TECHNICAL FIELD

The invention, is about broadband (~3 octaves) flow silencers and sound insulation structures, which are used to reduce noise carried by and emitted from fluid flows whilst enabling through-flow with minimal pressure losses in: cooling, ventilation and air conditioning systems, fans, ducts, vents and chimneys, exhausts of land, air and sea vehicles, high-speed and/or pressurized gas and liquid flows, and in blowers and similar structures, through which multi-phase flows composed of solid and liquid phases such as ore, water present. The invention utilizes acoustic meta-material technology with smaller size and lihter weight compared to conventional silencers and mufflers.

The invention is also about the broadband silencer/insulating structure, which can be manufactured only from a single raw material by well-known production methods such as sheet metal forming, plastic extrusion, plastic injection and 3D printing; whose dimensions, geometry and internal structure are determined for acoustic performance with an accuracy of 0.1 mm; and composed of a single muffler, at least one complementary muffler and auxiliary parts. The invention relates to structures, which do not contain any foam, wool, or felt-made material, are of high acoustic-performance, non-flammable, low-cost, environmentally friendly and recyclable, and have minimal effect on the flow passing through them.

STATE OF THE ART

Currently, glass wool, rock wool, felt, foam, and similar materials are the most widely employed materials in noise control applications. By time and intensive use and through external factors such as rapid air flow, friction, vibration, and impact; these conventional materials tend to lose their integrity, posing a threat to human health (by respiration and contact) through spread fibers and particles of different sizes (IO ^-6 ~ 10 ^-2 m) into the surroundings. Additionally, they can contain moisture, dirt, dust, harmful chemicals, and host a suitable environment for adhesion and reproduction of bacteria and viruses. After first few years of use, they lose their function (all or partially) and finally, they are not usually recycled at the end of their life cycle, due to the difficulty in and/or cost of recycling.

Silencers and mufflers are typically used to control the noise emitted to the environment at the inlets/outlets of fans, ducts, vents and chimneys, exhaust of land, air and sea vehicles in cooling, ventilation, and air conditioning systems. Typically, complex, heavy and high-cost solutions that are composed of various layers and parts composed of, auxiliary solid resonators and channels, mineral wools (rock wool and glass wool), polymer foams, and auxiliary structures and coatings to cover the acoustic materials and to prevent the dispersion of particles such as, metal/ceramic foams perforated (perforated) and micro-perforated structures. These the internal structures of such silencers and mufflers, which are aimed to reduce flow noise, induces high pressure losses in fluid flow, affecting the system performances and bringing additional burden and cost to the systems.

As an alternative to the conventional mufflers and silencer mentioned above, these materials can also be applied to channels through which fluids flow. Insulation layers as low- thickness mattresses and plates made of mineral wool and foam materials can be applied to the inner and outer surfaces of the ducts pose a danger to human health and the environment, especially in late portion of their life as mentioned in the previous paragraphs. Any deterioration or malfunction is difficult to repair and creates difficulties in application.

In the state of the art, US10947876, mentions acoustic meta-material silencers with spiral air channel structure located between the inner and outer duct walls. In this document, Fano resonance principle and spiral channels are disclosed. As can be clearly seen in Figures 2A, 2B, 4B and 9C of the figures of US10947876 , acoustic performance can be achieved at and around a single resonance frequency, not being broadband effective in the solving the problem of flow noise, which typically includes noise in a wide frequency range of up to 3 octaves. Although this document discloses shapes showing the channels of variable height in alternative structures; it does not present any broadband noise control performance.

In US9500108, a similar silencer is separated by a solid wall within the structure and divides the flow into two; It is mentioned that the acoustic silencing feature is obtained from the difference in length of the path of the acoustic waves between these two separate flow paths. US9500108 discloses of one or more spiral structures with constant channel height that spans along the silencer in cylindrical channels, is in an air-permeable and elongated cylindrical structure. The performance offered by US9500108, as can be seen in Figure 3 of this document, provides silencing either in high frequencies (indicated by the number 40) or in a narrow frequency range (1100-1600 Hz; ~0.5 octaves). This document also does not show any performance in the typical broadband frequency range encountered in flow noise.

