# Prediction of insertion loss of plenums above the plane wave cutoff frequency.

INTRODUCTION

The dominant source of noise in ducted air distribution systems is fan noise. Selecting an appropriate and well-designed fan can reduce the sound power at the source. In addition, sound is absorbed by the duct wall, and is reflected back towards the source due to the abrupt change in area at the termination (Kingsbury, 2007). Nevertheless, the addition of a plenum or silencer is often required to realize a suitable reduction in noise.

The metric most commonly used to characterize the sound attenuation of plenums is insertion loss. Insertion loss is defined as the decrease in sound pressure or sound intensity level measured at a receiver location when a plenum is inserted into the path between the source and the receiver. It is notable that insertion loss depends on the attenuating element or plenum itself as well as the source and termination. The other commonly used metric to assess muffler and silencer systems is transmission loss. It is defined as the difference between the incident ([W.sub.i]) and transmitted power ([W.sub.t]) assuming an anechoic termination. Insertion loss of a plenum will only be equal to the transmission loss of the system if both the source and termination impedances are anechoic.

Transmission loss is not typically used by the ASHRAE community because it is problematic as a metric above the cutoff frequency. Measurement and especially prediction of transmission loss is complicated by cross modes in the inlet and/or outlet ducts. As a result, transmission loss depends on the nature of the source (plane wave, diffuse field, etc.) above the cutoff. Insertion loss is not necessarily easier to predict using mathematical or computational approaches because source and termination impedances may in fact vary across the duct cross-section. However, insertion loss is generally preferred for large duct systems because it is easier to measure.

For a number of years, the equation in the ASHRAE Handbook (1999) was that developed by Wells (1958), which was based on room acoustics theory. The ASHRAE committee on sound and vibration (TC 2.6) recognized that the equation was useable to provide a rough estimate, but inadequate to accurately predict plenum attenuation. Consequently, TC 2.6 sponsored two research projects aimed at eventually improving the ASHRAE Handbook. In the first project, Mouratidis and Becker (2003) measured an assortment of different plenums with varying sizes and wall constructions. The plenum was constructed such that the base was open and rested on concrete. For some cases, absorbing lining (fiber) was placed on all sides of the plenum except the base. Plenum configurations included examples with the inlet and outlet ducts inline and at right angles to one another.

The measurements were made in accordance with ASTM E477 (ASTM, 2006). A loudspeaker array was used to generate pink noise and was positioned in a source room. The source room was connected to a 12.5 m x 8.5 m x 5.2 m (41 ft x 28 ft x 17 ft) reverberation room via an unlined 38 m (125 ft) duct in which plenums could be inserted. Insertion loss was determined by measuring the sound pressure in one-third octave bands in the receiving room with and without the plenum inserted into the duct connecting source and receiving rooms.

Mouratidis and Becker (2003) were unable to measure insertion loss for inlet and outlet ducts at right angles. Measuring insertion loss, while preferable, requires acoustic facilities with reverberation rooms positioned appropriately. Instead, Mouratidis and Becker measured the sound power of the inlet duct without the plenum installed and the sound power at the outlet of the plenum using sound intensity scanning. They defined a noise reduction as being the difference between these two sound powers. Above the cutoff frequency, the measured noise reduction was comparable to the measured insertion loss for cases where both measurement techniques were applied. Over 40 cases were compared, and both metrics agreed well above the plane wave cutoff. However, there were significant deviations in insertion loss and noise reduction below the cutoff frequency.

Based on the measured data, Mouratidis and Becker (2003) developed separate empirical equations for below and above the plane wave cutoff frequency respectively. However, the suitability of these equations for a wider range of cases has not been tested. The equations currently in the ASHRAE Handbook (2011) are based on these results.

