Sound Attenuation in Ducts 497
Table 14.8
Silencer Self Noise Octave Band Corrections dB
Freq. Hz 63
125 250
500 1k
2k 4k
8k Correction
4 4
6 8
13 18
23 28
for the actual area of the silencer being used. In most cases S
o
= 4 ft
2
0.37 sq m, but when the measurements were made on a different unit, the actual face area must be utilized.
Self noise is the power radiated from the receiver end of the silencer and is combined with the sound power levels from other sources exiting the silencer. Most of the high-frequency
self noise is generated at the air inlet, so it is attenuated in its passage through the silencer in the downstream direction but not in the upstream direction. Hence high-frequency 1k Hz self
noise levels are greater on the return-air side of an HVAC system. Low-frequency self-noise levels do not vary significantly with flow direction.
When self-noise data are not available, they can be estimated using Fry, 1988 L
w
∼ = 55 log
V V
+ 10 log N + 10 log H
H − 45
14.31 where
L
w
= sound power level generated by the silencer dB V = velocity in the splitter airway m s or ft min
V = reference velocity 1 for m s and 196.8 for ft min
N = number of air passages H = height or circumfrence round of the silencer mm or in
H = reference height 1 for mm or 0.0394 for in
The spectrum of noise generated by the silencer is calculated by subtracting octave band corrections given in Table 14.8 from the overall sound power level.
Back Pressure
Silencers create some additional back pressure or flow resistance due to the constriction they present. Silencers that minimize this pressure loss are available but there is generally a
trade off between back pressure and low-frequency attenuation. Sometimes it is necessary to expand the duct to increase the silencer face area and reduce the pressure loss. It is desirable
to minimize the silencer back pressure, usually limiting it to less than 10 of the total rated fan pressure. The position of the silencer in the duct, relative to other components, also affects
the back pressure. Figure 14.9 shows published data IAC Corp. that give the multiplier of the standard back pressure for various silencer positions.
14.4 BREAKOUT
The phenomenon known as breakout describes the transmission of sound energy from the interior of the duct out through its walls and into an occupied space. The analysis of the
process combines elements of duct attenuation as well as the transmission loss through the duct walls.
498 Architectural Acoustics
Figure 14.9 Duct Silencer Back Pressure Multipliers Industrial Acoustics
Corp., 1989
Transmission Theory
The breakout transmission loss defines the relationship between the sound power level enter- ing an incremental slice of the duct at position z and that radiating out through the walls of
that slice. Figure 14.10 illustrates the geometry following Ver, 1983. The breakout transmission loss at a given point is
L
TLio
= 10 log d W
i
z d W
io
z 14.32
Figure 14.10 Duct Breakout Geometry
Sound Attenuation in Ducts 499
where the sound power incident on the increment of duct length, dz is d W
i
z = I
i
z P d z = W
i
z P
S d z
14.33 and the radiated power emanating from this slice on the outside of the duct is
d W
io
z = d W
i
z 10
−0.1 LTLio
= W
i
z S
10
−0.1 LTLio
P d z 14.34
If the internal sound power decreases with distance along the duct due to radiation through the duct walls and interaction with the interior surface, according to the relationship
W
i
z = W
i
0 e
−τ +2β z
14.35 the sound power radiated by a length l of duct is given by
W
io
z =
l
d W
i
z d z
= W
i
P S
10
−0.1 LTLio l
e
−τ +2β z
d z 14.36
which yields W
io
z = W
i
P ℓ S
10
−0.1 LTLio
1 − e
−τ +2β l
τ + 2β l 14.37
and converting to levels L
wio
= L
wi
− L
TLio
+ 10 log P l
S + D
14.38 where
L
wio
= sound power radiated out of the duct dB L
wi
= sound power level entering the duct dB L
TLio
= sound transmission loss from the inside to the outside of the duct dB P = perimeter of the duct m or ft
l = length of the duct m or ft S = cross sectional area of the duct m
2
or ft
2
D, the duct loss term, is defined as D = 10 log
1 − e
−τ +2β l
τ + 2β l 14.39
where β =
L
duct
8.68 Nepers ft or Nepers m
L
duct
= attenuation per unit length inside the duct dB τ =
P S
10
−0.1 LTLio
500 Architectural Acoustics
The D term in Eq. 14.39 can be ignored in short sections of duct, particularly when the duct is unlined and unwrapped. However, for lined duct it should be included. In internally lined
ducts the attenuation term is usually larger than the breakout term. In flex or fiberglass ducts the breakout term may dominate, though transmission loss data for these products is difficult
to obtain. The breakout sound power can never exceed the internal sound power.
Once the sound has penetrated the duct walls it radiates into the room at high frequencies as a normal line source. Equation 8.85 can be used to predict the expected sound pressure
level in the room. Alternatively if the duct is long and unlined it radiates like a line source at high frequencies and the sound pressure level is given by
L
p
= L
wio
+ 10 log Q
2 π r l +
4 R
+ K 14.40
where K is 0.1 for SI units and 10.5 for FP units. At low frequencies if the duct is oriented perpendicular to two parallel walls it may
excite resonant modes in the room, in which case the simple diffuse field condition does not exist Ver, 1984.