Acoustic meta-materials are designed to show acoustic properties that cannot be obtained with standard acoustic materials in use, and/or they exhibit significantly higher acoustic performance compared to standard materials of similar dimensions. Meta-material structures, which show acoustic insulation and absorption properties through design and geometry, are mostly resonant in nature thus typically operate in a narrow frequency range while being feasible in terms of cost and manufacturability, i.e., composed of single or a few unit cells with simple inner structure. For the acoustic meta-materials to show broadband acoustic performance they should either contain many structures of different dimensions (that will perform in different frequency bands) or composed of periodic structures with many layers and several unit cells in the order of tens to hundreds. Typical economically feasible acoustic meta-material structures show high acoustic performance in narrow bands, in a certain frequency range (~0.5 octave). Especially to control the noise in a wide frequency range (-3-3.5 octaves). They are insufficient in controlling broadband noise (that cover more than 3-3.5 octaves) such flow noise. Acoustic meta-material technology was first introduced in 2001 . Although there are several acoustic meta-material noise control and silencer structures present in academic literature, there is no or very few commercial applications and products are available in market with their high design and most importantly manufacturing costs.

The structures and the inventions disclosed in the publications and related patents; “Sun, M. et al. “Broadband Acoustic Ventilation Barriers”, Xinsheng Fang, Dongxing Mao, Xu Wang, and Yong Li Phys. Rev. Applied 13, 044028” and “Ghaffarivardavagh R. et al. “Ultra-open acoustic meta-material silencer based on Fano-like interference”, Phys. Rev. B 99, 024302” (same structure included in US patent application with number US20160201530) also shows silencing utilizing a similar solution. The invention of this application focuses on the broadband noise problem, meanwhile the prior art structures are only effective in a limited range. In the works of Ghaffarivardavagh R. et al., the acoustic performance is very narrow band (-0.2 octaves) around a single resonance frequency, as presented in Figure 3.c. In the work of Sun, M. et. al., only a slightly wider but still narrow band (-0.5 octave) performance is shown, as seen in Figure 3.b. The invention of present application provide a broadband performance covering 3 to 3.5 octaves by utilizing a two silencer approach. In addition to broadband performance obtained by this approach, the performances of the first and second silencers are tuned to be effective 1 to

1.5 octaves and 2 to 2.5 octaves respectively; showing an 2 to 5 fold increase in effective frequency range of individual silencers and 7 fold when combined. This performance is achieved carefully tuning the dimensions, mainly: the inner channel and outer wall dimensions, the ratio and the actual values of First/second silencer spiral shaped intermediate channel constant helix angle section half height (si, ₂ ) and first/second silencer spiral shaped intermediate channel exponentially varying helix angle section half height pi, ₂ , and first/second silencer exponential variation factor exponential value, the factor ni, ₂ for spiral intermediate wall structures . Staircase shaped intermediate wall structure, is completely new in literature to show broadband acoustic performance.

BRIEF DESCRIPTION OF THE INVENTION

The present invention is related to acoustic meta-material silencer structures that meets requirements, eliminates all disadvantages and brings some additional advantages.

The main purpose of the invention; is to reduce flow noise with acoustic performance in 3-

3.5 octaves, starting from low frequencies (for example: between 400-3000 Hz), providing >15 dB insulation by means of transmission/insertion loss in all 1/6 octave bands in the frequency range in: cooling, ventilation and air conditioning systems, fans, ducts, vents and chimneys, exhausts of land, air and sea vehicles, high-speed and/or pressurized gas and liquid flows, and in blowers and similar structures, through which multi-phase flows composed of solid and liquid phases such as ore, water present. The invention offers very small volume and weight (~1/5- 1/10), low cost, and very low pressure loss, compared to conventional silencer and mufflers, with structures made from single raw material utilizing traditional manufacturing processes. In addition to providing a high-performance solution, the invention aims to provide an environmentally friendly solution to noise. It provides noise control without using traditionally used standard materials such as felt, foam and mineral wools, which are not recyclable or not recycled due to high costs. The material, from which the invention is manufactured, can be a completely recyclable material such as metals and recyclable polymers or and already recycled material.

