The transmission loss of double expansion chamber mufflers with unequal size chambers

Apphed Acoustws 24 (1988) 15-32

The Transmission Loss of Double Expansion Chamber MufHers with Unequal Size Chambers

J S Lamancusa Mechamcal Engineering Department, The Pennsylvania State Umverslty, Umverslty Park, Pennsylvanm 16802, USA (Received 24 June 1987, revised version recewed and accepted 30 November 1987)

ABSTRACT The double expanston chamber muffler ts a commonly used sdencmg element m duct notse apphcattons The transmts~ton loss for the case of equal stze chambers ts known and eastly calculated A closed form expresston for the transmtsston loss for the case of chambers of unequal length and area ts presented here It ts also shown that slgntficant tunmg benefits result wtth chambers of unequal stze For evenness of response, wtth a mmtmum passband, chamber length ratios of 2 1 and connectmg tube lengths equal to the shortest chamber are supertor

INTRODUCTION The double expansion chamber muffler illustrated in Fig 1 is a c o m m o n l y used element for the attenuation of no~se transmitted through ducts The theoretical transmission loss o f such a device is defined as the logarithmic ratlo of incident to transmitted acoustic power F o r the case of reflecUonfree terminations, this can be expressed in terms of sound pressure as TL = 10log10 [ R e ( A I / A 5 ) 2 + I m ( A l / A s ) 2]

(1)

where AI Is the complex pressure a m p h t u d e o f the wave m o d e n t on the muffler, and A 5 ~s the complex pressure amplitude o f the transmitted wave exiting the muffler The classic reference by Davis et al 1 provides the plane 15 Apphed Acousttcs 0003-682X/88/$03 50 © 1988 Elsevter Apphed Science Pubhshers Ltd, England Printed m Great Bratam

16

]

% [ ~lmanc l~s~l

S = C r o s s S e c t i o n a l Area

L

A ~

Sa

F~g 1

I

S

I

b

n = Sa/S ~

.j-Lc-i.- L Sc

Sd

.=

E ~A

BI~_ I ~__B2 1

1

m = Sb/S~

-~A II ~ B s

~A II1 ,__B 4

3

4

2

I'v ---,A

S 5

"

~" e

_~ o

G e o m e l r ~ o f d o u b l e e x p a n s i o n c h a m b e r m u f f l e r w lth u n e q u a l l y sized c h a m b e r s a n d reflecnonless termmaUon

wave derivation for the case of equal chambers (L1 Fig 1), and a closed form solution for A1/4

=

]-'2, m

=

n, as defined in

1 A l 45 - 16m 2 I~[4m(m + 1)2 cos k ( 2 L 1 + Lc) - 4 m l m - 1)2 cos k ( 2 L x - Lc)]

+ l[2{m 2 + l){m + 1)2 sm k ( 2 L l + Lc} - 2{m 2 + l)0n - lj 2 x sm k(2L 1 - Lc) - 4(m 2 -

lj 2 sin kLc] I

(2}

where m is the area ratio ( = S b / S . = Sd/Sc), L~ IS the length of expansions (meters), k is the wavenumber ( = u)/~), L~ Is the connecting tube length (meters), ~ is the sound velocity (m s-~), and ~o is the frequency (rads s -i} Typxcal results shown an Fig 2 (for L L = L ~ = L z = 2 4 1 n ) show the desirable p r o p e m e s of this type of silencer, namely its high attenuation over a wide frequency range An inherent problem, however, ~s the uneven nature of the response, with a deep passband occurring at a frequency at which one half wavelength corresponds to the length of each chamber The length of the connecting tube (L,) affects the width of this passband, with longer lengths producing a wider passband While the transmlssmn loss of devices with equally sized chambers is easily calculated and well understood, many physical lmplementauons of double expansion chambers necessarily fall into the category of having unequal size chambers An automobile intake system, with ~ts combination of relatively large diameter air cleaner and intake manifold, joined by a smaller bore carbureton is such an example One might logically wish to analyze and minimize the noise transmlssmn of such a system O f particular interest are the possible tuning benefits of making the chambers unequal m size

Transmtsston loss of mufflers

17

45

4O

\

35 eel

"0

I

3O

.J

25

0

Z 0 U3 O3

z <

2O

/

\

1

i

15

/

QC I,,-

10

V l/v 0

200

400

FREQUENCY

Fig. 2.

