2=0.091875 m (see Fig. 2). In addition, the values Li=Lo=0.1 m are used in the finite element discretizations to guarantee plane wave propagation conditions in the inlet/outlet sections [33]. Transmission loss computations have been carried out using 8-noded axisymmetric quadrilateral elements with quadratic interpolation, the FE meshes having an approximate element size of
Conclusions
A finite element model has been derived for the acoustic analysis of perforated dissipative silencers with high temperature and thermal gradients in the presence of mean flow. The spatial variations of the temperature field have been shown to generate heterogeneities in the mean flow as well as in the properties of the sound propagation media (air and absorbent material). For the central passage, an acoustic velocity potential-based wave equation has been considered, valid for non-uniform mean
Acknowledgements
Authors gratefully acknowledge the financial support of Ministerio de Economía y Competitividad and the European Regional Development Fund (projects DPI2010-15412 and TRA2013-45596-C2-1-R), Generalitat Valenciana (project Prometeo/2012/023) and Universitat Politècnica de València (PAID-05-12, project SP20120452).
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