The invention provide a silencer structure with specific external and internal dimensions to provide performance within different frequencies ranges utilizing at least 3-3.5 octave uninterrupted broadband acoustic performance, which consists of a first silencer and a second silencer, each with an outer channel structure comprised of specific channel base profiles, number of channels/layers, channel heights and structures, an inner channel; and a connecter that closes the gap between these two silencers ensuring the continuity of the flow and without loss of acoustic performance.

The silencer structure of the inventiondoes not shed particles harmful to the environment and human health, and that will not show reduction in performance due to usage and age, or impregnation or coverage with dust, grease, mold, water and other similar environmental agents compared to traditional acoustic insulation materials. Additionally, the silencers would be easily cleaned only with water or compressed air, if needed only for safety/visual purposes, not to restore the performance.

The invention provides noise control by application of silencers directly to flow channels and/or to vents/outlets of air conditioning and ventilation systems with rectangular and/or circular profiles, in industrial, residential, commercial and other common settings.

Another aim of the invention is to develop a silencer that can be produced from preferably recycled and/or recyclable raw materials.

The structural and characteristic features of the invention and all its advantages will be understood more clearly thanks to the figures given below and the detailed explanations provided related to these figures.

FIGURES

To best understand the structure of the present invention and its advantages with additional elements, the invention should be evaluated together with the figures described below.

Figure 1. Broadband flow noise acoustic meta-material silencer

Figure 2. Parts of Broadband flow noise acoustic meta-material silencer

Figure 3. a The first silencer, the second silencer and flow noise silencer representative acoustic performance graph Figure 3.b Transmission loss presented in the work of Sun, M. et al. “Broadband Acoustic Ventilation Barriers”

Figure 3.c Transmission loss presented in the work of Ghaffarivardavagh R. et al. “Ultra-open acoustic meta-material silencer based on Fano-like interference

Figure 4 Silencer with circular profile outer wall, circular profile inner wall, and spiral shaped intermediate channel structure

Figure 5. Silencer with rectangular profile outer wall, rectangular profile inner wall, and stairwell shaped intermediate channel structure

