Numerical assessment of two-chamber mufflers ...

Original Article

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

Journal of Low Frequency Noise, Vibration and Active Control 2017, Vol. 36(1) 3–26 ! The Author(s) 2017 DOI: 10.1177/0263092317693477 journals.sagepub.com/home/lfn

Min-Chie Chiu

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 nonperforated tubes), mufflers with internally extended non-perforated tubes and non-perforated tubes have been evaluated. 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.1 Studies of simple expansion mufflers based on plane wave theory were also completed.2,3 In order to increase the acoustical performance of lower frequency sound energy, Sullivan and Crocker4 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 coupled equations.5,6 Jayaraman and Yam,7 presuming an unreasonable inner and outer duct, used a method for finding an analytical solution. Additionally, Rao and Munjal8 provided a generalized decoupling method. Now, regarding the flowing effect, Peat,9 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: [email protected] 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).

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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 internally 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 numerical method such as the finite element method10,11 will be predictably time-consuming when doing a broadband 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 work12–15 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) As derived in previous work,12–15 for the acoustical element (I) shown in Figure 2, the four-pole matrix between nodes 1 and 2 is

p1 o co u1

¼ f1 ðLi , Di , Mi Þ

TS11,1

TS11,2

TS12,1

TS12,2

p2 o c o u2

ð1Þ

For the acoustical element (II), the four-pole matrix between nodes 2 and 3 is

p2 o co u2

¼

TSE11,1

TSE11,2

TSE12,1

TSE12,2

p3

ð2Þ

o co u3

Similarly, the four-pole matrix between nodes 3 and 4 is

p3 o co u3


TS21,1 TS22,1

TS21,2 TS22,2

Figure 1. Noise elimination on an air compressor room within a limited space.

p4 o c o u4

ð3Þ

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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).

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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

p4

¼

o co u4

TSC11,1

TSC11,2

TSC12,1

TSC12,2

p5

ð4Þ

o c o u5

As shown in previous work12–15 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

p5

¼

o co u5

p6a o co u6a

TPPE11,1

TPPE11,2

TPPE12,1

TPPE12,2

¼

TPPC11,1 TPPC12,1

TPPC11,2 TPPC12,2

p6a

ð5Þ

o co u6a

p7 o co u7

ð6Þ

Combining equations (5)–(6), the four-pole matrix between nodes 5 and 7 yields

p5

¼

o co u5

TPP11,1

TPP11,2

TPP12,1

TPP12,2

p7

ð7Þ

o co u7

Likewise, the four-pole matrix between nodes 7 and 8 for a sudden expanded duct is

p7 o co u7

¼

TSE21,2 TSE22,2

TSE21,1 TSE22,1

p8

ð8Þ

o co u8

The four-pole matrix between nodes 8 and 9 in a straight duct is

p8 o co u8


TS31,1

TS31,2

TS32,1

TS32,2

p9 o c o u9

ð9Þ

The four-pole matrix between nodes 9 and 10 for a sudden contracted duct is

p9 o co u9

¼

TSC21,1

TSC21,2

TSC22,1

TSC22,2

p10

ð10Þ

o co u10

Moreover, the four-pole matrix between nodes 10 and 11 in a straight duct is

p10 o co u10


TS41,1 TS42,1

TS41,2 TS42,2

p11 o co u11

ð11Þ

The total transfer matrix assembled by multiplication is

p1 o co u1

¼ f1 ðLi , Di , Mi Þ f2 ðLi , Di , Mi Þf3 ðLi , Di , Mi Þ f4 ðLi , Di , Mi Þ f5 ðLi , Di , Mi Þ

TS11,1

TS12,1 TS12,2 TSE12,1 TSE12,2 TS22,1 TS22,2 TSC11,1 TSC11,2 TPP11,1 TPP11,2 TSE21,1 TSE21,2

TSC12,1 TS31,1 TS32,1 TS41,1 TS42,1

TS11,2

TSE11,1

TSE11,2

TS21,1

TSC12,2 TPP12,1 TPP12,2 TS31,2 TSC21,1 TSC21,2 TS32,2 TSC22,1 TSC22,2 TS41,2 p11 TS42,2 o co u11

TS21,2

TSE22,1

TSE22,2

ð12Þ

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A simplified form is expressed in a matrix as

p1 o co u1

¼

T11

T12

T21

T22

p11 o co u11

ð13Þ

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(1)7(1) and nodes 5(2)7(2) can be combined to an equivalent matrix

p5 o co u5

"

