Electronic Journal «Technical Acoustics» http://webcenter.ru/~eeaa/ejta/

2004, 9

Long-Jyi Yeh*, Ying-Chun Chang, Min-Chie Chiu

Department of Mechanical Engineering, Tatung University, Taipei, Taiwan 104, R.O.C.

Design optimization of double-chamber mufflers on constrained venting system by GA method

Received 18.06.2004, published 11.08.2004

Whilst the space volume of mufflers in a noise control system is often constrained for maintenance in practical engineering work, the maximization on muffler’s performance becomes important and essential. To efficiently depress the venting noise, a high performance of double-chamber muffler is then proposed and investigated in this paper. To assess the optimal solution in the muffler design, the genetic algorithm (GA), a stochastic algorithm, is also applied, accordingly.

This paper presents the GA application for the size optimal design of doublechamber muffler under space constraints and dealing with broadband noise. Using technique of four-pole matrix for sound transmission loss (STL) calculation in conjunction with the GA technique, the optimisation was carried out.

Before GA operation, a single-chamber muffler is simulated and compared with the experimental data for accuracy check of mathematical model. Thereafter, a simple program of noise control at the pure tone of 500 Hz has been pre-run to verify the correctness of genetic algorithm before the optimal design of broadband noise was performed. Results show that both the accuracy of mathematical model and the correctness of GA method are acceptable. Consequently, the GA optimization on double-chamber muffler proposed in this study may provide a quick and correct approach.

Keywords: double-chamber muffler; four-pole matrix method; sound transmission loss; space constraints; GA optimization

INTRODUCTION

As high noise levels can be harmful to workers and can lead not only to psychological but also physiological ailment, the attention of noise control on equipment is then focused gradually. To eliminate the noise value of venting system, muffler is habitually in use [1]. However, the space of muffler is often limited as required by operation and maintenance. And even if many researches of muffler designs have been well addressed, the discussion of optimal design under space constraints is rarely emphasized. In the previous work by Yeh et al. [2], the graphical analysis of optimal shape design to improve the performance of sound transmission loss (STL) on a constrained single expansion muffler was discussed.

To efficiently depress the noise emitted from venting system, the sizing optimization of constrained double-chamber muffler with extended tubes by mathematical gradient methods

*

Corresponding author, e-mail: ljyeh@ttu.edu.tw, mailing address: Department of Mechanical Engineering, Tatung University, 40 Chungshan N. Rd., 3rdSec., Taipei, Taiwan 104, R.O.C.

was explored and discussed by Yeh et al. [3]. However it is troublesome to seek for a good starting point during the different gradient-based optimal processes even in the exterior penalty function method or the interior penalty function method. Therefore, the new optimizer of genetic algorithm is thus introduced.

GA optimizers are robust, stochastic search methods modeled on the concepts of natural selection and evolution [4]. Unlike the traditional gradient-based method, which needs the derivatives and the good starting point in the objective function, GA optimizers are able to locate easily the global optimum in a near optimal manner. In this paper, GA is coupled with the transfer matrix method, based on the plane wave theory, to optimize the performance of muffler on constrained venting system.

1. THEORETICAL BACKGROUND

The whole inside flow condition of the double-chamber muffler represented by ten chosen nodes (pt1~pt10) is shown in Figure 1. As deduced by Prasad, Crocker and Munjal [5, 6, 7], the individual transfer matrix in each element is simply expressed as (designations are in nomenclature section)

/ - \

Pi

PoCoU1

= - jMikLi /(1-Mi )

b11 b12 b21 b22

/ \ P2

yPoCoU2 j

(1a)

where

b11 — cos

P2

1 - M i

; bi2 = j sin

1 0 0 SJS^

; b21 = j sin

' P3 ^

yPoCoU3 y

; b22 — COS

(1b)

(2)

f P3 '

yP0C0U3 y

= e- jM2kL2 /(1-M/)

C11 C12

C21 C22

f Pa '

yPoCoU4 y

(3a)

where

c11 — COS

' kL2 ' 1 - M 22

; C12 = jsin

1 - m 22

;c 21 = jsin

1 - m 22

;c 22 =cos

(3b)

' Pa ' "1 0 " ' P5 ^

yPoCoU4 , 0 S 3/ S 2 _ P C U5 ^ r o o 5 ^

' P5 ' = e - jM 3kL3/(1-M32) d11 d12 P Os

P C u5 \^ r o o 5 <N 2 2 d P C U6 y' o o 6 J

(4)