Figure 6. Spiral shaped intermediate channel detail

Figure 7. Example spiral shaped intermediate channel helix angle profile

Figure 8-a. Stairwell shaped intermediate channel detail

Figure 8-b. Stairwell shaped intermediate channel mid region layer detail

Figure 9. Spacer part detail

Figure 10. Partially flow transparent flow silencer panel structure

Figure 11. Alternative First/Second Silencer Embodiments

Figure 12. Effects of changing ni, ₂ on single First/Second Silencer performance

REFERENCE NUMBERS

1. Silencer assembly

2. Flow direction

3. Silencer inlet

4. Silencer outlet

5. Silencer flow channel

6. Silencer outer wall

7. Inner channel

8. Inner channel wall

9. Staircase shaped intermediate channel wall

10. Spiral shaped intermediate channel wall with constant helix angle

11. Spiral shaped intermediate channel wall with exponentially changing helix angle

12. First Silencer

13. Second Silencer

14. Spacer flow channel

15. Spacer outer wall

16. Spacer

17. Spiral shaped intermediate channel wall

18. Mutual plane of constant and exponentially varying intermediate channels

19. Spacer inlet

20. Spacer outlet

21. Panel with first silencer

22. Panel with second silencer panel

23. Silencer alignment direction

24. First silencer acoustic transmission loss

25. Second silencer acoustic transmission loss

26. Silencer assembly acoustic transmission loss

27. Staircase shaped intermediate channel wall mid region

28. Staircase shaped intermediate channel wall angled part

29. Staircase shaped intermediate channel wall straight part

30. Top and bottom layer of staircase shaped intermediate channel wall

31. Flat plates of panel with first silencer

32. Flat plates of panel with first silencer

33. Broadband effective partially flow transparent silencer panel

34. Staircase shaped intermediate channel wall mid region layer DIMENSIONS

RI ,2. First, second silencer circular profile outer wall radius n,2. First, second silencer circular profile inner wall radius

AI, ₂ . First, second silencer rectangular profile outer wall width

BI ,2. First, second silencer rectangular profile outer wall length ai, ₂ . First, second silencer rectangular profile inner wall width bi,2. First, second silencer rectangular profile inner wall length tdi,d2. First, second silencer rectangular profile outer wall thickness tji 2. First, second silencer rectangular profile inner wall thickness tai,a2. First, second silencer rectangular intermediate channel wall thickness

HI ,2. First, second silencer height

L. Spacer height

L _p . Partially flow enabling silencer panel spacing ti_. Spacer wall thickness

Pi,2. Staircase angle cisi,s2. . First, second silencer spiral shaped intermediate channel constant helix angle

□di,d2. First, second silencer spiral shaped intermediate channel exponentially varying helix angle

Si,2 . First, second silencer spiral shaped intermediate channel constant helix angle section half height pi,2. First, second silencer spiral shaped intermediate channel exponentially varying helix angle section half height u_fi,2- First, second silencer exponential variation factor r , 2. First, second silencer exponential variation factor base value zi,2. First, second silencer exponential variation factor elevation coordinate , ₂ . First, second silencer exponential variation factor exponential value

DETAILED DESCRIPTION OF THE INVENTION

Preferred structure of the Broadband Acoustic Meta-material Flow Silencer (1) of the invention is described hereafter for better understanding of the topic without any limiting effect.

Figure 1 shows the internal structure of the acoustic meta-material flow noise silencer of the invention, effective in the wide frequency range.

Flow noise silencer (1) with acoustic meta-material structure, which is developed using acoustic meta-material technology, effective in a wide frequency range, is basically as presented Figure 2 as: positioned in the direction of flow (2), and composed of the first silencer (12), the second silencer (13) and a spacer (16) that connects the two silencers in a way that not allowing flow leakage and not affecting of the acoustic performance of the two silencers.

Flow noise silencer (1) of the invention can perform in three different alternative profiles for inner channel walls (8), outer walls (6), and in two different intermediate channel wall (9,17) structures. For each first/second silencer: the inner wall can have a circular, rectangular or hexagonal profile, the outer wall can have a circular, rectangular or a hexagonal profile, and the intermediate channel wall can be comprised either of staircase shaped or spiral shaped wall structure; leading to 18 different possible configurations for each first/second silencer. These 18 configurations are alternatives to each other to solve the technical problem of broadband noise to achieve a broadband performance and to apply the silencer to various different fields and systems while covering 3 to 3.5 octaves. Alternative embodiments are presented in Figure 11.

For the silencer to show high performance in a wide frequency range; It is important that the internal structures of the First Silencer (12), the Second Silencer (13) and the spacer (16) and the dimensions of all the parts that make up the internal and external structure are determined with a precision of 0.01 mm depending on the need, possible application space/destination, desired frequency range and aim.

