Source: https://jmrplens.github.io/phonometry/devices/noise-control/silencer-measurement/ # Measuring a silencer (ISO 7235 and ISO 11691) Everything a silencer model computes comes from geometry. The figure a supplier publishes does not. It is an **insertion loss measured by substitution**: the same rig run twice, once with a plain duct where the silencer will go and once with the silencer in it, and the difference between the two receiving-side levels, band by band. Reading a catalogue without knowing that is how a computed 8,9 dB and a published "25 dB" end up in the same sentence. Two standards describe the measurement, and they differ in scope as much as in what they ask of the laboratory. **ISO 7235:2003** is the full procedure, with a modal filter decoupling the source, a qualified receiving side and a stated measurement uncertainty, and it covers silencers, air-terminal units and other duct elements, with flow and without. **ISO 11691:1995** is six printed pages carrying two equations, and it measures silencers and nothing else, without flow and with none in the answer, up to a design velocity of 15 m/s. A measurement that needs flow, or an object that is not a silencer, is outside ISO 11691 and belongs to ISO 7235. *The three things a measurement report carries beyond the numbers themselves: how its octaves were folded, what its duct mouth was keeping in, and how far any of it can be trusted.* ## 1. One subtraction, two numberings *The two series are the whole method: everything else in both standards is about making sure nothing but the test object changed between them.* The measurement is the same in both: $$ D_\mathrm{i} = L_{W\mathrm{II}} - L_{W\mathrm{I}} $$ with $\mathrm{I}$ the series that had the test object and $\mathrm{II}$ the series that had the substitution duct. ISO 11691 writes the identical thing as $D = L_{p1} - L_{p2}$, and numbers the two series **the other way round**: its 1 is the substitution duct. Two standards for one measurement, with opposite subscripts, is exactly the sort of thing that gets entered backwards, so the arguments here are named for what was in the duct rather than for either numbering. ```python from phonometry import noise_control substitution = [88.0, 90.0, 91.0, 92.0, 92.0, 91.0] # dB, empty duct with_silencer = [84.0, 83.0, 79.0, 72.0, 66.0, 63.0] # dB, silencer fitted d_i = noise_control.substitution_insertion_loss(substitution, with_silencer) print(d_i) # [ 4. 7. 12. 20. 26. 28.] dB ``` If the receiving room's absorption moved between the two series, that difference is not yet the insertion loss. Clause 6.3 puts it right with $10\lg(T_2/T_1)$, where $T_2$ is the reverberation time measured with the test object installed. A room that got **deader** while the silencer was in it was flattering the silencer, and the correction takes that back: ```python corrected = noise_control.substitution_insertion_loss( substitution, with_silencer, reverberation_times=(2.1, 1.8), ) print(corrected.round(2)) # [ 3.33 6.33 11.33 19.33 25.33 27.33] dB ``` Clause 6.3 also allows $T_2 = T_1$ outright when the test object sits outside the room, and then the pair can be left out. ## 2. What the number is not It is not a transmission loss. It is measured against a particular substitution duct in a particular rig, and it carries that rig with it in two ways worth naming. The first is the **limiting insertion loss**: sound flanks along the duct walls rather than through the silencer, and no arrangement can measure past what its own flanking lets round. ISO 7235 has the laboratory measure that ceiling with the substitution duct acoustically blocked and record it as a function of frequency (7.4). A very large catalogue figure is a claim about the test arrangement as much as about the device. The second is the receiving side, which decides how much of the sound the microphones see at all. ISO 7235 allows three (5.2.4): a reverberation room to ISO 3741 qualified at least down to the 125 Hz one-third octave, which is preferred; a test duct with an anechoic termination whose reflection coefficient is no greater than 0,3; or essentially free-field conditions at the open end. ISO 11691 keeps only the reverberation room and the free-field alternatives, and asks for 3,5 m of duct on each side of the silencer. None of that is arithmetic, which is why the rest of this