(5a)

where

d 11 — cos

1 - M 32

; d12 — j sin

1 - M 32

; d21 — j sin

1 - m 32

; d 22 — cos

' kL3 A

1 - m 32

(5b)

/ P6 '

p c u 6

y r o o 6 y f \

Pi

yPoC0Ui ,

1 0

0 S J S 3_

= e

- jM 4kL4/(1-M 42)

ei1 ei2

,e21 e22.

f \ P8

yPoCoU8 ,

where

e11 = cos

1 - M

v ’ y

10

; ei2 = jsin

1 - M 42

;e 2i = jsin

P oC oU8

P9

yPoCoU9,

where

/11 = cos

= e

-jM5kL5 /(1-M52)

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/ P9 '

yPoCoU9 y

/11 /12

/21 ./22

r kL 5 ^

1 - M 52

; /12 = j sin

; /21 = j sin

; /22 COs

^ kL5 A 1 - M 52

(6)

(ia)

(ib) (8) (9a) (9b)

By using the matrix substitution on equations (1)-(9), the complete system matrix is

yPoCoU1

= e

M1L 2L2 M3L3 4L4 M 5L5

1-M12 1-M22 1-M32 1-M42 1-M52

b11 b12

b21 b22.

1 0

0 S2 S,

dn d]2

d21 d22

10 0 Si

S3

e11 e12

e21 e22

10

0 —

S 4

/11 /12 r P10 ^

/21 /22

yPoCoU 10 J

C11 C12

C21 2 2 O

E-T T 12

T _ 21 T 22

10

0 S3 S2

/ P10 ^

yPoCoU 10 J

Consequently, the STL of a muffler is [7]

STL(f, Q, Li, L2, L3, L4, Ls, , A, D3, D5) = 20log

+10log

_Sl '

VS.« ,

(10)

(11)

To reduce the parameters in STL function, not only the constraint condition but also the new parameter rt1 are introduced as

rt 1 = ; L2 = Lo - L1 - L3 - L4 - L5; Do = constant; Lo = constant. (12)

L 2

An alternative form of Eq. (11) is expressed as

(\TU + T12 + T21 + T22,

STL(f, Q, L1, L3, L5, rt,, D17 D3, D5) = 20log 1 11 12 21 221

2

+10log

A'

S10

(13)

The sound pressure level (SPLj) at muffler outlet with respect to the octave frequency i, yields

SPLi = SPLoi - STLi, (14)

where SPLoi is the SPL at the muffler inlet (or pipe outlet) without reflection effect, and STLi is the muffler’s STL with respect to the relative octave band frequency i.

; e 22 cos

Finally, the overall sound pressure level, SPLT , at muffler outlet without reflection is expressed as:

By using the formula of Eq. (13), the objective function used in GA optimization with respect to muffler is

2. MODEL CHECKS

Before performing the GA optimal simulation on mufflers, the accuracy check of mathematical model on the fundamental element of a single-chamber muffler is performed by experimental data [8]. As depicted in Figure 2, the accuracy comparison between the theoretical and experimental data for the models is in good agreement. Therefore, the proposed fundamental mathematical model is acceptable. Consequently, the model linked with numerical method is applied for the sizing optimization in the following section.

3. GENETIC ALGORITHM

The concept of Genetic Algorithms, first formalized by Holland [9] and extended to functional optimization by Jong [10] later, involves the use of optimization search strategies patterned after Darwinian notion of natural selection and evolution. During a GA optimization, a set of trial solutions is chosen and “evolves” toward an optimal solution. In the following, we give a short description of the genetic algorithm, which is applied as the optimizer in the sizing optimization of double-chamber muffler.

A. Populations and Chromosomes

The initial population is built up by randomization. The parameter set is encoded to form a string, which represents the chromosome. As the bit length of the chromosome, was chosen as bitn, the interval of the kth design parameter with [Lb,Ub]k was thereafter mapped to the band of binary value. The mapping system between the variable interval of [Lb,Ub]k and the kh binary chromosome of [000...000 - 111...111]k was then built.

By evaluation of the objective function (OBJ), each chromosome is assigned with the fitness.

B. Parents

By using the probabilistic computation weighted by the relative fitness, pairs of chromosome are selected as the candidate parents. The weighted roulette wheel selection is then applied. Each individual in the population is assigned space on the roulette wheel, which is proportional to the individual relative fitness. For n set of candidate parent, the weighted

Individuals with the largest portion on the wheel have the greatest probability to be selected as parent generation for the next generation.