TC 2.6 directed a follow-on investigation where numerical simulation was used to predict the transmission loss of plenums (Herrin et al., 2007). In that study, boundary element simulation was combined with transfer matrix theory (Munjal, 1987) to develop a suitable model. Plane wave propagation was assumed in the inlet and outlet ducts though the model was not limited to plane wave behavior in the plenum chamber itself. The model compared favorably with measurement and proved superior to the empirical model developed by Mouratidis and Becker. However, the approach used was not suitable above the inlet and outlet duct plane wave cutoff frequency.

This paper details a numerical simulation approach that can be extended beyond the cutoff frequency for inlet and outlet ducts as well as plenums. The approach meets an important need since the plane wave cutoff frequency is quite low for duct and plenum systems. For instance, the cutoff frequency for a 0.61 m x 0.61 m (2 ft x 2 ft) square duct is only 280 Hz. The suggested technique is a modal approach where modes are first determined using a finite element (FE) model. Stochastic boundary conditions at the source and termination are then applied in modal coordinates. The simulation approach is compared to measurement, the ASHRAE Handbook, and to Wells' equation.

ASHRAE HANDBOOK AND EMPIRICAL EQUATIONS FOR PLENUM ATTENUATION

Wells (1958) developed an expression to estimate plenum attenuation based on conservation of energy. The total sound energy density at the outlet is assumed to be the summation of the direct and reverberant field energy densities (Wells, 1958). The plenum attenuation was defined as the difference between the sound power entering and exiting the plenum. This definition of attenuation is broad and is neither a transmission or insertion loss. Nevertheless, at high frequencies, insertion loss will be less sensitive to the boundary conditions at the source and termination and should be roughly the same as the attenuation. Wells' equation is written as:

IL = 10log[[[S.sub.e](Qcos[theta]/4[pi][d.sup.2] + [1-a]/a[S.sub.w])].sup.-1] (1)

where the average sound absorption coefficient (a) can be determined from

a[S.sub.w] = [n.summation over (i =1)][a.sub.i][S.sub.i] (2)

Mouratidis and Becker (2003) developed separate expressions for the insertion loss below and above the cutoff frequency for the plenum. Below the cutoff frequency insertion loss was expressed as

IL = A * [S.sub.w] + [W.sub.e] + Offset Angle Effect. (3)

Above the cutoff frequency, insertion loss was expressed as

IL = b[S.sub.e][[Q/4[pi][d.sup.2] + [(1 - a)]/[S.sub.w]a].sup.n] + Offset Angle Effect (4)

FINITE ELEMENT SIMULATION

The method described below is not fundamentally new, and is described in greater depth by Herrin et al. (2011). The technique is illustrated in Figure 1. A finite element model is used to simulate the plenum, and inlet and outlet ducts. The input is a diffuse acoustic field. Additionally, a baffled termination is assumed which corresponds closely to a reverberation chamber being used as the receiving room. A finite element model of a plenum is shown in Figure 1. This model was compared to a straight duct having the same total length and cross-sectional dimensions. For both models, the output power was calculated in 2.5 Hz increments up to 1125 Hz. The narrowband sound power results were then summed in one-third octave bands. Insertion loss was calculated directly by taking the difference between the output power for a straight duct, and for a plenum inserted into the system.

The VA-One software (ESI Group, 2010) was used for the predictions. The input was applied using a diffuse acoustic field loading. Shorter and Langley (2005) developed this loading by relating the cross-spectral matrix of the force to the imaginary part of the direct field dynamic stiffness matrix via a reciprocity relationship. This loading is defined in modal coordinates and assumes a reverberant field at the inlet, which should compare closely to a source room with a reverberant field.

The termination or radiation impedance is specified as a connection to a semi-infinite fluid in VA-One. This boundary condition is equivalent to a baffled termination. The correct radiation impedance could be computed using either the boundary element method, or the Rayleigh integral method. However, VA-One opts to approximate the radiation impedance using a wavelet approach developed by Langley (2007), which is computationally faster. Herrin et al. (2011) showed a validation case for this boundary condition.