page is short. The arithmetic that remains is the part a reader can get wrong on paper. ## 3. Octaves are folded on the energy, not on the decibels A measurement is made in one-third octaves and often reported in octaves. ISO 11691 Equation (2) says how, and it is not an average of the three numbers: $$ D_\mathrm{oct} = -10 \lg\left[\frac{1}{3}\left( 10^{-D_1/10} + 10^{-D_2/10} + 10^{-D_3/10} \right)\right]\ \text{dB} $$ The average is taken on what the silencer **lets through**. That matters because a silencer is rarely flat across an octave, and the band that leaks decides the answer: ```python thirds = [4.0, 7.0, 12.0, 20.0, 26.0, 28.0] print(noise_control.octave_insertion_loss(thirds).round(2)) # # [ 6.57 23.28] dB ``` The plain arithmetic means of the same two groups are 7,67 and 24,67 dB, so reading the decibels rather than the energy would have overstated both octaves by about a decibel. On a steeper silencer the gap is larger: three thirds of 30, 30 and 5 dB give **9,7 dB** over the octave, not 21,7. Almost all the transmitted sound is coming through the one band that does not work, and Equation (2) is written out rather than described precisely so that this cannot be got wrong. ISO 11691 states the assumption it rests on: the sound pressure levels of the three one-third octaves are taken to be equal in the series run with the substitution duct, which is what lets their energies be weighted equally. ## 4. Three microphone positions, or five In a test duct the spatial average comes from at least three microphone positions equally spaced on a line across the duct, spanning at least a quarter wavelength of the band, about half way along the duct. Three is enough only if the three agree. ISO 7235 Table 6 says how closely, and if the highest and lowest differ by more than that, five positions shall be used: ```python levels = [70.0, 74.0, 79.0] # dB at the three key positions print(noise_control.microphone_spread_limit(125.0)) # 7.0 dB print(noise_control.microphone_positions_required(levels, 50.0)) # 3 print(noise_control.microphone_positions_required(levels, 125.0)) # 5 ``` The same three levels are acceptable at 50 Hz, where the limit is 10 dB, and not at 125 Hz, where it is 7. The limit falls with frequency because a duct at low frequency has a standing-wave pattern that three points sample badly, and at high frequency does not. One thing to know about Table 6: its rows read 50, 63, 80, 100, 125 and then `> 160` Hz, so the **160 Hz one-third octave belongs to no row** and is given no limit at all. Every other row names a single band, and 160 Hz is a one-third-octave centre like the rest, so it is read here as belonging to the last row. The gap is in the [errata register](https://jmrplens.github.io/phonometry/reference/errata/). ## 5. How repeatable any of this is Both standards answer, and neither answer is flattering. ISO 11691 says outright that exact information on the precision of its method cannot be given, that interlaboratory tests would be needed for a real reproducibility standard deviation, and that this is what makes it a survey standard. Its Table 1 offers an estimate only: 2 dB up to the 1,25 kHz one-third octave and 3 dB above it. ISO 7235 Table 7 has three columns, and the disagreement between them is the useful part: ```python for band in (50.0, 250.0, 1000.0, 4000.0): print(band, [ noise_control.measurement_reproducibility(band, quantity=q) for q in ("insertion_loss", "transmission_loss", "intensity") ]) # 50.0 [1.5, 3.0, 3.0] # 250.0 [1.0, 3.0, 1.5] # 1000.0 [2.0, 3.0, 1.0] # 4000.0 [3.0, 3.0, 1.0] ``` Insertion loss is measured best in the middle of the range and worst at the top; the sound-intensity route runs the other way; transmission loss is a flat 3 dB everywhere. Clause 7.9 explains why: only the insertion-loss column came from tests, on 1 m long parallel-baffle silencers, and the other two rest on experience. A column that does not move with frequency is the shape of an estimate, not of a measurement. What goes on the report is twice the table value, for a coverage probability of 95 %: ```python print(noise_control.measurement_expanded_uncertainty(250.0)) # 2.0 dB print(noise_control.measurement_expanded_uncertainty(4000.0)) # 6.0 dB ``` A silencer quoted at 25 dB in the 4 kHz band is being quoted to within 6 dB. ## 6. The scope ISO 11691 draws round