Computation of a reactive silencer (muffler) with a double expansion chamber (EC2) using SILDIS^® software (Excel): an example

The SILDIS^® software (Excel) enables the calculation of the acoustic and aerodynamic performance of a reactive silencer (muffler), such as a double expansion chamber, as illustrated by an example relating to a typical component of an internal combustion engine exhaust system (Example 1B.4.3 in the user manual). It comes to a device which attenuates noise:

• without the recourse to sound-absorbing porous materials
• through sound wave reflection phenomena caused by changes in waveguide cross-section (e.g. sudden expansion or contraction) and/or the presence of a rigid back causing a gas flow reversal (resonator effect)

Whether used alone or combined with other reactive or dissipative stages (which can also be modeled using SILDIS^®), such a simple expansion chamber exhibits frequency-dependent acoustic performance. To make the best possible use of it, the chamber must be "tuned" - meaning its geometry is selected to maximize attenuation within a specific frequency band of interest, for what axial dimensions (lengths) play a crucial role, while the ratio of the chamber diameter to the inlet (and outlet) diameter significantly impacts the magnitude of the attenuation provided by the resulting acoustic filter. Such an optimization is achieved very easily and very quiclky using the software SILDIS^®, as illustrated below.

Problem statement (envisaged application relating to the computation of a reactive silencer (muffler) with a double expansion chamber using software SILDIS^®)

The aim is to calculate the acoustic and aerodynamic performance of a purely reactive silencer (muffler) [1] featuring a double expansion chamber, such a mounting being referred to as EC2 in the software [2], with a circular cross-section, with an inlet and outlet diameter dS = 40 mm [3], a chamber diameter dL = 160 mm [4], and a length for each chamber stage L = 200 mm [5] using the software's optimization features for the extended lengths at the chamber's inlets [6] and outlets [7] (i.e. lx_tun tuning values automatically displayed by SILDIS^®); the lengths of the tubes outside the expansion chamber are considered negligible i.e. are set to zero [8]. The silencer is intended for use with a fluid close to clean, dry air [9] with a flow rate of 0.01 kg/s [10] at 350°C [11] and a pressure of 100000 Pa [12]. The reference spectrum is assumed to be "pink noise" with a sound power level of 130 dB/octave [13]. Total pressure loss is calculated using the IDE2 model for uniform tubes and the MUN1 model for extended inlets and outlets [14]. Silencer self-noise is calculated according to the general 3733c* model [15], the 3733c thermodynamic correction model [16], and the 3733c spectral correction model [17]. Tube roughness is set to 0.00015 m [18]. The TCC1 added-length model is used [19]. Transmission loss ("TL") is the chosen indicator of acoustic performance (excluding self-noise) [20]. The language used is English [21].

Reminder: the geometric tuning length Lx_tun (m) depends on the choice of added length model (this choice must therefore be made first)..

The calculations involved are of practical interest, possibly in the context of engineering or Research and Development (R&D) projects, for the design of various noise-attenuating devices: 

• for internal combustion engine exhaust systems
• for compressors (then with a lower gas temperature)

Input data for the computation of a reactive silencer (muffler) with a double expansion chamber (to be entered in Excel worksheets of SILDIS^® software Module 1)

The input data required for the computation are listed hereafter in reference with the above data (see figures in brackets in the previous §, used as placemarks for explaining the selection below). The input cells are referred to thanks to their Excel’s coordinates (column / line) in the following part extracted from user’s manual.

Worksheet [in COALA] 

Worksheet [in COSIL-1B] COmputation of SILencers

Worksheet [in-out CODIS-1B] COmputation DISplay

Computation results for a reactive silencer (muffler) with a double expansion chamber with Module 1B of software SILDIS^®

Results: main performance indicators for a reactive silencer (muffler) with a double expansion chamber

Upon completion of the modeling and simulations, and after all calculations, various performance indicators are obtained for the silencer.

Sound power level with silencer (muffler) 117.5 dB(A), after consideration of regenerated noise which limits sound attenuation

Silencer (muffler) sound transmission loss 19.6 dB(A)

Silencer (muffler) total pressure loss 158 Pa

In the present case, the results obtained in terms of sound power level with the silencer (muffler) and in terms of sound transmission loss - expressed as overall A-weighted values - are not accurate due to the consideration, for the evaluation of such overall values, of high-frequency performance characteristics - as required - , while the calculations rely on the plane-wave assumption, the agreement between simulation and measurement results - in case of a circular geometry - is optimal (indeed, excellent) up to frequency f1.0 = 1.84 * c / (π.d) * √ (1- M²) (Hz), but less good around frequency f2.0 = 3.05 * c / (π.d) * √ (1- M²) (Hz),  and even less good for frequency f0.1 = 3.83 * c / (π.d) * √ (1- M²) (Hz), formulas in which f1,0 ,  f2,0  , f01 correspond to the first higher-order modes (where d = diameter, c = speed of sound, and M = Mach number under operating conditions).

