The optimization of the acoustic silencer volume is very important to develop it and to get high-performance, the importance of the silencer was appeared in industrial field to eliminate the noise of the duct by efficient and economical method. The main goal of this research is to optimize the transmission loss (TL) by analytical method of the Double-Chamber Silencer (DCS), the TL has been selected as the main parameter in silencer because it does not based on the source or the termination impedances. First we calculated the power transmission coefficient (PTC) and the TL of an acoustic silencer, then used the Lagrange method to optimize the silencer length. All calculation of silencer data is obtained by solving the governing equations in commercial software Matlab^{®}. A several calculations for different silencer length at many frequency ranges were performed simultaneously. Finally, this research supports the efficient and rapid techniques for DCS optimal design under narrow space. The results show that the acoustic TL is maximized at the desired frequency.

A silencer is considered a type elimination devices of passive noise. It is generally used in ventilation equipment such as a diesel engine and internal combustion engines [

For single entry/exit expansion chambers, the predictions of the transmission loss boundary elements show good agreement with the analytical and experimental results [

The PTCs will be Calculated for the silencer over [0–3 KHz] range of frequencies, and optimize the length B for higher level of sound attenuation. Assume an anechoic termination at the right end of the silencer.

The flow condition and location of the silencer is specified in _{1}~pt_{8}), as shown in

The system with equal size chambers shown in _{j} and a backward traveling wave of amplitude _{j} are assumed. The Helmholtz equation [

where

The continuity conditions at each junctions can be obtained by applying continuity of pressure and volume velocity, the continuity conditions at the first junction (1–1) are:

And at the second junction (1–2) are:

And at the third junction (2–1) are:

And at the fourth junction (2–2) are:

where

where

The algebraic system of the DCS model can be obtained from

The following set of eight simultaneous complex equations:

where

The PTC is:

The attenuation TL is:

For the most of calculations were considered in the computational parts of the work the radius of the chamber and the inlet/outlet pipe are

Geometry | Chamber radius |
Inlet/Outlet pipe radius |
Total chamber length |
A silencer length ratio ( |
---|---|---|---|---|

1 | 0.1 | 0.05 | 0.2 | 0 |

2 | 0.1 | |||

3 | 0.2 | |||

4 | 0.4 | |||

5 | 0.6 | |||

6 | 0.8 | |||

7 | 1.0 |

Typical results shown a

Octave band center frequency (Hz) | Silencer length ratio ( |
||||||
---|---|---|---|---|---|---|---|

0 | 0.1 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | |

210 | 0 | 13 | 30 | 29 | 13 | 13 | |

275 | 13 | 13 | 30 | 13 | 30 | 36 | |

327 | 8.5 | 25 | 37 | 33 | 25 | 8.5 | |

415 | 0 | 35 | 0 | 0 | |||

500 | 8.5 | 37 | 33 | 25 | 8.5 | ||

550 | 13 | 36 | 30 | 13 | 30 | 36 | |

615 | 0 | 13 | 30 | 29 | 13 | 13 | |

1035 | 0 | 13 | 30 | 29 | 13 | 13 | |

1102 | 13 | 36 | 30 | 13 | 30 | 36 | |

1152 | 8.5 | 37 | 33 | 25 | 8.5 | ||

1242 | 0 | 35 | 0 | 0 | |||

1324 | 8.5 | 25 | 37 | 33 | 25 | 8.5 | |

1375 | 13 | 13 | 30 | 13 | 30 | 36 | |

1440 | 0 | 13 | 30 | 29 | 13 | 13 | |

1860 | 0 | 13 | 30 | 29 | 13 | 13 | |

1926 | 13 | 13 | 30 | 13 | 30 | 36 | |

1982 | 8.5 | 25 | 37 | 33 | 25 | 8.5 | |

2057 | 0 | 35 | 0 | 0 | |||

2145 | 8.5 | 37 | 33 | 25 | 8.5 | ||

2200 | 13 | 36 | 30 | 13 | 30 | 36 | |

2265 | 0 | 13 | 30 | 29 | 13 | 13 | |

2685 | 0 | 13 | 30 | 29 | 13 | 13 | |

2751 | 13 | 36 | 30 | 13 | 30 | 36 | |

2805 | 8.5 | 37 | 33 | 25 | 8.5 | ||

2890 | 0 | 35 | 0 | 0 | |||

2972 | 8.5 | 25 | 37 | 33 | 25 | 8.5 | |

3000 | 11 | 19 | 34 | 2.2 | 37 | 31.6 |

Remark: The cell filled with grey color is the Peak value of TL, and unfilled cell for Narrow dome value of TL.

For all silencer geometry configuration the effect of silencer on acoustic TL was completely vanished sequently at frequencies 825 Hz, 1650 Hz, 2475 Hz respectively.

The acoustic TL with respect to silencer length ratio under full periodicity from 0 Hz to 800 Hz is shown in

Moreover, acoustic TL becomes larger when

In order to maximize the value of acoustic TL, the minimal value of length of B is planned and carried out. The three parameter

From

It should be noted that this derivation of the optimization equation under full periodicity from 0 Hz to 800 Hz could be solved by computer using Matlab^{®} software and the results of derivation are shown in

It is possible to observe in

Length of chamber |
|||
---|---|---|---|

Frequency (Hz) | Acoustic TL (dB) | Optimized variables | |

Silencer length ratio (B/L) | Silencer length |
||

288 | 39.41 | 0.095 | 0.019 |

450 | 39.12 | 0.149 | 0.0298 |

612 | 41.1 | 0.204 | 0.0408 |

948 | 38.96 | 0.318 | 0.0636 |

1278 | 38.93 | 0.423 | 0.0846 |

1764 | 38.92 | 0.592 | 0.1184 |

1938 | 38.81 | 0.646 | 0.1292 |

2256 | 38.81 | 0.755 | 0.151 |

2421 | 38.81 | 0.809 | 0.1618 |

2925 | 38.79 | 0.977 | 0.1954 |

From this optimization process we conclude the best value of

In this work we presented complete mathematically study for optimal design of in Double-Chamber Silencer (DCS). The exact solution of Helmholtz equation was performed to calculate the power transmission coefficient (PTC) and the TL of silencer via Matlab^{®} software for different silencer length. The Lagrange method were used for maximizing the acoustic TL using.

In this paper where defined the optimum silencer length is 0.0408 m with fixed pipe radius, encountered the maximum acoustic TL of 41.1 dB in the resonance frequency of DCS (612 Hz) estimated by the analytical equation. From the results we can conclude that the acoustic TL is maximized at the desired frequency. This work reinforces the rapid and efficient approach to an optimal design for DCS under narrow space.