itself The survey method is deliberately narrow, and its limits are published rather than implied. The design velocity may not exceed 15 m/s, because the method runs the rig with no flow at all and so includes none of the self-generated noise. It is written for circular silencers from 80 mm to 2 m in diameter, or rectangular ones of comparable area. And the test ducts have to be close in cross section to what they feed, between 0,6 and 1,7 times the area of the silencer or the substitution duct (4.5). Outside that, the joints reflect more than the method allows for, and the library says so: ```python print(noise_control.substitution_area_ratio(0.0962, 0.0962)) # 1.0 print(noise_control.SURVEY_MAX_VELOCITY_M_S) # 15.0 print(noise_control.SURVEY_AREA_RATIO_RANGE) # (0.6, 1.7) ``` ## 7. The open end, and the two quantities that need it A duct radiating into a room does not hand the room everything that reaches its mouth. Well below the frequency at which the mouth is a wavelength across it is a poor radiator, and most of the energy turns round and travels back up the duct. Annex B.3 puts a number on it: $$ D_\mathrm{td} = 10 \lg\left[1 + \frac{\Omega}{\left(\dfrac{4\pi f \sqrt{S}}{c}\right)^{2}}\right]\ \text{dB} $$ The group $4\pi f \sqrt{S} / c$ is the mouth measured in wavelengths, and $\Omega$ is the solid angle it radiates into. A 350 mm duct flush with a wall holds back 11 dB at 63 Hz and 0,05 dB at 2 kHz, which is nothing: ```python import numpy as np bands = np.array([63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0]) area = 0.0962 # m2, a 350 mm circular duct d_td = noise_control.open_end_transmission_loss(bands, area) print(d_td.round(2)) # [11.23 6.14 2.5 0.77 0.21 0.05] dB ``` *The solid angle is the only term of Equation (B.3) a laboratory chooses rather than measures, and the five choices are the same in both standards.* The solid angle is Table B.1, and the same five values are Table 1 of ISO 5135. It works the way round that surprises people: $\Omega$ sits in the numerator, so a duct ending in the middle of the room keeps **more** sound in than one flush with a wall, not less. A baffle is what makes an opening a good radiator, because it stops the pressure relieving round the rim of the mouth, and an unbaffled end of the same size sends more of the sound back up the duct: ```python free = noise_control.open_end_transmission_loss( bands, area, solid_angle_sr=noise_control.RADIATION_SOLID_ANGLES["C"], ) print(free.round(2)) # [14.07 8.59 4.08 1.43 0.4 0.1 ] dB ``` Three more decibels at 63 Hz, and the reflection coefficient rises with it. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry-assets/main/images/anim_fdtd_open_end.webm) *Both carriers are below the 857 Hz cut-on of a duct that wide, so the field inside it is the plane wave the whole of ISO 7235's duct arithmetic assumes. The reflection coefficients read off the standing wave are the simulated geometry's own, and not Equation (B.4): that closed form is a piston approximation for a circular mouth, where this is a slit in a screen. What the two share is the mechanism, and it is the mechanism that decides whether a level measured in a room is the level in the duct behind it.* Equation (B.4) says the same fact the other way round, as a pressure reflection coefficient, and the two close exactly on the energy: $D_\mathrm{td} = -10\lg(1 - r^2)$ at every frequency, area and solid angle. That identity is the conformance anchor for both, because neither is printed with a worked value. It also has a use of its own: 5.2.4 qualifies a test duct as anechoic only below $r = 0{,}3$, which this bare open end reaches somewhere between 500 Hz and 1 kHz. ```python r = noise_control.open_end_reflection_coefficient(bands, area) print(r.round(3)) # [0.962 0.87 0.662 0.404 0.215 0.11 ] ``` The library carries a second closed form for the same physics, Reynolds' as given by Long, in [`end_reflection_loss_closed_form`](https://jmrplens.github.io/phonometry/devices/noise-control/noise-control/). It raises the same argument to 1,88 rather than to 2, and for a circular duct in free space the two read $10\lg[1 + (c/\pi f d)^2]$ against $10\lg[1 + (c/\pi f d)^{1,88}]$. They agree closely where the argument is near 1 and part company at the ends of the range. Two quantities need it. Equation (6) turns the measured insertion loss of