600

-- H z

Transmission loss of double expansion chamber muffler with equally sized chambers of 061 m (24") and 061 m (24") connecting tube, area ratio = 16

D E R I V A T I O N F O R U N E Q U A L SIZE C H A M B E R S The system with unequal size chambers shown in Fig 1 IS analyzed using the method and notation of Ref 1 Plane wave propagation is assumed since the transverse system dimensions are smaller than one wavelength over the frequency range of interest A forward traveling wave of complex amplitude A and a backward traveling wave of amplitude B are assumed By applying continuity of pressure and volume velocity at eachjunctlon in the system, the following set of eight simultaneous complex equations are derived At junction 1 A1 + B1

= A2

+ B2

A I - B1 = rn(A2

--

B 2)

(continuity of pressure)

(3)

(continuity of volume velocity)

(4)

At junction 2 A2 e-'kL~ + B2 e+~kL~ = A 3 + B 3

re(A2 e -'kL' -- B2 e +'kL~) =

A3 -

B3

(5I (6)

At junction 3 A a e -'kLc + B 3 e +lkLc = A4 + B4 A3 e - ' k L ~ - B3 e +,kL~ = n ( A 4 -

BJ

(7) (8)

J S Lamancusa

18

At junctmn 4 A 4 e-,kL2 + B4 e ~ |kL2 = A5

(9)

n ( A 4 e -'kL2 - B4 e +|kL~) = A s

(10)

After the extremely tedmus task of back subst~tutmn, the closed form solution for the ratm of incident to exit pressure ( A ~ / A 5) is derived 1

A 1 / A 5 - 1_ 6_n m {[C1 + C 2 - - C 3 ) c ° s k ( L 1 + Lc + L2) + ( - C 4

× cosk(L 1 + L c - L 2 ) + ( C 6 "1- ( - - C 6 --t- C 7 -

- C5 + C 3 )

C7 + C s ) c o s k ( L I

- L~ + L2)

C 9 ) c o s k ( L 1 - L~ - L2)]

+l[(C1 + C2 + C 3 ) s m k ( L x

+ Lc + L2) + ( - C 4 -

C5 - C3)

x s m k ( L 1 + Lc - L2) + (C6 - C7 - Ca) s i n k ( L 1 - L~ + L z ) + ( - C 6 + C 7 + C9) s l n k ( L 1 - L~ - Lz)]}

(11)

where C1 = 2(m + 1)(n + 1)z, C 3 =

(m + 1)(m - 1)(n + 1)(n - 1),

C2 = (m + 1)(m - 1)(n + 1)2 C4 =

2(m + 1)(n -- 1)2

C5 = (m + 1)(m - 1)(n - 1)2,

C6 = 2(m + 1)(n + 1)(n - 1)

Cv = (m + 1)2(n + 1)(n - i),

C8 = ( m - 1)2(n- 1)2

C9 = (m - 1)2(n + 1)2 from which the transmission loss ( T L ) is calculated by eqn (1) It can be shown that eqn (11) reduces to eqn (2) for the case of equal size chambers (L 1 = L2, m = rt) It should be noted that these same equations could be solved by computer using a complex, simultaneous equation solver Reference 2 outhnes a semi-automated manner of analyzing plane wave systems consisting of multiple elements including expansion chambers, junctmns, resonators, etc Transm~sslon loss, insertion loss, or the radiated pressure at a system exit with fimte radiation impedance, may all be calculated The four-pole matrix method may also be used 3-4 Higher order simulations utlhzmg fimte element analysis 5 and b o u n d a r y element methods 6 may also be employed, to examine behavior both within and beyond the plane wave region An excellent compilation o f analysis methods apphcable to ducts and mufflers is to be found m Ref. 7 However, the closed form solutxon presented here (eqn (11)) provides an easier and computatlonally more effioent method for this c o m m o n l y encountered configuration While the transmission loss cannot

Transmtsston loss of mufflers

19

be directly related to the performance in a real system with for instance, a flanged or unflanged exit tube, it is still a very useful quantity for a quahtatlve or 'first look' analysis Experimental verifications by Davis et al 1 and higher order modeling by Seybert and Cheng, 6 indicate that the assumption of plane wave propagatmn is generally vahd up to a frequency where one wavelength corresponds to the length of the chambers