SPLT(Q,L„L3,L5,rtx,DlsD3,D5) = 10log ^10SPLi/10

(15)

OBJ = SPLT (Q, L1S L3, L5, rt1, D1S D3, D5).

(16)

roulette wheel for the ££th individual was represented as

C Offspring

One pair of offspring is generated from the selected parent (mating pool) by crossover. Crossover occurs with a probability of pc. Both the random selection of a crossover and combination of the two parent’s genetic data are then proceeded. The scheme of single-point crossover is chosen in GA’s optimization.

Recombination and parent selection is the principle method for the evolution in GA.

D. Mutation

Genetically, mutation occurs with a probability of pm of which the new and unexpected point will be brought into the GA optimizer’s search domain. It’s an essential operation to keep the diversity of GA population and then improve the accuracy of GA’s optimization.

E. Elitism

To keep the best gene and improve the accuracy of optimization during reproduction, the elitism scheme in the parent generations is thus presented and developed.

F New Generation

Reproduction includes selection, crossover, mutation and elitism. The reduplication continues until a new generation is constructed and the original generation is substituted. Highly fit characteristics produce more copies of themselves in subsequent generation resulting in a movement of the population towards an optimal direction. The process can be terminated when a number of generations exceed a pre-selected value of genno.

The operation in GA method is pictured as Figure 3. In addition, the GA optimizer developed for SPL minimization at exit of muffler is depicted in Figure 4.

4. GA OPTIMIZATION

4.1. Case Study

A noise control of diesel engine noise at the exhausted outlet shown in Figure 5 is introduced as the numerical case. The overall sound pressure level (SPL) inside the diesel engine’s outlet pipe is 105.6 dB. In addition, the sound pressure spectrum in octave band is shown as below:

/ Hz 31.5 63 125 250 500 1k 2k 4k 8k

SPL, dB 86 90 94 93 104 95 91 88 64

The available space for muffler is 0.3 meter in width, 0.3 meter in height and 1.0 meter in length. By equation (16), seven design parameters are chosen in GA optimization. To avoid the larger pressure drop and the flow-generated noise to occur in muffler [11], the minimal diameters (venting device) at D1, D3 and D5 are specified to no less than 0.0762 (m). In addition, for the ease of manufacture in mufflers, each segment of muffler is limited to not less than 0.1 (m).

A series of assumptions of the constrained condition in design are illustrated as 0.0762 (m) < D1 < 0.3 (m); 0.0762 (m) < D3 < 0.3 (m); 0.0762 (m) < D5 < 0.3 (m);

0.1 (m) < L1 < 0.2 (m); 0.1 (m) < L3 < 0.2 (m); 0.1 (m) < L5 < 0.2 (m); 0.5 < rt1 < 2;

D2 =D4=Do =0.3 (m).

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The design volume flow rate (Q) is confined to 0.8 (m /s).

4.2. Correctness on GA

Before the minimization of SPL in full band noise reduction is performed, a simple optimal program in maximizing the STL at the desired pure tone of 500 Hz has been pre-run. To obtain the accurate design by GA optimization, the larger number of population (popuSize), generation (gen no) and bit length (bit_n) are set to 60, 500 and 40 respectively. Both the typical ratio crossover (pc) and mutation ratio (pm) used in following GA optimization are chosen as 0.8 and 0.05 independently [12].

The optimal results are listed and plotted in Table 1 and Figure 6 respectively. As the result indicated in Figure 6, the optimal STL of double-chamber mufflers is tuned and maximized at the desired frequency of 500 Hz by the GA optimization. Consequently, the correctness by using GA optimization technique is acceptable.

5. RESULTS AND DISCUSSION

5.1. Results

5.1.1. Effect on GA operators

As described in Section 4.2, the larger number of population (popuSize), generation (gen no) and bit length (bit_n) are also set to 60, 500 and 40 individually. To identify the effects among GA operators, four trial cases with different values of control parameters (pc, pm and eltno) are thus varied and discussed. Four cases chosen are described as follows:

A. Case 1:pc=0.8, pm=0.05, elt_no=1

By using the crossover of 0.8 and mutation of 0.05, the GA optimization is proceeded and accompanied with an elitism of 1. The result shows that the best generation occurred at generation #463. The best values of design parameters - D1, D3, D5, L1, L3, L5, rt1 - are found to be 0.0762 (m), 0.2998 (m), 0.0762 (m), 0.1001 (m), 0.1717 (m), 0.1000 (m) and 1.2283 individually. The optimal value of SPL on muffler is 93.6 dB(A) with respect to these design parameters. In addition, the computation time of optimization process in personal computer is 3.30 minutes. GA optimization response with respect to generations is shown in Figure 7. As indicated in Figure 7, it reveals that the optimal process is obviously stable and more aggressive.

