3.1. Validation of Theoretical Prediction
The finite element method (FEM) simulations are performed by commercial software COMSOL Multiphysics to validate the accuracy of the transmission loss results of the analytical model. The parameters of the ITEC model in this validation case are listed in
Table 1. The transmission loss results of the frequency domain between 1 and 1000 Hz are shown in
Figure 2. It can be seen that the ITEC has three resonance frequencies in the selected frequency domain. The TMM results match the FEM results well, especially near the first transmission loss peak. The frequencies corresponding to transmission loss peaks are listed in
Table 2. It could be seen that the transmission loss peaks have a maximum error of 10 Hz near the second peak and a minimum error of 4 Hz near the first peak. In general, the analytical model has relatively high accuracy.
The analytical model could also be used to predict the transmission loss of the infinity tube. According to Lato et al. [
19], for an infinity tube, as shown in
Figure 1c, the transmission loss is:
where
SM represents the cross-section area of the main duct,
S2 denotes the cross-section area of IT, and
L2 represents the length of IT. In
Figure 1b, if
SEC has the same value as
SN, the ITEC would become an infinity tube. Therefore, the analytical model of the ITEC could also be used to predict the transmission loss of IT. In this case, matrices
TBC and
TDE in Equation (10) turn into identity matrices, which indicates that Equation (10) becomes:
Using the expression of TT in Equation (17), the transmission loss of the infinity tube could be deduced. In the following research, the analytical model based on Equation (15) is used to calculate the transmission loss of IT and then is compared with the results from Equation (16).
As shown in
Figure 3, the solid lines represent the transmission loss of ITs with
L2 = 1.15 m and various
S2/SM ratios. At the same time, ITs with the same geometries are used to examine the transmission loss by Equation (15).
Figure 3 illustrates a good agreement between the analytical model from the second part and from the research conducted by Lato et al. [
19]. The results indicate that Equation (15) could predict the IT transmission loss.
3.2. Noise Attenuation Ability of the ITEC
A comparison of the transmission loss between the IT and ITEC is carried out to examine the noise attenuation ability of the ITEC. The parameters of the ITEC are the same as the geometric model of
Table 1, and the IT parameters are selected as the
S2/SM = 1/4 case in
Figure 3, which indicates that IT has the same cross-section area as the neck of the ITEC.
Figure 4 shows the analytical transmission loss between the IT and ITEC. The transmission loss peaks of the ITEC are 115, 403, and 733 Hz, while the transmission loss peaks of IT are 149, 447, and 745 Hz. The results show that an expansion chamber could lead to a decrease of 34, 46, and 12 Hz in resonance frequency. On the other hand, compared with IT, the noise attenuation bands of the ITEC under three transmission loss peaks are non-uniform. In the lower frequency (1–350 Hz), the attenuation band of the ITEC is significantly wider than IT. In medium frequency (350–650 Hz), they are approximately close to each other. In the higher frequency (650–1000 Hz), the attenuation band of the ITEC is narrower than IT. This feature indicates that the ITEC is more efficient in reducing low-frequency noise. In addition, since the ITEC has decreased resonance frequency, it has an advantage in low-frequency noise control compared with IT.
The Helmholtz resonator is widely used as a muffler for ductwork systems in industry [
8]. To further examine the noise attenuation ability and assess the potential in the industrial application of the ITEC, a comparison of the transmission loss between the ITEC and Helmholtz resonator system is conducted here. As illustrated in
Figure 5a, if a sharable sidewall is placed at the midpoint of the ITEC, the whole system could be regarded as two curved Helmholtz resonators mounted on the same cross-section of the main duct. Cai and Mak [
21] have examined the transmission loss of parallel HRs system, which is shaped as shown in
Figure 5b.
Figure 5a,b indicate that curved HRs are geometrically similar to the parallel HRs system. However, the ductwork system is always located in a limited space. A curved HRs system could save more space if it has the same cavity volume as a straight HRs system.
In this study, the parameters of curved HRs are the same as the ITEC in
Table 1. The neck parameters of parallel HRs are
LN = 95.91 mm and
SN = 1418.6 mm
2. The cavity volume of parallel HRs is the same as half of the chamber volume of curved HRs, which is easy to obtain from
Table 1. As shown in
Figure 6, the transmission loss of curved HRs is the same as the ITEC, with the same resonance frequencies and noise attenuation bandwidths. This indicates that the sharable sidewall has no impact on the noise attenuation mechanism of the ITEC. However, the transmission loss of the parallel HRs system is different. It has only two resonance frequencies from 1 to 800 Hz, while the ITEC and curved HRs have three. In addition, under the lower-frequency domain (1–350 Hz), the resonance frequency of parallel HRs is 133 Hz, which has an increase of 14 Hz compared with the ITEC and curved HRs; under the moderate frequency domain (350–650 Hz), the resonance frequency of parallel HRs is 473 Hz, which has an increase of 64 Hz compared with the ITEC and curved HRs. The characteristic of resonance frequencies shows that the ITEC and curved HRs are entirely different from the parallel HRs system, although they are geometrically similar. The ITEC has a lower resonance frequency than parallel HRs, which indicates that the ITEC is more suitable for reducing low-frequency noise. Furthermore, the curved shape of the ITEC and curved HRs system has an advantage in a constrained space. These characteristics indicate that the ITEC would have potential in ductwork systems.