Absorption can be added to the finite element model in one of two ways. For fiber lining, the absorption is modeled using a complex impedance matrix expressed in modal coordinates to describe the impedance of the lining on the acoustic cavity. For unlined plenums, the procedure suggested by Herrin et al. (2007) was adopted to determine an appropriate loss factor. The sound absorbing coefficient for the plenum walls was chosen using the ASHRAE Handbook (2011), and a loss factor for the plenum was determined by using the expression (Lyon and DeJong, 1995)

[eta] = C[S.sub.w]/4[pi]fV a/[2-a] (5)

One important difference between the simulation and the measurement is the length of inlet and outlet ducts. The combined 38 m (125 ft) length of inlet and outlet ducts was not modeled in the FEM. Instead, the inlet and outlet duct lengths were shortened to 1.52 m (5 ft) to decrease the FEM model size. The effect of length will be most evident at low frequencies where the modes are well separated. However, measurement of insertion loss at very low frequencies is also problematic because signal to noise ratio is often low due to the low source strength, and also environmental or background noise.

RESULTS

The finite element approach was validated for plenums by comparing the predicted insertion loss to the measured insertion loss and noise reduction from Mouratidis and Becker (2003). Additionally, simulation results were also compared to Wells (Equation 1) and the empirical formula currently used in the ASHRAE Handbook (Equations 3 and 4) discussed earlier. Table 1 summarizes the five plenum configurations analyzed.
```
Table 1 Test Configurations

Case  Width  Height  Length   Duct       Duct       Lining
Side   Orientation
Length

A     1.2 m  1.2 m   0.9 m   0.3 m   Inline       Unlined
(4     (4      (3      (1
ft)    ft)     ft)     ft)

B     1.2 m  1.2 m   0.9 m   0.3 m   Inline       4 inch
(4     (4      (3      (1                   fiber
ft)    ft)     ft)     ft)

C     1.2 m  1.8 m   1.5 m   0.6 m   Inline       4 inch
(4     (6      (5      (2                   fiber
ft)    ft)     ft)     ft)

D     1.2 m  1.8 m   1.5 m   0.6 m   Right Angle  8 inch
(4     (6      (5      (2                   fiber
ft)    ft)     ft)     ft)

E     1.2 m  1.8 m   3.0 m   0.6 m   Right Angle  8 inch
(4     (6      (10     (2                   fiber
ft)    ft)     ft)     ft)
```

A typical insertion loss comparison for an unlined plenum with inline inlet and outlet ducts is shown in Figure 2. Notice that the FEM simulation agrees well with the measured insertion loss over the entire frequency range. Similar results for lined plenums are shown in Figures 3 (Case B) and 4 (Case C). The FEM simulation results agree with measurement above 200 Hz while the ASHRAE Handbook under predicts the insertion loss at frequencies below 400 Hz. It is also noteworthy that the empirical equation in the ASHRAE Handbook (2011) appears to be inferior to the acoustic FEM for most cases.

The inlet and outlet ducts are oriented at right angles to each other for Cases D (Figure 5) and E (Figure 6). Simulation results are compared to the measured noise reduction because insertion loss is unavailable. The FEM simulation compares well to measurement above 200 Hz. It is also notable that the ASHRAE Handbook results only modestly improve Wells' equation at low frequencies.

CONCLUSIONS

The suggested FEM approach is a trustworthy approach for predicting the insertion loss of plenums including both a) unlined and lined plenums, and b) plenums with inlet and outlet ducts inline or at right angles to one another. The methodology has several important uses for the ASHRAE community. Most importantly, the results demonstrate that simulation can be used in the place of expensive testing. Moreover, simulation can be used to further improve equations in the ASHRAE Handbook at both low and high frequencies, and the effect of outlet duct placement can be thoroughly investigated.

Further work should entail validation of the approach for more complicated duct and plenum systems. For example, multiple inlet and outlet ducts, and duct splits can be easily simulated. Though structureborne sound was not considered in the present study, it is likely that the method could be extended to include breakout noise prediction.