¼

TPP11,1

1 2

2 TPP12,1

TPP11,2

#

TPP12,2

p7 o co u7

ð14Þ

where [TPP1i,j] is the four-pole matrix for a single perforated plug tube. The equivalent acoustical field of muffler B is shown in Figure 4. Consequently, the total transfer matrix assembled by multiplication is

p1 o co u1

¼ f1 ðLi , Di , Mi Þ f2 ðLi , Di , Mi Þ f3 ðLi , Di , Mi Þ f4 ðLi , Di , Mi Þ f5 ðLi , Di , Mi Þ

TS11,1

TS11,2

TS12,1

TS12,2

TSC11,1 TSC12,1 TSE21,1

TSE11,2

TS21,1

TS21,2

TSE12,1 TSE12,2 TS22,1 TS22,2 # " 1 TSC11,2 TPP11,1 2 TPP11,2 TSC12,2 2 TPP12,1 TPP12,2 TSE21,2 TS31,1 TS31,2 TSC21,1 TSC21,2

TSE22,1 TSE22,2 TS41,1 TS41,2 TS42,1

TSE11,1

TS42,2

TS32,1 p11

TS32,2

TSC22,1

ð15Þ

TSC22,2

o co u11


p1 o co u1

¼

T 11

T 12

T 21

T 22

p11 o co u11

Figure 4. Equivalent acoustical field for a muffler hybridized with perforated plug tubes (mufflers B and C).

ð16Þ

8


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(1)–7(1), 5(2)–7(2), 5(3)– 7(3), and 5(4)–7(4) nodes can be combined to an equivalent matrix

p5 o co u5

"

¼

TPP11,1

1 4

4 TPP12,1

TPP11,2

#

TPP12,2

p7 o c o u7

ð17Þ

where [TPP1i,j] is the four-pole matrix for a single perforated tube. 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

p1 o c o u1

¼ f1 ðLi , Di , Mi Þ f2 ðLi , Di , Mi Þ f3 ðLi , Di , Mi Þ f4 ðLi , Di , Mi Þ f5 ðLi , Di , Mi Þ

TS11,1

TS11,2

TS12,1

TS12,2

TSE11,2

TS21,1

TS21,2

TSE12,1 TSE12,2 TS22,1 TS22,2 # " 1 TSC11,2 TPP11,1 TPP1 1,2 4 TSC12,2 4 TPP12,1 TPP12,2 TSE21,2 TS31,1 TS31,2 TSC21,1 TSC21,2

TSC11,1 TSC12,1 TSE21,1

TSE22,1 TSE22,2 TS41,1 TS41,2 TS42,1

TSE11,1

TS42,2

TS32,1 p11

TS32,2

TSC22,1

ð18Þ

TSC22,2

o co u11


p1 o c o u1

¼

T 11 T 21

T 12 T 22

p11

o co u11

ð19Þ

Overall sound power level The STL of mufflers A–C are defined as16 T11 þ T12 þ T21 þ T22 S1 STL1 ðQ, f, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ ¼ 20 log þ 10 log 2 S11

ð20aÞ

T11 þ T S1 12 þ T21 þ T22 þ 10 log STL2 ðQ, f, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ ¼ 20 log 2 S11

ð20bÞ

T11 þ T S1 12 þ T21 þ T22 þ 10 log STL3 ðQ, f, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ ¼ 20 log 2 S11

ð20cÞ

where RT1 ¼ D2; RT2 ¼ D3=D2; RT3 ¼ D4; RT4 ¼ L3; RT5 ¼ dH; RT6 ¼

ð20dÞ

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The silenced octave sound power level emitted from a muffler’s outlet is SWLi ¼ SWLOðfi Þ STLðfi Þ

ð21Þ

where SWLOð fi Þ is the original SWL at the inlet of a muffler (or pipe outlet), and fi is the relative octave band frequency; STLð fi Þ is the muffler’s STL with respect to the relative octave band frequency (fi); SWLi is the silenced SWL at the outlet of a muffler with respect to the relative octave band frequency. Finally, the overall SWLT silenced by a muffler at the outlet is ( SWLTK ¼ 10 log ¼ 10

n X

) SWLi =10

10

i¼1 n ½SWLOðf Þ 1 log 10STLK ðf1 Þ=10

þ

½SWLOðf2 Þ 10STLK ðf2 Þ=10

þ

½SWLOðf3 Þ 10STLK ðf3 Þ=10

þ þ

½SWLOðfn Þ 10STLK ðfn Þ=10

o

ð22Þ

Objective function STL maximization for a tone (f) noise. The objective functions in maximizing the STL at a pure tone (f) are OBJ11 ¼ STL1 ðQ, f, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ

