Numerical assessment of two-chamber mufflers hybridized with multiple parallel perforated plug tubes using simulated annealing method:

Min-Chie Chiu

doi:10.1177/0263092317693477

Numerical assessment of two-chamber mufflers hybridized with multiple parallel perforated plug tubes using simulated annealing method:

Chiu, Min-Chie 2017-03-01 00:00:00 Enormous effort has been applied to research on mufflers hybridized with a single perforated plug tube; nonetheless, mufflers conjugated with multiple parallel perforated plug tubes that disperse venting fluid and reduce secondary noise have been overlooked. To this end, an analysis of the sound transmission loss of two-chamber mufflers with multiple parallel perforated plug tubes that are optimally designed to perform within a limited space will be presented. Here, using a decoupled numerical method, a four-pole system matrix for evaluating acoustic performance (sound transmission loss) is derived. During the optimization process, a simulated annealing method, which is a robust scheme utilized to search for the global optimum by imitating a physical annealing process, is used. Prior to dealing with a broadband noise, the sound transmission loss’s maximization relative to a one-tone noise (200 Hz) is produced to check the simulated annealing method’s reliability. The mathematical model is also confirmed for accuracy. To understand the acoustical effects brought about by the various tubes (perforated tubes, internally extended non-perforated tubes, and non- perforated tubes), mufflers with internally extended non-perforated tubes and non-perforated tubes have been evalu- ated. The optimization of three kinds of two-chamber mufflers hybridized with one, two, and four perforated plug tubes have also been compared. The results are revealing: the acoustical performance of mufflers conjugated with more perforated plug tubes decreases as a result of the decrement of the acoustical function for acoustical elements (II) and (III). Accordingly, in order to design a better muffler, an advanced presetting of the maximum (allowable) flowing velocity is necessary before an appropriate number of perforated plug tubes can be chosen for the optimization process. Keywords Multiple parallel perforated plug tube, two-chamber, simulated annealing Introduction Research on reactive muﬄers was initiated in 1054 by Davis et al. Studies of simple expansion muﬄers based on 2,3 plane wave theory were also completed. In order to increase the acoustical performance of lower frequency sound energy, Sullivan and Crocker introduced a muﬄer equipped with an internal perforated tube. Also, a series of theories and numerical techniques for decoupling the acoustical problems have been posited to solve the 5,6 7 coupled equations. Jayaraman and Yam, presuming an unreasonable inner and outer duct, used a method for ﬁnding an analytical solution. Additionally, Rao and Munjal provided a generalized decoupling method. Now, regarding the ﬂowing eﬀect, Peat, by ﬁnding the eigen value in transfer matrices, proposed a numerical decoupling method. However, still neglected was research work on muﬄers conjugated with multiple parallel perforated plug tubes, which disperse venting ﬂuid and reduce secondary ﬂowing noise. Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology, Taiwan, Republic of China Corresponding author: Min-Chie Chiu, Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology, Taiwan, Republic of China. Email: minchie.chiu@msa.hinet.net Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/ by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 4 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Therefore, to analyze the sound transmission loss (STL) of two-chamber muﬄers equipped with multiple parallel perforated plug tubes that are optimally designed to perform within a limited space, three kinds of two-chamber muﬄers linked together by multiple perforated plug tubes (muﬄer A: a two-chamber muﬄer intern- ally equipped with one perforated plug tube; muﬄer B: a two-chamber muﬄer internally equipped with two perforated and parallel plug tubes; muﬄer C: a two-chamber muﬄer internally equipped with four perforated and parallel plug tubes) are introduced. Because the geometric shapes of muﬄers A–C are complicated, the 10,11 numerical method such as the ﬁnite element method will be predictably time-consuming when doing a broad- band noise reduction assessment. In order to quickly and eﬃciently approach best design values, the numerical decoupling methods used in forming a four-pole matrix are in line with the simulated annealing method and presented in the article. Theoretical background Three kinds of two-chamber muﬄers connected with multiple perforated plug tubes have been adopted for noise elimination in the air compressor room shown in Figure 1. Before the acoustical ﬁelds of the muﬄers were analyzed, the acoustical elements had been identiﬁed. As shown in Figure 2, four kinds of muﬄer components, including four straight ducts, two sudden expanded ducts, two sudden contracted ducts, and multiple perforated plug ducts, are identiﬁed and marked as I, II, III, and IV. Additionally, the acoustical ﬁeld within the muﬄer is represented by 11 points. The outline dimension of the muﬄers is shown in Figure 3. As derived in previous 12–15 work and shown in Appendices 14, individual transfer matrices with respect to straight ducts, perforated plug ducts, and sudden expanded/contracted ducts are described below. Muffler A (a two-chamber muffler connected with one perforated plug tubes) 12–15 As derived in previous work, for the acoustical element (I) shown in Figure 2, the four-pole matrix between nodes 1 and 2 is p TS1 TS1 p 1,1 1,2 1 2 ¼ fðÞ L , D , M ð1Þ 1 i i i c u TS1 TS1 c u o o 1 2,1 2,2 o o 2 For the acoustical element (II), the four-pole matrix between nodes 2 and 3 is p TSE1 TSE1 p 2 1,1 1,2 3 ¼ ð2Þ c u TSE1 TSE1 c u o o 2 2,1 2,2 o o 3 Similarly, the four-pole matrix between nodes 3 and 4 is p TS2 TS2 p 3 1,1 1,2 4 ¼ fðÞ L , D , M ð3Þ 2 i i i TS2 TS2 c u c u o o 3 2,1 2,2 o o 4 Figure 1. Noise elimination on an air compressor room within a limited space. Chiu 5 Figure 2. Acoustical elements in three kinds of two-chamber mufflers hybridized with perforated plug tubes (mufflers A to C). Figure 3. The outline dimension of three kinds of two-chamber mufflers internally hybridized with multiple parallel and perforated plug tubes (mufflers A to C). 6 Journal of Low Frequency Noise, Vibration and Active Control 36(1) For the acoustical element (III), the four-pole matrix between nodes 4 and 5 is p TSC1 TSC1 p 4 1,1 1,2 5 ¼ ð4Þ c u TSC1 TSC1 c u o o 4 2,1 2,2 o o 5 12–15 As shown in previous work and derived in Appendices 1 and 2, for an acoustical element (IV), which is composed of two acoustical parts (one, an expanded perforated plug tube; and the other, a contracted perforated plug tube), the four-pole matrix between nodes 5 and 6a and nodes 6a and 7 is p TPPE1 TPPE1 p 5 1,1 1,2 6a ¼ ð5Þ c u TPPE1 TPPE1 c u o o 5 2,1 2,2 o o 6a p TPPC1 TPPC1 p 6a 1,1 1,2 7 ¼ ð6Þ c u TPPC1 TPPC1 c u o o 6a 2,1 2,2 o o 7 Combining equations (5)–(6), the four-pole matrix between nodes 5 and 7 yields p TPP1 TPP1 p 5 1,1 1,2 7 ¼ ð7Þ c u TPP1 TPP1 c u o o 5 2,1 2,2 o o 7 Likewise, the four-pole matrix between nodes 7 and 8 for a sudden expanded duct is p TSE2 TSE2 p 7 1,1 1,2 8 ¼ ð8Þ c u TSE2 TSE2 c u o o 7 2,1 2,2 o o 8 The four-pole matrix between nodes 8 and 9 in a straight duct is p TS3 TS3 p 1,1 1,2 8 9 ¼ fðÞ L , D , M ð9Þ 3 i i i c u TS3 TS3 c u o o 8 2,1 2,2 o o 9 The four-pole matrix between nodes 9 and 10 for a sudden contracted duct is p TSC2 TSC2 p 9 1,1 1,2 10 ¼ ð10Þ c u TSC2 TSC2 c u o o 9 2,1 2,2 o o 10 Moreover, the four-pole matrix between nodes 10 and 11 in a straight duct is p TS4 TS4 p 10 1,1 1,2 11 ¼ fðÞ L , D , M ð11Þ 4 i i i TS4 TS4 c u c u o o 10 2,1 2,2 o o 11 The total transfer matrix assembled by multiplication is ¼ fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M 1 i i i 2 i i i 3 i i i 4 i i i 5 i i i c u o o 1 TS1 TS1 TSE1 TSE1 TS2 TS2 1,1 1,2 1,1 1,2 1,1 1,2 TS1 TS1 TSE1 TSE1 TS2 TS2 2,1 2,2 2,1 2,2 2,1 2,2 TSC1 TSC1 TPP1 TPP1 TSE2 TSE2 1,1 1,2 1,1 1,2 1,1 1,2 ð12Þ TSC1 TSC1 TPP1 TPP1 TSE2 TSE2 2,1 2,2 2,1 2,2 2,1 2,2 TS3 TS3 TSC2 TSC2 1,1 1,2 1,1 1,2 TS3 TS3 TSC2 TSC2 2,1 2,2 2,1 2,2 TS4 TS4 p 1,1 1,2 11 TS4 TS4 c u 2,1 2,2 o o 11 Chiu 7 A simpliﬁed form is expressed in a matrix as p T T p 1 11 11 12 ¼ ð13Þ c u T T c u o o 1 o o 11 21 22 Muffler B (a two-chamber muffler connected with two perforated plug tubes) Similarly, the acoustical four-pole matrix between nodes 12, nodes 23, nodes 34, nodes 89, nodes 910, and nodes 1011 is the same as equations (1)–(3) and equations (7)–(11). As derived in Appendix 3, for two parallel perforated plug tubes connected to two chambers, the four-pole matrices between nodes 5 7 and nodes (1) (1) 5 7 can be combined to an equivalent matrix (2) (2) "# p TPP1 TPP1 p 5 1,1 1,2 7 ¼ ð14Þ c u 2 TPP1 TPP1 c u o o 5 o o 7 2,1 2,2 where [TPP1 ] is the four-pole matrix for a single perforated plug tube. i,j The equivalent acoustical ﬁeld of muﬄer B is shown in Figure 4. Consequently, the total transfer matrix assembled by multiplication is ¼ fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M 1 i i i 2 i i i 3 i i i 4 i i i 5 i i i c u o o 1 TS1 TS1 TSE1 TSE1 TS2 TS2 1,1 1,2 1,1 1,2 1,1 1,2 TS1 TS1 TSE1 TSE1 TS2 TS2 2,1 2,2 2,1 2,2 2,1 2,2 "# TSC1 TSC1 TPP1 TPP1 1,1 1,2 1,1 1,2 ð15Þ TSC1 TSC1 2 TPP1 TPP1 2,1 2,2 2,1 2,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 1,1 1,2 1,1 1,2 1,1 1,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 2,1 2,2 2,1 2,2 2,1 2,2 TS4 TS4 p 1,1 1,2 11 TS4 TS4 c u 2,1 2,2 o o 11 A simpliﬁed form is expressed in a matrix as T T p p 1 11 11 12 ¼ ð16Þ c u T T c u o o 1 o o 11 21 22 Figure 4. Equivalent acoustical field for a muffler hybridized with perforated plug tubes (mufflers B and C). 8 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Muffler C (a two-chamber muffler connected with four perforated plug tubes) The acoustical four-pole matrix between nodes 1–2, nodes 2–3, nodes 3–4, nodes 8–9, nodes 9–10, and nodes 10–11 is the same as equations (1)–(3) and equations (9)–(11). As derived in Appendix 4, for a two chambers muﬄer connected with four parallel perforated plug tubes, the four-pole matrices between nodes 5 –7 ,5 –7 ,5 – (1) (1) (2) (2) (3) 7 , and 5 –7 nodes can be combined to an equivalent matrix (3) (4) (4) "# p p TPP1 TPP1 5 1,1 1,2 7 ¼ ð17Þ c u c u 4 TPP1 TPP1 o o 5 2,1 2,2 o o 7 where [TPP1 ] is the four-pole matrix for a single perforated tube. i,j The related equivalent acoustical ﬁeld of muﬄer C is also presented and shown in Figure 4. Consequently, the total transfer matrix assembled by multiplication is ¼ fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M 1 i i i 2 i i i 3 i i i 4 i i i 5 i i i c u o o 1 TS1 TS1 TSE1 TSE1 TS2 TS2 1,1 1,2 1,1 1,2 1,1 1,2 TS1 TS1 TSE1 TSE1 TS2 TS2 2,1 2,2 2,1 2,2 2,1 2,2 "# TSC1 TSC1 TPP1 TPP1 1,1 1,2 1,1 1,2 ð18Þ TSC1 TSC1 4 TPP1 TPP1 2,1 2,2 2,1 2,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 1,1 1,2 1,1 1,2 1,1 1,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 2,1 2,2 2,1 2,2 2,1 2,2 TS4 TS4 p 1,1 1,2 11 TS4 TS4 c u 2,1 2,2 o o 11 A simpliﬁed form is expressed in a matrix as p T T p 1 11 11 12 ¼ ð19Þ c u T T c u o o 1 o o 11 21 22 Overall sound power level The STL of muﬄers A–C are deﬁned as T þ T þ T þ T 11 12 21 22 ð20aÞ STLðÞ Q, f, RT , RT , RT , RT , RT , RT¼ 20 log þ 10 log 1 1 2 3 4 5 6 2 S T þ T þ T þ T 11 12 21 22 ð20bÞ STLðÞ Q, f, RT , RT , RT , RT , RT , RT¼ 20 log þ 10 log 2 1 2 3 4 5 6 2 S T þ T þ T þ T 11 12 21 22 STLðÞ Q, f, RT , RT , RT , RT , RT , RT¼ 20 log þ 10 log ð20cÞ 3 1 2 3 4 5 6 2 S where RT1 ¼ D2; RT2 ¼ D3=D2; RT3 ¼ D4; ð20dÞ RT4 ¼ L3; RT5 ¼ dH; RT6 ¼ Chiu 9 The silenced octave sound power level emitted from a muﬄer’s outlet is SWL ¼ SWLOðÞ f STLðÞ f ð21Þ i i i where SWLOð f Þ is the original SWL at the inlet of a muﬄer (or pipe outlet), and f is the relative octave band i i frequency; STLð f Þ is the muﬄer’s STL with respect to the relative octave band frequency (f ); SWL is the silenced i i i SWL at the outlet of a muﬄer with respect to the relative octave band frequency. Finally, the overall SWL silenced by a muﬄer at the outlet is () SWL =10 SWL ¼ 10 log 10 TK i¼1 ð22Þ no ½SWLOðÞ f ½SWLOðÞ f ½SWLOðÞ f ½SWLOðÞ f 1 2 3 n ¼ 10 log STL ðÞ f =10 STL ðÞ f =10 STL ðÞ f =10 STL ðÞ f =10 K 1 K 2 K 3 K n 10 þ 10 þ 10 þ þ 10 Objective function STL maximization for a tone (f) noise. The objective functions in maximizing the STL at a pure tone (f) are OBJ ¼ STLðÞ Q, f, RT , RT , RT , RT , RT , RT ð23aÞ 11 1 1 2 3 4 5 6 OBJ ¼ STLðÞ Q, f, RT , RT , RT , RT , RT , RT ð23bÞ 12 2 1 2 3 4 5 6 OBJ ¼ STLðÞ Q, f, RT , RT , RT , RT , RT , RT ð23cÞ 13 3 1 2 3 4 5 6 The related ranges of the parameters are Q ¼ 0:01 m =s ; L ¼ 1:2mðÞ; D ¼ 0:6mðÞ; o o L ¼ 0:05ðÞ m ; L ¼ 0:05ðÞ m ; D ¼ 0:2mðÞ; 1 5 1 RT1 :½ 0:1, 0:25 ; RT2 :½ 0:3, 0:7 ; ð23dÞ RT3 :½ 0:1, 0:5 ; RT4 :½ 0:3, 0:5 ; RT5 :½ 0:00175, 0:007 ; RT6 :½ 0:01, 0:3 SWL minimization for a broadband noise. To minimize the overall SWL , the objective function is OBJ ¼ SWLðÞ Q, RT , RT , RT , RT , RT , RT ð24aÞ 21 T1 1 2 3 4 5 6 OBJ ¼ SWL ðQ, RT , RT , RT , RT , RT , RTÞð24bÞ 22 T2 1 2 3 4 5 6 OBJ ¼ SWL ðQ, RT , RT , RT , RT , RT , RTÞð24cÞ 23 T3 1 2 3 4 5 6 Model check Before performing the simulated annealing (SA) optimal simulation on muﬄers, an accuracy check of the mathem- atical model on a one-chamber muﬄer with one perforated plug tube is performed by Sullivan and Crocker. As indicated in Figure 5, comparisons between theoretical data and experimental data are in agreement. Therefore, the model of two muﬄers connected with multiple perforated plug tubes is adopted in the following optimization process. Case studies The noise reduction of an air compressor room within a space-constrained room is exempliﬁed and shown in Figure 1. The sound power level (SWL) inside the air compressor’s outlet is shown in Table 1 where the overall SWL reaches 107.4 dB. 10 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Figure 5. Performance of an acoustical element of a one perforated plug tube with the mean flow (M ¼ M ¼ 0.05, D ¼ 0.0493 (m), 1 2 1 Do ¼ 0.1016 (m), L ¼ L ¼ 0.1286 (m), L ¼ L ¼ 0.1 (m), L ¼ L ¼ 0.0 (m), t ¼ 0.081 (m), dh ¼ dh ¼ 0.00249 (m), C1 C2 1 2 A1 B2 1 2 Z ¼Z ¼ 0.037), experimental data is from Sullivan and Crocker. 1 2 Table 1. Unsilenced SWL of an air compressor inside a duct outlet. f (Hz) 125 250 500 1k 2k Overall SWL-dB(A) 90 95 104 102 100 107.4 It is obvious that the octave band frequencies (125 Hz, 250 Hz, 500 Hz, 1000 Hz, and 2000 Hz) have higher noise levels (90104 dB). To reduce the huge venting noise emitted from the air compressor’s outlet, the noise elimin- ation on the ﬁve primary noises using the two-chamber muﬄers connected with multiple perforated plug tubes (muﬄers AC) that disperse venting ﬂuid and decrease the secondary ﬂowing noise is considered. To obtain the best acoustical performance within a ﬁxed space, numerical assessments linked to a SA optimizer are applied. Before the minimization of a broadband noise is performed, a reliability check of the SA method by maximization of the STL at a targeted tone (200 Hz) is performed. As shown in Figure 1, the available space for a muﬄer is 0.6 m in width, 0.6 m in height, and 1.2 m in length. The ﬂow rate (Q) and thickness of a perforated tube (t) are preset at 0.01 (m /s) and 0.001 (m), respectively. The corresponding OBJ functions, space constraints, and the ranges of design parameters are summarized in equations (23)(24). Simulated annealing method Evolutionary Algorithms (EAs), which search for appropriate global solution in engineering problems, have, for two decades, been laboriously elaborated upon. It should also be noted that as a stochastic search method, SA is recognized as one of the best. Further, prior to SA optimization, the need to choose starting data which is necessary for the classical gradient methods of EPFM, IPFM and FDM has been eliminated. As a result, for use in a muﬄer’s shape optimization, SA is designated as an optimizer. The fundamental idea underlying SA was ﬁrst introduced by Metropolis et al. and further developed by Kirkpatrick et al. In the SA method where each point X of the search space is compared to a state of some physical system, the function F(X) to be minimized is interpreted as the internal energy of that state. And so, bringing the system from an arbitrary initial state to one with minimum possible energy is the desired objective. Because of annealing, which is a heating process that stabilizes a metal’s temperature while slowing cooling it, the particles remain close to the minimum energy state (see Table 2 for the pseudo-code that implements the SA heuristic). Now, in order to emulate the SA’s evolution, a new random solution (X ) is chosen from the neigh- borhood of the current solution. If, perchance, there is a negative change in the objective function ( F 0), Chiu 11 Table 2. The pseudo-code implementing the simulated annealing heuristic. T: ¼ To X: ¼ Xo F : ¼ F(X) k: ¼ 0 while n < iter Xn’ :¼ neighbor (Xn) if P < P then F(Xn’): ¼ F(Xn’)*w ; F ¼ F(Xn’)-F(Xn) max 1 i else F ¼ F(Xn’)-F(Xn) if F P then Xn’ ¼ Xn; T’n ¼ kk*Tn; n : ¼ n þ 1 else if random() < pb(F /C*Tn) then Xn’ ¼ Xn; T’n ¼ kk*Tn; n : ¼ n þ 1 return the new solution will be recognized as the new current solution with the transition property (pb(X )) of 1. If, on the other hand, there is not a negative change, the probability of a transition to the new state will be the function pbðÞ F=CT . As shown in equation (25), the new transition property (pb(X )) varied from 0 to 1 will be calculated using the Boltzmann’s factor (pb(X ) ¼ expðF=CTÞ), where C and T are the Boltzmann constant and the current temperature. 1, F 0 pbðÞ X ¼ ð25aÞ exp , f 4 0 CT F ¼ FXðÞFXðÞ ð25bÞ For the purpose of escaping the local optimum, SA also permits movement that results in solutions that are inferior (uphill movement) to the current solution. Hence, if the transition property (pb(X )) is greater than a random number of rand(0,1), the new inferior solution which results in a higher energy condition will be accepted; otherwise, it will be rejected. Each successful substitution of the new current solution will conduct to the decay of the current temperature as T ¼ kk T ð26Þ new old where kk is the cooling rate. This process is repeated until the preset (iter) of the outer loop is reached. Results and discussion Results The accuracy of the SA optimization depends on two kinds of SA parameters including kk (cooling rate) and iter 20,21 (maximum iteration). To achieve good optimization, the following parameters are varied step by step: kk (0.91, 0.93, 0.95, 0.97, 0.99); iter (25, 50, 100, 500, 1000, 2000). Two results of optimization (one, pure tone noises used for SA’s accuracy check; and the other, a broadband noise occurring in an air compressor room) are described below. Pure tone noise optimization Muffler A. Before dealing with a broadband noise for muﬄer AC, the STL’s maximization for muﬄer A with respect to a one-tone noise (200 Hz) is introduced for a reliability check on the SA method. By using equations (20a) and (23b), the maximization of the STL with respect to muﬄer A (a two-chamber muﬄer hybridized with one perforated plug tube) at the speciﬁed pure tone (200 Hz) was performed ﬁrst. As indicated in Table 3, ten sets of SA parameters are tried in the muﬄer’s optimization. Obviously, the optimal design data can be obtained from 12 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Table 3. Optimal STL for muffler A (equipped with one perforated plug tube) at various SA parameters (targeted tone of 200 Hz). SA parameter Design parameters Result iter kk RT1 RT2 RT3 RT4 RT5 RT6 STL dB 200 Hz 25 0.91 0.2326 0.6536 0.4536 0.4768 0.006391 0.2664 21.1 25 0.93 0.2075 0.5866 0.3866 0.4433 0.005512 0.2178 24.7 25 0.95 0.1889 0.5371 0.3371 0.4186 0.004862 0.1819 27.8 25 0.97 0.1537 0.4431 0.2431 0.3715 0.003628 0.1137 34.9 25 0.99 0.1472 0.4257 0.2257 0.3629 0.003400 0.1012 36.5 50 0.99 0.1372 0.3992 0.1992 0.3496 0.003051 0.08189 39.3 100 0.99 0.1300 0.3801 0.1801 0.3401 0.002801 0.06808 41.9 500 0.99 0.1171 0.3455 0.1455 0.3227 0.002347 0.04299 51.4 1000 0.99 0.1107 0.3286 0.1286 0.3143 0.002126 0.03075 60.8 2000 0.99 0.1042 0.3112 0.1112 0.3056 0.001897 0.01810 80.1 Note. The gray shades mean the SA parameters that are adjusted during the the optimization serarching process. Figure 6. STL with respect to various kk (muffler A: iter ¼ 25, target tone ¼ 200 Hz). the last set of SA parameters at (kk, iter) ¼ (0.99, 2000). Using the optimal design in a theoretical calculation, the optimal STL curves with respect to various SA parameters (kk, iter) are plotted and depicted in Figures 6 and 7. As revealed in Figures 6 and 7, the STL is precisely maximized at the desired frequency of 200 Hz. Consequently, the SA optimizer is reliable in the optimization process. Mufflers D and E. As indicated in Figure 8, to appreciate the acoustical performance of two-chamber muﬄers equipped with various tubes (muﬄer D: a muﬄer equipped with one non-perforated and internally extended tube; muﬄer E: a muﬄer equipped with one non-perforated tube), the shape optimization of muﬄers D and E at 200 Hz is also performed using the same SA parameters of (kk, iter) ¼ (0.99, 2000). The optimal design parameters for muﬄers A, D, and E are summarized in Table 4 and plotted in Figure 9. As indicated in Figure 9, the optimal STLs for muﬄers A, D, and E have been precisely located at the speciﬁed frequency of 200 Hz. As indicated in Table 4 and Figure 9, the STL of muﬄer A at 200 Hz reaches 80.1 dB. In addition, the STLs with respect to muﬄers D and E are 43.8 dB and 46.2 dB. Consequently, muﬄer A with a perforated plug tube reach is superior to other muﬄers. Moreover, the acoustical performances of both muﬄers D and E at a speciﬁed 200 Hz are almost the same. Broadband noise optimization Mufflers A–C. Similarly, adopting the same SA parameters of (kk, iter) ¼ (0.99, 2000) and doing the minimiza- tion of the SWL , SWL , and SWL with respect to muﬄers AC in broadband noise, the resulting design T-1 T-2 T-3 Chiu 13 Figure 7. STL with respect to various iter (muffler A: kk ¼ 0.99, target tone ¼ 200 Hz). Figure 8. The outline dimension of three kinds of mufflers hybridized with various tube (mufflers A, D, and E). Table 4. Comparison of the optimal STL for mufflers A, D, and E (targeted tone of 200 Hz). Muffler Design parameters Result Muffler A RT1 RT2 RT3 RT4 RT5 RT6 STL dB 200 Hz 0.1042 0.3112 0.1112 0.3056 0.001897 0.01810 80.1 Muffler D RT1* RT2* RT3* RT4* STL dB 200 Hz 0.1042 0.1112 0.3028 0.05140 43.8 Muffler E RT1** RT2** RT3** STL dB 200 Hz 0.1042 0.1112 0.3056 46.2 Note: Muffler D: RT1* ¼ D2; RT2* ¼ D3; RT3* ¼ L2; RT4* ¼ L4. Muffler E: RT1** ¼ D2; RT2** ¼ D3; RT3** ¼ L2. 