While enabling flow with low pressure drop (example: < 1 kPa pressure drop at < 3 m/s air flow); First Silencer (12) and Second Silencer (13) dimensions are determined to show high acoustic performance in the representative target 3-3.5 octave frequency range, when used together, as presented in Figure 3-a. For the broadband acoustic performance, it is essential to use two silencers in harmony to cover the frequency range of 3 to 3.5 octaves:

• the First Silencer (12) is tuned to provide high acoustic transmission loss (24) in the middle portion of this -3.5 octave frequency range

• the Second Silencer (13) provides high acoustic transmission loss (25) at the lower and upper limits of the aforementioned range,

• and when they are used together (1), the silencer provides >15 dB attenuation/insulation performance (26) in the entire target 3-3.5 octave frequency range with several further-expressed peaks at the resonance frequencies.

The geometric design of the two silencers (12,13) are done, individually to cover a frequency range as wide as possible, and in harmony to cover the broadband frequency span, without performance falling below 15 dB within the range. Opposed to the acoustic performances of the works of Ghaffarivardavagh R. et al., presented in Figure 3-b, whose the acoustic performance is very narrow band (~0.2 octaves) around a single resonance frequency and presented in Figure 3-c of the work of Sun, M. et. al., only a slightly wider (compared to Ghaffarivardavagh R. et al.) but still narrow band (-0.5 octave), the invention presents high performance in 1-1.5 octave bands for First Silencer (12) and in 2-2.5 octave bands for Second Silencer (13) ultimately leading to a performance, whose span is 3-3.5 octaves.

The acoustic properties of the flow noise silencer with an acoustic meta-material structure effective in a wide frequency range are independent of the material properties of the solid material forming the structure. It is possible to manufacture the silencers (12), (13) and spacer (16) from any material with higher strength than acrylic plastic and similar hard polymers without any loss of performance.

The First Silencer (12) and the Second Silencer (13) (from now on, 1. ₂ subscripts will represent First and Second Silencer respectively) the acoustic meta-material flow noise silencer effective in the wide frequency range are defined by the structures and dimensions having a height of HI ,2 as:

• Outer wall (6): preferably with thickness of tdi,d2 <3 mm; either with a circular profile having an outer radius RI ,2 as presented in Figure 4, or with a rectangular profile having outer dimensions of AI, ₂ width and BI ,2 length as presented in Figure 5

• Inner channel wall (8), preferably with thickness of t _d i,d2 <3 mm; either with a circular profile having an inner radius n, ₂ as presented in Figure 4, or with a rectangular profile having inner dimensions of ai, ₂ width and bi , ₂ length as presented in Figure 5

• Intermediate channel wall (17), preferably with thickness tai, _a2 <2 mm; either staircase shaped (9) with the staircase angle of Pi , ₂ as presented in Figure 5, or spiral shaped (17) composed of: constant helix angle portion with the half height of Si, ₂ (10) and constant helix angle of a _s i, _S 2, and an exponentially varying helix angle portion (11) with the half height of pi, ₂ and exponentially varying helix angle of a _d i,d2 as presented in Figure 4.

The outer shape and dimensions of the First Silencer (12) and the Second Silencer (13) are, either a cylinder with an outer radius Ri, ₂ and height of HI , ₂ or a rectangular prism with a width, length, and height of: AI, _2I BI , ₂ , and HI , ₂ , respectively.

The inner channel shape and dimensions of the First Silencer (12) and the Second Silencer (13) are, either a cylinder with an inner radius n, ₂ and height of HI , ₂ or a rectangular prism with a width, length, and height of: ai, ₂ , bi, ₂ , and HI , ₂ , respectively.