3.3. Parametric Study of the ITEC
In this section, ITECs with different geometric parameters are analyzed to discuss the influence of geometrics on noise attenuation performance.
Figure 7 shows the transmission loss results of ITECs with different length ratios. The total length of ITECs (
LEC + 2
LN) is fixed (1150 mm), while the neck and expansion chamber lengths have different values. The length values are shown in
Table 3. Both FEM simulation and TMM analysis are conducted to validate the accuracy of transmission loss results. According to research in the previous parts, the ITEC would have three peaks of 1–350 Hz, 350–650 Hz, and 650–1000 Hz, respectively. Therefore, the frequency domains are divided into three sub-domains to distinguish 1st, 2nd, and 3rd TL peaks. Under the lower-frequency domain, the peaks of ITECs with different length ratios are close, while they have more significant differences under moderate and higher-frequency domains. We could summarize the following principles for ITECs with different length ratios:
- (1)
Under the lower-frequency domain, length ratios have little influence on resonance frequency and attenuation bandwidth. ITECs with a higher length ratio would slightly decrease transmission loss peaks and have slightly narrower attenuation bands.
- (2)
Under the moderate frequency domain, the length ratio significantly influences transmission loss performance. ITECs with higher length ratios have a higher resonance frequency and narrower attenuation bands. Compared with the low-frequency condition, ITECs have significantly narrower attenuation bands under the moderate frequency domain, which indicates that ITECs with higher length ratios are not suitable for medium-frequency noise attenuation.
- (3)
Length ratio has a significant influence on transmission loss under the higher-frequency domain. With the length ratio changing from 1/10 to 1/4, the transmission loss peak has increased by nearly 100 Hz. In addition, ITECs with a length ratio equal to 1/4 have a significantly wider bandwidth than the other length ratios, which indicates that the increase in neck length of the ITECs would be good for high-frequency noise attenuation.
Furthermore, the influence of the cross-section area ratio is investigated. As listed in
Table 4, three
SN/SEC ratios correspond to three different expansion chamber radii and a fixed neck radius. The transmission loss results are shown in
Figure 8. It could be obtained from
Figure 8 that a higher cross-section area ratio would be better for low-frequency noise attenuation. The ITEC with
SN/SEC = 1/4 has the lowest peak frequency and widest noise attenuation bandwidth under the lower-frequency domain. On the contrary, the lower cross-section area ratio would be better for high-frequency noise attenuation. The ITEC with
SN/SEC = 1/1.44 has the highest peak frequency and widest noise attenuation band under the higher-frequency domain.
In
Figure 9, we perform the transmission loss results of the ITECs with different total lengths. The total length of the ITECs is changed from 0.6 and 0.75 times to the original length (
L =
LEC + 2
Ln = 1.15 m);
LEC and
Ln are also scaled down simultaneously, whereas the radii of the neck and expansion chamber have remained unchanged. As shown in
Figure 9, the shorter total length would have a broader sound attenuation bandwidth. At the same time, the transmission loss peak would shift to the higher-frequency domain, even exceeding 1000 Hz, the upper limit of this research. In addition, the TMM results of 0.6
L would lead to a more significant error than the FEM results, which is due to the fact that the neck length of 0.6
L is very short. According to Ingard [
22], an end correction is non-negligible for the aperture neck to improve transmission loss accuracy. For this reason, TMM in this study is not suitable for the short
Ln case. Both TMM and FEM results show that the ITECs with shorter lengths would have a broader noise attenuation band. Meanwhile, the transmission loss peaks tend to shift to the higher-frequency domain. Therefore, the ITECs with shorter lengths would have better noise attenuation performance but are not suitable for low-frequency noise reduction. The transmission loss peak (
TLmax) and the resonance frequency (
f0) of ITECs with different geometric parameters are summarized in
Table 5.
Figure 10 illustrates the influence of different geometric parameters of ITEC on the
TLmax. It can be seen that changing the length ratio leads to a change of 19 dB in the
TLmax in higher-frequency domain and a change of 11.8 dB in the
TLmax in moderate frequency domain. Changing the cross-section area ratio has a change of 12.1 dB in higher-frequency domain. Changing the total length has a change of 7.8 dB in lower-frequency domain. Therefore, adjusting the length ratio and the cross-section area ratio are beneficial for improving the higher- and medium-frequency noise attenuation ability, and adjusting the total length is useful for improving the lower-frequency noise attenuation ability.
Figure 11 illustrates the influence of different geometric parameters of ITEC on
f0. Changing the total length has more of a significant impact on the resonance frequency than changing the length ratio and the cross-section area ratio. The
f0 under three peaks has an increase of 98, 302, and 254 Hz. This indicates that adjusting the total length is an effective way to control the frequency of noise reduction.