In this regard, it should be noted that significant high-frequency performance is generally not expected from a reactive silencer stage, being rather assigned to a dissipative silencer stage, with which it can generally be quite easily obtained.

Results: screenshots of worksheet [in-out CODIS1B] (COmputation of DISplay) of Module 1B of software SILDIS^® for a reactive silencer (muffler) with a double expansion chamber

Acoustics & Aerodynamics/aeraulics

In what follows, performance indicators are linked by the formulas Lw1 = 10 * log [10^ (0.1 * (Lw0 – ΔLi’)) + 10^ (0.1 * Lw)] ; ΔL = Lw1 - Lw0

Results: screenshots of worksheet [out EC2] of Module 1B of software SILDIS^® for a reactive silencer (muffler) with a double expansion chamber

The displayed dimensions are the result of the entered data, allowing for verification of the input data's suitability (with the software, other sketches are available showing added lengths and total lengths - including added lengths - that are taken into account).

Remarks regarding the performance of a reactive silencer (muffler) with a double expansion chamber simulated with Module 1B of software SILDIS^®

The input and output data (simulation results) above are those of the version of Module 1B of the SILDIS^® software normally sold; some cells (e.g. model selections for different calculation steps) are pre-filled by default with robust selections (which can be modified by the user) for the considered silencer mounting. In order to limit questions from possibly inexperienced users in relation to the input data to be considered (explanations being provided in user's manual foreseen to be completed by a training), it is possible to restrict the functionalities of the software package by fixing (instead of allowing them to vary) some input data/some models for different steps of the calculations (depending on applications), so as to further simplify the use. 

The Module 1 of software SILDIS^® allows reactive silencer with a double expansion chamber simuled performance results:

• not necessarily using english language: french is possible too (also for entering input data)
• possibly comparable with some laboratory measurement standards e.g. NF EN ISO 7235 Acoustics - Laboratory measurement procedures for ducted silencers and air terminal units- Insertion loss, flow noise and total pressure loss (when appropriate models are selected in software drop-down menus)
• which can be obtained in less than a quarter of an hour, including time necessary for data entry (cf. tutorial of which link is provided in present post bottom)

Acoustic performance - without the consideration of self-noise - is determined using the transfer matrix method to model sound transmission with fluid flow through a cascade of basic silencer elements, the arrangement of which (i.e. their specific sequence) is pre-programmed for various configurations (depending on element to be considered for the modeling, to which corresponds a matrix relevant for its group^[1], for which the - geometric - characteristic length is adjusted by a length correction^[2] that accounts for the specific geometry. Consequently, the modeling approach is not limited to the case of the simple expansion chamber as considered in this example.

The Module 1B of the SILDIS^® software also enables the prediction of silencer performance for more complex case studies - such as those encountered in certain industrial applications - involving modeling based on various configurations (more or less usual)^[3] for which the necessary calculations are fully programmed:

• involving up to 3 expansion chambers in series:
  • possibly purely reactive, as in the example featured in this publication (possibly accounting for gas flow direction reversal(s), e.g. using overlapping connecting tubes)
  • possibly including a dissipative chamber^[4]
  • possibly including 1 or 2 Helmholtz resonators (with short or long neck)
  • possibly including 1 or 2 concentric resonators - with perforated tube(s) -
• possibly featuring a radial silencer inlet
• possibly featuring a radial silencer outlet

With Module 1B of the SILDIS^® software, acoustic performance excluding self-noise (this later being calculated separetely with the same software module) can be evaluated using various indicators:

• sound transmission loss (TL) (as in this example)
• insertion loss (involving source and load impedances)
• sound pressure level difference at specified locations (sometimes referred to as "noise reduction")

These acoustical performance indicators are evaluated using an analytical method, whereas the evaluation of other reactive silencers (mufflers) performance indicators is based on regressions.

Overview of the methodology used for the evaluation of acoustic & aerodynamic performance indicators of reactive silencers (mufflers) with Module 1B of software SILDIS^® 

Returning to reactive silencers (muffler) with expansion chamber(s), as modeled in this calculation example, there are numerous applications in the power generation sector (compressors, internal combustion engines for propulsion or electricity generation, i.e. generator sets) and for test benches:

• regarding workers protection (employees and external contractors handling supervision and maintenance, as well as engineers and technicians conducting laboratory tests who must be protected from hearing damage)
• regarding environmental protection (ensuring peace and quiet for neighbors of facilities housing engines, pumps, other turbomachinery, or other process equipment)

Thus, Module 1B of the SILDIS^® software enables the calculation of the aerodynamic and acoustic performance of reactive silencers (muffler) - including self-generated noise (due to flow) - using a single Excel workbook^[4] and a single data entry step, while employing different models depending on the element group^[1].