14 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Figure 9. Comparison of the optimal STLs of three kinds of mufflers (mufflers A, D, and E) optimization at pure tone of 200 Hz. Table 5. Comparison of the minimized SWL of three kinds of mufflers (mufflers AC) (broadband noise). Design parameters Result Muffler type RT1 RT2 RT3 RT4 RT5 RT6 SWL – dB(A) Muffler A 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 46.5 Muffler B 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 52.3 Muffler C 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 58.9 Figure 10. Comparison of the optimal STLs of three kinds of mufflers (mufflers A, B, and C) and the original SWL. parameters are shown in Table 5. Using the optimal design parameters in a theoretical calculation, the optimal STL curves with respect to various muﬄers AC are plotted and depicted in Figure 10. As illustrated in Table 5, the resultant sound power levels with respect to three kinds of muﬄers have been reduced from 107.4 dB to 46.5 dB, 52.3 dB, and 58.9 dB. Mufflers F and G. As indicated in Figure 11, to appreciate the acoustical performance of two-chamber muﬄers equipped with various tubes (muﬄer F: a muﬄer equipped with four non-perforated and internally extended tubes; muﬄer G: a muﬄer equipped with four non-perforated tubes), the shape optimization of muﬄers F and G for a broadband noise is also performed using the same SA parameters of (kk, iter) ¼ (0.99, 2000). The optimal design parameters for muﬄers C, F, and G are summarized in Table 6 and plotted in Figure 12. As illustrated in Chiu 15 Figure 11. The outline dimensions of three kinds of mufflers hybridized with various tubes (mufflers C, F, and G). Table 6. Comparison of the optimal STL for mufflers C, F, and G (broadband noise). Muffler Design parameters Result Muffler C RT1 RT2 RT3 RT4 RT5 RT6 SWL – dB(A) 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 58.9 Muffler F RT1*** RT2*** RT3*** RT4*** SWL – dB(A) 0.1042 0.1112 0.3028 0.05140 68.2 Muffler G RT1**** RT2**** RT3**** SWL – dB(A)B 0.1042 0.1112 0.3056 80.3 Note: Muffler F: RT1*** ¼ D2; RT2*** ¼ D3; RT3*** ¼ L2; RT4* ¼ L4. Muffler G: RT1**** ¼ D2; RT2**** ¼ D3; RT3**** ¼ L2. Figure 12. Comparison of the optimal STLs of three kinds of mufflers (mufflers C, F, and G) and the original SWL. 16 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Table 6, the resultant sound power levels with respect to three kinds of muﬄers (muﬄers C, F, and G) have been reduced from 107.4 dB to 58.9 dB, 68.2 dB, and 80.3 dB. As indicated in Figure 12, muﬄer A with four perforated plug tubes is superior to other muﬄers. Moreover, muﬄer G with four non-perforated tubes has the worst acoustical performance. Discussion In order to decrease the secondary ﬂowing noise generated from the higher speed ﬂow, new muﬄer designs with multiple perforated plug tubes are presented. To achieve a suﬃcient optimization, the selection of the appropriate SA parameter set is essential. As indicated in Table 3, the best SA set of muﬄer A at the targeted pure tone noise of 200 Hz has been shown. The related STL curves with respect to various SA parameters are plotted in Figures 6 and 7. Figures 6 and 7 reveal that the predicted maximal value of the STL is located at the desired frequency. Similarly, in dealing with pure tone noise (200 Hz) in muﬄers D and E, the proﬁle shown in Figure 9 indicates that the maximum STLs of the muﬄers are also located at the speciﬁed frequency. In dealing with the broadband noise, the acoustical performance among three kinds of two-chamber muﬄers connected with multiple perforated plug tube (muﬄers A, B, and C) are shown in Table 5 and Figure 10. As can be observed in Table 5, the overall STL of the two-chamber muﬄer equipped with one perforated plug tube (muﬄer A) reaches 60.9 dB. However, the overall STLs of the two-chamber muﬄers equipped with two perforated plug tubes (muﬄer B) and four perforated plug tubes (muﬄer C) are 55.1 dB and 48.5 dB. 1:84c Here, for the broadband noise optimization in Table 5, the related cutoﬀ frequencies f5 for muﬄers A– C at the internally parallel perforated plug tubes (D3 ¼ 0.032427 m) and the outlet tube (D4 ¼ 0.1112) are 6145 Hz and 1800 Hz, respectively. The maximum frequency range for muﬄers A–C used in Figure 10 is 2000 Hz which closes to the valid frequency range of 1800 Hz. To appreciate the acoustical eﬀect of the perforated plug tubes (muﬄers AC), the internally extended non- perforated tubes (muﬄers D and F), and the non-perforated tubes (muﬄers E and G), a comparison of the muﬄers’ acoustical performance is carried out and shown in Tables 4 and 6 and Figures 9 and 12. As illustrated in Table 4 and Figure 9, when dealing with a pure tone noise, muﬄer A with a perforated plug tube is superior to those muﬄers having a non-perforated tube and an internally expanded non-perforated tube. Similarly, when dealing with a broadband noise, the overall STLs with respect to muﬄers C, F, and G shown in Table 6 and Figure 12 are 48.5 dB, 39.2 dB, and 27.1 dB. It is obvious that a two-chamber muﬄer equipped with multiple perforated plug tubes is also superior to these muﬄer equipped with multiple expended/ non-extended non-perforated tubes. Moreover, the results shown in Table 5 and Figure 10 indicate that the two-chamber muﬄer hybridized with fewer perforated plug tubes is superior to muﬄers that are equipped with more perforated plug tubes. It is obvious that for the muﬄers equipped with more perforated plug tubes, the acoustical performances of the acoustical element (II) between nodes 4 and 5 and acoustical element (III) between nodes 7 and 8 will largely decrease due to the decrement of the area ratio. Therefore, the overall noise reduction of the muﬄers with more perforated plug tubes will decrease. Conclusion It has been shown that two-chamber muﬄers hybridized with multiple perforated plug tubes can be easily and eﬃciently optimized within a limited space by using a decoupling technique, a plane wave theory, a four-pole transfer matrix, and a SA optimizer. As indicated in Table 3 and Figures 6 and 7, two kinds of SA parameters (kk and iter) play essential roles in the solution’s accuracy during SA optimization. Figures 6, 7, and 9 indicate that the tuning ability established by adjusting design parameters of muﬄers A, D, and E is reliable. Additionally, as indicated in Tables 4 and 6 and Figures 9 and 12, the acoustical eﬀect of various tubes has been discussed. Results reveal that the acoustical performance of the perforated plug tubes is superior to other tubes (internally extended non-perforated tubes and non-perforated tubes). Moreover, appropriate design parameters of three kinds of the muﬄers hybridized with multiple perforated plug tubes (muﬄers AC) has been assessed. As indicated in Table 5, the resultant SWL with respect to these muﬄers is 46.7 dB, 52.3 dB, and 58.9 dB. Obviously, the two-chamber muﬄer hybridized with fewer perforated plug tubes is superior to the muﬄers equipped with more perforated plug tubes. It can be seen that more perforated plug tubes installed between the two-chamber muﬄer will disperse the venting ﬂuid and reduce the secondary ﬂowing noise; however, the acoustical perform- ances of the acoustical element (II) between nodes 4 and 5 and acoustical element (III) between nodes 7 and 8 will decrease even though the parallel perforated plug tubes for the muﬄers have increased. Therefore, before choosing Chiu 17 the appropriate number of perforated plug tubes and performing the muﬄer’s shape optimization, presetting the maximum ﬂowing velocity within the muﬄer is required. Declaration of conflicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: The author acknowledges the ﬁnancial support of the National Science Council (NSC 99-2622-E-235-002-CC3, ROC). References 1. Davis DD, Stokes JM and Moorse L. Theoretical and experimental investigation of mufflers with components on engine muffler design. NACA Report, 1192, 1954. 2. Prasad MG and Crocker MJ. Studies of acoustical performance of a multi-cylinder engine exhaust muffler system. J Sound Vib 1983; 90: 491–508. 3. Prasad MG. A note on acoustic plane waves in a uniform pipe with mean flow. J Sound Vib 1984; 95: 284–290. 4. Sullivan JW and Crocker MJ. Analysis of concentric tube resonators having unpartitioned cavities. J Acoust Soc Am 1978; 64: 207–215. 5. Sullivan JW. A method of modeling perforated tube muffler components I: Theory I. J Acoust Soc Am 1979; 66: 772–778. 6. Sullivan JW. A method of modeling perforated tube muffler components I: Theory II. J Acoust Soc Am 1979; 66: 779–788. 7. Jayaraman K and Yam K. Decoupling approach to modeling perforated tube muffler component. J Acoust Soc Am 1981; 69: 390–396. 8. Rao KN and Munjal ML. A generalized decoupling method for analyzing perforated element mufflers. In: Nelson acoustics conference, Madison, 1984. 9. Peat KS. A numerical decoupling analysis of perforated pipe silencer elements. J Sound Vib 1988; 123: 199–212. 10. Chiu MC. Numerical assessment of rectangular side inlet/outlet plenums internally hybridized with two crossed baffles using a FEM, neural network, and GA method. J Low Freq Noise Vib Active Control 2014; 33: 271–288. 11. Yang FJ, Guo ZH, Fu X, et al. Computation of acoustic transfer matrices of swirl burner with finite element and acoustic network method. J Low Freq Noise Vib Active Control 2015; 34: 169–184. 12. Chang YC and Chiu MC. Shape optimization of one-chamber perforated plug/non-plug mufflers by simulated annealing method. Int J Numer Methods Eng 2008; 74: 1592–1620. 13. Chiu MC. Shape optimisation of multi-chamber mufflers with plug-inlet tube on a venting process by genetic algorithms. Appl Acoust 2010; 71: 495–505. 14. Chiu MC. Optimal design of multi-chamber mufflers hybridized with perforated intruding inlets and resonated tube using simulated annealing. ASME J Vib Acoust 2010; 132: 9. 15. Chiu MC. GA optimization on a venting system with three-chamber hybrid mufflers within a constrained back pressure and space. ASME J Vib Acoust 2012; 134: 11. 16. Munjal ML. Acoustics of ducts and mufflers with application to exhaust and ventilation system design. New York: John Wiley & Sons, 1987. 17. Chang YC, Yeh LJ, Chiu MC, et al. Shape optimization on constrained single-layer sound absorber by using GA method and mathematical gradient methods. J Sound Vib 2005; 286: 941–961. 18. Metropolis A, Rosenbluth W, Rosenbluth MN, et al. Equation of static calculations by fast computing machines. J Chem Phys 1953; 21: 1087–1092. 19. Kirkpatrick S, Gelatt CD and Vecchi MP. Optimization by simulated annealing. Science 1983; 220: 671–680. 20. Chiu MC. Numerical assessment of hybrid mufflers on a venting system within a limited back pressure and space using simulated annealing. J Low Freq Noise Vib Active Control 2011; 30: 247–275. 21. Chiu MC. Acoustical treatment of multi-tone broadband noise with hybrid side-branched mufflers using a simulated annealing method. J Low Freq Noise Vib Active Control 2014; 33: 79–112. 22. Rao KN and Munjal ML. Experimental evaluation of impedance of perforates with grazing flow. J Sound Vib 1986; 108: 283–295. Appendix 1 Transfer matrix of expanded perforated plug duct As indicated in Figure 13, the expansion perforated duct is composed of an outer duct and an inner open-ended (one end only) perforated tube. Based on Sullivan and Crocker’s derivation, the continuity equations and momentum equations with respect to the inner and outer tubes at nodes 5 and 5 a are 18 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Figure 13. Acoustical field of an expanded perforated plug duct. Inner tube Continuity equation @ @u 4 @ 5 5 o 5a V þ þ u þ ¼ 0 ð27Þ @x @x D @t Momentum equation @ @ @p þ V u þ ¼ 0 ð28Þ o 5 @t @x @x Outer tube Continuity equation @u 4D @ 5a 1 o 5a u þ ¼ 0 ð29Þ 2 2 @x D D @t 2 3 @u @p 5a 5a þ ¼ 0 ð30Þ @t @x Assuming that the acoustic wave is a harmonic motion under the isentropic processes in ducts, then 2 j!t pxðÞ , t ¼ ðÞ x c e ð31Þ The acoustic impedance of the perforation ( c ) is expressed as o o p ðÞ x p ðÞ x 5 5a c ¼ ð32Þ o o uxðÞ For perforates with a stationary medium ¼½ 0:006 þ jkðÞ t þ 0:75dH = ð33aÞ 1 1 For perforates with a grazing ﬂow ¼½ 0:514D M =ðÞ L þ j0:95ktðÞ þ 0:75dH = ð33bÞ 3 5 c1 1 1 1 Chiu 19 where dH is the diameter of a perforated hole on an inner tube, t is the thickness of the inner perforated tube, and is the porosity of the perforated tube. Substituting equations (31) and (32) for equations (27)(30) and eliminating u and u yield 5 5A d d 2 2 1 M 2jM k þ k p 5 5 dx dx ð34aÞ 4 d M þ jkðÞ p p ¼ 0 5 5 5a D dx d 4D þ k p þ j ðÞ p p ¼ 0 ð34bÞ 5a 5 5a 2 2 2 dx D D 2 3 Alternatively, equations (34a) and (34b) can also be expressed as 00 0 0 p þ p þ p þ p þ p ¼ 0 ð35aÞ 1 2 5 3 4 5a 5 5 5a 0 00 0 p þ p þ p þ p þ p ¼ 0 ð35bÞ 5 6 7 8 5a 5 5 5a 5a dp dp 5 5a 0 0 Let p ¼ ¼ y , p ¼ ¼ y , p ¼ y , p ¼ y ð36Þ 1 2 5 3 5a 4 5 5a dx dx According to equations (35) and (36), the new matrix between {y } and {y} is 2 3 2 32 3 y y 1 3 2 4 1 6 7 6 76 7 y y 5 7 6 8 2 6 2 7 6 76 7 6 7 ¼ 6 76 7 ð37aÞ 4 5 4 54 5 y 100 0 y y 010 0 y or y ¼½ H y ð37bÞ Let y ¼½ fg ð38Þ Plugging equations (38) into (37) and then multiplying ½ by both sides yield ¼½ Kfg ð39aÞ where 2 3 00 0 6 7 0 00 6 7 ½ K ¼½ ½ H½ ¼ ; : 6 7 i 4 5 00 0 ð39bÞ the eigen value of½ H;½ : the model matrix 4x4 formed by four sets of eigen vectors of½ H 4x1 4x4 The related solution of equation (39) then obtained is ¼ c e ð40Þ i i 20 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Using equations (28), (30), (39), and (40), the relationship of the acoustic pressure and the acoustical particle velocity yields 2 3 2 32 3 p ðÞ x E E E E c 5 1,1 1,2 1,3 1,4 1 6 7 6 76 7 p ðÞ x E E E E c 6 5a 7 6 2,1 2,2 2,3 2,4 76 2 7 6 7 ¼ 6 76 7 ð41Þ 4 5 4 54 5 c u ðÞ x E E E E c o o 5 3,1 3,2 3,3 3,4 3 c u ðÞ x E E E E c o o 5a 4,1 4,2 4,3 4,4 4 Plugging x ¼ 0 and x ¼ L into equation (41) and rearranging it, the resultant relationship of acoustic pressure c1 and particle velocity between x ¼ 0 and x ¼ L becomes c1 2 3 2 3 p ðÞ 0 pðÞ L 5 5 c1 6 7 6 7 p ðÞ 0 p ðÞ L 5a 5a c1 6 7 6 7 ¼½ ð42aÞ 6 7 6 7 4 5 4 5 c u ðÞ 0 c uðÞ L o o 5 o o 5 c1 c u ðÞ 0 c u ðÞ L o o 5a o o 5a c1 where ½ ¼½ E0ðÞ½ EðÞ L ð42bÞ c1 To obtain the transform matrix between the inlet (x ¼ 0) and the outlet (x ¼ L ) of the inner tubes, the two c1 boundary conditions for the outer tuber at x ¼ 0 and x ¼ L are calculated and listed below: c1 p ðÞ 0 5a ¼j c cotðÞ kL ð43aÞ o o a1 u ðÞ 0 5a pðÞ L 2 c1 ¼j c cotðÞ kL ð43bÞ o o b1 uðÞ L 2 c1 Plugging equation (43) into equation (42) and developing them, the transfer matrix deduced is p TPPE TPPE p 5 1,1 1,2 6a ¼ ð44aÞ c u TPPE TPPE c u o o 5 2,1 2,2 o o 6a where p ¼ p ðÞ 0 ; u ¼ u ðÞ 0 ; p ¼ p ðÞ L ; u ¼ u ðÞ L ; 5 5 5 5 6a 5a c1 6a 5a c1 j cotðÞ kL 2,2 4,2 a1 TPPE ¼ þ c j c cotðÞ kL ; 1,1 1,1 1,3 o o 1,1 o o b1 j cotðÞ kL 2,4 4,4 a1 TPPE ¼ þ c j c cotðÞ kL ; 1,2 1,4 1,3 o o 1,1 o o b1 j cotðÞ kL 2,2 4,2 a1 ð44bÞ TPPE ¼ þ c c cotðÞ kL ; 2,1 3,2 3,3 o o 3,1 o o b1 j cotðÞ kL 2,4 4,4 a1 TPPE ¼ þ c c cotðÞ kL ; 2,2 3,4 3,3 o o 3,1 o o b1 j cotðÞ kL þ cotðÞ kL cotðÞ kL 4,3 a1 4,1 a1 b1 G ¼ c E o o þ j cotðÞ kL 2,3 2,1 b1 Chiu 21 Appendix 2 Transfer matrix of contracted perforated plug duct As indicated in Figure 14, the contracted perforated duct is composed of an inner open-ended (one end only) perforated tube and an outer contracted duct. As deduced in Appendix 1, the continuity equations and momentum equations with respect to the inner and outer tubes at nodes 6 and 6 b are Inner tube Continuity equation @ @u 4 @ 6 6 o 6a V þ þ u þ ¼ 0 ð45Þ @x @x D @t Momentum equation @ @ @p þ V u þ ¼ 0 ð46Þ o 6 @t @x @x Outer tube Continuity equation @u 4D @ 6a 3 o 6a u þ ¼ 0 ð47Þ 2 2 @x D D @t 2 3 @u @p 6a 6a þ ¼ 0 ð48Þ @t @x Likewise, as derived in equations (49)(55), the new matrix between {y } and {y} is 2 3 2 32 3 y y 1 3 2 4 1 6 0 7 6 76 7 y y 5 7 6 8 2 6 2 7 6 76 7 ¼ ð49aÞ 6 7 6 76 7 4 5 4 54 5 y 100 0 y y 010 0 y or y ¼½ y ð49bÞ Figure 14. Acoustic field of a contracted perforated plug duct. 22 Journal of Low Frequency Noise, Vibration and Active Control 36(1) where dp dp 6 6a 0 0 y ¼ p ¼ , y ¼ p ¼ , y ¼ p , y ¼ p ð49cÞ 1 2 3 6 4 6a 6 6a dx dx Let y ¼½ fg ð50aÞ which is 2 3 2 32 3 dp =dx 6 1,1 1,2 1,3 1,4 1 6 7 6 76 7 dp =dx 6a 2,1 2,2 2,3 2,4 2 6 7 6 76 7 ¼ ð50bÞ 6 7 6 76 7 4 5 4 54 5 6 3,1 3,2 3,3 3,4 3 6a 4,1 4,2 4,3 4,4 4 ½ is the model matrix formed by four sets of eigen vectors of½ HH . 4x4 4x1 4x4 Plugging equation (50) into (49) and then multiplying½ by both sides yield ¼½ KKfg ð51aÞ where 2 3 6 7 0 00 6 2 7 ½ KK ¼½ ½ HH½ ¼ 6 7; 4 5 00 0 ð51bÞ : the eigen value of½ HH Obviously, equation (51) is a decoupled equation. The related solution then obtained is ¼ cc e ð52Þ i i Using equations (46), (48), (51), and (52), the relationship of acoustic pressure and particle velocity obtained is 2 3 2 32 3 p ðÞ x EE EE EE EE cc 6 1,1 1,2 1,3 1,4 1 6 7 6 76 7 p ðÞ x EE EE EE EE cc 6a 2,1 2,2 2,3 2,4 2 6 7 6 76 7 ¼ ð53Þ 6 7 6 76 7 4 5 4 EE EE EE EE 54 5 c u ðÞ x cc o o 6 3,1 3,2 3,3 3,4 3 EE EE EE EE c u ðÞ x cc o o 6a 4,1 4,2 4,3 4,4 4 Taking two cases of x ¼ 0 and x ¼ L and rearranging them, the resultant relationship of the acoustic pressure c2 and the acoustic particle velocity between x ¼ 0 and x ¼ L becomes c2 2 3 2 3 p ðÞ 0 pðÞ L 6 6 c2 6 7 6 7 p ðÞ 0 p ðÞ L 6 6a 7 6 6a c2 7 6 7 ¼½ 6 7 ð54aÞ 4 5 4 5 c u ðÞ 0 c uðÞ L o o 6 o o 6 c2 c u ðÞ 0 c u ðÞ L o o 6a o o 6a c2 where ½ ¼½ EEðÞ 0½ EEðÞ L ð54bÞ c2 Chiu 23 To obtain the transform matrix between inlet (x ¼ 0) and outlet (x ¼ L ) of the inner tubes, two boundary c2 conditions for the outer tube at x ¼ 0 and x ¼ L are taken into calculation and listed below: c2 p ðÞ 0 ¼j c cotðÞ kL ð55aÞ o o a2 u ðÞ 0 p ðÞ L 6a c2 ¼j c cotðÞ kL ð55bÞ o o b2 u ðÞ L 6a c2 By plugging equations (55) into equation (54) and developing them, the transfer matrix deduced is p TPPC TPPC p 6a 1,1 1,2 7 ¼ ð56aÞ c u TPPC TPPC c u o o 6a 2,1 2,2 o o 7 where p ¼ p ðÞ 0 ; u ¼ u ðÞ 0 ; p ¼ pðÞ L ; u ¼ uðÞ L ; 6a 6a 6a 6a 7 6 c2 7 6 c2 j cotðÞ kL 1,1 3,1 a2 TPPC ¼ þ c j c cotðÞ kL ; 1,1 2,1 2,4 o o 2,2 o o b2 j cotðÞ kL 1,3 3,3 a2 TPPC ¼ þ c j c cotðÞ kL ; 1,2 2,3 2,4 o o 2,2 o o b2 j cotðÞ kL 1,1 3,1 a2 ð56bÞ TPPC ¼ þ c c cotðÞ kL ; 2,1 4,1 4,4 o o 4,2 o o b2 j cotðÞ kL 1,3 3,3 a2 TPPC ¼ þ c c cotðÞ kL ; 2,2 4,3 4,4 o o 4,2 o o b2 j cotðÞ kL þ cotðÞ kL cotðÞ kL 3,4 a2 3,2 a2 b2 G ¼ c c o o þ j cotðÞ kL 1,4 1,2 b2 Appendix 3 Transfer matrix of two parallel perforated plug tubes As indicated in Figure 3, the acoustical four-pole matrix between node 5 and 7 is deduced by combining (1) (1) equation (44) and equation (56) yields p TPPE TPPE TPPC TPPC p 5ð1Þ 1,1 1,2 1,1 1,2 7ð1Þ c u c u TPPE TPPE TPPC TPPC o o 5ð1Þ 2,1 2,2 2,1 2,2 o c 7ð1Þ ð57Þ TPP1 TPP1 p 7ð1Þ 1,1 1,2 TPP1 TPP1 c u o o 71ðÞ 2,1 2,2 Developing equation (57) yields p ¼ TPP1 p þ TPP1 c u ð58aÞ 5ð1Þ 1,1 7ð1Þ 1,2 o c 71ðÞ c u ¼ TPP1 p þ TPP1 c u ð58bÞ o o 5ð1Þ 2,1 7ð1Þ 2,2 o c 7ð1Þ Similarly, the acoustical four-pole matrix between node 5 and 7 is (2) (2) p TPP1 TPP1 p 5ð2Þ 7ð2Þ 1,1 1,2 ¼ ð59Þ c u TPP1 TPP1 c u o o 5ð2Þ 2,1 2,2 o o 7ð2Þ 24 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Developing equation (59) yields p ¼ TPP1 p þ TPP1 c u ð60aÞ 5ð2Þ 1,1 7ð2Þ 1,2 o c 7ð2Þ c u ¼ TPP1 p þ TPP1 c u ð60bÞ o o 5ð2Þ 2,1 7ð2Þ 2,2 o c 7ð2Þ Combining equation (58a) and equation (60a) yields p þ p ¼ TPP1 p þ p þ TPP1 c u þ u ð61Þ 5ð1Þ 5ð2Þ 1,1 7ð1Þ 7ð2Þ 1,2 o c 7ð1Þ 7ð2Þ Likewise, combining equation (58b) and equation (60b) yields c u þ u ¼ TPP1 p þ p þ TPP1 c u þ u ð62Þ o o 5ð1Þ 5ð2Þ 2,1 7ð1Þ 7ð2Þ 2,2 o c 7ð1Þ 7ð2Þ where p ¼ p ¼ p ; p ¼ p ¼ p ; 5 5ð1Þ 5ð2Þ 7 7ð1Þ 7ð2Þ ð63Þ u ¼ u þ u ; u ¼ u þ u 5 5ð1Þ 5ð2Þ 7 7ð1Þ 7ð2Þ Plugging equation (63) into equations (61) and (62) yields 2 p ¼ 2 TPP1 p þ TPP1 c u ð64aÞ 5 1,1 7 1,2 o c 7 c u ¼ 2 TPP1 p þ TPP1 c u ð64bÞ o o 5 2,1 7 2,2 o c 7 Rearranging equation (64) in a matrix form, the equivalent four-pole matrix between nodes 5 and 7 shown in Figure 4 is "# p p TPP1 TPP1 5 1,1 1,2 7 ¼ ð65Þ c u c u 2 TPP1 TPP1 o o 5 2,1 2,2 o o 7 Appendix 4 Transfer matrix of four parallel perforated plug tubes As indicated in Figure 3, muﬄer C’s acoustical four-pole matrix between node 5 and 7 is (1) (1) p TPP1 TPP1 p 5ð1Þ 1,1 1,2 7ð1Þ ¼ ð66Þ c u c u TPP1 TPP1 o o 5ð1Þ 2,1 2,2 o o 7ð1Þ Developing equation (66) yields p ¼ TPP1 p þ TPP1 c u ð67aÞ 5ð1Þ 1,1 7ð1Þ 1,2 o c 7ð1Þ c u ¼ TPP1 p þ TPP1 c u ð67bÞ o o 5ð1Þ 2,1 7ð1Þ 2,2 o c 7ð1Þ Similarly, the acoustical four-pole matrices between node 5 and 7 ,node5 and 7 , and node 5 and 7 ,is (2) (2) (3) (3) (4) (4) p TPP1 TPP1 p 5ð2Þ 1,1 1,2 7ð2Þ ¼ ð68aÞ c u TPP1 TPP1 c u o o 5ð2Þ o o 7ð2Þ 2,1 2,2 Chiu 25 p TPP1 TPP1 p 5ð3Þ 1,1 1,2 7ð3Þ ¼ ð68bÞ c u TPP1 TPP1 c u o o 5ð3Þ 2,1 2,2 o o 7ð3Þ p TPP1 TPP1 p 5ð4Þ 1,1 1,2 7ð4Þ ¼ ð68cÞ c u c u TPP1 TPP1 o o 5ð4Þ 2,1 2,2 o o 7ð4Þ Developing equations (68a)(68c) yields p ¼ TPP1 p þ TPP1 c u ð69aÞ 5ð2Þ 1,1 7ð2Þ 1,2 o c 7ð2Þ c u ¼ TPP1 p þ TPP1 c u ð69bÞ o o 5ð2Þ 2,1 7ð2Þ 2,2 o c 7ð2Þ p ¼ TPP1 p þ TPP1 c u ð69cÞ 5ð3Þ 1,1 7ð3Þ 1,2 o c 7ð3Þ c u ¼ TPP1 p þ TPP1 c u ð69dÞ o o 5ð3Þ 2,1 7ð3Þ 2,2 o c 7ð3Þ p ¼ TPP1 p þ TPP1 c u ð69eÞ 5ð4Þ 1,1 7ð4Þ 1,2 o c 7ð4Þ c u ¼ TPP1 p þ TPP1 c u ð69fÞ o o 5ð4Þ 2,1 7ð4Þ 2,2 o c 7ð4Þ Combining equation (67a) and equations (69a), (69c), and (69e) yields p þ p þ p þ p ¼ TPP1 p þ p þ p þ p 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ 1,1 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ ð70Þ þ TPP1 c u þ u þ u þ u 1,2 o c 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ Likewise, combining equation (67b) and equation (69b), (69d), and (69f) yields c u þ u þ u þ u ¼ TPP1 p þ p þ p þ p o o 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ 2,1 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ ð71Þ þ TPP1 c u þ u þ u þ u 2,2 o c 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ where p ¼ p ¼ p ¼ p ¼ p ; 5 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ p ¼ p ¼ p ¼ p ¼ p ; 7 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ ð72Þ u ¼ u þ u þ u þ u ; 5 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ u ¼ u þ u þ u þ u 7 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ Plugging equation (72) into equations (70) and (71) yields 4 p ¼ 4 TPP1 p þ TPP1 c u ð73aÞ 5 1,1 6 1,2 o c 7 c u ¼ 4 TPP1 p þ TPP1 c u ð73bÞ o o 5 2,1 7 2,2 o c 7 Rearranging equation (73) in a matrix form, the equivalent four-pole matrix between nodes 5 and 7 shown in Figure 4 is "# p p TPP1 TPP1 5 1,1 1,2 7 ¼ ð74Þ c u c u 4 TPP1 TPP1 o o 5 2,1 2,2 o o 7 26 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Appendix 5 Notation C sound speed (m s ) x x i i c , c c , c cc , cc cc , cc coefficients in function c e and cc e 1 2, 3 4, 1 2, 3 4 i i dH the diameter of a perforated hole on the inner perforated tube (m) D diameter of the i-th perforated tubes (m) D diameter of the outer tube (m) f cyclic frequency (Hz) iter maximum iteration pﬃﬃﬃﬃﬃﬃﬃ j imaginary unit (¼ 1) k wave number (¼ ) kk cooling rate L , L lengths of perforate part for the perforated plug ducts (m) c1 c2 L , L , L , L lengths of non-perforate part for the perforated plug ducts (m) a1 a2 b1 b2 L length of the i-th perforated tubes (m) L total length of the muffler (m) M mean flow Mach number at the i-th node OBJ objective function (dB) p acoustic pressure at the i-th node (Pa) pb(T) transition probability 3 1 Q volume flow rate of venting gas (m s ) RT1, RT2, RT3, RT4, RT5, RT6 design parameter for mufflers AC RT1*, RT2*, RT3*, RT4* design parameter for mufflers D and F RT1**, RT2**, RT3** design parameter for mufflers E and G S section area at the i-th node(m ) STL sound transmission loss (dB) SWLO unsilenced sound power level inside the muffler’s inlet (dB) SWL overall sound power level inside the muffler’s output (dB) t the thickness of the i-th inner perforated tube (m) TS1 , TS2 , TS3 , TS4 components of four-pole transfer matrices for an acoustical mechanism with ij ij ij ij straight ducts TPP1 components of a four-pole transfer matrix for an acoustical mechanism with a ij perforated plug tube TSC1 , TSC2 components of a four-pole transfer matrix for an acoustical mechanism with a ij ij contracted perforated intruding tube TSE1 , TSE2 components of a four-pole transfer matrix for an acoustical mechanism with ij ij an expanded perforated intruding tube T components of a four-pole transfer system matrix ij T current temperature ( C) T initial temperature ( C) u acoustic particle velocity at the i-th node (m s ) u acoustical particle velocity passing through a perforated hole from the i-th ij node to the j-th node (m s ) V mean flow velocity at the inner perforated tube (m s ) the i-th eigen value of½ H the i-th eigen value of½ HH specific acoustical impedance of the inner perforated tube Z the porosity of the i-th inner perforated tube. 3 1 acoustical density at the i-th node (m s ) ½ the model matrix formed by four sets of eigen vectors of½ H 4x1 4x4 4x4 ½ the model matrix formed by four sets of eigen vectors of½ HH 4x1 4x4 4x4 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE http://www.deepdyve.com/lp/sage/numerical-assessment-of-two-chamber-mufflers-hybridized-with-multiple-6XIT0Pg0oN