an air-terminal unit into its transmission loss by putting back what the mouth was keeping in anyway, so the two are the same number at the top of the range and eleven decibels apart at the bottom: ```python d_i = np.array([4.0, 7.0, 12.0, 20.0, 26.0, 28.0]) print(noise_control.measured_transmission_loss(d_i, d_td).round(2)) # # [15.23 13.14 14.5 20.77 26.21 28.05] dB ``` And Equation (7) makes the flow noise a sound power, $L_W = \overline{L_p} + D_\mathrm{td} + C$, where $C$ is the ISO 3741 level difference between the power radiated into the room and the average pressure in it. Clause 6.4 is explicit that $\overline{L_p}$ goes in **without** a background correction: the two series are reported separately and the reader subtracts them. ## 8. Where higher-order modes start The modal filter between the source and the test object exists to stop higher-order modes reaching the silencer, and its requirement steps at the frequency where those modes can propagate in the connected ducts: at least 3 dB of longitudinal attenuation of the fundamental at the low-frequency end, and at least 5 dB above that frequency (5.2.2.3). NOTE 2 prints where it is: ```python print(round(noise_control.modal_filter_cut_on(diameter_m=0.4), 1)) # 505.9 Hz print(round(noise_control.modal_filter_cut_on(larger_dimension=0.5), 1)) # 343.0 Hz ``` The rectangular form, $0{,}5\,c/H$, is exact: the first mode of a rigid rectangular duct is a half wavelength across the larger dimension. The circular one, $0{,}59\,c/d$, is rounded. The exact coefficient is the first zero of $J_1'$ over $\pi$, which is 0,58607, so Equation (4) sits 0,67 % high: on the 0,4 m duct of the ISO 11691 sound source that is 505,9 Hz where [`circular_duct_cut_on`](https://jmrplens.github.io/phonometry/devices/noise-control/duct-path/) gives 502,6 Hz. Three and a half hertz does not matter for choosing a modal filter, and it is worth knowing which of the two numbers is the physics. ## 9. What the object costs to push air through The third thing ISO 7235 measures has nothing to do with sound. A silencer that works and costs a fan half its pressure is not a good silencer, so 6.5 measures the **total pressure loss coefficient**, and the whole point of the coefficient rather than the loss is that a loss means nothing without the flow it was measured at. Start with the air. Equation (10) is the ideal gas law with the standard's own constants, and the static pressure it takes is a gauge pressure against the ambient, so the two add: ```python rho = noise_control.normal_air_density(200.0, 101325.0, 20.0) print(round(rho, 4)) # 1.2073 kg/m3 q_v = noise_control.volume_flow_rate(1.2, rho) print(round(q_v, 4)) # 0.9939 m3/s ``` ISO 7235 prints $R = 287$ and writes the absolute temperature as $\theta + 273\ ^\circ\text{C}$, neither of which is the accurate figure. The offset alone puts the density 0,051 % high at 20 °C, and the gas constant adds 0,017 % to that, for 0,069 % in all. It does not cancel: the same density is in the dynamic pressure of both test series, so the coefficient is **scaled** by that one factor rather than shifted, and comes out 0,069 % low. That is far under the uncertainty of a pressure-loss test, and using the printed constants is what reproduces a result computed to the standard, so the library keeps both. It is recorded in the [errata register](https://jmrplens.github.io/phonometry/reference/errata/) as a property of the source rather than as a defect. Equation (9) rather than (8) is used when the flow meter and the test object are far enough apart in temperature or pressure that their density ratio leaves 0,98 to 1,02: outside that window the meter is not measuring the flow the test object sees. The velocity head is Equation (13), and the coefficient is the loss divided by it: ```python p_d1 = noise_control.dynamic_pressure(q_v, 0.0962, rho) print(round(p_d1, 2)) # 64.44 Pa delta_p_t = noise_control.total_pressure_loss(45.0, p_d1, 0.0962, 0.0962) print(round(noise_control.pressure_loss_coefficient(delta_p_t, p_d1), 3)) # # 0.698 ``` That number belongs to the object rather than to the test point, at least to the extent the flow is dynamically similar: a loss grows as the square of the velocity and so does the head it is divided by, so the algebra returns the same coefficient at twice the flow. Real flow is not exactly similar, because