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

The SILDIS^® software (Excel) enables the computation of the acoustic and aerodynamic performance of a reactive silencer, such as a simple expansion chamber, as illustrated by an example realting to a typical component of an internal combustion engine exhaust system (Example 1B.4.2 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 with a simple expansion chamber using software SILDIS^®)

The aim is to calculate the acoustic and aerodynamic performance of a purely reactive silencer [1] featuring a simple expansion chamber, such a mounting being referred to as EC1 in the software [2], with a circular cross-section, with an inlet and outlet diameter dS = 100 mm [3], a chamber diameter dL = 300 mm [4], and a length L = 1200 mm [5] suing the software's optimization features for the extended lengths at the chamber inlet [6] and outlet [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 - flow arte [9] of 0.1 kg/s [10] at 500°C [11] and a pressure of 100,000 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].

Sketch of a simple expansion chamber (EC1) as modeled using SILDIS® software; element 6 is the uniform inlet tube (with the smallest diameter), element 5 is an extended inlet, element 4 is a uniform tube (with the largest diameter), element 3 is an extended outlet, and element 2 is the uniform outlet tube (with the smallest diameter).

Input data (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] 

Item	Cell for input	Foreseen action	Input	See placemark / comment
Language	Z1	For english, select E, for french select F	F	[21]
Temperature (°C)	D7	Enter a real number	500	[11]
Pressure (Pa)	D8	Enter a positive real number	100000	[12]
Lw0 (dB ref. 1pW)	C15 to L15	Enter a positive real number for the sound power level per 1/1 octave frequency band	130	[13]

Worksheet [in COSIL-1B] COmputation of SILencers

Item	Cell for input	Foreseen action	Input	See placemark / comment
Mass flow rate (kg/s)	B5	Enter a positive real number	0.1	[10]
Inlet diameter (m)	B7	Enter a positive real number	0.1	[3]
General model of flow acoustic power	S5	Select a model (in a list)	3c*	[15]
Model of thermodynamic correction	S6	Select a model (in a list)	3733c	[16]
Model of spectral correction	S7	Select a model (in a list)	3733c	[17]
Mounting	B19	Select a model (in a list)	EC1	[2]
Chamber model (stage 1)	AC19	Select a model (in a list)	REA	[1]
dS (stage 1) (m)	O88	Enter a positive real number	0.1	[3] selection by default
dL (stage 1) (m)	O89	Enter a positive real number	0.3	[4]
Chamber total length L (m)	O93	Enter a positive real number	1.2	[5]
Tube's roughness Δ (m)	M95	Enter a positive real number	0.00015	[18]
Geometric length Lx ≠ 0 (m)	H137, J137, P137, Q137	Enter a positive real number	0.57507, 0.27507, 0.57507, 0.27507	[6] cf. cell P136, [6] cf. cell P137
Model for total pressure loss	H141, I141, J141, P141, Q141	Select a model (in a list)	IDE2, IDE2, IDE2, MUN1, MUN1	[14] selection by default
Model for added length	K250, N250	Select a model (in a list)	TCC1	[19]

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

Item	Cell for input	Foreseen action	Input	See placemark / comment
Model pour ΔL	AN48	Select a model (in a list) for acoustic performance without flow noise indicator	TL	[20]

Outcome: main results of simulation with Module 1B of software SILDIS^®

Sound power level with silencer 125.0 dB(A), after consideration of regenerated noise which limits sound attenuation

Silencer sound transmission loss 12.1 dB(A)

Silencer total pressure loss 230 Pa

In the present case, the results obtained in terms of sound power level with the silencer 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.

Outcome: screenshots of worksheet [in-out CODIS1B] (COmputation of DISplay) of Module 1B of software SILDIS^®

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

Result of the acoustic and aeraulic/aerodynamic performance simulation of a reactive silencer with a simple expansion chamber optimized (via tuning) using SILDIS® software

Outcome: screenshots of worksheet [out EC1] of Module 1B of software SILDIS^®

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).

Summary of dimensions entered for modeling the reactive silencer with a single expansion chamber optimized (via tuning) using SILDIS® software.

Remarks regarding the performance of a reactive silencer with a simple 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 simple 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) with a simple expansion chamber performance indicators is based on regressions.

Overview of the methodology used for the evaluation of acoustic & aerodynamic performance indicators of reactive silencers with siingle expansion chamber with Module 1B of software SILDIS^® 

Performance indicator	Performance without flow noise ΔL'	Flow noise Lw	Total pressure loss Δpt
Methodology	Analytic	Regression based on measurements	Regression based on measurements

Returning to reactive silencers 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 - 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 expansion chambers via extended inlet and outlet tubes makes it possible to maximize acoustic performance; compared to the "untuned" configuration, this yields a highly significant improvement in both noise reduction levels and frequency range in which noise reduction device is efficient. It should be noted that the recommended values for the geometric lengths upstream and downstream of the perforated section, provided by the software, are directly usable to eliminate dips in the acoustic filter's frequency response curve. The figure below illustrates the following main results:

• for the "without" tuning configuration, the sound transmission loss curve exhibits dramatic dips near frequencies that are multiples of c/2L = 229 Hz (where c is the speed of sound under the silencer's operating conditions in m/s, and L is the expansion chamber length in m); such performance weaknesses are, of course, problematic for the practical overall effectiveness of this type of noise control device
• for the "with" tuning configuration, these dips no longer appear near frequencies that are multiples of c/2L = 229 Hz, and the sound transmission loss is significantly higher than that of the "without" tuning configuration across a wide frequency range (the first dip in the noise attenuation curve appears at the frequency 2c/L = 915 Hz, i.e. slightly below the frequency f1,0 = 1072 Hz); a reactive silencer sized in this way - following the tuning rules programmed into the SILDIS^® software - offers performance of great practical value, as it covers several adjacent octave bands

Comparative acoustic performance of a reactive silencer with a simple expansion chamber using SILDIS® software: sound transmission loss "with" and "without" tuning via extended tubes

The phenomenon causing the disappearance - upon tuning - of three-quarters of the dips (troughs) observed in the frequency response curve of the resulting acoustic filter is the resonance of the cavities with an impervious rigid back formed by the tubes extending from the inlet and outlet of the expansion chamber, yielding performance peaks at frequencies that would otherwise correspond to dips (throughs).

Of course, the software accounts for the influence of the flow, not only regarding the silencer's self-noise but also when evaluating the transfer matrices that model the sound transmission of each element of the cascade making the noise-attenuation device; that its impact on transmission loss is low - as for the present example, with a Mach number close to 0.05, what the figure below is illustrating - does not hold true in all cases, particularly with higher Mach numbers e.g. resulting from silencer operating conditions with higher gas velocity and/or lower temperature.

Comparison of the sound transmission loss of a simple expansion chamber: with or without flow (calculation results with software SILDIS®)

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 1 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.

Tutorial

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