快速多極展開法於消音器性能分析之應用

Abstract

本研究主要目的為消音器傳輸損失之性能研究。邊界元素法相較於有限元素法更適合應用於消音器之性能分析，因為有限元素法必須分割其內部的領域。利用邊界元素法其係數矩陣為滿矩陣且常具有非對稱之性質。為了降低電腦運算時間之需求，本文將利用快速多極展開預測消音器之性能。此數值方法利用加法定理分離場點及源點為兩部分。因此，快速多極展開相較於邊界元素法可以降低電腦運算時間從 到 ，其中N為未知數之量且 為一常數。 四埠參數法廣泛應用於消音器性能之探討，僅能分析具有單一入出口型消音器為其限制。為了使研究更加完整，本文引入轉換函數法來分析具有單一出口及雙出口型消音器之傳輸損失。分析消音器性能之數值方法一般為有限元素法，邊界元素法或平面波法。平面波法是一方便且快速之數值方法，但其分析之有效頻率有限。此外，有限元素法特別耗用大量之運算時間及記憶體於三維問題或是高頻問題。因此，快速多極展開配合轉換函數法將被使用於本文預測二維及三維消音器之傳輸損失。 本文分析大量設計不相同之消音器:平行之單膨脹管型消音器，具有轉角入口之單膨脹管型消音器，槽孔型消音器及具有吸音材之消音器，豆莢型消音器等等。此外，也考量了不同之參數:入出口管半徑，膨脹管長度，槽孔管孔隙率及半徑。數值結果也與實驗及相關數值方法比較，提高了數值結果之正確性。 數值結果與實驗相較下進一步證實了轉換函數法可以成功應用於分析具有單一或是雙出口型消音器。於已公開發表之文獻中尚未發現轉換函數法被應用於分析具有雙出口型消音器之傳輸損失。值得一提的是轉換函數法亦可應用於預測具有多出口型之消音器。並聯型消音器及具有垂直轉角型之消音器其傳輸損失於特定的頻率表現優異。另外，可以根據波長及入口管半徑之調整，進而移動此優異性能出現之頻率範圍。豆莢型消音器利用內砍之吸音圓柱使聲音能量大量被消散，故常使用於排氣及散熱系統中，提升其傳輸損失之表現。此外，槽孔型消音器其孔隙率若大於20%，由數值結果可以發現槽孔管之特性將不明顯。

The purpose of this study is to investigate the transmission losses of mufflers. As a numerical scheme for analyzing a muffler, the boundary element method is more suitable than domain methods, such as the finite element method, which has to mesh the domain. However, the coefficient matrices established by the boundary element method are full and often non-symmetrical. In order to decrease the computational time required, the fast multipole method has been developed to predict attenuations of mufflers. This numerical scheme takes advantage of the addition theorem to separate the field points and source points into two terms. Therefore, the fast multipole method, when compared with the boundary element method, reduces CPU time from an order of to , where N is the number of unknowns and is a constant. While the four-pole parameter representation has been widely established in previous literatures, this technique was applied only to single-inlet/single-outlet mufflers in those studies. For a more comprehensive investigation, the transfer function is utilized to predict the transmission losses of mufflers with either double or single outlets in this work. Performance figures can mainly be calculated by different numerical simulations, such as the finite element method, the boundary element method and plane wave theory. The plane wave theory is a convenient and fast approach but it is limited in that it can only predict muffler attenuation below the cut-off frequency. In addition, finite element method, in particular, requires high computational and memory resources for 3D acoustic problems or at high frequencies. For these reasons, fast multipole method combined with transfer function method is adopted to analyze 2D and 3D muffler performance in the present work. The transmission losses of mufflers with different designs, such as parallel simple expansion chamber mufflers, simple expansion chamber mufflers with a right-angle inlet, perforated mufflers with or without absorbent material, pod mufflers, etc, are analyzed in this work. Parameters also taken into account in this work include: the radii of the inlet and outlet pipes, the length and radius of the simple expansion chamber, the flow resistivity of absorbent material, the porosity of perforated ducts, the perforated chamber radius, and so on. The numerical results are compared to experiments and previous published data, and the results clearly show that the agreements are good. It stands to reason that the transfer function can successfully predict the attenuation of mufflers with single or with double outlets. It is worth mentioning that this scheme has not been adopted in previous research to determine the transmission losses of mufflers with two outlets. Above all, the present method can also be extended to the study of single-inlet/multi-outlet mufflers. The parallel mufflers and mufflers with a right angle inlet in which without any absorbent material attached show a better performance at particular frequencies. The better attenuation can also be shifted according to the relations of the wavelength and the radius of inlet pipe. Pod muffler is also a prevalent setup in exhaust system or in heating, ventilation and air conditioning (HVAC) systems. Since absorbent pods or plates are embedded into a lined duct to make a split flow passage, thereby increasing transmission losses. From the numerical results one can also see the contribution of perforated pipe can be ignored when the porosity is chosen to be greater than 20%.