Staircase shaped intermediate channel structure (9) is located between the outer wall (6) and the inner channel wall (7), as presented in Figure 5 and in detail at Figure 8. The staircase shape is composed of two distinct regions: staircase shaped intermediate channel wall mid layer region (27) that is located in the middle of identical top and bottom layer of staircase shaped intermediate channel wall regions (30). The staircase shaped intermediate channel wall mid layer region (27) is composed of layers (34) (Figure 8-b). Each layer is composed of eight inclined rectangular staircase shaped intermediate channel wall angled parts (28) and four (or multiples of four) Staircase shaped intermediate channel wall straight parts (29) that are positioned between outer wall (6) and inner channel wall (7). The staircase shaped intermediate channel wall angled parts are rotated by an angle of Pi , ₂ with respect to plane of silencer inlet (3), and the staircase shaped intermediate channel wall straight parts (29) positioned parallel to silencer inlet (3), connecting consecutive staircase shaped intermediate channel wall angled parts (28) that ensures the continuity of the channel structure. Top and bottom layers of staircase shaped intermediate channel walls (30) are identical and actually formed by cutting each layer (X) of the staircase shaped intermediate channel wall mid region (27) in half along the flow direction (2). Spiral shaped intermediate channel (17) is located between the outer wall (6) and the inner channel wall (7), as presented in Figure 4 and in Figure 6 in detail, the spiral shape is composed of: constant helix angle portion (10) with the half height of Si,2 (total height: 2SI ,2) and constant helix angle of a _s i. _S 2, and an exponentially varying helix angle portion (11) with the half height of pi,2 and exponentially varying helix angle of Odi,d2. The constant helix angle portion (10) is symmetrical with respect to the plane defined normal to the flow direction (2) at the point zi,2=0. The exponentially varying helix angle portion (11) is composed of two parts, again, which is symmetrical with respect to the plane defined normal to the flow direction (2) at the point zi,2=0, and placed on top and bottom of the constant helix angle portion (10) as presented in Figure 6. The are assembled such as the spiral shaped intermediate channel wall is continuous forming continuous flow channels. The helix angle 0di,d20f the exponentially varying helix angle portion (11), is equal to the helix angle a _s i, _S 2 of the constant helix angle portion (10) at the mutual plane of constant and exponentially varying intermediate channels (18). The helix angle 0di,d20f the exponentially varying helix angle portion (11), increases with the formula Odi,d2 = a _s i, _S 2 x u_fi,2, starting from the mutual plane of constant and exponentially varying intermediate channels (18) to the inlet (3) and outlet (4) symmetrically. u_fi,2 is the exponential growth factor equal to m ^{111 2 Z1 2} ’, while m, being a real number greater than 1 , zi,2 being the coordinate value in the direction of the flow (2) valued between 0 and pi ,2, m,2 being a real constant number having a value greater than 1/ zi,2 for all zi,2 values. In Figure 7, representative graphs of the helix angles are shown from mid plane of the silencer (12), (13) to either inlet (3) or outlet (4) (symmetrical for each half of the silencer (12), (13) for values 0.5, 0.75, 1 , 1.25 and 1.5 of ni,2. The total height of Hi,2 is twice the sun of the values Si,2 and pi,2 (HI,2= 2 x [si,2 +pi,2]). In each of the cases presented in Figure 6, the helix angles at the constant helix angle portion (10) are smaller than the helix angles exponentially varying helix angle portion (11), and greater ni, ₂ values lead to higher helix angles at the silencer inlet (3) and outlet (4). The heights of the constant helix angle portion (10) and the exponentially varying helix angle portion (11) (si, ₂ and pi, ₂ ), their ratio and the value of ni, ₂ are key factors in widening the effective frequency ranges of the First/Second Silencer (12,13)

The effects of changing the value of ni,2 for the optimized structures is presented in Figure 12. The transmission loss profile of a single First/second silencer with dimensions Ri= 63.5000 mm, H=25.3752 mm, tdi=tji=t _a i= 2.00 mm, Si=6.5253 mm, pi=7.6985 mm with ni=1.2895 optimized for being effective 1200 to 2200 Hz frequency range, along with silencer TL profiles obtained keeping all the dimensional parameters same, but varying m as 1 , 1.25 and 1.5. As seen from the figure, even for close values (1.2500 and 1.2508) of the m parameter, the optimized transmission loss profiles changes, and with large variations there is significant change in frequency ranges and values obtained.