B. Case 2: pc=0.8, pm=0.05, elt_no=0

By using the crossover of 0.8 and mutation of 0.05, the GA optimization is proceeded and accompanied with an elitism of 0. The result shows that the best generation occurred at generation #463. The best values of design parameters - D1, D3, D5, L1, L3, L5, rt1 - are found to be 0.0792 (m), 0.2811 (m), 0.0771 (m), 0.1011 (m), 0.1349 (m), 0.1315 (m) and 0.7588 individually. The optimal value of SPL on muffler is 95.0 dB(A) with respect to corresponding design parameters. In addition, the computation time of optimization process in personal computer is 3.41 minutes. GA optimization response with respect to generations is shown in Figure 8. As the result of Figure 8, the fluctuation of best solution is found to be violent. By no means, the GA optimal process becomes unstable with the lack of elitism scheme.

C. Case 3: pc=0.8, pm=0, elt_no=1

By using the crossover of 0.8 and mutation of 0.0, the GA optimization is proceeded and accompanied with an elitism of 1. The result shows that the best generation occurred at generation #108. The best values of design parameters - D1, D3, D5, L1, L3, L5, rt1 - are found to be 0.0762 (m), 0.2930 (m), 0.0763 (m), 0.1467 (m), 0.1269 (m), 0.1019 (m) and 0.7832 individually. The optimal value of SPL on muffler is 94.5 dB(A) with respect to these design parameters. In addition, the computation time of optimization process in personal computer is 3.40 minutes. GA optimization response with respect to generations is shown in Figure 9. It is obvious that the diversity of population is insufficient after a long term of generation. Therefore, a worse solution will be obtained due to the lack of mutation scheme.

D. Case 4: pc=0, pm=0.05, elt_no=1

By using the crossover of 0.0 and mutation of 0.05, the GA optimization is proceeded and accompanied with an elitism of 1. The result shows that the best generation occurred at generation #365. The best values of design parameters - D1, D3, D5, L1, L3, L5, rt1 - are found to be 0.0836 (m), 0.2597 (m), 0.0767 (m), 0.1540 (m), 0.1848 (m), 0.1006 (m) and 1.5509 individually. The optimal value of SPL on muffler is 95.5 dB(A) with respect to corresponding design parameters. In addition, the computation time of optimization process in personal computer is 3.25 minutes. GA optimization response with respect to generations is shown in Figure 10. It is found that the optimization is inert with the lack of crossover scheme.

The comparison of optimization for four cases is illustrated in Table 2. As indicated in Table 2, the first case in which crossover and mutation/elitism were applied has the minimal value of SPL compared with other cases.

5.1.2. Simulations

As discussed in Section 5.1.1, the GA parameters of crossover mutation and elitism with respect to 0.8, 0.05 and 1 are applied in the following optimization.

Because of the randomization of initial populations in GA optimization, ten times of simulation are carried out for the more accuracy purpose. The optimal SPL with respect to different random initial populations are depicted in Table 3.

5.2. Discussion

As indicated in Table 3, even though the initial populations are different, the deviations of the resultant SPL are trivial. Besides, it is found in each simulation that the design parameters of D1, D3 and D5 are almost to be consistent with the values of 0.0762, 0.2996 and 0.0762 respectively in each simulation. It implied that they are the primary design parameters and was converged in the muffler optimization. Among them, the third set with resultant SPL of 93.60 dB(A) is identified as the best simulation. The related optimal STL with respect to spectrum is plotted and shown in Figure 11. In addition, a performance curve of muffler at the muffler’s pipe outlet is shown in Figure 12. As indicated in Figure 12, the noise reduction of muffler with respect to spectrum is obvious. Consequently, the optimal shape of muffler with respect to the best design set is shown in Figure 13.

In addition, an average of the ten sets of design data were made and shown in Table 3. By taking the averaged design data into calculation, the corresponding SPL of 93.61 dB was then obtained. Obviously, it is still within the region of the tenth simulated results.

6. CONCLUSION

It has been shown that GA can be used in the optimization of noise control on venting system by adjusting the size of muffler under the space constraints. As indicated in Table 3, the difference of resultant SPL with respect to different random initial population is trivial. Therefore, GA becomes easier to use. The case study reveals that each of crossover, mutation and elitism plays an essential role in GA optimization.