APPLICATION AND RESULTS The effect on transmission loss predicted by eqn (11), as the second chamber length is varied, can be nicely visualized as a three-dimensional surface or contour plot Figure 3(a) shows a surface plot, as a function of dimensionless frequency (Fr=fL1/c) and chamber length ratio (L2/L1) , for a fixed connecting tube length of L c = 0 0 The same data is also shown in contour plot form with hnes of Iso-transmlsslon loss, in Fig 3(b) Similar results are plotted In Figs 4-9 for connecting tube lengths up to 1 5 x L~

Effect of length ratio on passbands Of note IS the consistent presence of deep passbands at FL1/c = 0, 0 5 and 1 0 for the equal chamber case Careful observation of these results indicates that as the second chamber length ratio is decreased to ½, the width of the second passband decreases, independent of connecting tube length Further decrease in the length ratio causes an increase in the passband width Very small changes in the first or third passband are seen

Effect of length ratio on stopband amplitude Two major stopbands typically occur in the range of the dimensionless frequency up to 1 0 for the equal chamber case The exception is when the connector length approaches zero These bands are centered around frequency ratios of 025 and 075 The height of the first stopband progressively decreases as the length ratio decreases The second stopband at first decreases, reaching a minimum at approximately LE/L 1 = 3, and then increases to a maximum in the vicinity of L2/L 1 = ½ A length ratio of ½ produces stopbands of nearly equal height

Effect of connecting tube length The effect of connecting tube length is to modify the depth and width of the second passband (at F r = ½) The effect of connecting tube length IS illustrated

~

0-.

,,,.o

z~

0

~o

0

0

__

I

I

0

l

Chamber

I

0

length

F

ratio

I

0

(L2/L1)

F

o

I

0 [

0

Transmission loss (dB)

e~

Transmlsston loss of mufflers

21

~_~ 4 0 ]

t '%.

"W/Y V

~

04 _.~,o t~~'''~'

<"" 10

0

0 2 Fre(~enCV r~tsu

02

04 06 Frequency ratio (fLl/c)

..J ..J

o o r-e-

_o

to E

t~ .c

(.9

0

08

10

Fig. 4. Transmission loss with unequally sized chambers for a fixed connecting tube length r a t i o Lc/L 1 = 0 25

22

1 5; Lamamu~a

c

4oL

i~ 2° o

/e

lO

\ o8~

o

~ ~04 ratio 6"k~'~J 02 f r e ~ e ~ c~

c)

o

02

/

L

0

F~g 5

02

04 Frequency

06 ratio (fLl/c)

08

10

T r a n s m i s s i o n loss with unequally sized c h a m b e r s for a fixed connecting tube length raho I~'L~=05

Transmtsston loss of mufflers

c:

23

401

"~"4 o6~

o 0

0 4 r~%~o £IL'~Ic) Fre~ue~C~

02 _1 _J 0 .,u04

l-

e~

E o 08

0

02

O4 O6 Frequency rat,o (fLl/C)

O8

10

Fig 6. Transmxss~on loss with unequally sized chambers for a fixed connecting tube length raho Lc/L I = 0 75

24

J

~,~ L a m a n ¢ u s a

40~

%

/

06 04

r a t i o ('~L~Ic)

Freckler~cV 0

/

S

o2-

/I

d v

004I_

}o ~o 5 08

0

F~g 7

02

04 Frequency

ratio

06 (fLl/C)

08

10

Transmission loss wath unequally sized chambers for a fixed connecting tube length ratio L 'L~ = I O

Transmission loss of mufflers

25

= 4o]

O4

\

o '"o.oa"
~

~//~/

/

~"<'zj 10

0 6 .... ,~ 0 4

02

_,, ra,,.~o " ' ~ '

Freo,Uer~CY "

0

0

02 _J ..A

904 L. e-

E e-

08

0

Fig. 8.

02

04 06 Frequency ratio (fLl/C)

O8

10

Transmission loss with unequally sized chambers for a fixed connecting tube length rauo L¢/L 1 = 1 25

26

J 5 Lamancusa

40] ~ °-2 o 1

i 04 0

0

02 J d v

~o4 0

£ t~

~06 E tU

08

0

F~g 9

02

04 06 Frequency ratJo (fLl/C)

08

10

Transmission loss with unequally sized chambers for a fixed connecting tube length ratio L c~LI = l 5

Transmtsston loss of mufflers

27

~, 3ol

~-,,otJ o~ 000,9"~, "Ot-,ta ?

