Download - Sound in Ducts

Noise and Vibration Control

4. SOUND IN DUCTS

Indian Institute of Technology Roorkee 1/46


Many technical systems involve the transport of a liquid or a gas, i.e., pipe, duct, or channel flow.

Examples :cooling water in a nuclear power plant, exhaust gases from an internal combustion engine, or fresh air in a ventilation system.

Because the source is primarily designed to perform a more meaningful task than the generation of sound

(e.g., to drive an automobile in the case of an automobile engine), and is usually already designed and built when the noise problem is “discovered”,

the required solution is a modification to the duct system. Hindrances provided to counter the transmission of sound

are commonly called mufflers or silencers, depending on the context.



Silencer - an element in the flow duct that acts to reduce the sound transmitted along the duct while allowing free flow of the gas through the flow passage.



Passive silencer : the sound is attenuated by reflection and absorption of the acoustic energy within the element.

– reactive: reflection of sound waves(i) side branch muffler or resonator chamber muffler and (ii) expansion chamber muffler

-dissipative : dissipation/ absorption of acoustic energy

Active silencer : the noise is canceled by electronic feed forward and feedback techniques.



Indian Institute of Technology Roorkee

Muffler design– coupled problem treatment:(i) sound source (fan, internal combustion engine, compressor);(ii) duct system (ventilation channel, exhaust gas system, air pressure duct);(iii) termination (register, exhaust gas outlet, nozzle).

note : in cases where the source and termination are reflection-free, the coupling is small - design can be based on an analysis of the transmission properties of the muffler.transmission isolation (DTL), - ratio of the incident to the transmitted sound power when the muffler is connected to a reflection-free termination ( )tiTL WWD log10⋅=

insertion loss (DIL) - where the sound pressure at some point in the duct system (outlet), is compared for two different muffler configurations A and B,

ABILD pplog20 ⋅=

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Muffler performance requirements:(i) “right” outer geometry;(ii) low pressure drop;(iii) sufficient sound attenuation

(i) demands on the outer geometry normally, besides limiting the total volume, also imply a specific outer form. example- passenger cars, in which the muffler is often to be located behind the rear axle, adjacent to the reserve tire. (ii) The demand for a small pressure drop, which is proportional to the square of the average flow velocity, U2, is driven by operating costs of the complete system. For passenger car, the pressure drop across the entire exhaust system is of the order of magnitude 300 mbar, and since the corresponding volume flow is 10 m3/min, the power loss is about 7.5 hp .



Figure 1 Modern luxury class passenger cars have mufflers with very complex structures, in order to be able to fulfill the stringent comfort demands



Both of these demands are, of course, secondary to the last-mentioned, primary demand for adequate sound attenuation.

That can be formulated in a more or less detailed way, as a specification for a maximum A-weighted sound level measured at the outlet, or as a narrow band specification curve.

From these specifications, the muffler design task can be formulated as follows:

arrive at a construction that, within a certain volume of space, gives the desired attenuation without overloading the source.



4.1 Sound Propagation In Ducts

sound in channels - straight, cylindrical tube. L>>D; Therefore, below a certain frequency the sound

pressure is nearly constant over the cross section. plane wave region: Frequency region where the only

modes or propagating waves, that can exist are those that can vary longitudinally.

At frequencies above the plane wave region, more complicated wave forms arise;

For each of these higher modes, there is a bounding frequency (cut-on frequency), below which the wave is strongly damped.

In the plane wave region, the higher modes can only exist locally, in the vicinity of geometric discontinuities.



Figure 2 Reactive muffler with 2 cavities and no flow restriction

-Low cost--Lower back pressure



Figure 3 Reactive muffler with no direct passage between the inlet and exit

-More attenuation--High back pressure affects engine performance



Typical property of sound propagation in a duct - the difference in phase velocity between waves traveling upstream and waves traveling downstream.

x

yr

U

z

D

Figure 4 Cross section of a straight duct with its coordinate system. The average flow velocity of the fluid is U.



The modified wave equation

Assumptions:i. Assume small disturbancesii. Ignore the viscosity and heat transfer within the fluid

assuming that the steady flow rate is constant over the cross section,The particle velocity is :

( ) zzyyxxt eueueuUtru rrrrr+++= )(,

The linearized continuity equation :

00 =⋅∇+∂∂

+∂∂ u

xU

tr

ρρρ

the linearized equation of motion:

00 =∇+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ p

xuU

tu rr

ρ



modified wave equation for the flowing fluid within channel

01 2

22 =⎟

⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∇ px

Utc

p

- can be solved by separation of variables.- therefore assume a solution analogous to

( ) tixi eezyp x ωψ kp ,ˆ=in which the exponential factors correspond to wave propagation along the channel.From above equations:

0)2()( 22222

2

2

2

=−+++∂∂

+∂∂ ψψ xxx MMkk

zykkk

Set the expression inside the second parentheses to a2; then,

0)( 22

2

2

2=+

∂

∂+

∂

∂ ψαψzy



For every higher mode, there is a limiting frequency, beneath which the longitudinal wave number becomes complex, and the mode is therefore exponentially damped;

That frequency is the mode’s cut-on frequency. At low enough frequencies, only the plane wave

propagates in the channel; all higher modes are strongly attenuated, i.e., “cut-off”. As the frequency increases and the sound wavelength

falls beneath the channel cross-sectional dimensions, more and more higher modes can propagate.