REFERENCES

[1] Edward B. Magrab. Environmental Noise Control. John Wiley & Sons, New York, 1975.

[2] L. J. Yeh, M. C. Chiu, G. J. Lay. Computer aided design on single expansion muffler under space constraints. Proceedings of the 19th National Conference on Mechanical Engineering (The Chinese Society of Mechanical Engineers), C7: pp. 625-633, Yun Lin, R.O.C., 2002.

[3] L. J. Yeh, YC.Chang M. C. Chiu, G. J. Lai, M. G. Her. Shape optimization of constrained double-chamber muffler with extended tubes by mathematical gradient methods. Proceedings of the 3rd Conference on S.M.E., pp. 625-633, Kao hsiung, R.O.C., 2002.

[4] D. E. Goldberg. Genetic Algorithm in Search, Optimization and Machine Learning. Addison-Wesley, Massachusets, 1988.

[5] M. G. Prasad, M. J. Crocker. Studies of acoustical performance of a multi-cylinder engine exhaust muffler system. Journal of Sound and Vibration. 1983, 90(4), pp. 491-508.

[6] M. G. Prasad, A note on acoustic plane waves in a uniform pipe with mean flow, Journal of Sound and Vibration. 1984, 95(2), pp. 284-290.

[7] M. L. Munjal. Acoustics of Ducts and Mufflers with Application to Exhaust and Ventilation System Design. John Wiley & Sons, New York, 1987.

[8] Y. H. Kim, J. W. Choi and B. D. Lim. Acoustic characteristics of an expansion chamber with constant mass flow and steady temperature gradient (theory and numerical simulation). Journal of Vibration and Acoustics. 1990, 112, pp. 460-467.

[9] J. H. Holland. Adaptation in Natural Artificial System, University of Michigan Press, 1992.

[10] D. Jong. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. Doctoral thesis, Dept. Computer and Communication Sciences, University of Michigan, Ann Arbor, 1975.

[11] Mark E. Schaffer. A Practical Guide to Noise and Vibration Control for HVAC Systems. ASHRAE, 1991.

[12] L. J. Yeh, Y. C. Chang and M. C. Chiu. Shape optimization on constrained linearly expanded tubes by using genetic algorithm. Electronic Journal “Technical Acoustics“ <http:/ webcenter.ru / ~eeaa/ejta/> 2003, 18.

Figure 1. Flow condition for a double-chamber muffler

Figure 2. Performance of a single-chamber muffler without the mean flow D1=D2=0.0365 (m), Do=0.15 (m), L1=L3=0.1 (m), L2=0.3 (m)

GA

Population

Mating Pool

Randomly

■=>

Selection By Roulette Wheel

Parents

i 1 W 1 ô ô 1

|0|0 1 0|1|0 1 1 0 0

lOllllllllllllllIl 111

lllllllôioioillllllll

'C.*

«mu

IfflUcIfllMhlôlùlol

lililqMMPHIAwfol

■llili IIII010101 OIQlli

ML'JMil UUf IL'JL'JM

Elitism

New Offspring

I1I0I1I1I 11 11 1I0IQI1I

iok)iiiiiai6iiiaioiii

111011HI6I0111110101

IllllOlOlllOlllllfllOl

lolOlildolûlilolôlil

01

1I0I0I0I1I0I0I0I

lilolililolMlololol

liliIdddQlôlolôlol

Mutation

£

-----li

r—Œ

------I?

------1!

—i —[

DQDQEEDQQD

ULUDOLLlLiJilÎlïlL!]

-ILILI0IQI1IQIII llQIOl -KHOI1IQI1IQI1IOIOI1I J0IÔIIIÛIÙIÛI1IÔIÔI0I

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<=

-lUolllllOIOIOlOlOlOl

-lilLldoldOIOIfllOIOI

UUMIIWMnWI'lH

r^i

—1 =3'

0 UD

i/>

1/) rD

0 1

< “O

rt) O

-3 ZJ

Figure 3. Operations in GA method

Figure 4. Block diagram of the GA optimization on muffler

R.C. Wall

Available Space for Muffler

U////////////////////////////////////A

Figure 5. Elevation plan of noise control system in diesel engine room

Figure 6. STL of muffler with respect to frequency domain (GA Optimization at 500 Hz)

Figure 7. Response of GA optimization with respect to generations at case 1

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Figure 8. Response of GA optimization with respect to generations at case 2