10

~Oor~ ~ r ~ o ~ , ~ , ~ %,.~~

02

,

i=re~ eoc'j ' ' -

15

~1C ..Y 0 ¢.-

¢-

~o5 t~ tO c tO

0

Fig. 10

02

04 06 Frequency ratio (fLl/c)

08

10

Transrmsslon loss with unequally sized chambers for a fixed chamber length ratio L2/L 1 = 0 5

J S Lamancusa

28

in Fig 10 for a constant chamber length ratio of L2/L 1 = ½ The minimum passband width results from a connecting tube length of L J2 (see Fig 5) A connecting tube length equal to the first chamber length causes the widest passband Longer connecting tube lengths cause a more uneven response characteristic, with additional stopband lobes as seen in Figs 8 and 9

C O M P A R I S O N O F T R A N S M I S S I O N LOSS TO B E H A V I O R W I T H TAlL PIPE AND PARTIALLY REFLECTING TERMINATION The actual performance of a double expansion chamber muffler In a real system will obviously differ from theoretical predictions of transmission loss depending on the termination impedance A typical configuration with a tail pipe and partially reflecting exit is shown in Fig 11 C o m m o n l y used measures of real muffler performance include transmission loss, insertion loss Lit, and noise reduction LN~ 3 Noise reduction is defined as the difference between the sound pressure levels measured at the input of a muffler and at its output Both insertion loss and noise reduction are dependent on the termination impedance In the case of our double expansion muffler, with an exit pipe terminating in a flanged end, one can write the additional plane wave equations needed as follows At junction 4, eqns (9) and (i0) become

4~e ,kl.,_+

B4e+,kt.2 =

As + Bs

n ( A 4 e -'kL2 - - B 4 e +'kL2) = A 5 -

(12)

B5

(13)

At the end of exit pipe of length = L e, junction 5 A 5 e -'kL° + B 5 e +'kLe =

A6 =

exit pressure

As(1 -- ZSJp~le-,kL~ + Bs(1 + ZSe/pc)e +,gLo = 0

I14) (15)

where pc is the characteristic impedance of air, and Z is the radiation impedance at the exit The radiation impedance (Z) for a plane piston in an lnfimte baffle 4 5 ~s approximated by the first two terms of the infinite Bessel function series as Z=

p' {[(k~ )2 rra 2 _

(ka)4~ ~8ka 12 J + I [ ~

32(ka) 3 ]'~ 45rt J J

(Nsm-3)

(16)

where a is the radius of the piston (exit pJpe) in meters Owing to the increased number of equations, a numerical, complex simultaneous equation solution was employed The noise reduction, 20 loglo (P,,,/Pou,), for the equal chamber case with L¢/L~ = 0 5 and an exit pipe equal to one half of the first chamber length is shown in Fig 12 A

Transmtsston loss of mufflers S = Cross S e c t i o n a l A r e a

m = Sb/S a

i A -*

F,g 11.

n -- S d / S c

r

--*A

-.A

B~*- I +--B;

S

29

-*A

-,A

II,__BaaIII,_B44

~"

IV *--B55

V--*Ae ~=

Geometry of double expansion chamber muffler with tall pipe and radiation impedance at exit

20

10

m I

II

z o a i.

~'/

/

II

rl

II

--10

II

-2O

II

Z

/J

it , ~

\

~"

/' //

' ',

/~

/ \

/

-30

--40

;

0

I

02 EQUAL CHAMBERS

Fig. 12

1

r

04

I

r

06

I

p

'l

08

(Thousonds) FREQUENCY RATIO UNEOUAL CHAMBERS

Comparison of equal and unequal (L 2 = L1/2) chamber mufflers with identical connecting and exit pipes (L c = L1/2, L~ = LI/2)

J S Laman¢u~a

30

-20

-10

/v

I tt

it

I

it

)l

7

)t

o

f

t~

10

O I.d W

tn o z

I

,o/

//

)

L

I

t

I t

I I

I

'

)

/ I

/ / ,¢

3O

40

I

0 - --

Fig. 13

I

O2 EQUAL CHAMBERS

r

i

04

)