ð23aÞ

OBJ12 ¼ STL2 ðQ, f, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ

ð23bÞ

OBJ13 ¼ STL3 ðQ, f, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ

ð23cÞ

The related ranges of the parameters are Q ¼ 0:01 m3 =s ; Lo ¼ 1:2 ðmÞ; Do ¼ 0:6 ðmÞ; L1 ¼ 0:05 ðmÞ; L5 ¼ 0:05 ðmÞ; D1 ¼ 0:2 ðmÞ; RT1 : ½0:1, 0:25; RT2 : ½0:3, 0:7; RT3 : ½0:1, 0:5; RT4 : ½0:3, 0:5;

ð23dÞ

RT5 : ½0:00175, 0:007; RT6 : ½0:01, 0:3

SWL minimization for a broadband noise. To minimize the overall SWLT, the objective function is OBJ21 ¼ SWLT1 ðQ, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ

ð24aÞ

OBJ22 ¼ SWLT2 ðQ, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ

ð24bÞ

OBJ23 ¼ SWLT3 ðQ, RT1 , RT2 , RT3 , RT4 , RT5 , RT6 Þ

ð24cÞ

Model check Before performing the simulated annealing (SA) optimal simulation on mufflers, an accuracy check of the mathematical model on a one-chamber muffler with one perforated plug tube is performed by Sullivan and Crocker.4 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.

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Figure 5. Performance of an acoustical element of a one perforated plug tube with the mean flow (M1 ¼ M2 ¼ 0.05, D1 ¼ 0.0493 (m), Do ¼ 0.1016 (m), LC1 ¼ LC2 ¼ 0.1286 (m), L1 ¼ L2 ¼ 0.1 (m), LA1 ¼ LB2 ¼ 0.0 (m), t ¼ 0.081 (m), dh1 ¼ dh2 ¼ 0.00249 (m), Z1 ¼ Z2 ¼ 0.037), experimental data is from Sullivan and Crocker.4

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 elimination 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 (m3/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 FDM17 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.18 and further developed by Kirkpatrick et al.19 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 (X0 ) is chosen from the neighborhood of the current solution. If, perchance, there is a negative change in the objective function ( F 0),

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11 Table 2. The pseudo-code implementing the simulated annealing heuristic. T: ¼ To X: ¼ Xo F : ¼ F(X) k:¼0 while n < iter Xn’ :¼ neighbor (Xn) if Pmax < P1 then F(Xn’): ¼ F(Xn’)*wi; F ¼ F(Xn’)-F(Xn) else F ¼ F(Xn’)-F(Xn) if F P1 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(X0 )) 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 pbðF=CTÞ. As shown in equation (25), the new transition property (pb(X0 )) varied from 0 to 1 will be calculated using the Boltzmann’s factor (pb(X0 ) ¼ expðF=CTÞ), where C and T are the Boltzmann constant and the current temperature. 0 pbðX0 Þ ¼ @

1, F 0 F exp , f 40 CT

F ¼ FðX0 Þ FðXÞ

ð25aÞ

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(X0 )) 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 Tnew ¼ kk Told

ð26Þ

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 (maximum iteration). To achieve good optimization, the following parameters are varied step by step:20,21 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

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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

iter

kk

RT1

RT2

RT3

RT4

RT5

RT6

Result STL200

25 25 25 25 25 50 100 500 1000 2000

0.91 0.93 0.95 0.97 0.99 0.99 0.99 0.99 0.99 0.99

0.2326 0.2075 0.1889 0.1537 0.1472 0.1372 0.1300 0.1171 0.1107 0.1042

0.6536 0.5866 0.5371 0.4431 0.4257 0.3992 0.3801 0.3455 0.3286 0.3112

0.4536 0.3866 0.3371 0.2431 0.2257 0.1992 0.1801 0.1455 0.1286 0.1112

0.4768 0.4433 0.4186 0.3715 0.3629 0.3496 0.3401 0.3227 0.3143 0.3056

0.006391 0.005512 0.004862 0.003628 0.003400 0.003051 0.002801 0.002347 0.002126 0.001897

0.2664 0.2178 0.1819 0.1137 0.1012 0.08189 0.06808 0.04299 0.03075 0.01810

21.1 24.7 27.8 34.9 36.5 39.3 41.9 51.4 60.8 80.1

Hz

dB

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 minimization of the SWLT-1, SWLT-2, and SWLT-3 with respect to mufflers AC in broadband noise, the resulting design