That is of great significance to the design of reactive sound mufflers, since a large number of higher propagating modes afford a greater opportunity for sound energy to pass through the muffler.



D

b

hRectangular cross section

Circular cross section

bcf c 210 = + _ Dcf c π841.101 = +

_

hcf c 201 =+

_ Dcf c π054.302 = ++__

21

221111

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

hbcf c

+ + _

_ Dcf c π832.310 = +

_

bcf c =02+ +_ Dcf c π201.403 = +

+_+

_

_

hcf c =20 ++_

Dcf c π318.504 = +

++

+

__

__

21

222114

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

hbcf c

+ ++_

__Dcf c π331.511 = +_

_+

Table 1 Cut-on-frequencies of the lowest modes of a channel with rigid walls, and rectangular or circular cross section. The dashed lines are nodal lines for sound pressure.



4.2 Reactive silencers and side branch resonators

Assumption: i. for simplicity, the influence of the steady flow is ignoredii. The analysis is limited to the plane wave region

Area discontinuity:

an incident wave is partially reflected, and the energy transmitted downstream thereby reduced



Duct

Duct

No reflection

Reflection at area change

Reflection at duct branch

Figure 5 A sudden change in cross sectional area reflects a portion of the acoustic energy



S1 S2

x

Figure 6 A simple area discontinuity (expansion) is the simplest way to bring about a reflection. The reflection properties of the area discontinuity depend only on the relative cross sectional areas of the incoming and outgoing channels.



Ignoring any reflections that might occur at discontinuities downstream of the area change, the expressions for the fields upstream and downstream, respectively, are:

ikxikx epep −−+ += 111 ˆˆp

ikxep −+= 22 ˆp

in which the time factor eiwt is left out Because there are only plane waves, the corresponding expression for particle velocity is:

cepep ikxikxx 0111, )ˆˆ( ρ−−+ −=u

cep ikxx 022, ˆ ρ−+=u

where the fluid’s properties, i.e., c and r are assumed to not change across the area discontinuity.



Continuity of acoustic pressure and volume flow rate serve as coupling conditions across the discontinuity

21 pp =22,11, SS xx uu =

+−+ =+ 211 ˆˆˆ ppp+−+ =− 22111 ˆ)ˆˆ( pSppS

From these equations, the reflection and transmission coefficients, defined for the area discontinuity are respectively

)()( 2121 SSSSR +−=

)(2 211 SSST +=

the transmission isolation for an area discontinuity is obtained as )4)(log(10 21

221 SSSSDTL +⋅=



Reflected sound

Silencer

Reflected sound

Fan

Fan

Figure 6 Replacing the expensive and space-demanding resistive muffler by a series of reflections, provided in a natural way atexisting area discontinuities and corners.


Noise and Vibration ControlExpansion chamber

S1

x

S2

L

Figure 7 Expansion chamber- Here the Inlet and outlet ducts have the same cross sectional areas.



Derivation of the transmission isolation is carried out in the same way as for the area change. Using a 1-dimensional analysis, and assuming continuity of volume flow rate and pressure, the expression for the transmission isolation of an expansion chamber may be found to be

))(sin)22(1log(10 221221 kLSSSSDTL −+⋅=

maximal damping is obtained when ,4Lncf = n = 1, 3, 5 …,

i.e., when the length of the chamber coincides with an odd multiple of a quarter wavelength

The maximal attenuation increases with increasing magnitude of the area jump, and can, for large jumps S2 / S1 » 1, be estimated by

)2log(20 12 SSDTL ⋅≈



TL

996 Hz 1652 Hz 2073 Hz Cross section

D

[dB]

0 500 1000 1500 20000

5

10

15

20

25

30

Frequency [Hz]

Figure 8 Transmission -isolation DTL for an expansion chamber with an eccentrically located inlet and outlet. 1-D analysis – bold line; 3-D analysis –thin line. thin vertical dotted lines - cut-on frequencies of different higher modes.



TL

2073 Hz Cross section

D

[dB]

0 500 1000 1500 20000

5

10

15

20

25

30

Frequency [Hz]

Figure 9 Transmission isolation DTL for an expansion chamber with a centrally located inlet and outlet. 1-D analysis – bold line; 3-D analysis –thin line.



Side branches

a

L S s

S

p 1+p 1

- p 2

+

p s

Figure 10 Common practical realization of a side branch muffler: a quarter wave resonator



General expression for the transmission isolation of a side branch in the plane wave region:

Assume that the dimensions of the branch inlet, in the wall of the main channel, are much less than the wavelength ka<<1.