Transmission Loss of Rectangular Ducts
The duct transmission loss for breakout of rectangular ducts is divided into regions by fre- quency that are similar to those discussed in Chapt. 9 for flat panels ASHRAE, 1987. The
transmission loss behavior with frequency is first stiffness controlled, then mass controlled, and finally coincidence controlled. Figure 14.11 shows the general structure of the loss for
rectangular ducts.
For all but very small ducts the fundamental wall resonance falls below the frequency range of interest. In this region there is a minimum transmission loss that is dependent on
the duct dimensions a, b in inches and l in feet
L
TLio
min = 10 log 24 l
1 a
+ 1
b 14.41
At higher frequencies, in the mass controlled region, there is a crossover frequency, below which the transmission loss is affected by the duct dimensions, and above which it follows
Figure 14.11 Interior to Exterior Transmission Loss for Rectangular Ducts
ASHRAE, 1987
Sound Attenuation in Ducts 501
normal mass law. The crossover frequency is given by f
L
= 24120
√ a b
14.42 where a is the larger and b the smaller duct dimension in inches. Below this frequency the
transmission loss is given by
L
TLio
= 10 log f m
2 s
a + b + 17
14.43 where f is the frequency, and m
s
is the duct wall surface mass in lbssq ft. Above the crossover frequency where the normal mass law holds, the transmission loss
for steel ducts is given as in Eq. 9.21 by L
TLio
= 20 log f m
s
− K
TL
14.44 where K
TL
= 47.3 in SI units and = 33.5 in FP units. At very high frequencies side walls exhibit the normal behavior at the coincidence frequency, but for thin sheet metal this lies
above 10 kHz and is of little practical interest.
Transmission Loss of Round Ducts
When sound breaks out of round ducts the definitions in Eq. 14.38 are the same as those used in rectangular ducts. The inside area is S =
π d
2
4 and the outside transmitting area is
P l = 12 π d l, where d is in inches and l is in feet. The behavior of round ducts is less well understood and more complicated than with rectangular ducts; however, if the analysis is
confined to octave bands, the transmission loss can be approximated by a curve, shown in Fig. 14.12. The low-frequency transmission loss is theoretically quite high because the hoop
strength of the ducts in their fundamental breathing mode is very large. In practice we do not achieve the predicted theoretical maximum, which may be as high as 80 dB, and a practical
limit of 50 dB is used. As the frequency increases the transmission loss is dependent on the localized bending of the duct walls.
Figure 14.12 Interior to Exterior Transmission Loss for Round Ducts Reynolds,
1990
502 Architectural Acoustics
Reynolds 1990 has given three formulas to approximate the curve segments shown in Fig. 14.12. The transmission loss is given by the larger of the following two formulas:
L
TLio
= 17.6 log m
s
− 49.8 log f − 55.3 log d + C
o
14.45 L
TLio
= 17.6 log m
s
− 6.6 log f − 36.9 log d + 97.4 14.46
where L
TLio
= sound transmission loss from the inside to the outside of the duct dB m
s
= massunit area lb sq ft d = inside diameter of the duct in
C
o
= 230.4 for long - seam ducts or 232.9 for spiral - wound ducts In the special case where the frequency is 4000 Hz and the duct is greater than or equal to
26 inches in diameter there is a coincidence effect and L
TLio
= 17.6 log m
s
− 36.9 log f + 90.6 14.47
Since the maximum allowable level is 50 dB, if the calculated level exceeds this limit the transmission loss is set to 50.
Transmission Loss of Flat Oval Ducts
The transmission loss of flat oval or obround ducts falls in between the behavior of square and rectangular ducts. The lower limit of the transmission loss is given by
L
TLio
= 10 log P l
S 14.48
since if it were any less, according to the definition of transmission loss in Eq. 14.38, it would imply amplification. For obround ducts the areas are given by
S = b a − b + π
b
2
4 14.49
and P l = 12 l [2 a − b + πb]
14.50 At low to mid frequencies the wall strength is close to a rectangular duct because of bending
of the flat sides. Assuming the radiation is entirely through the flat sides the transmission loss is given by Reynolds, 1990
L
TLio
= 10 log f m
2 s
δ
2
P + 20
14.51 where
L
TLio
= sound transmission loss from the inside to the outside of the duct dB m
s
= massunit area lbs sq ft f = octave band center frequency Hz
P = perimeter in = 2 a − b + πb δ = fraction of the perimeter taken up by the flat sides
δ = 1
1 + π
b 2 a − b
Eq. 14.51 holds up to a limiting frequency f
L
= 8115
b .
Sound Attenuation in Ducts 503
Figure 14.13 Duct Break-in Geometry
14.5 BREAK-IN