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DOI: 10.1177/0263092317693477
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Abstract

Enormous effort has been applied to research on mufflers hybridized with a single perforated plug tube; nonetheless, mufflers conjugated with multiple parallel perforated plug tubes that disperse venting fluid and reduce secondary noise have been overlooked. To this end, an analysis of the sound transmission loss of two-chamber mufflers with multiple parallel perforated plug tubes that are optimally designed to perform within a limited space will be presented. Here, using a decoupled numerical method, a four-pole system matrix for evaluating acoustic performance (sound transmission loss) is derived. During the optimization process, a simulated annealing method, which is a robust scheme utilized to search for the global optimum by imitating a physical annealing process, is used. Prior to dealing with a broadband noise, the sound transmission loss’s maximization relative to a one-tone noise (200 Hz) is produced to check the simulated annealing method’s reliability. The mathematical model is also confirmed for accuracy. To understand the acoustical effects brought about by the various tubes (perforated tubes, internally extended non-perforated tubes, and non- perforated tubes), mufflers with internally extended non-perforated tubes and non-perforated tubes have been evalu- ated. The optimization of three kinds of two-chamber mufflers hybridized with one, two, and four perforated plug tubes have also been compared. The results are revealing: the acoustical performance of mufflers conjugated with more perforated plug tubes decreases as a result of the decrement of the acoustical function for acoustical elements (II) and (III). Accordingly, in order to design a better muffler, an advanced presetting of the maximum (allowable) flowing velocity is necessary before an appropriate number of perforated plug tubes can be chosen for the optimization process. Keywords Multiple parallel perforated plug tube, two-chamber, simulated annealing Introduction Research on reactive muﬄers was initiated in 1054 by Davis et al. Studies of simple expansion muﬄers based on 2,3 plane wave theory were also completed. In order to increase the acoustical performance of lower frequency sound energy, Sullivan and Crocker introduced a muﬄer equipped with an internal perforated tube. Also, a series of theories and numerical techniques for decoupling the acoustical problems have been posited to solve the 5,6 7 coupled equations. Jayaraman and Yam, presuming an unreasonable inner and outer duct, used a method for ﬁnding an analytical solution. Additionally, Rao and Munjal provided a generalized decoupling method. Now, regarding the ﬂowing eﬀect, Peat, by ﬁnding the eigen value in transfer matrices, proposed a numerical decoupling method. However, still neglected was research work on muﬄers conjugated with multiple parallel perforated plug tubes, which disperse venting ﬂuid and reduce secondary ﬂowing noise. Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology, Taiwan, Republic of China Corresponding author: Min-Chie Chiu, Department of Mechanical and Automation Engineering, Chung Chou University of Science and Technology, Taiwan, Republic of China. Email: minchie.chiu@msa.hinet.net Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/ by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 4 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Therefore, to analyze the sound transmission loss (STL) of two-chamber muﬄers equipped with multiple parallel perforated plug tubes that are optimally designed to perform within a limited space, three kinds of two-chamber muﬄers linked together by multiple perforated plug tubes (muﬄer A: a two-chamber muﬄer intern- ally equipped with one perforated plug tube; muﬄer B: a two-chamber muﬄer internally equipped with two perforated and parallel plug tubes; muﬄer C: a two-chamber muﬄer internally equipped with four perforated and parallel plug tubes) are introduced. Because the geometric shapes of muﬄers A–C are complicated, the 10,11 numerical method such as the ﬁnite element method will be predictably time-consuming when doing a broad- band noise reduction assessment. In order to quickly and eﬃciently approach best design values, the numerical decoupling methods used in forming a four-pole matrix are in line with the simulated annealing method and presented in the article. Theoretical background Three kinds of two-chamber muﬄers connected with multiple perforated plug tubes have been adopted for noise elimination in the air compressor room shown in Figure 1. Before the acoustical ﬁelds of the muﬄers were analyzed, the acoustical elements had been identiﬁed. As shown in Figure 2, four kinds of muﬄer components, including four straight ducts, two sudden expanded ducts, two sudden contracted ducts, and multiple perforated plug ducts, are identiﬁed and marked as I, II, III, and IV. Additionally, the acoustical ﬁeld within the muﬄer is represented by 11 points. The outline dimension of the muﬄers is shown in Figure 3. As derived in previous 12–15 work and shown in Appendices 14, individual transfer matrices with respect to straight ducts, perforated plug ducts, and sudden expanded/contracted ducts are described below. Muffler A (a two-chamber muffler connected with one perforated plug tubes) 12–15 As derived in previous work, for the acoustical element (I) shown in Figure 2, the four-pole matrix between nodes 1 and 2 is p TS1 TS1 p 1,1 1,2 1 2 ¼ fðÞ L , D , M ð1Þ 1 i i i c u TS1 TS1 c u o o 1 2,1 2,2 o o 2 For the acoustical element (II), the four-pole matrix between nodes 2 and 3 is p TSE1 TSE1 p 2 1,1 1,2 3 ¼ ð2Þ c u TSE1 TSE1 c u o o 2 2,1 2,2 o o 3 Similarly, the four-pole matrix between nodes 3 and 4 is p TS2 TS2 p 3 1,1 1,2 4 ¼ fðÞ L , D , M ð3Þ 2 i i i TS2 TS2 c u c u o o 3 2,1 2,2 o o 4 Figure 1. Noise elimination on an air compressor room within a limited space. Chiu 5 Figure 2. Acoustical elements in three kinds of two-chamber mufflers hybridized with perforated plug tubes (mufflers A to C). Figure 3. The outline dimension of three kinds of two-chamber mufflers internally hybridized with multiple parallel and perforated plug tubes (mufflers A to C). 6 Journal of Low Frequency Noise, Vibration and Active Control 36(1) For the acoustical element (III), the four-pole matrix between nodes 4 and 5 is p TSC1 TSC1 p 4 1,1 1,2 5 ¼ ð4Þ c u TSC1 TSC1 c u o o 4 2,1 2,2 o o 5 12–15 As shown in previous work and derived in Appendices 1 and 2, for an acoustical element (IV), which is composed of two acoustical parts (one, an expanded perforated plug tube; and the other, a contracted perforated plug tube), the four-pole matrix between nodes 5 and 6a and nodes 6a and 7 is p TPPE1 TPPE1 p 5 1,1 1,2 6a ¼ ð5Þ c u TPPE1 TPPE1 c u o o 5 2,1 2,2 o o 6a p TPPC1 TPPC1 p 6a 1,1 1,2 7 ¼ ð6Þ c u TPPC1 TPPC1 c u o o 6a 2,1 2,2 o o 7 Combining equations (5)–(6), the four-pole matrix between nodes 5 and 7 yields p TPP1 TPP1 p 5 1,1 1,2 7 ¼ ð7Þ c u TPP1 TPP1 c u o o 5 2,1 2,2 o o 7 Likewise, the four-pole matrix between nodes 7 and 8 for a sudden expanded duct is p TSE2 TSE2 p 7 1,1 1,2 8 ¼ ð8Þ c u TSE2 TSE2 c u o o 7 2,1 2,2 o o 8 The four-pole matrix between nodes 8 and 9 in a straight duct is p TS3 TS3 p 1,1 1,2 8 9 ¼ fðÞ L , D , M ð9Þ 3 i i i c u TS3 TS3 c u o o 8 2,1 2,2 o o 9 The four-pole matrix between nodes 9 and 10 for a sudden contracted duct is p TSC2 TSC2 p 9 1,1 1,2 10 ¼ ð10Þ c u TSC2 TSC2 c u o o 9 2,1 2,2 o o 10 Moreover, the four-pole matrix between nodes 10 and 11 in a straight duct is p TS4 TS4 p 10 1,1 1,2 11 ¼ fðÞ L , D , M ð11Þ 4 i i i TS4 TS4 c u c u o o 10 2,1 2,2 o o 11 The total transfer matrix assembled by multiplication is ¼ fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M 1 i i i 2 i i i 3 i i i 4 i i i 5 i i i c u o o 1 TS1 TS1 TSE1 TSE1 TS2 TS2 1,1 1,2 1,1 1,2 1,1 1,2 TS1 TS1 TSE1 TSE1 TS2 TS2 2,1 2,2 2,1 2,2 2,1 2,2 TSC1 TSC1 TPP1 TPP1 TSE2 TSE2 1,1 1,2 1,1 1,2 1,1 1,2 ð12Þ TSC1 TSC1 TPP1 TPP1 TSE2 TSE2 2,1 2,2 2,1 2,2 2,1 2,2 TS3 TS3 TSC2 TSC2 1,1 1,2 1,1 1,2 TS3 TS3 TSC2 TSC2 2,1 2,2 2,1 2,2 TS4 TS4 p 1,1 1,2 11 TS4 TS4 c u 2,1 2,2 o o 11 Chiu 7 A simpliﬁed form is expressed in a matrix as p T T p 1 11 11 12 ¼ ð13Þ c u T T c u o o 1 o o 11 21 22 Muffler B (a two-chamber muffler connected with two perforated plug tubes) Similarly, the acoustical four-pole matrix between nodes 12, nodes 23, nodes 34, nodes 89, nodes 910, and nodes 1011 is the same as equations (1)–(3) and equations (7)–(11). As derived in Appendix 3, for two parallel perforated plug tubes connected to two chambers, the four-pole matrices between nodes 5 7 and nodes (1) (1) 5 7 can be combined to an equivalent matrix (2) (2) "# p TPP1 TPP1 p 5 1,1 1,2 7 ¼ ð14Þ c u 2 TPP1 TPP1 c u o o 5 o o 7 2,1 2,2 where [TPP1 ] is the four-pole matrix for a single perforated plug tube. i,j The equivalent acoustical ﬁeld of muﬄer B is shown in Figure 4. Consequently, the total transfer matrix assembled by multiplication is ¼ fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M 1 i i i 2 i i i 3 i i i 4 i i i 5 i i i c u o o 1 TS1 TS1 TSE1 TSE1 TS2 TS2 1,1 1,2 1,1 1,2 1,1 1,2 TS1 TS1 TSE1 TSE1 TS2 TS2 2,1 2,2 2,1 2,2 2,1 2,2 "# TSC1 TSC1 TPP1 TPP1 1,1 1,2 1,1 1,2 ð15Þ TSC1 TSC1 2 TPP1 TPP1 2,1 2,2 2,1 2,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 1,1 1,2 1,1 1,2 1,1 1,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 2,1 2,2 2,1 2,2 2,1 2,2 TS4 TS4 p 1,1 1,2 11 TS4 TS4 c u 2,1 2,2 o o 11 A simpliﬁed form is expressed in a matrix as T T p p 1 11 11 12 ¼ ð16Þ c u T T c u o o 1 o o 11 21 22 Figure 4. Equivalent acoustical field for a muffler hybridized with perforated plug tubes (mufflers B and C). 8 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Muffler C (a two-chamber muffler connected with four perforated plug tubes) The acoustical four-pole matrix between nodes 1–2, nodes 2–3, nodes 3–4, nodes 8–9, nodes 9–10, and nodes 10–11 is the same as equations (1)–(3) and equations (9)–(11). As derived in Appendix 4, for a two chambers muﬄer connected with four parallel perforated plug tubes, the four-pole matrices between nodes 5 –7 ,5 –7 ,5 – (1) (1) (2) (2) (3) 7 , and 5 –7 nodes can be combined to an equivalent matrix (3) (4) (4) "# p p TPP1 TPP1 5 1,1 1,2 7 ¼ ð17Þ c u c u 4 TPP1 TPP1 o o 5 2,1 2,2 o o 7 where [TPP1 ] is the four-pole matrix for a single perforated tube. i,j The related equivalent acoustical ﬁeld of muﬄer C is also presented and shown in Figure 4. Consequently, the total transfer matrix assembled by multiplication is ¼ fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M fðÞ L , D , M 1 i i i 2 i i i 3 i i i 4 i i i 5 i i i c u o o 1 TS1 TS1 TSE1 TSE1 TS2 TS2 1,1 1,2 1,1 1,2 1,1 1,2 TS1 TS1 TSE1 TSE1 TS2 TS2 2,1 2,2 2,1 2,2 2,1 2,2 "# TSC1 TSC1 TPP1 TPP1 1,1 1,2 1,1 1,2 ð18Þ TSC1 TSC1 4 TPP1 TPP1 2,1 2,2 2,1 2,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 1,1 1,2 1,1 1,2 1,1 1,2 TSE2 TSE2 TS3 TS3 TSC2 TSC2 2,1 2,2 2,1 2,2 2,1 2,2 TS4 TS4 p 1,1 1,2 11 TS4 TS4 c u 2,1 2,2 o o 11 A simpliﬁed form is expressed in a matrix as p T T p 1 11 11 12 ¼ ð19Þ c u T T c u o o 1 o o 11 21 22 Overall sound power level The STL of muﬄers A–C are deﬁned as T þ T þ T þ T 11 12 21 22 ð20aÞ STLðÞ Q, f, RT , RT , RT , RT , RT , RT¼ 20 log þ 10 log 1 1 2 3 4 5 6 2 S T þ T þ T þ T 11 12 21 22 ð20bÞ STLðÞ Q, f, RT , RT , RT , RT , RT , RT¼ 20 log þ 10 log 2 1 2 3 4 5 6 2 S T þ T þ T þ T 11 12 21 22 STLðÞ Q, f, RT , RT , RT , RT , RT , RT¼ 20 log þ 10 log ð20cÞ 3 1 2 3 4 5 6 2 S where RT1 ¼ D2; RT2 ¼ D3=D2; RT3 ¼ D4; ð20dÞ RT4 ¼ L3; RT5 ¼ dH; RT6 ¼ Chiu 9 The silenced octave sound power level emitted from a muﬄer’s outlet is SWL ¼ SWLOðÞ f STLðÞ f ð21Þ i i i where SWLOð f Þ is the original SWL at