doubling the rate doubles the Reynolds number too, and that is the reason 6.5.2 measures at five rates and averages rather than trusting one. The drift over a test range is small, and it is not zero. Equation (12) is the part worth reading twice. Measuring static pressures on both sides is not enough when the two sides are different sizes, because an object that widens the duct converts velocity head back into static pressure and a static difference alone would credit it with a recovery that is only bookkeeping. The bracket $1 - (S_1/S_2)^2$ puts it back, and the NOTE to Equation (14) says what usually happens to it: as a rule $S_1 = S_2$, and it vanishes. Where it does not, it is not small: ```python widening = noise_control.total_pressure_loss(45.0, p_d1, 0.0962, 2 * 0.0962) print(round(noise_control.pressure_loss_coefficient(widening, p_d1), 3)) # # 1.448, from the same 45 Pa of static loss ``` ## 10. The substitution trick, again The fundamental method of 6.5.2.2 measures the coefficient the way the acoustic half measures insertion loss: run the rig with the test object, run it again with the substitution duct, and the difference belongs to the object. The computational route of 6.5.2.2.3 does the subtraction on the coefficients rather than on the pressures, which means the two series need share neither their flow rates nor even their number of points: ```python import numpy as np heads = np.array([20.0, 40.0, 60.0, 80.0, 100.0]) # Pa, five airflow rates with_object = 2.5 * heads without = 0.6 * heads zeta = noise_control.average_pressure_loss_coefficient( with_object, heads, without, heads, ) print(round(zeta, 3)) # 1.9 ``` Five rates per series, spread evenly over the range, and the lowest has to produce more than 10 Pa so that the smallest number in the average is still a measurement rather than the resolution of the manometer. The library says so on both counts: `average_pressure_loss_coefficient` warns below five points, and `pressure_loss_coefficient` warns on a loss of 10 Pa or less, the boundary included, because the clause reads *greater than*. What the flow has to be before any of that counts is a matter of geometry. The upstream duct is straight for five equivalent diameters or two metres, whichever is greater, so that the velocity profile has settled; it must be uniform to ±10 % of the mean over the section, excluding the 15 mm nearest the walls, surveyed ten points along each of two perpendicular axes about $1{,}5\,d_e$ upstream. The two length rules cross at a 0,4 m equivalent diameter: ```python print(round(noise_control.upstream_straight_length(0.0962), 2)) # 2.0 m print(round(noise_control.upstream_straight_length(0.5), 2)) # 3.99 m ``` ## 11. The third standard, and the formula it shares ISO 5135:1999 measures something else again: the sound power an air-terminal device, air-terminal unit, damper or valve radiates, determined in a reverberation room to ISO 3741. What a designer needs is not that but what the device puts into the duct behind it, and Equation (1) is the step between them: $$ L_{W\mathrm{duct}} = L_W + \Delta L_\mathrm{r} $$ The correction $\Delta L_\mathrm{r}$ is Equation (2) of ISO 5135, and it is worth writing both printings side by side: $$ \Delta L_\mathrm{r} = 10\lg\left[1 + \left(\frac{c}{4\pi f}\right)^{2}\frac{\Omega}{S}\right] \qquad D_\mathrm{td} = 10\lg\left[1 + \frac{\Omega}{\left(\dfrac{4\pi f\sqrt{S}}{c}\right)^{2}}\right] $$ Expand either and both become $10\lg[1 + \Omega c^2 / (16\pi^2 f^2 S)]$. They are one formula, their two solid-angle tables agree entry for entry, and `open_end_transmission_loss` is both. That is why nothing new appears here for the correction itself: ```python d_lr = noise_control.open_end_transmission_loss(bands, area) lw_room = np.array([58.0, 60.0, 61.0, 59.0, 55.0, 50.0]) print(noise_control.duct_sound_power_level(lw_room, d_lr).round(2)) # # [69.23 66.14 63.5 59.77 55.21 50.05] dB ``` A device measured in a room is understated in the duct by eleven decibels at the bottom of the range and by nothing at the top. The NOTE to Table 1 offers a way round the correction rather than a second formula for it: fit a transmission element to ISO 7235 and no correction is applied at all. ## 12. Reading a level at a duty nobody measured A device is not tested at the one operating point a designer will use it at. ISO 