I

06

O8

I

(Thousands) F R E Q U E N C Y RATIO UNEQUAL CHAMBERS

Comparison of equal and unequal (L2/L 1 = 0 5) chamber mufflers with the same total length (2 66 x L1) and equal connecting tube lengths (Lc = L1/2)

constant pressure source was employed A similar plot for the unequal chamber case judged the most superior based on ItS transmission loss, (L2/L l = 0 5, exit pipe length = Lz/2) is also shown m Fig 12 The unequal chamber design produces more attenuation in the middle frequencies, shghtly less attenuation at low and high frequencies Of note is the absence of the passband at a dimensionless frequency or ratio o f L 1 to wavelength of 0 5 when the chambers are not equal Figure 13 illustrates the result for the two systems when the total length is kept constant For this case, it can be seen that the unequal chamber design provides more noise reduction (up to a frequency ratio of about 0 6) The passband at a frequency ratio of 0 5 has once again been eliminated

CONCLUSIONS The transmission loss of a double expansion chamber muffler with unequal chambers has been derived and used to study the potential benefits of this configuration If evenness o f response over a wide frequency range, with a minimum passband, is the criteria, the second chamber of a double expansion chamber muffler should be half the length of the first This

Transmtsswn loss o f mufflers

31

produces the narrowest passband (at a frequency ratio of 0 5) and two stopbands of approximately equal height A second chamber length ratio of 0 75 produces the most uneven response The depth and width of the stopband at a frequency ratio of 0 5 is minimized by a connecting tube length ofLx/2 A connecting tube length equal to the length of the longest chamber (L 0 produces an undesirably wide passband The physical order of the chambers has no effect on the transmission loss Identlfical results would be obtained if the order was switched, and the second chamber was the longer of the two Figure 14 compares the transmission loss for the equal chamber case with a connecting tube length of L1/2 to the unequal chamber case with a second chamber half as long as the first The superior performance in the vicinity of a frequency ratio of 0 5, is clearly evident The smaller size of the muffler with unequal chambers would also be a desirable attribute Predictions of attenuation with a tall pipe and partially reflecting termination show similar improvements in performance when one chamber is again half the length of the other The closed form solution for the transmission loss of double expansion chamber mufflers with unequal chambers which is presented here allows the

60

50 m

I

40 /

0 ..J

z 0 t/1

30

z

20

\x\

\

\

tlI

IL

IlO

i

o

0.2

i

i

"(

04

i

06

i

i y

OB

(Thousonds)

FREQUENCY ---

Fig. 14.

EQUAL CHAMBERS

RATIO ., UNEQUAL

CHAMBERS

Comparison of transmission loss for equal and unequal chamber cases with Lc/ L l = 0 5

32

J 5 Lamancusa

efficient analysis of a c o m m o n l y encountered physical configuration Th~s solution also enables the designer to utd~ze an additional parameter and potentmlly tune the transmission loss to obtain more desirable frequency characteristics

REFERENCES 1 Daws, D, Stokes, G, Moore, D and Stevens, G, Theoretwal and e~perlmental mvesttgatlon oj mufflers wtth ~omments on engme-e>chaust muffler destgn, NACA Report 1192, 1954 2 Eversman, W, A systematic procedure for the analysis of multiply branched acoustic transmission hnes, ASME J Vlb A¢ ou~t Stre~ Rehab Des, 109 (1987), pp 168-77 3 Crocker, M J, Internal combustion engine exhaust muffling, Norse-Con 77, pp 331-58, 1977 Presented at Nasa Langley, Noise Control Foundation, Poughkeepsie, NY 4 Prasad, M G and Crocker, M J, Insertion loss studies on models of automotive exhaust systems, J A~oust Soc A m , 70(5) (1981), pp 1339~,4 5 Young, C I and Crocker, M J, Prediction of transmission loss In mufflers by the finite-element method, J Atouvt So~ A m , 57 (1975), pp 144-8 6 Seybert, A F and Cheng, C Y, A boundary element for modehng duct, piping and muffler systems, Norse-Con 87, pp 319-24, 1987 Presented at Pennsylvama State University, Noise Control Foundation, Poughkeepsie, NY 7 Munjal, M L, A~ou~tl~s o/du~ts and mufftos, Wlley-lntersclence, New York, 1987