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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

Muffler A

RT1 0.1042 RT1* 0.1042 RT1** 0.1042

Muffler D Muffler E

RT2 0.3112 RT2* 0.1112 RT2** 0.1112

Result RT3 0.1112 RT3* 0.3028 RT3** 0.3056

RT4 0.3056 RT4* 0.05140

RT5 0.001897

RT6 0.01810

STL200 80.1 STL200 43.8 STL200 46.2

Hz

dB

Hz

dB

Hz

dB

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.

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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 SWLT of three kinds of mufflers (mufflers AC) (broadband noise). Design parameters Muffler type

RT1

RT2

RT3

RT4

RT5

RT6

Result SWLT – dB(A)

Muffler A Muffler B Muffler C

0.1042 0.1042 0.1042

0.3112 0.3112 0.3112

0.1112 0.1112 0.1112

0.3056 0.3056 0.3056

0.001897 0.001897 0.001897

0.0180 0.0180 0.0180

46.5 52.3 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

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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

Muffler C

RT1 0.1042 RT1*** 0.1042 RT1**** 0.1042

Muffler F Muffler G

RT2 0.3112 RT2*** 0.1112 RT2**** 0.1112

Result RT3 0.1112 RT3*** 0.3028 RT3**** 0.3056

RT4 0.3056 RT4*** 0.05140

RT5 0.001897

RT6 0.0180

SWLT – dB(A) 58.9 SWLT – dB(A) 68.2 SWLT – dB(A)B 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


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. Here, for the broadband noise optimization in Table 5, the related cutoff frequencies f 5 1:84c D 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 nonperforated 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 SWLT 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 performances 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,4 the continuity equations and momentum equations with respect to the inner and outer tubes at nodes 5 and 5 a are

18


Figure 13. Acoustical field of an expanded perforated plug duct.

Inner tube Continuity equation V

@5 @u5 4o @5a þ o þ ¼0 uþ @x @x D3 @t

ð27Þ

Momentum equation

@ @ @p5 þV ¼0 u5 þ @t @x @x

ð28Þ

@u5a 4D1 o @5a 2 ¼0 uþ 2 @x @t D2 D3

ð29Þ

@u5a @p5a þ ¼0 @t @x

ð30Þ

o

Outer tube Continuity equation o

o

Assuming that the acoustic wave is a harmonic motion under the isentropic processes in ducts, then pðx, tÞ ¼ ðxÞ c2o ej!t

ð31Þ

The acoustic impedance of the perforation (o co ) is expressed as o co ¼

p5 ðxÞ p5a ðxÞ uðxÞ

ð32Þ

For perforates with a stationary medium22 ¼ ½0:006 þ jkðt þ 0:75dH1 Þ=1

ð33aÞ

¼ ½0:514D3 M5 =ðLc1 1 Þ þ j0:95kðt þ 0:75dH1 Þ=1

ð33bÞ

For perforates with a grazing flow22

Chiu

19

where dH1 is the diameter of a perforated hole on an inner tube, t is the thickness of the inner perforated tube, and 1 is the porosity of the perforated tube. Substituting equations (31) and (32) for equations (27)(30) and eliminating u5 and u5A yield d2 d 2 þ k 1 M25 2jM k p5 5 dx dx2 4 d M5 þ jk ðp5 p5a Þ ¼ 0 D3 dx

ð34aÞ

d2 4D3 2 ðp5 p5a Þ ¼ 0 þ k p5a þ j 2 2 dx D2 D23

ð34bÞ

Alternatively, equations (34a) and (34b) can also be expressed as9 p005 þ 1 p05 þ 2 p5 þ 3 p05a þ 4 p5a ¼ 0