the inlet of a muﬄer (or pipe outlet), and f is the relative octave band i i frequency; STLð f Þ is the muﬄer’s STL with respect to the relative octave band frequency (f ); SWL is the silenced i i i SWL at the outlet of a muﬄer with respect to the relative octave band frequency. Finally, the overall SWL silenced by a muﬄer at the outlet is () SWL =10 SWL ¼ 10 log 10 TK i¼1 ð22Þ no ½SWLOðÞ f ½SWLOðÞ f ½SWLOðÞ f ½SWLOðÞ f 1 2 3 n ¼ 10 log STL ðÞ f =10 STL ðÞ f =10 STL ðÞ f =10 STL ðÞ f =10 K 1 K 2 K 3 K n 10 þ 10 þ 10 þ þ 10 Objective function STL maximization for a tone (f) noise. The objective functions in maximizing the STL at a pure tone (f) are OBJ ¼ STLðÞ Q, f, RT , RT , RT , RT , RT , RT ð23aÞ 11 1 1 2 3 4 5 6 OBJ ¼ STLðÞ Q, f, RT , RT , RT , RT , RT , RT ð23bÞ 12 2 1 2 3 4 5 6 OBJ ¼ STLðÞ Q, f, RT , RT , RT , RT , RT , RT ð23cÞ 13 3 1 2 3 4 5 6 The related ranges of the parameters are Q ¼ 0:01 m =s ; L ¼ 1:2mðÞ; D ¼ 0:6mðÞ; o o L ¼ 0:05ðÞ m ; L ¼ 0:05ðÞ m ; D ¼ 0:2mðÞ; 1 5 1 RT1 :½ 0:1, 0:25 ; RT2 :½ 0:3, 0:7 ; ð23dÞ RT3 :½ 0:1, 0:5 ; RT4 :½ 0:3, 0:5 ; RT5 :½ 0:00175, 0:007 ; RT6 :½ 0:01, 0:3 SWL minimization for a broadband noise. To minimize the overall SWL , the objective function is OBJ ¼ SWLðÞ Q, RT , RT , RT , RT , RT , RT ð24aÞ 21 T1 1 2 3 4 5 6 OBJ ¼ SWL ðQ, RT , RT , RT , RT , RT , RTÞð24bÞ 22 T2 1 2 3 4 5 6 OBJ ¼ SWL ðQ, RT , RT , RT , RT , RT , RTÞð24cÞ 23 T3 1 2 3 4 5 6 Model check Before performing the simulated annealing (SA) optimal simulation on muﬄers, an accuracy check of the mathem- atical model on a one-chamber muﬄer with one perforated plug tube is performed by Sullivan and Crocker. As indicated in Figure 5, comparisons between theoretical data and experimental data are in agreement. Therefore, the model of two muﬄers connected with multiple perforated plug tubes is adopted in the following optimization process. Case studies The noise reduction of an air compressor room within a space-constrained room is exempliﬁed and shown in Figure 1. The sound power level (SWL) inside the air compressor’s outlet is shown in Table 1 where the overall SWL reaches 107.4 dB. 10 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Figure 5. Performance of an acoustical element of a one perforated plug tube with the mean flow (M ¼ M ¼ 0.05, D ¼ 0.0493 (m), 1 2 1 Do ¼ 0.1016 (m), L ¼ L ¼ 0.1286 (m), L ¼ L ¼ 0.1 (m), L ¼ L ¼ 0.0 (m), t ¼ 0.081 (m), dh ¼ dh ¼ 0.00249 (m), C1 C2 1 2 A1 B2 1 2 Z ¼Z ¼ 0.037), experimental data is from Sullivan and Crocker. 1 2 Table 1. Unsilenced SWL of an air compressor inside a duct outlet. f (Hz) 125 250 500 1k 2k Overall SWL-dB(A) 90 95 104 102 100 107.4 It is obvious that the octave band frequencies (125 Hz, 250 Hz, 500 Hz, 1000 Hz, and 2000 Hz) have higher noise levels (90104 dB). To reduce the huge venting noise emitted from the air compressor’s outlet, the noise elimin- ation on the ﬁve primary noises using the two-chamber muﬄers connected with multiple perforated plug tubes (muﬄers AC) that disperse venting ﬂuid and decrease the secondary ﬂowing noise is considered. To obtain the best acoustical performance within a ﬁxed space, numerical assessments linked to a SA optimizer are applied. Before the minimization of a broadband noise is performed, a reliability check of the SA method by maximization of the STL at a targeted tone (200 Hz) is performed. As shown in Figure 1, the available space for a muﬄer is 0.6 m in width, 0.6 m in height, and 1.2 m in length. The ﬂow rate (Q) and thickness of a perforated tube (t) are preset at 0.01 (m /s) and 0.001 (m), respectively. The corresponding OBJ functions, space constraints, and the ranges of design parameters are summarized in equations (23)(24). Simulated annealing method Evolutionary Algorithms (EAs), which search for appropriate global solution in engineering problems, have, for two decades, been laboriously elaborated upon. It should also be noted that as a stochastic search method, SA is recognized as one of the best. Further, prior to SA optimization, the need to choose starting data which is necessary for the classical gradient methods of EPFM, IPFM and FDM has been eliminated. As a result, for use in a muﬄer’s shape optimization, SA is designated as an optimizer. The fundamental idea underlying SA was ﬁrst introduced by Metropolis et al. and further developed by Kirkpatrick et al. In the SA method where each point X of the search space is compared to a state of some physical system, the function F(X) to be minimized is interpreted as the internal energy of that state. And so, bringing the system from an arbitrary initial state to one with minimum possible energy is the desired objective. Because of annealing, which is a heating process that stabilizes a metal’s temperature while slowing cooling it, the particles remain close to the minimum energy state (see Table 2 for the pseudo-code that implements the SA heuristic). Now, in order to emulate the SA’s evolution, a new random solution (X ) is chosen from the neigh- borhood of the current solution. If, perchance, there is a negative change in the objective function ( F 0), Chiu 11 Table 2. The pseudo-code implementing the simulated annealing heuristic. T: ¼ To X: ¼ Xo F : ¼ F(X) k: ¼ 0 while n < iter Xn’ :¼ neighbor (Xn) if P < P then F(Xn’): ¼ F(Xn’)*w ; F ¼ F(Xn’)-F(Xn) max 1 i else F ¼ F(Xn’)-F(Xn) if F P then Xn’ ¼ Xn; T’n ¼ kk*Tn; n : ¼ n þ 1 else if random() < pb(F /C*Tn) then Xn’ ¼ Xn; T’n ¼ kk*Tn; n : ¼ n þ 1 return the new solution will be recognized as the new current solution with the transition property (pb(X )) of 1. If, on the other hand, there is not a negative change, the probability of a transition to the new state will be the function pbðÞ F=CT . As shown in equation (25), the new transition property (pb(X )) varied from 0 to 1 will be calculated using the Boltzmann’s factor (pb(X ) ¼ expðF=CTÞ), where C and T are the Boltzmann constant and the current temperature. 1, F 0 pbðÞ X ¼ ð25aÞ exp , f 4 0 CT F ¼ FXðÞFXðÞ ð25bÞ For the purpose of escaping the local optimum, SA also permits movement that results in solutions that are inferior (uphill movement) to the current solution. Hence, if the transition property (pb(X )) is greater than a random number of rand(0,1), the new inferior solution which results in a higher energy condition will be accepted; otherwise, it will be rejected. Each successful substitution of the new current solution will conduct to the decay of the current temperature as T ¼ kk T ð26Þ new old where kk is the cooling rate. This process is repeated until the preset (iter) of the outer loop is reached. Results and discussion Results The accuracy of the SA optimization depends on two kinds of SA parameters including kk (cooling rate) and iter 20,21 (maximum iteration). To achieve good optimization, the following parameters are varied step by step: kk (0.91, 0.93, 0.95, 0.97, 0.99); iter (25, 50, 100, 500, 1000, 2000). Two results of optimization (one, pure tone noises used for SA’s accuracy check; and the other, a broadband noise occurring in an air compressor room) are described below. Pure tone noise optimization Muffler A. Before dealing with a broadband noise for muﬄer AC, the STL’s maximization for muﬄer A with respect to a one-tone noise (200 Hz) is introduced for a reliability check on the SA method. By using equations (20a) and (23b), the maximization of the STL with respect to muﬄer A (a two-chamber muﬄer hybridized with one perforated plug tube) at the speciﬁed pure tone (200 Hz) was performed ﬁrst. As indicated in Table 3, ten sets of SA parameters are tried in the muﬄer’s optimization. Obviously, the optimal design data can be obtained from 12 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Table 3. Optimal STL for muffler A (equipped with one perforated plug tube) at various SA parameters (targeted tone of 200 Hz). SA parameter Design parameters Result iter kk RT1 RT2 RT3 RT4 RT5 RT6 STL dB 200 Hz 25 0.91 0.2326 0.6536 0.4536 0.4768 0.006391 0.2664 21.1 25 0.93 0.2075 0.5866 0.3866 0.4433 0.005512 0.2178 24.7 25 0.95 0.1889 0.5371 0.3371 0.4186 0.004862 0.1819 27.8 25 0.97 0.1537 0.4431 0.2431 0.3715 0.003628 0.1137 34.9 25 0.99 0.1472 0.4257 0.2257 0.3629 0.003400 0.1012 36.5 50 0.99 0.1372 0.3992 0.1992 0.3496 0.003051 0.08189 39.3 100 0.99 0.1300 0.3801 0.1801 0.3401 0.002801 0.06808 41.9 500 0.99 0.1171 0.3455 0.1455 0.3227 0.002347 0.04299 51.4 1000 0.99 0.1107 0.3286 0.1286 0.3143 0.002126 0.03075 60.8 2000 0.99 0.1042 0.3112 0.1112 0.3056 0.001897 0.01810 80.1 Note. The gray shades mean the SA parameters that are adjusted during the the optimization serarching process. Figure 6. STL with respect to various kk (muffler A: iter ¼ 25, target tone ¼ 200 Hz). the last set of SA parameters at (kk, iter) ¼ (0.99, 2000). Using the optimal design in a theoretical calculation, the optimal STL curves with respect to various SA parameters (kk, iter) are plotted and depicted in Figures 6 and 7. As revealed in Figures 6 and 7, the STL is precisely maximized at the desired frequency of 200 Hz. Consequently, the SA optimizer is reliable in the optimization process. Mufflers D and E. As indicated in Figure 8, to appreciate the acoustical performance of two-chamber muﬄers equipped with various tubes (muﬄer D: a muﬄer equipped with one non-perforated and internally extended tube; muﬄer E: a muﬄer equipped with one non-perforated tube), the shape optimization of muﬄers D and E at 200 Hz is also performed using the same SA parameters of (kk, iter) ¼ (0.99, 2000). The optimal design parameters for muﬄers A, D, and E are summarized in Table 4 and plotted in Figure 9. As indicated in Figure 9, the optimal STLs for muﬄers A, D, and E have been precisely located at the speciﬁed frequency of 200 Hz. As indicated in Table 4 and Figure 9, the STL of muﬄer A at 200 Hz reaches 80.1 dB. In addition, the STLs with respect to muﬄers D and E are 43.8 dB and 46.2 dB. Consequently, muﬄer A with a perforated plug tube reach is superior to other muﬄers. Moreover, the acoustical performances of both muﬄers D and E at a speciﬁed 200 Hz are almost the same. Broadband noise optimization Mufflers A–C. Similarly, adopting the same SA parameters of (kk, iter) ¼ (0.99, 2000) and doing the minimiza- tion of the SWL , SWL , and SWL with respect to muﬄers AC in broadband noise, the resulting design T-1 T-2 T-3 Chiu 13 Figure 7. STL with respect to various iter (muffler A: kk ¼ 0.99, target tone ¼ 200 Hz). Figure 8. The outline dimension of three kinds of mufflers hybridized with various tube (mufflers A, D, and E). Table 4. Comparison of the optimal STL for mufflers A, D, and E (targeted tone of 200 Hz). Muffler Design parameters Result Muffler A RT1 RT2 RT3 RT4 RT5 RT6 STL dB 200 Hz 0.1042 0.3112 0.1112 0.3056 0.001897 0.01810 80.1 Muffler D RT1* RT2* RT3* RT4* STL dB 200 Hz 0.1042 0.1112 0.3028 0.05140 43.8 Muffler E RT1** RT2** RT3** STL dB 200 Hz 0.1042 0.1112 0.3056 46.2 Note: Muffler D: RT1* ¼ D2; RT2* ¼ D3; RT3* ¼ L2; RT4* ¼ L4. Muffler E: RT1** ¼ D2; RT2** ¼ D3; RT3** ¼ L2. 