The following coupling conditions can be formulated for pressure and flow rate, between the sound field upstream and downstream of the branch, assuming that the cross sectional area of the channel is the same before and after the side branch :

+−+ ==+ 211 pppp sPressure:

+−+ +=+ 2,1,1, xssxx SS uuuuFlow rate:

where index s refers to quantities pertaining to the side branch



Making use of the equation of motion, above equation can be modified into a condition relating pressures, as

+−+ +=− 2011 pZppp sss ScS ρ

in which the properties of the side branch are also contained in the specific impedance

sss upZ =

Substituting provides the following expression for the transmission isolation

20 21log10 ssTL ScSD Zρ+⋅=

Above formula describes the transmission behavior of a duct with a side branch.



Quarter wave resonatorA quarter wave resonator is the simplest kind of closed side

branch; It consists of a duct with a constant cross sectional area,

terminated by a hard wall;thus, the boundary condition is

0)( =LsuThe side branch sound pressure and particle velocity in the plane wave region are

ikxs

ikxss ee −−+ += ppp ˆˆ

cee ikxs

ikxss ρ)ˆˆ(1,

−−+ −= ppu

where the boundary condition implies that

kLiss e 2ˆˆ −+ = pp




From that, the impedance of the quarter wave resonator is found to be

)cot(0 kLcis ρ−=Z

As noted earlier, maximal transmission isolation occurs when Zs = 0,

i.e., when n = 1, 3, 5, …2πnkL =

i.e., when the length of the resonator coincides with odd multiples of a quarter wavelength (hence the name);

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20

0

10

400 Frequency [Hz]

200 600 800

DTL [dB]

5

15

25

0 1000

Figure 11 Calculated transmission isolation DTL for a quarter wave resonator, L = 0.86 m, at standard temperature and pressure.

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Helmholtz resonator- acoustic equivalent of the mechanical mass-spring system. It consists of a closed volume that communicates with the duct system by a throat, with area SS and length L.

L, S

V0

s

"Spring"

"Mass"

Figure 12 Schematic illustration of a Helmholtz resonator



Helmholtz resonator inlet impedance

02

00 ViScLi ss ωρωρ +=Z

V0 - resonator volume,

Ssρ0L - air column mass

Maximal attenuation is obtained, when Zs = 0, i.e., when

02 LVScf s

r π=

-eigenfrequency of the Helmholtz resonator



4.4 Resistive silencersResistive mufflers convert acoustic energy to heat. Walls of the channel are covered by porous absorbents ( mineral wool

or glass fibers )

Figure 12 Resistive muffler - based on dissipation of acoustic energy



Because damping is mainly obtained from work done by viscous forces, resistive mufflers should be designed to maximize the particle velocities in the porous material.

It is often advantageous to locate absorbent a small distance away from the wall, since the normal velocity at the wall itself is zero.

To damp fan noise in ventilation ducts, a resistive-type muffler, called a baffle muffler, is often used.

The reason for that is that ventilation noise is primarily of a broad-band, hissing variety, and the typically large dimensions of the duct permit such a muffler.



Figure 13 Baffle mufflers for ventilation systems.

By working with absorbents of different thicknesses, and distributing these in different ways throughout the cross section, different sound attenuation characteristics can be obtained



For rectangular ducts clad with porous material on all walls,the following approximate empirical relations for the transmission isolation, valid in the plane wave region for low steady flow rates, are formulated as

4.105.1 αS

LPDTL =

WhereP - free perimeter clad with absorbent, S - free cross sectional area, L - length of the absorbent, along the muffler,α - absorption factor at the frequency under consideration;



-represents the simplest possible model of a resistive muffler; the porous material is replaced by a bounding surface which does not have any influence on the sound field other than the dissipation of acoustic energy

-Assumes infinite in length of dissipative channel

- ignores the reactive attenuation that arises due to the area discontinuities at the inlet and outlet of the dissipative channel

- ratio of the perimeter to area is of great significance for thedamping properties of the resistive muffler



The above equation completely ignores the reactive attenuation that arises due to the area discontinuities at the inlet and outlet of the dissipative channel;

on the contrary, it is regarded as infinite in length.

It is evident that the ratio of the perimeter to the area is of great significance for the damping properties of the resistive muffler.

a) b)

Absorbent

Figure 14 Typical cross sectional dimensions of resistive mufflers: a) automobile muffler; b) ventilation muffler



The resistive muffler has many practical advantages over the reactive type.

For example, it is often possible to make use of any unallocated spaces for the purpose of sound mitigation.

By filling such spaces with absorbent, and directing the flow through them, they become simple resistive mufflers.

Approach is effectively used to attenuate ventilation noise.



Figure 14 an example where unused space, when clad with absorbent material, can be used as resistive mufflers



Note:Because of the non-reflecting character of resistive

mufflers, the coupling effects due to other system elements are not as strong as for reactive mufflers.

The performance of a resistive muffler, consequently, is not as sensitive to the muffler’s placement in the system as is the performance of a resonator.

That, together with its insignificant frequency dependence, has made the resistive muffler very widely used.

The situation is rarely made worse by the presence of a resistive muffler, and its damping properties also correspond, to some extent, to hearing sensitivity.



Frequency [Hz] 100 2170

60

40

20

80

0

DTL

[dB]

2400 1940 1710 1480 1250 1020 790 560 330

Figure 15 Measured transmission isolation DTL for the resistive rear muffler of a Saab 9000 Turbo.



Thank you