The spacer (16) that connects the First Silencer (12) and the Second Silencer (13), presented in Figure 9, is composed of the spacer outer wall (15) that covers all surrounding surfaces between the spacer inlet (19) and spacer outlet (20) preferably with a thickness t <3 mm. The profile of the spacer inlet (19) (circular or rectangular) of the spacer outer walls (15) should have the same profile with the First Silencer outlet (4); and the profile spacer outlet (20) (circular or rectangular) of the spacer outer walls (15) should have the same profile with Second Silencer inlet (3); ensuring continuity of the flow. The height L of the spacer (16) should allow flow with minimum pressure drop between spacer inlet (19) and spacer outlet (20) and not interfering with the acoustic performance of the broadband effective flow noise silencer (1).

The invention, addition to its use as a flow noise silencer, can be utilized as partially flow enabling sound insulating/silencing panel structures (33), as presented in Figure 10, by using the First Silencer (12) and Second Silencer (13) structures that constitutes the essential of the broadband effective flow noise silencer (1); without using any spacer (16) with:

• Panel with the First Silencer (21), which is composed of multiple First Silencers (12), and flat plates (31) that covers all the void between multiple First Silencers (12)

• Panel with the Second Silencer (22), which is composed of multiple Second Silencers (13), and flat plates (32) that covers all the void between multiple Second Silencers (13). The Panel with the First Silencer (21) and the Panel with the Second Silencer (22) should be positioned parallel to each other at a distance of Lp from each other ensuring same or similar performance obtained for the broadband effective flow noise silencer (1) by using the First Silencer (12), the Second Silencer (13), and the spacer (16). Additionally, for best performance the center lines (23) of the First Silencers (12) and the Second Silencers (13) should be coincident.

The most important outcome of the invention is, enabling effective broadband silencing performance covering 3 to 3.5 octaves, starting possibly from the low frequency of 250 Hz, possibly to the high frequency of 3 to 4 kHz, which corresponds to the upper frequency limit where human hearing is most sensitive, with continuous and high insertion/transmission loss (>15 dB) by tailoring the dimensions of the structure with 0.01 mm precision, extremely low (AP<1 kPa) pressure drop, highly durable (<5 % performance loss over 10 years of use and/or being covered with dirt/mud/grease/oil), light weight (< 1 kg), and can be manufactured out of one single solid state material, preferably from recycled material and/or that can be recyclable after use.

The dimensions of the First Silencer, the Second Silencer and the spacer can be within the limits of, depending on the application and frequency range requirements, as: RI, ₂ , Ai, ₂ and BI, ₂ between 4 and 50 cm; n, ₂ , ai, ₂ and bi, ₂ between 2 and 45 cm, (while being at least 1 cm smaller than RI, ₂ , AI, ₂ and BI, ₂ ); tdi.d2, tii 2, tai,a2 between 0.5 and 5 mm; L, and L _p between 1 and 10 cm; HI, ₂ between 0.5 and 20 cm; Si, ₂ and pi, ₂ . between 0.5 and 19.5 cm twice their sum being equal to Hi , ₂ ); Pi, ₂ and a _s i, _S 2 between 4 and 20 degrees, a _d i,d2 between a _s i, _S 2 and 8 to 60 degrees, and ni, ₂ ratio between 0.5 and 1.5.

The dimensions of a representative silencer effective in 250-3000 Hz range can be obtained as: RI=R ₂ =30.00 cm, ri=14.9223 cm, r ₂ =11.2345 cm, tdi=td2=tii=tj ₂ =tai=ta2= ti_=2.00 mm, Hi=48.22 mm, H2=38.16 mm, L=40.00 mm, Si=12.56 mm, pi=36.76 mm, S2=14.06 mm, p2=24.10 mm, a _s i=8.4°, a _si =7.6°, mi=m2=e (natural log base), ni=0.9435 and n2=1.2121. A ±5% change in dimensions of a single parameter could lead to performance decline up to 70%.