14 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Figure 9. Comparison of the optimal STLs of three kinds of mufflers (mufflers A, D, and E) optimization at pure tone of 200 Hz. Table 5. Comparison of the minimized SWL of three kinds of mufflers (mufflers AC) (broadband noise). Design parameters Result Muffler type RT1 RT2 RT3 RT4 RT5 RT6 SWL – dB(A) Muffler A 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 46.5 Muffler B 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 52.3 Muffler C 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 58.9 Figure 10. Comparison of the optimal STLs of three kinds of mufflers (mufflers A, B, and C) and the original SWL. parameters are shown in Table 5. Using the optimal design parameters in a theoretical calculation, the optimal STL curves with respect to various muﬄers AC are plotted and depicted in Figure 10. As illustrated in Table 5, the resultant sound power levels with respect to three kinds of muﬄers have been reduced from 107.4 dB to 46.5 dB, 52.3 dB, and 58.9 dB. Mufflers F and G. As indicated in Figure 11, to appreciate the acoustical performance of two-chamber muﬄers equipped with various tubes (muﬄer F: a muﬄer equipped with four non-perforated and internally extended tubes; muﬄer G: a muﬄer equipped with four non-perforated tubes), the shape optimization of muﬄers F and G for a broadband noise is also performed using the same SA parameters of (kk, iter) ¼ (0.99, 2000). The optimal design parameters for muﬄers C, F, and G are summarized in Table 6 and plotted in Figure 12. As illustrated in Chiu 15 Figure 11. The outline dimensions of three kinds of mufflers hybridized with various tubes (mufflers C, F, and G). Table 6. Comparison of the optimal STL for mufflers C, F, and G (broadband noise). Muffler Design parameters Result Muffler C RT1 RT2 RT3 RT4 RT5 RT6 SWL – dB(A) 0.1042 0.3112 0.1112 0.3056 0.001897 0.0180 58.9 Muffler F RT1*** RT2*** RT3*** RT4*** SWL – dB(A) 0.1042 0.1112 0.3028 0.05140 68.2 Muffler G RT1**** RT2**** RT3**** SWL – dB(A)B 0.1042 0.1112 0.3056 80.3 Note: Muffler F: RT1*** ¼ D2; RT2*** ¼ D3; RT3*** ¼ L2; RT4* ¼ L4. Muffler G: RT1**** ¼ D2; RT2**** ¼ D3; RT3**** ¼ L2. Figure 12. Comparison of the optimal STLs of three kinds of mufflers (mufflers C, F, and G) and the original SWL. 16 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Table 6, the resultant sound power levels with respect to three kinds of muﬄers (muﬄers C, F, and G) have been reduced from 107.4 dB to 58.9 dB, 68.2 dB, and 80.3 dB. As indicated in Figure 12, muﬄer A with four perforated plug tubes is superior to other muﬄers. Moreover, muﬄer G with four non-perforated tubes has the worst acoustical performance. Discussion In order to decrease the secondary ﬂowing noise generated from the higher speed ﬂow, new muﬄer designs with multiple perforated plug tubes are presented. To achieve a suﬃcient optimization, the selection of the appropriate SA parameter set is essential. As indicated in Table 3, the best SA set of muﬄer A at the targeted pure tone noise of 200 Hz has been shown. The related STL curves with respect to various SA parameters are plotted in Figures 6 and 7. Figures 6 and 7 reveal that the predicted maximal value of the STL is located at the desired frequency. Similarly, in dealing with pure tone noise (200 Hz) in muﬄers D and E, the proﬁle shown in Figure 9 indicates that the maximum STLs of the muﬄers are also located at the speciﬁed frequency. In dealing with the broadband noise, the acoustical performance among three kinds of two-chamber muﬄers connected with multiple perforated plug tube (muﬄers A, B, and C) are shown in Table 5 and Figure 10. As can be observed in Table 5, the overall STL of the two-chamber muﬄer equipped with one perforated plug tube (muﬄer A) reaches 60.9 dB. However, the overall STLs of the two-chamber muﬄers equipped with two perforated plug tubes (muﬄer B) and four perforated plug tubes (muﬄer C) are 55.1 dB and 48.5 dB. 1:84c Here, for the broadband noise optimization in Table 5, the related cutoﬀ frequencies f5 for muﬄers A– C at the internally parallel perforated plug tubes (D3 ¼ 0.032427 m) and the outlet tube (D4 ¼ 0.1112) are 6145 Hz and 1800 Hz, respectively. The maximum frequency range for muﬄers A–C used in Figure 10 is 2000 Hz which closes to the valid frequency range of 1800 Hz. To appreciate the acoustical eﬀect of the perforated plug tubes (muﬄers AC), the internally extended non- perforated tubes (muﬄers D and F), and the non-perforated tubes (muﬄers E and G), a comparison of the muﬄers’ acoustical performance is carried out and shown in Tables 4 and 6 and Figures 9 and 12. As illustrated in Table 4 and Figure 9, when dealing with a pure tone noise, muﬄer A with a perforated plug tube is superior to those muﬄers having a non-perforated tube and an internally expanded non-perforated tube. Similarly, when dealing with a broadband noise, the overall STLs with respect to muﬄers C, F, and G shown in Table 6 and Figure 12 are 48.5 dB, 39.2 dB, and 27.1 dB. It is obvious that a two-chamber muﬄer equipped with multiple perforated plug tubes is also superior to these muﬄer equipped with multiple expended/ non-extended non-perforated tubes. Moreover, the results shown in Table 5 and Figure 10 indicate that the two-chamber muﬄer hybridized with fewer perforated plug tubes is superior to muﬄers that are equipped with more perforated plug tubes. It is obvious that for the muﬄers equipped with more perforated plug tubes, the acoustical performances of the acoustical element (II) between nodes 4 and 5 and acoustical element (III) between nodes 7 and 8 will largely decrease due to the decrement of the area ratio. Therefore, the overall noise reduction of the muﬄers with more perforated plug tubes will decrease. Conclusion It has been shown that two-chamber muﬄers hybridized with multiple perforated plug tubes can be easily and eﬃciently optimized within a limited space by using a decoupling technique, a plane wave theory, a four-pole transfer matrix, and a SA optimizer. As indicated in Table 3 and Figures 6 and 7, two kinds of SA parameters (kk and iter) play essential roles in the solution’s accuracy during SA optimization. Figures 6, 7, and 9 indicate that the tuning ability established by adjusting design parameters of muﬄers A, D, and E is reliable. Additionally, as indicated in Tables 4 and 6 and Figures 9 and 12, the acoustical eﬀect of various tubes has been discussed. Results reveal that the acoustical performance of the perforated plug tubes is superior to other tubes (internally extended non-perforated tubes and non-perforated tubes). Moreover, appropriate design parameters of three kinds of the muﬄers hybridized with multiple perforated plug tubes (muﬄers AC) has been assessed. As indicated in Table 5, the resultant SWL with respect to these muﬄers is 46.7 dB, 52.3 dB, and 58.9 dB. Obviously, the two-chamber muﬄer hybridized with fewer perforated plug tubes is superior to the muﬄers equipped with more perforated plug tubes. It can be seen that more perforated plug tubes installed between the two-chamber muﬄer will disperse the venting ﬂuid and reduce the secondary ﬂowing noise; however, the acoustical perform- ances of the acoustical element (II) between nodes 4 and 5 and acoustical element (III) between nodes 7 and 8 will decrease even though the parallel perforated plug tubes for the muﬄers have increased. Therefore, before choosing Chiu 17 the appropriate number of perforated plug tubes and performing the muﬄer’s shape optimization, presetting the maximum ﬂowing velocity within the muﬄer is required. Declaration of conflicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: The author acknowledges the ﬁnancial support of the National Science Council (NSC 99-2622-E-235-002-CC3, ROC). References 1. Davis DD, Stokes JM and Moorse L. Theoretical and experimental investigation of mufflers with components on engine muffler design. NACA Report, 1192, 1954. 2. Prasad MG and Crocker MJ. Studies of acoustical performance of a multi-cylinder engine exhaust muffler system. J Sound Vib 1983; 90: 491–508. 3. Prasad MG. A note on acoustic plane waves in a uniform pipe with mean flow. J Sound Vib 1984; 95: 284–290. 4. Sullivan JW and Crocker MJ. Analysis of concentric tube resonators having unpartitioned cavities. J Acoust Soc Am 1978; 64: 207–215. 5. Sullivan JW. A method of modeling perforated tube muffler components I: Theory I. J Acoust Soc Am 1979; 66: 772–778. 6. Sullivan JW. 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Shape optimization of one-chamber perforated plug/non-plug mufflers by simulated annealing method. Int J Numer Methods Eng 2008; 74: 1592–1620. 13. Chiu MC. Shape optimisation of multi-chamber mufflers with plug-inlet tube on a venting process by genetic algorithms. Appl Acoust 2010; 71: 495–505. 14. Chiu MC. Optimal design of multi-chamber mufflers hybridized with perforated intruding inlets and resonated tube using simulated annealing. ASME J Vib Acoust 2010; 132: 9. 15. Chiu MC. GA optimization on a venting system with three-chamber hybrid mufflers within a constrained back pressure and space. ASME J Vib Acoust 2012; 134: 11. 16. Munjal ML. Acoustics of ducts and mufflers with application to exhaust and ventilation system design. New York: John Wiley & Sons, 1987. 17. Chang YC, Yeh LJ, Chiu MC, et al. Shape optimization on constrained single-layer sound absorber by using GA method and mathematical gradient methods. J Sound Vib 2005; 286: 941–961. 18. Metropolis A, Rosenbluth W, Rosenbluth MN, et al. Equation of static calculations by fast computing machines. J Chem Phys 1953; 21: 1087–1092. 19. Kirkpatrick S, Gelatt CD and Vecchi MP. Optimization by simulated annealing. Science 1983; 220: 671–680. 20. Chiu MC. Numerical assessment of hybrid mufflers on a venting system within a limited back pressure and space using simulated annealing. J Low Freq Noise Vib Active Control 2011; 30: 247–275. 21. Chiu MC. Acoustical treatment of multi-tone broadband noise with hybrid side-branched mufflers using a simulated annealing method. J Low Freq Noise Vib Active Control 2014; 33: 79–112. 22. Rao KN and Munjal ML. Experimental evaluation of impedance of perforates with grazing flow. J Sound Vib 1986; 108: 283–295. Appendix 1 Transfer matrix of expanded perforated plug duct As indicated in Figure 13, the expansion perforated duct is composed of an outer duct and an inner open-ended (one end only) perforated tube. Based on Sullivan and Crocker’s derivation, the continuity equations and momentum equations with respect to the inner and outer tubes at nodes 5 and 5 a are 18 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Figure 13. Acoustical field of an expanded perforated plug duct. Inner tube Continuity equation @ @u 4 @ 5 5 o 5a V þ þ u þ ¼ 0 ð27Þ @x @x D @t Momentum equation @ @ @p þ V u þ ¼ 0 ð28Þ o 5 @t @x @x Outer tube Continuity equation @u 4D @ 5a 1 o 5a u þ ¼ 0 ð29Þ 2 2 @x D D @t 2 3 @u @p 5a 5a þ ¼ 0 ð30Þ @t @x Assuming that the acoustic wave is a harmonic motion under the isentropic processes in ducts, then 2 j!t pxðÞ , t ¼ ðÞ x c e ð31Þ The acoustic impedance of the perforation ( c ) is expressed as o o p ðÞ x p ðÞ x 5 5a c ¼ ð32Þ o o uxðÞ For perforates with a stationary medium ¼½ 0:006 þ jkðÞ t þ 0:75dH = ð33aÞ 1 1 For perforates with a grazing ﬂow ¼½ 0:514D M =ðÞ L þ j0:95ktðÞ þ 0:75dH = ð33bÞ 3 5 c1 1 1 1 Chiu 19 where dH is the diameter of a perforated hole on an inner tube, t is the thickness of the inner perforated tube, and is the porosity of the perforated tube. Substituting equations (31) and (32) for equations (27)(30) and eliminating u and u yield 5 5A d d 2 2 1 M 2jM k þ k p 5 5 dx dx ð34aÞ 4 d M þ jkðÞ p p ¼ 0 5 5 5a D dx d 4D þ k p þ j ðÞ p p ¼ 0 ð34bÞ 5a 5 5a 2 2 2 dx D D 2 3 Alternatively, equations (34a) and (34b) can also be expressed as 00 0 0 p þ p þ p þ p þ p ¼ 0 ð35aÞ 1 2 5 3 4 5a 5 5 5a 0 00 0 p þ p þ p þ p þ p ¼ 0 ð35bÞ 5 6 7 8 5a 5 5 5a 5a dp dp 5 5a 0 0 Let p ¼ ¼ y , p ¼ ¼ y , p ¼ y , p ¼ y ð36Þ 1 2 5 3 5a 4 5 5a dx dx According to equations (35) and (36), the new matrix between {y } and {y} is 2 3 2 32 3 y y 1 3 2 4 1 6 7 6 76 7 y y 5 7 6 8 2 6 2 7 6 76 7 6 7 ¼ 6 76 7 ð37aÞ 4 5 4 54 5 y 100 0 y y 010 0 y or y ¼½ H y ð37bÞ Let y ¼½ fg ð38Þ Plugging equations (38) into (37) and then multiplying ½ by both sides yield ¼½ Kfg ð39aÞ where 2 3 00 0 6 7 0 00 6 7 ½ K ¼½ ½ H½ ¼ ; : 6 7 i 4 5 00 0 ð39bÞ the eigen value of½ H;½ : the model matrix 4x4 formed by four sets of eigen vectors of½ H 4x1 4x4 The related solution of equation (39) then obtained is ¼ c e ð40Þ i i 20 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Using equations (28), (30), (39), and (40), the relationship of the acoustic pressure and the acoustical particle velocity yields 2 3 2 32 3 p ðÞ x E E E E c 5 1,1 1,2 1,3 1,4 1 6 7 6 76 7 p ðÞ x E E E E c 6 5a 7 6 2,1 2,2 2,3 2,4 76 2 7 6 7 ¼ 6 76 7 ð41Þ 4 5 4 54 5 c u ðÞ x E E E E c o o 5 3,1 3,2 3,3 3,4 3 c u ðÞ x E E E E c o o 5a 