5135 5.5.2 fits a straight line by least squares through the levels against $\lg q_V$ when the tests were made at a constant pressure loss coefficient, or against $\lg \Delta p_\mathrm{t}$ when they were made at a constant flow rate. The same fit serves the band levels and the A-weighted one: ```python duty = np.array([0.05, 0.1, 0.2, 0.4, 0.8]) # m3/s levels = np.array([38.0, 44.5, 50.0, 56.5, 62.0]) # dB(A) line = noise_control.fit_operating_line(duty, levels) print(round(line.slope, 2)) # 19.93 dB per decade print(round(line.maximum_deviation, 2)) # 0.3 dB print(round(line.level_at(0.3), 1)) # 53.7 dB(A) ``` Two rules keep that honest, and the library says so on both rather than refusing: a fit that is not a straight line is still a fit, and a level read past the range is still a number, and what the standard asks is that a report be honest about them. The measured points have to sit within 3 dB of the line, because past that the levels are not a straight line in this variable and reading the line off means nothing. And the line may be extended down to half the smallest duty measured and up to twice the largest, and no further, which for these five points is 0.025 to 1.6 m³/s. Either one out of range raises a `SilencerMeasurementWarning`: ```python print(line.valid_range) # (0.025, 1.6) ``` Clause 8 k) closes the loop: a report gives the fully corrected levels to the nearest half decibel and has to state which of them were extrapolated rather than measured directly. `.plot()` draws that distinction, shading the two ends of the range that are extrapolation. ## Standards ISO 7235:2003, ISO 11691:1995 and ISO 5135:1999, read from BS EN ISO 7235:2009, BS EN ISO 11691:2009 and BS EN ISO 5135:1999, which endorse them without modification. Implemented, from ISO 7235 and ISO 11691: Equation (1) of both with the reverberation-time correction of ISO 7235 6.3; the octave fold of ISO 11691 Equation (2); ISO 7235 Table 6 and the three-or-five rule of 6.2.1; ISO 11691 Table 1 and all three columns of ISO 7235 Table 7, with the coverage factor of 7.9; the scope of ISO 11691 1.1 and 4.5; the open-end transmission loss and reflection coefficient of Equations (B.3) and (B.4) with the solid angles of Table B.1; the transmission loss of Equation (6) and the flow-noise sound power of Equation (7); the cut-on frequencies of Equations (4) and (5); and the flow half of 6.5, from the gas law of Equations (10), (21) and (22) to the substitution average of Equation (18) and the settling length of 6.5.2.2.1. From ISO 5135: the duct sound power level of Equation (1), Equation (2), which is ISO 7235 (B.3) written out again, and the least-squares operating line of 5.5.2 with its 3 dB fit limit and its half-to-twice reading range. Checked in the [conformance report](https://jmrplens.github.io/phonometry/reference/conformance/); the gap Table 6 leaves at 160 Hz and the two printed gas-law constants are in the [errata register](https://jmrplens.github.io/phonometry/reference/errata/). Not implemented: the facility requirements themselves, the flow measurement of ISO 5167-1, the ISO 3741 determination the receiving side is handed to, and ISO 11820, which measures a silencer in situ. ## See also - [Silencers](https://jmrplens.github.io/phonometry/devices/noise-control/silencers/): the reactive four-pole models this measurement is the counterpart of, and the rig it is made on, described in full. - [Duct-Borne Noise: Fan to Room](https://jmrplens.github.io/phonometry/devices/noise-control/duct-path/): where a measured insertion loss goes once it has been bought. - [HVAC Noise the German Way (VDI 2081)](https://jmrplens.github.io/phonometry/devices/noise-control/vdi2081-air-systems/): the splitter-silencer insertion loss predicted from geometry, for comparison with what a laboratory would measure. - [Errata in published sources](https://jmrplens.github.io/phonometry/reference/errata/): the gap Table 6 of ISO 7235 leaves at 160 Hz, and the two printed gas-law constants. - API reference: [`noise_control.silencer_measurement`](https://jmrplens.github.io/phonometry/reference/api/noise_control/silencer-measurement/). - [Industrial noise control](https://jmrplens.github.io/phonometry/devices/noise-control/noise-control/): the rival closed form for the end reflection, and the air-terminal corrections a diffuser is selected with.