ð35aÞ

5 p05 þ 6 p5 þ p005a þ 7 p05a þ 8 p5a ¼ 0

ð35bÞ

Let p05 ¼

dp5 dp5a ¼ y1 , p05a ¼ ¼ y2 , p5 ¼ y3 , p5a ¼ y4 dx dx

ð36Þ

According to equations (35) and (36), the new matrix between {y0 } and {y} is 2

y01

3

2

1 6 y0 7 6 5 6 27 6 6 0 7¼6 4 y3 5 4 1 y04

3 7

2 6

0

0

1

0

0

32 3 y1 4 6y 7 8 7 76 2 7 76 7 0 5 4 y3 5 0

ð37aÞ

y4

or y0 ¼ ½H y

ð37bÞ

Let y ¼ ½fg

ð38Þ

Plugging equations (38) into (37) and then multiplying ½1 by both sides yield 0

¼ ½Kfg

ð39aÞ

where 2

1

0

0

0

3

60 6 ½K ¼ ½1 ½H½ ¼ 6 40

2 0

0 3

0 0

7 7 7; i : 5

0

0

0

4

ð39bÞ

the eigen value of ½H; ½4x4 : the model matrix formed by four sets of eigen vectors 4x1 of ½H4x4 The related solution of equation (39) then obtained is i ¼ ci ei x

ð40Þ

20


Using equations (28), (30), (39), and (40), the relationship of the acoustic pressure and the acoustical particle velocity yields 2

3

E1,1

E1,2

E1,3

6 6 6 4 o co u5 ðxÞ

7 6E 7 6 2,1 7¼6 5 4 E3,1

E2,2 E3,2

E2,3 E3,3

3 c1 6 7 E2,4 7 76 c2 7 76 7 E3,4 54 c3 5

o co u5a ðxÞ

E4,1

E4,2

E4,3

E4,4

p5 ðxÞ p5a ðxÞ

2

E1,4

32

ð41Þ

c4

Plugging x ¼ 0 and x ¼ Lc1 into equation (41) and rearranging it, the resultant relationship of acoustic pressure and particle velocity between x ¼ 0 and x ¼ Lc1 becomes 2

3

p5 ð 0Þ

2

3

p5 ðLc1 Þ

6 p ð0Þ 5a 6 6 4 o co u5 ð0Þ

7 6 p ðL Þ 5a c1 7 6 7 ¼ ½6 5 4 o co u5 ðLc1 Þ

o co u5a ð0Þ

o co u5a ðLc1 Þ

7 7 7 5

ð42aÞ

where ½ ¼ ½Eð0Þ½EðLc1 Þ1

ð42bÞ

To obtain the transform matrix between the inlet (x ¼ 0) and the outlet (x ¼ Lc1) of the inner tubes, the two boundary conditions for the outer tuber at x ¼ 0 and x ¼ Lc1 are calculated and listed below: p5a ð0Þ ¼ jo co cotðkLa1 Þ u5a ð0Þ

ð43aÞ

p2 ðLc1 Þ ¼ jo co cotðkLb1 Þ u2 ðLc1 Þ

ð43bÞ

Plugging equation (43) into equation (42) and developing them, the transfer matrix deduced is

p5 o co u5

¼

TPPE1,1

TPPE1,2

TPPE2,1

TPPE2,2

p6a

o co u6a

ð44aÞ

where p5 ¼ p5 ð0Þ; u5 ¼ u5 ð0Þ; p6a ¼ p5a ðLc1 Þ; u6a ¼ u5a ðLc1 Þ; 2,2 j4,2 cotðkLa1 Þ 1,3 o co j1,1 o co cotðkLb1 Þ ; TPPE1,1 ¼ 1,1 þ GE 2,4 j4,4 cotðkLa1 Þ 1,3 o co j1,1 o co cotðkLb1 Þ ; TPPE1,2 ¼ 1,4 þ GE 2,2 j4,2 cotðkLa1 Þ TPPE2,1 ¼ 3,2 þ 3,3 o co 3,1 o co cotðkLb1 Þ ; GE 2,4 j4,4 cotðkLa1 Þ TPPE2,2 ¼ 3,4 þ 3,3 o co 3,1 o co cotðkLb1 Þ ; GE ! 4,3 j cotðkLa1 Þ þ 4,1 cotðkLa1 Þ cotðkLb1 Þ GE ¼ o co 2,3 þ 2,1 j cotðkLb1 Þ

ð44bÞ

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 V

@6 @u6 4o @6a þ o þ ¼0 uþ @x @x D3 @t

ð45Þ

Momentum equation

@ @ @p6 þV ¼0 u6 þ @t @x @x

ð46Þ

@u6a 4D3 o @6a 2 ¼0 uþ 2 @x @t D2 D3

ð47Þ

@u6a @p6a þ ¼0 @t @x

ð48Þ

o

Outer tube Continuity equation o

o

Likewise, as derived in equations (49)(55), the new matrix between {y0 } and {y} is 2

y01

3

2

1 6 y0 7 6 5 6 27 6 6 0 7¼6 4 y3 5 4 1 y04

0

3 7

2 6

0

0

1

0

or y0 ¼ ½ y

Figure 14. Acoustic field of a contracted perforated plug duct.