4,1 4,2 4,3 4,4 4 Plugging x ¼ 0 and x ¼ L into equation (41) and rearranging it, the resultant relationship of acoustic pressure c1 and particle velocity between x ¼ 0 and x ¼ L becomes c1 2 3 2 3 p ðÞ 0 pðÞ L 5 5 c1 6 7 6 7 p ðÞ 0 p ðÞ L 5a 5a c1 6 7 6 7 ¼½ ð42aÞ 6 7 6 7 4 5 4 5 c u ðÞ 0 c uðÞ L o o 5 o o 5 c1 c u ðÞ 0 c u ðÞ L o o 5a o o 5a c1 where ½ ¼½ E0ðÞ½ EðÞ L ð42bÞ c1 To obtain the transform matrix between the inlet (x ¼ 0) and the outlet (x ¼ L ) of the inner tubes, the two c1 boundary conditions for the outer tuber at x ¼ 0 and x ¼ L are calculated and listed below: c1 p ðÞ 0 5a ¼j c cotðÞ kL ð43aÞ o o a1 u ðÞ 0 5a pðÞ L 2 c1 ¼j c cotðÞ kL ð43bÞ o o b1 uðÞ L 2 c1 Plugging equation (43) into equation (42) and developing them, the transfer matrix deduced is p TPPE TPPE p 5 1,1 1,2 6a ¼ ð44aÞ c u TPPE TPPE c u o o 5 2,1 2,2 o o 6a where p ¼ p ðÞ 0 ; u ¼ u ðÞ 0 ; p ¼ p ðÞ L ; u ¼ u ðÞ L ; 5 5 5 5 6a 5a c1 6a 5a c1 j cotðÞ kL 2,2 4,2 a1 TPPE ¼ þ c j c cotðÞ kL ; 1,1 1,1 1,3 o o 1,1 o o b1 j cotðÞ kL 2,4 4,4 a1 TPPE ¼ þ c j c cotðÞ kL ; 1,2 1,4 1,3 o o 1,1 o o b1 j cotðÞ kL 2,2 4,2 a1 ð44bÞ TPPE ¼ þ c c cotðÞ kL ; 2,1 3,2 3,3 o o 3,1 o o b1 j cotðÞ kL 2,4 4,4 a1 TPPE ¼ þ c c cotðÞ kL ; 2,2 3,4 3,3 o o 3,1 o o b1 j cotðÞ kL þ cotðÞ kL cotðÞ kL 4,3 a1 4,1 a1 b1 G ¼ c E o o þ j cotðÞ kL 2,3 2,1 b1 Chiu 21 Appendix 2 Transfer matrix of contracted perforated plug duct As indicated in Figure 14, the contracted perforated duct is composed of an inner open-ended (one end only) perforated tube and an outer contracted duct. As deduced in Appendix 1, the continuity equations and momentum equations with respect to the inner and outer tubes at nodes 6 and 6 b are Inner tube Continuity equation @ @u 4 @ 6 6 o 6a V þ þ u þ ¼ 0 ð45Þ @x @x D @t Momentum equation @ @ @p þ V u þ ¼ 0 ð46Þ o 6 @t @x @x Outer tube Continuity equation @u 4D @ 6a 3 o 6a u þ ¼ 0 ð47Þ 2 2 @x D D @t 2 3 @u @p 6a 6a þ ¼ 0 ð48Þ @t @x Likewise, as derived in equations (49)(55), the new matrix between {y } and {y} is 2 3 2 32 3 y y 1 3 2 4 1 6 0 7 6 76 7 y y 5 7 6 8 2 6 2 7 6 76 7 ¼ ð49aÞ 6 7 6 76 7 4 5 4 54 5 y 100 0 y y 010 0 y or y ¼½ y ð49bÞ Figure 14. Acoustic field of a contracted perforated plug duct. 22 Journal of Low Frequency Noise, Vibration and Active Control 36(1) where dp dp 6 6a 0 0 y ¼ p ¼ , y ¼ p ¼ , y ¼ p , y ¼ p ð49cÞ 1 2 3 6 4 6a 6 6a dx dx Let y ¼½ fg ð50aÞ which is 2 3 2 32 3 dp =dx 6 1,1 1,2 1,3 1,4 1 6 7 6 76 7 dp =dx 6a 2,1 2,2 2,3 2,4 2 6 7 6 76 7 ¼ ð50bÞ 6 7 6 76 7 4 5 4 54 5 6 3,1 3,2 3,3 3,4 3 6a 4,1 4,2 4,3 4,4 4 ½ is the model matrix formed by four sets of eigen vectors of½ HH . 4x4 4x1 4x4 Plugging equation (50) into (49) and then multiplying½ by both sides yield ¼½ KKfg ð51aÞ where 2 3 6 7 0 00 6 2 7 ½ KK ¼½ ½ HH½ ¼ 6 7; 4 5 00 0 ð51bÞ : the eigen value of½ HH Obviously, equation (51) is a decoupled equation. The related solution then obtained is ¼ cc e ð52Þ i i Using equations (46), (48), (51), and (52), the relationship of acoustic pressure and particle velocity obtained is 2 3 2 32 3 p ðÞ x EE EE EE EE cc 6 1,1 1,2 1,3 1,4 1 6 7 6 76 7 p ðÞ x EE EE EE EE cc 6a 2,1 2,2 2,3 2,4 2 6 7 6 76 7 ¼ ð53Þ 6 7 6 76 7 4 5 4 EE EE EE EE 54 5 c u ðÞ x cc o o 6 3,1 3,2 3,3 3,4 3 EE EE EE EE c u ðÞ x cc o o 6a 4,1 4,2 4,3 4,4 4 Taking two cases of x ¼ 0 and x ¼ L and rearranging them, the resultant relationship of the acoustic pressure c2 and the acoustic particle velocity between x ¼ 0 and x ¼ L becomes c2 2 3 2 3 p ðÞ 0 pðÞ L 6 6 c2 6 7 6 7 p ðÞ 0 p ðÞ L 6 6a 7 6 6a c2 7 6 7 ¼½ 6 7 ð54aÞ 4 5 4 5 c u ðÞ 0 c uðÞ L o o 6 o o 6 c2 c u ðÞ 0 c u ðÞ L o o 6a o o 6a c2 where ½ ¼½ EEðÞ 0½ EEðÞ L ð54bÞ c2 Chiu 23 To obtain the transform matrix between inlet (x ¼ 0) and outlet (x ¼ L ) of the inner tubes, two boundary c2 conditions for the outer tube at x ¼ 0 and x ¼ L are taken into calculation and listed below: c2 p ðÞ 0 ¼j c cotðÞ kL ð55aÞ o o a2 u ðÞ 0 p ðÞ L 6a c2 ¼j c cotðÞ kL ð55bÞ o o b2 u ðÞ L 6a c2 By plugging equations (55) into equation (54) and developing them, the transfer matrix deduced is p TPPC TPPC p 6a 1,1 1,2 7 ¼ ð56aÞ c u TPPC TPPC c u o o 6a 2,1 2,2 o o 7 where p ¼ p ðÞ 0 ; u ¼ u ðÞ 0 ; p ¼ pðÞ L ; u ¼ uðÞ L ; 6a 6a 6a 6a 7 6 c2 7 6 c2 j cotðÞ kL 1,1 3,1 a2 TPPC ¼ þ c j c cotðÞ kL ; 1,1 2,1 2,4 o o 2,2 o o b2 j cotðÞ kL 1,3 3,3 a2 TPPC ¼ þ c j c cotðÞ kL ; 1,2 2,3 2,4 o o 2,2 o o b2 j cotðÞ kL 1,1 3,1 a2 ð56bÞ TPPC ¼ þ c c cotðÞ kL ; 2,1 4,1 4,4 o o 4,2 o o b2 j cotðÞ kL 1,3 3,3 a2 TPPC ¼ þ c c cotðÞ kL ; 2,2 4,3 4,4 o o 4,2 o o b2 j cotðÞ kL þ cotðÞ kL cotðÞ kL 3,4 a2 3,2 a2 b2 G ¼ c c o o þ j cotðÞ kL 1,4 1,2 b2 Appendix 3 Transfer matrix of two parallel perforated plug tubes As indicated in Figure 3, the acoustical four-pole matrix between node 5 and 7 is deduced by combining (1) (1) equation (44) and equation (56) yields p TPPE TPPE TPPC TPPC p 5ð1Þ 1,1 1,2 1,1 1,2 7ð1Þ c u c u TPPE TPPE TPPC TPPC o o 5ð1Þ 2,1 2,2 2,1 2,2 o c 7ð1Þ ð57Þ TPP1 TPP1 p 7ð1Þ 1,1 1,2 TPP1 TPP1 c u o o 71ðÞ 2,1 2,2 Developing equation (57) yields p ¼ TPP1 p þ TPP1 c u ð58aÞ 5ð1Þ 1,1 7ð1Þ 1,2 o c 71ðÞ c u ¼ TPP1 p þ TPP1 c u ð58bÞ o o 5ð1Þ 2,1 7ð1Þ 2,2 o c 7ð1Þ Similarly, the acoustical four-pole matrix between node 5 and 7 is (2) (2) p TPP1 TPP1 p 5ð2Þ 7ð2Þ 1,1 1,2 ¼ ð59Þ c u TPP1 TPP1 c u o o 5ð2Þ 2,1 2,2 o o 7ð2Þ 24 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Developing equation (59) yields p ¼ TPP1 p þ TPP1 c u ð60aÞ 5ð2Þ 1,1 7ð2Þ 1,2 o c 7ð2Þ c u ¼ TPP1 p þ TPP1 c u ð60bÞ o o 5ð2Þ 2,1 7ð2Þ 2,2 o c 7ð2Þ Combining equation (58a) and equation (60a) yields p þ p ¼ TPP1 p þ p þ TPP1 c u þ u ð61Þ 5ð1Þ 5ð2Þ 1,1 7ð1Þ 7ð2Þ 1,2 o c 7ð1Þ 7ð2Þ Likewise, combining equation (58b) and equation (60b) yields c u þ u ¼ TPP1 p þ p þ TPP1 c u þ u ð62Þ o o 5ð1Þ 5ð2Þ 2,1 7ð1Þ 7ð2Þ 2,2 o c 7ð1Þ 7ð2Þ where p ¼ p ¼ p ; p ¼ p ¼ p ; 5 5ð1Þ 5ð2Þ 7 7ð1Þ 7ð2Þ ð63Þ u ¼ u þ u ; u ¼ u þ u 5 5ð1Þ 5ð2Þ 7 7ð1Þ 7ð2Þ Plugging equation (63) into equations (61) and (62) yields 2 p ¼ 2 TPP1 p þ TPP1 c u ð64aÞ 5 1,1 7 1,2 o c 7 c u ¼ 2 TPP1 p þ TPP1 c u ð64bÞ o o 5 2,1 7 2,2 o c 7 Rearranging equation (64) in a matrix form, the equivalent four-pole matrix between nodes 5 and 7 shown in Figure 4 is "# p p TPP1 TPP1 5 1,1 1,2 7 ¼ ð65Þ c u c u 2 TPP1 TPP1 o o 5 2,1 2,2 o o 7 Appendix 4 Transfer matrix of four parallel perforated plug tubes As indicated in Figure 3, muﬄer C’s acoustical four-pole matrix between node 5 and 7 is (1) (1) p TPP1 TPP1 p 5ð1Þ 1,1 1,2 7ð1Þ ¼ ð66Þ c u c u TPP1 TPP1 o o 5ð1Þ 2,1 2,2 o o 7ð1Þ Developing equation (66) yields p ¼ TPP1 p þ TPP1 c u ð67aÞ 5ð1Þ 1,1 7ð1Þ 1,2 o c 7ð1Þ c u ¼ TPP1 p þ TPP1 c u ð67bÞ o o 5ð1Þ 2,1 7ð1Þ 2,2 o c 7ð1Þ Similarly, the acoustical four-pole matrices between node 5 and 7 ,node5 and 7 , and node 5 and 7 ,is (2) (2) (3) (3) (4) (4) p TPP1 TPP1 p 5ð2Þ 1,1 1,2 7ð2Þ ¼ ð68aÞ c u TPP1 TPP1 c u o o 5ð2Þ o o 7ð2Þ 2,1 2,2 Chiu 25 p TPP1 TPP1 p 5ð3Þ 1,1 1,2 7ð3Þ ¼ ð68bÞ c u TPP1 TPP1 c u o o 5ð3Þ 2,1 2,2 o o 7ð3Þ p TPP1 TPP1 p 5ð4Þ 1,1 1,2 7ð4Þ ¼ ð68cÞ c u c u TPP1 TPP1 o o 5ð4Þ 2,1 2,2 o o 7ð4Þ Developing equations (68a)(68c) yields p ¼ TPP1 p þ TPP1 c u ð69aÞ 5ð2Þ 1,1 7ð2Þ 1,2 o c 7ð2Þ c u ¼ TPP1 p þ TPP1 c u ð69bÞ o o 5ð2Þ 2,1 7ð2Þ 2,2 o c 7ð2Þ p ¼ TPP1 p þ TPP1 c u ð69cÞ 5ð3Þ 1,1 7ð3Þ 1,2 o c 7ð3Þ c u ¼ TPP1 p þ TPP1 c u ð69dÞ o o 5ð3Þ 2,1 7ð3Þ 2,2 o c 7ð3Þ p ¼ TPP1 p þ TPP1 c u ð69eÞ 5ð4Þ 1,1 7ð4Þ 1,2 o c 7ð4Þ c u ¼ TPP1 p þ TPP1 c u ð69fÞ o o 5ð4Þ 2,1 7ð4Þ 2,2 o c 7ð4Þ Combining equation (67a) and equations (69a), (69c), and (69e) yields p þ p þ p þ p ¼ TPP1 p þ p þ p þ p 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ 1,1 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ ð70Þ þ TPP1 c u þ u þ u þ u 1,2 o c 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ Likewise, combining equation (67b) and equation (69b), (69d), and (69f) yields c u þ u þ u þ u ¼ TPP1 p þ p þ p þ p o o 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ 2,1 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ ð71Þ þ TPP1 c u þ u þ u þ u 2,2 o c 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ where p ¼ p ¼ p ¼ p ¼ p ; 5 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ p ¼ p ¼ p ¼ p ¼ p ; 7 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ ð72Þ u ¼ u þ u þ u þ u ; 5 5ð1Þ 5ð2Þ 5ð3Þ 5ð4Þ u ¼ u þ u þ u þ u 7 7ð1Þ 7ð2Þ 7ð3Þ 7ð4Þ Plugging equation (72) into equations (70) and (71) yields 4 p ¼ 4 TPP1 p þ TPP1 c u ð73aÞ 5 1,1 6 1,2 o c 7 c u ¼ 4 TPP1 p þ TPP1 c u ð73bÞ o o 5 2,1 7 2,2 o c 7 Rearranging equation (73) in a matrix form, the equivalent four-pole matrix between nodes 5 and 7 shown in Figure 4 is "# p p TPP1 TPP1 5 1,1 1,2 7 ¼ ð74Þ c u c u 4 TPP1 TPP1 o o 5 2,1 2,2 o o 7 26 Journal of Low Frequency Noise, Vibration and Active Control 36(1) Appendix 5 Notation C sound speed (m s ) x x i i c , c c , c cc , cc cc , cc coefficients in function c e and cc e 1 2, 3 4, 1 2, 3 4 i i dH the diameter of a perforated hole on the inner perforated tube (m) D diameter of the i-th perforated tubes (m) D diameter of the outer tube (m) f cyclic frequency (Hz) iter maximum iteration pﬃﬃﬃﬃﬃﬃﬃ j imaginary unit (¼ 1) k wave number (¼ ) kk cooling rate L , L lengths of perforate part for the perforated plug ducts (m) c1 c2 L , L , L , L lengths of non-perforate part for the perforated plug ducts (m) a1 a2 b1 b2 L length of the i-th perforated tubes (m) L total length of the muffler (m) M mean flow Mach number at the i-th node OBJ objective function (dB) p acoustic pressure at the i-th node (Pa) pb(T) transition probability 3 1 Q volume flow rate of venting gas (m s ) RT1, RT2, RT3, RT4, RT5, RT6 design parameter for mufflers AC RT1*, RT2*, RT3*, RT4* design parameter for mufflers D and F RT1**, RT2**, RT3** design parameter for mufflers E and G S section area at the i-th node(m ) STL sound transmission loss (dB) SWLO unsilenced sound power level inside the muffler’s inlet (dB) SWL overall sound power level inside the muffler’s output (dB) t the thickness of the i-th inner perforated tube (m) TS1 , TS2 , TS3 , TS4 components of four-pole transfer matrices for an acoustical mechanism with ij ij ij ij straight ducts TPP1 components of a four-pole transfer matrix for an acoustical mechanism with a ij perforated plug tube TSC1 , TSC2 components of a four-pole transfer matrix for an acoustical mechanism with a ij ij contracted perforated intruding tube TSE1 , TSE2 components of a four-pole transfer matrix for an acoustical mechanism with ij ij an expanded perforated intruding tube T components of a four-pole transfer system matrix ij T current temperature ( C) T initial temperature ( C) u acoustic particle velocity at the i-th node (m s ) u acoustical particle velocity passing through a perforated hole from the i-th ij node to the j-th node (m s ) V mean flow velocity at the inner perforated tube (m s ) the i-th eigen value of½ H the i-th eigen value of½ HH specific acoustical impedance of the inner perforated tube Z the porosity of the i-th inner perforated tube. 3 1 acoustical density at the i-th node (m s ) ½ the model matrix formed by four sets of eigen vectors of½ H 4x1 4x4 4x4 ½ the model matrix formed by four sets of eigen vectors of½ HH 4x1 4x4 4x4

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"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Mar 1, 2017

Keywords: Multiple parallel perforated plug tube; two-chamber; simulated annealing

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Numerical assessment of two-chamber mufflers hybridized with multiple parallel perforated plug tubes using simulated annealing method:

Numerical assessment of two-chamber mufflers hybridized with multiple parallel perforated plug tubes using simulated annealing method:

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Numerical assessment of two-chamber mufflers hybridized with multiple parallel perforated plug tubes using simulated annealing method:

Numerical assessment of two-chamber mufflers hybridized with multiple parallel perforated plug tubes using simulated annealing method:

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