Figure 9. Response of GA optimization with respect to generations at case 3

Figure 10. Response of GA optimization with respect to generations at case 4

Figure 11. Optimal STL with respect to spectrum (case 1)

Figure 12. A comparison of SPL between muffler’s inlet and outlet (case 1)

0,127

0,1001

0,2244

OJ

Г'-

0,4482

0,1

OJ

JJ

Un

Meter

Figure 13. Optimal shape of muffler with respect to the best design set Boundary Constraints: Do=0.3 (m), Lo=1.0 (m)

Table 1. Optimal result in dealing with pure tone noise of 500 Hz

Common parameters Control parameters Results Elapsed time

popuSize gen no bit n pc pm elt no Di(m) D>3(m) D5(m) ¿1(m) L3(m) L5(m) rt1 SPL t(Min.)

60 500 40 0.8 0.05 1 0.0762 0.0764 0.2998 0.1004 0.1001 0.1037 0.5112 27.18 1.06

Table 2. Comparison of results for the variations of control parameters

Common parameters Control parameters Results Elapsed time

popuSize gen no bit n pc pm elt no A(m) D>3(m) D5(m) L1(m) L3(m) L5(m) rt1 SPL t(Min.)

Case1 60 500 40 0.8 0.05 1 0.0762 0.2998 0.0762 0.1001 0.1717 0.1000 1.2283 93.6 3.30

Case2 60 500 40 0.8 0.05 0 0.0792 0.2811 0.0771 0.1011 0.1349 0.1315 0.7588 95.0 3.41

Case3 60 500 40 0.8 0 1 0.0762 0.2930 0.0763 0.1467 0.1269 0.1019 0.7832 94.5 3.40

Case4 60 500 40 0 0.05 1 0.0836 0.2597 0.0767 0.1540 0.1848 0.1006 1.5509 95.5 3.25

Table 3. Results for multiple running (different random initial populations)

iteration Results Elapsed time

A(m) D>3(m) D5(m) L1(m) L3(m) L5(m) rt1 SPL t(Min.)

1st 0.0762 0.2997 0.0762 0.1002 0.1659 0.1008 1.3025 93.64 3.44

2nd 0.0762 0.2993 0.0763 0.1000 0.1772 0.1001 1.2169 93.65 3.38

3rd 0.0762 0.2996 0.0762 0.1001 0.1273 0.1000 1.9970 93.60 3.40

4th 0.0766 0.2999 0.0762 0.1017 0.1763 0.1001 1.2576 93.70 3.39

5th 0.0762 0.2998 0.0762 0.1008 0.1035 0.1001 1.9933 93.62 3.37

6th 0.0762 0.2987 0.0762 0.1002 0.1296 0.1004 1.9511 93.61 3.38

7th 0.0762 0.2960 0.0762 0.1003 0.1725 0.1004 1.3090 93.66 3.39

8th 0.0765 0.2999 0.0762 0.1032 0.1601 0.1006 1.2692 93.73 3.32

9th 0.0762 0.2927 0.0762 0.1001 0.1001 0.1015 1.5875 93.70 3.42

10th 0.0763 0.2982 0.0762 0.1001 0.1045 0.1000 1.9987 93.62 3.40

Average (1—10) 0.0762 0.2983 0.0762 0.1006 0.1417 0.1004 1.5882 93.61

Best (3rd) 0.0762 0.2996 0.0762 0.1001 0.1273 0.1000 1.9970 93.60 3.40

Note : popuSize=60; gen_no=500; bit_n=40; pc=0.8; pm=0.05; elt_no=l

NOMENCLATURE

bitn bit length

co sound speed (m s-1)

D diameter of the i-th segment of straight duct (m)

eltno selection of elite (1 for yes and 0 for no)

f frequency (Hz)

genno maximum no. of generation

j imaginary unit ( V-1)

k wave number.( o / co)

Li length of the i-th segment of straight duct (m)

mean flow Mach number at the i-th segment of straight duct pc crossover ratio

Pi pressure; acoustic pressure at the i-th point(Pa)

pm mutation ratio

popuSize no. of population

3 1

Q volume flow rate of venting gas (m s')

rt1 GA’s design parameter (L4/L2)

O 2

Si section area at the i-th point (m )

SPLo sound pressure level at the silencer inlet (dB)

SPLT sound pressure level at the silencer outlet (dB)

STL sound transmission loss (dB)

ui acoustic particle velocity at the i-th point (m s-1)

po air density (kg m-3)