32 3 y1 4 6y 7 8 7 76 2 7 76 7 0 5 4 y3 5 0

ð49aÞ

y4 ð49bÞ

22


where y1 ¼ p06 ¼

dp6 dp6a , y2 ¼ p06a ¼ , y3 ¼ p6 , y4 ¼ p6a dx dx

Let y ¼ ½fg

ð49cÞ ð50aÞ

which is 2

3

2

32

3

1,1

1,2

1,3

1,4

6 dp =dx 7 6 2,1 6 6a 7 6 6 7¼6 4 p6 5 4 3,1

2,2 3,2

2,3 3,3

6 7 2,4 7 76 2 7 76 7 3,4 54 3 5

4,1

4,2

4,3

4,4

dp6 =dx

p6a

1

ð50bÞ

4

½4x4 is the model matrix formed by four sets of eigen vectors 4x1 of ½HH4x4 . Plugging equation (50) into (49) and then multiplying ½1 by both sides yield 0

¼ ½KKfg

ð51aÞ

where 2

1 6 0 6 ½KK ¼ ½1 ½HH½ ¼ 6 4 0 0

0 2

0 0

0 0

3 0

3 0 0 7 7 7; 0 5

ð51bÞ

4

i : the eigen value of ½HH Obviously, equation (51) is a decoupled equation. The related solution then obtained is i ¼ cci ei x

ð52Þ

Using equations (46), (48), (51), and (52), the relationship of acoustic pressure and particle velocity obtained is 2

3

EE1,1

EE1,2

EE1,3

6 6 6 4 o co u6 ðxÞ

7 6 EE 2,1 7 6 7¼6 5 4 EE3,1

EE2,2 EE3,2

EE2,3 EE3,3

3 cc1 6 7 EE2,4 7 76 cc2 7 76 7 EE3,4 54 cc3 5

o co u6a ðxÞ

EE4,1

EE4,2

EE4,3

EE4,4

p6 ðxÞ p6a ðxÞ

2

EE1,4

32

ð53Þ

cc4

Taking two cases of x ¼ 0 and x ¼ Lc2 and rearranging them, the resultant relationship of the acoustic pressure and the acoustic particle velocity between x ¼ 0 and x ¼ Lc2 becomes 2

p6 ð 0Þ p6a ð0Þ

3

2

p6 ðLc2 Þ p6a ðLc2 Þ

6 6 6 4 o co u6 ð0Þ

7 6 7 6 7 ¼ ½6 5 4 o co u6 ðLc2 Þ

o co u6a ð0Þ

o co u6a ðLc2 Þ

3 7 7 7 5

ð54aÞ

where ½ ¼ ½EEð0Þ½EEðLc2 Þ1

ð54bÞ

Chiu

23

To obtain the transform matrix between inlet (x ¼ 0) and outlet (x ¼ Lc2) of the inner tubes, two boundary conditions for the outer tube at x ¼ 0 and x ¼ Lc2 are taken into calculation and listed below: p6 ð 0Þ ¼ jo co cotðkLa2 Þ u6 ð0Þ

ð55aÞ

p6a ðLc2 Þ ¼ jo co cotðkLb2 Þ u6a ðLc2 Þ

ð55bÞ

By plugging equations (55) into equation (54) and developing them, the transfer matrix deduced is

p6a o co u6a

¼

TPPC1,2 TPPC2,2

TPPC1,1 TPPC2,1

p7 o co u7

ð56aÞ

where p6a ¼ p6a ð0Þ;

u6a ¼ u6a ð0Þ; p7 ¼ p6 ðLc2 Þ; u7 ¼ u6 ðLc2 Þ; 1,1 j3,1 cotðkLa2 Þ 2,4 o co j2,2 o co cotðkLb2 Þ ; TPPC1,1 ¼ 2,1 þ Gc 1,3 j3,3 cotðkLa2 Þ 2,4 o co j2,2 o co cotðkLb2 Þ ; TPPC1,2 ¼ 2,3 þ Gc 1,1 j3,1 cotðkLa2 Þ TPPC2,1 ¼ 4,1 þ 4,4 o co 4,2 o co cotðkLb2 Þ ; Gc 1,3 j3,3 cotðkLa2 Þ 4,4 o co 4,2 o co cotðkLb2 Þ ; TPPC2,2 ¼ 4,3 þ Gc ! 3,4 j cotðkLa2 Þ þ 3,2 cotðkLa2 Þ cotðkLb2 Þ Gc ¼ o co 1,4 þ 1,2 j cotðkLb2 Þ