ACKNOWLEDGMENTS

The authors thank ESI Group for valuable suggestions and assistance with the VA-One software. Additionally, the authors appreciate the support of the Vibro-Acoustics Consortium.

NOMENCLATURE

IL = Insertion [loss

[S.sub.e] = Plenum exit [area

[S.sub.w] = Total wall surface area not including the inlet and exit

[S.sub.i] = Surface area for the [i.sup.th] plenum wall

Q = Directivity factor which is equals 2, 4, or 8 depending on whether the opening is at the center of the wall, a bihedral corner, or a trihedral corner respectively

d = Slant distance between the inlet and exit

[alpha] = Average sound absorption coefficient for the walls

[[alpha].sub.i] = Absorption coefficient for the [i.sup.th] plenum wall

[eta] = Loss factor for the plenum

[theta] = Angle of incidence for the direct sound field at exit

A = Surface area coefficient in dB/f[t.sup.2] or dB/[m.sup.2] defined by Mouratidis and Becker (2003)

[W.sub.e] = Wall effect defined by Mouratidis and Becker (2003)

b = Empirically determined constant found to be 3.505

n = Empirically determined constant found to be -0.359

c = Speed of sound

f = Frequency in Hz

V = Volume of the plenum

REFERENCES

ASHRAE. 1999. ASHRAE Handbook - Applications. Atlanta: American Society of Heating Refrigerating and Air-Conditioning Engineers.

ASHRAE. 2011. ASHRAE Handbook - Applications. Atlanta: American Society of Heating Refrigerating and Air-Conditioning Engineers.

ASTM E477-06A. 2006. Standard test method for measuring acoustical and airflow performance of duct liner materials and prefabricated silencers, Philadelphia: American Society for Testing and Materials.

ESI Group. 2010. VA One Users Guide.

Herrin, D. W., Cui, Z., and Liu, J. 2012. Predicting insertion loss of large duct systems above the plane wave cutoff frequency, Applied Acoustics, 73: 37-42.

Herrin, D. W., Tao, Z., Scalf, E. L., Allen, S. A., and Seybert, A. F. 2007. Using numerical acoustics to predict the attenuation of HVAC plenums, ASHRAE Transactions, 113(1): 10-18.

Kingsbury, H. F., 2007. Noise sources and propagation in ducted air distribution systems, Handbook of Noise and Vibration Control, Crocker, M. J. (ed.), pp. 1316-1322.

Langley, R. S. 2007. Numerical evaluation of the acoustic radiation from planar Structures with general baffle conditions using wavelets, Journal of the Acoustical Society of America, 121(2): 85-95.

Lyon, R. J. and DeJong, R. G. 1995. Theory and Application of Statistical Energy Analysis, 2nd Edition, Butterworth-Heinemann, Boston, MA.

Mouratidis, E. and Becker, J. 2003. The aero-acoustic properties of common HVAC plena, ASHRAE TRP-1026.

Munjal, M. L. 1982. Acoustics of Ducts and Mufflers, Wiley-Interscience, New York.

Shorter, P. J. and Langley, R. S. 2005. On the reciprocity relationship between direct field radiation and diffuse reverberant loading", Journal of the Acoustical Society of America, 117(1): 85-95.

Shorter, P. J. and Mueller, S. 2008. Modeling the mid-frequency response of poro-elastic materials in vibro-acoustics application," Symposium on the Acoustics of Poro-Elastic Materials, December 17-19, Bradford, United Kingdom.

Wells, R. J. 1958. Acoustical plenum chambers, Noise Control, 4: 9-15.

D. W. Herrin, PhD, PE

Member ASHRAE

S. Ramalingam

Student Member ASHRAE

Z. Cui

D. W. Herrin is an Associate Professor at the University of Kentucky, Lexington, KY. S. Ramalingam is a Senior Engineer at Emerson Climate Technologies, Sidney, OH. Z. Cui is a Senior Scientist at Livermore Software Technology Corporation, Livermore, CA.
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