The software feature allowing for the tuning of double expansion chambers via extended inlet and outlet tubes maximizes acoustic performance; compared to an untuned configuration, this yields a highly significant improvement in both the noise reduction level and the frequency range over which the noise reduction device is effective. It should be noted that the recommended geometric length of chamber inlet and outlet tubes provided by the software can be used directly to eliminate dips in the acoustic filter's frequency response curve.

In the case of a double expansion chamber, the length of each chamber stage can, of course, be varied without altering the total length (e.g. to adjust performance to meet specific objectives). Figure 5 below shows the following key results:

• for the "equal" configuration i.e. with double expansion chamber stages lengths being as mentioned above (0.2 m and 0.2 m, respectively) the sound transmission loss curve exhibits dips near the frequencies of 1640.5 Hz and 3281 Hz (performance is minimal for these frequencies, and no for others)
• for the "unequal" configuration i.e. with double expansion chamber stages lengths being not as mentioned above, but being instead of 0.23 m and 0.17 m, respectively, the sound transmission loss curve exhibits dips near the frequencies of 1752.5 Hz and 3505 Hz (performance is minimal for these frequencies, and no for others), rather than at 1640.5 Hz and 3281 Hz; this can be of great practical interest if the required attenuation at 1640.5 Hz is a significant issue, or even a critical factor for the silencer

In fact, the combination of such features makes the SILDIS^® software package - at the very least - a rarity, especially when the simulation capabilities it offers are accompanied by numerous other strengths:

• Module 1B of the SILDIS^® software eliminates the often tedious (complex, delicate, time-consuming, and costly) tasks associated with other calculation methods, thus offering several advantages (related to the fact that its use consists exclusively of filling in Excel spreadsheet cells with numerical values ​​and/or selecting models from drop-down menus):
  • no specific prerequisites for the user
  • no need for extensive training, as getting started is easy after a very quick initial support session
  • no need to import or create the silencer geometry, as it is simply configured by selecting dimensions
  • no need for meshing
• the calculation time is not counted in hours, or even minutes
• the basic investment is limited to a single (multi-user) license with initial onboarding support for getting started (advanced support is available as an option), at no additional cost for subsequent years
• the license cost is such that even occasional use is sufficient to recoup, all the more so when it comes to a software package providing - notably for parametric studies - silencer performance evaluations that would be more expensive to obtain otherwise (assuming they are then as complete and reliable ?)

The Module 1B of software SILDIS^® (based on Excel) is the outcome of decades of development and validation (in conditions meeting requirement of ISO 9001 relating to quality management systems) in the domain of acoustics (with a specialization in sound propagation and transmission in ducts and multilayer structures) and aeraulics by a human ressource able to combine theoretical considerations with field technical feedback (laboratories, measurements sites). It is a polyvalent, user friendly and reliable tool for sizing reactive silencer (possibly with the integration of a dissipative stage^[4]) in all contexts: from easiests cases (ventilation/air handling systems with ambiant thermodynamic conditions) up to most demanding applications in industry e.g. in energy production sector or for test benches (customizations are possible for a tailor-made tool, even for non specialized users).

Programming, marketing, training & hotline are available from one company: Isolation Technologie Services aka ITS. Being available with a near-perpetual licence (100 years), Module 1 of software SILDIS^® is a must-have for anyone involved in acoustical insulation/noise control, for engineering studies as well as for Research & Development (e.g. silencer manufacturers or integrators, engineering companies, acoustic consultants & architects offices).

[1] Uniform Tubes (UT), Extended Tubes (ET), Side Tubes (ST), Transverse Tubes (TT), Variable Tubes (VT), dissipative section (DIS), Helmholtz resonators (HR), Perforated Tubes (PT) for concentric resonators

[2] Termination Open Unflanged (TOU), Termination Open Flanged (TOF), Termination Connected-Coaxial, Termination Connected-Staggered (TCS), Termination Connected with Branch (TCB)

[3] Reactive silencers mountings for which computation is possible with software SILDIS^®: cf. link

[4] accounting for a dissipative stage requires a preliminary calculation using Module 1 of the SILDIS^® software, with some output data (the real and imaginary parts of the complex wavenumber in the airway) serving as input data for Module 1B.