ð56bÞ

Appendix 3 Transfer matrix of two parallel perforated plug tubes As indicated in Figure 3, the acoustical four-pole matrix between node 5(1) and 7(1) is deduced by combining equation (44) and equation (56) yields

p5ð1Þ o co u5ð1Þ

¼

TPPE1,1

TPPE1,2

TPPC1,1

TPPC1,2

TPPE2,1 TPPE2,2 TPPC2,1 TPPC2,2 p7ð1Þ TPP11,1 TPP11,2 ¼ TPP12,1 TPP12,2 o co u7ð1Þ

p7ð1Þ o cc u7ð1Þ

ð57Þ

Developing equation (57) yields p5ð1Þ ¼ TPP11,1 p7ð1Þ þ TPP11,2 o cc u7ð1Þ

ð58aÞ

o co u5ð1Þ ¼ TPP12,1 p7ð1Þ þ TPP12,2 o cc u7ð1Þ

ð58bÞ

Similarly, the acoustical four-pole matrix between node 5(2) and 7(2) is


¼

TPP11,1 TPP12,1

TPP11,2 TPP12,2


ð59Þ

24

Journal of Low Frequency Noise, Vibration and Active Control 36(1) Developing equation (59) yields p5ð2Þ ¼ TPP11,1 p7ð2Þ þ TPP11,2 o cc u7ð2Þ

ð60aÞ


ð60bÞ

Combining equation (58a) and equation (60a) yields p5ð1Þ þ p5ð2Þ ¼ TPP11,1 p7ð1Þ þ p7ð2Þ þ TPP11,2 o cc u7ð1Þ þ u7ð2Þ

ð61Þ

Likewise, combining equation (58b) and equation (60b) yields o co u5ð1Þ þ u5ð2Þ ¼ TPP12,1 p7ð1Þ þ p7ð2Þ þ TPP12,2 o cc u7ð1Þ þ u7ð2Þ

ð62Þ

where p5 ¼ p5ð1Þ ¼ p5ð2Þ ;

p7 ¼ p7ð1Þ ¼ p7ð2Þ ;

u5 ¼ u5ð1Þ þ u5ð2Þ ;

u7 ¼ u7ð1Þ þ u7ð2Þ

ð63Þ

Plugging equation (63) into equations (61) and (62) yields 2 p5 ¼ 2 TPP11,1 p7 þ TPP11,2 o cc u7

ð64aÞ

o co u5 ¼ 2 TPP12,1 p7 þ TPP12,2 o cc u7

ð64bÞ

Rearranging equation (64) in a matrix form, the equivalent four-pole matrix between nodes 5 and 7 shown in Figure 4 is

p5

"

o co u5

¼

TPP11,1 2 TPP12,1

1 2

TPP11,2 TPP12,2

#

p7 o c o u7

ð65Þ

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(1) and 7(1) is


¼

TPP11,1

TPP11,2

TPP12,1

TPP12,2


ð66Þ

Developing equation (66) yields p5ð1Þ ¼ TPP11,1 p7ð1Þ þ TPP11,2 o cc u7ð1Þ

ð67aÞ


ð67bÞ

Similarly, the acoustical four-pole matrices between node 5(2) and 7(2), node 5(3) and 7(3), and node 5(4) and 7(4), is


¼

TPP11,1 TPP12,1

TPP11,2 TPP12,2


ð68aÞ

Chiu

25

p5ð3Þ o co u5ð3Þ p5ð4Þ o co u5ð4Þ

¼

¼

TPP11,1

TPP11,2

TPP12,1

TPP12,2

TPP11,1

TPP11,2

TPP12,1

TPP12,2



ð68bÞ ð68cÞ

Developing equations (68a)(68c) yields p5ð2Þ ¼ TPP11,1 p7ð2Þ þ TPP11,2 o cc u7ð2Þ

ð69aÞ


ð69bÞ

p5ð3Þ ¼ TPP11,1 p7ð3Þ þ TPP11,2 o cc u7ð3Þ

ð69cÞ


ð69dÞ

p5ð4Þ ¼ TPP11,1 p7ð4Þ þ TPP11,2 o cc u7ð4Þ

ð69eÞ


ð69fÞ

Combining equation (67a) and equations (69a), (69c), and (69e) yields p5ð1Þ þ p5ð2Þ þ p5ð3Þ þ p5ð4Þ ¼ TPP11,1 p7ð1Þ þ p7ð2Þ þ p7ð3Þ þ p7ð4Þ þ TPP11,2 o cc u7ð1Þ þ u7ð2Þ þ u7ð3Þ þ u7ð4Þ

ð70Þ

Likewise, combining equation (67b) and equation (69b), (69d), and (69f) yields o co u5ð1Þ þ u5ð2Þ þ u5ð3Þ þ u5ð4Þ ¼ TPP12,1 p7ð1Þ þ p7ð2Þ þ p7ð3Þ þ p7ð4Þ þ TPP12,2 o cc u7ð1Þ þ u7ð2Þ þ u7ð3Þ þ u7ð4Þ

ð71Þ

where p5 ¼ p5ð1Þ ¼ p5ð2Þ ¼ p5ð3Þ ¼ p5ð4Þ ; p7 ¼ p7ð1Þ ¼ p7ð2Þ ¼ p7ð3Þ ¼ p7ð4Þ ;

ð72Þ

u5 ¼ u5ð1Þ þ u5ð2Þ þ u5ð3Þ þ u5ð4Þ ; u7 ¼ u7ð1Þ þ u7ð2Þ þ u7ð3Þ þ u7ð4Þ Plugging equation (72) into equations (70) and (71) yields 4 p5 ¼ 4 TPP11,1 p6 þ TPP11,2 o cc u7

ð73aÞ

o co u5 ¼ 4 TPP12,1 p7 þ TPP12,2 o cc u7

ð73bÞ

Rearranging equation (73) in a matrix form, the equivalent four-pole matrix between nodes 5 and 7 shown in Figure 4 is

p5 o co u5

"

¼

TPP11,1 4 TPP12,1

1 4

TPP11,2 TPP12,2

#

p7 o co u7

ð74Þ

26


Appendix 5 Notation Co c1, c2, c3, c4, cc1, cc2, cc3, cc4 dH Di Do f iter j k kk Lc1 , Lc2 La1 , La2 , Lb1 , Lb2 Li Lo Mi OBJ pi pb(T) Q RT1, RT2, RT3, RT4, RT5, RT6 RT1*, RT2*, RT3*, RT4* RT1**, RT2**, RT3** Si STL SWLO SWLT ti TS1ij, TS2ij, TS3ij, TS4ij TPP1ij TSC1ij, TSC2ij TSE1ij, TSE2ij Tij T To ui uij V i i Z i ½4x4 ½4x4

sound speed (m s1) coefficients in function ci ei x and cci ei x the diameter of a perforated hole on the inner perforated tube (m) diameter of the i-th perforated tubes (m) diameter of the outer tube (m) cyclic frequency (Hz) maximum iterationpffiffiffiffiffiffiffi imaginary unit (¼ 1) wave number (¼c!o ) cooling rate lengths of perforate part for the perforated plug ducts (m) lengths of non-perforate part for the perforated plug ducts (m) length of the i-th perforated tubes (m) total length of the muffler (m) mean flow Mach number at the i-th node objective function (dB) acoustic pressure at the i-th node (Pa) transition probability volume flow rate of venting gas (m3 s1) design parameter for mufflers AC design parameter for mufflers D and F design parameter for mufflers E and G section area at the i-th node(m2) sound transmission loss (dB) unsilenced sound power level inside the muffler’s inlet (dB) overall sound power level inside the muffler’s output (dB) the thickness of the i-th inner perforated tube (m) components of four-pole transfer matrices for an acoustical mechanism with straight ducts components of a four-pole transfer matrix for an acoustical mechanism with a perforated plug tube components of a four-pole transfer matrix for an acoustical mechanism with a contracted perforated intruding tube components of a four-pole transfer matrix for an acoustical mechanism with an expanded perforated intruding tube components of a four-pole transfer system matrix current temperature ( C) initial temperature ( C) acoustic particle velocity at the i-th node (m s1) acoustical particle velocity passing through a perforated hole from the i-th node to the j-th node (m s1) mean flow velocity at the inner perforated tube (m s1) the i-th eigen value of ½H the i-th eigen value of ½HH specific acoustical impedance of the inner perforated tube the porosity of the i-th inner perforated tube. acoustical density at the i-th node (m3 s1) the model matrix formed by four sets of eigen vectors 4x1 of ½H4x4 the model matrix formed by four sets of eigen vectors 4x1 of ½HH4x4