Plenum Attenuation

**Strutt|Mechanical Services|Duct Elements|Plenum** inserts the octave band attenuation from a plenum into the active row of the worksheet.

The **Arup Mechanical Services Guide** (Wells) method calculates attenuation from a plenum using:

Attenuation `= -10log_10 (S_2 (cos(theta)/(2 pi d^2) + 1/R_c))`

where,

`S_o` is the outlet area (m^{2})

`d` is the slant distance inlet/outlet (m)

`theta` is the angle `d` makes with the inlet axis (deg)

`R_c` is the Room Constant and is derived from:

`R_c = (S_o + S_i + S bar(alpha))/(1 - bar(alpha))`

where,

`S` is the total internal surface area of the plenum (m^{2}) excluding the inlet and outlet area

`S_i` is the plenum inlet area (mē)

`bar(alpha)` is the average absorption coefficient of the internal walls of the plenum *excluding the openings*

Strutt assumes that the inlet and outlet openings have `alpha = 1` and calculates the overall room constant accordingly

Inputs required in the calculation are:

- Length, width and height of plenum (not including inlet/outlet ducts), used to calculate total internal surface area of plenum.

- Inlet and Outlet dimensions, used to calculate total internal surface area of plenum.

- Outlet axis offsets from inlet axis, used to calculate slant distance and the angle of the slant distance with inlet axis.

- Absorption coefficients of the plenum lining (alternatively, coefficients for common lining materials may be selected from the drop-down menu.

Note: use both methods with caution where either the width or the height of the plenum is less than `1.5d`, in which case the plenum may not act as an expansion chamber

**ASHRAE 2007** calculates attenuation from a plenum as follows:

Plane wave propagation in a duct exists at frequencies below the cutoff, creating the need to consider two frequency ranges. The cutoff frequency `f_(co)` is the frequency above which plane waves no longer propagate in the duct.

`f_(co) = c/(2a)`

or

`f_(co) = 0.586 c/d`

where,

`f_(co)` is the cutoff frequency (Hz)

`c` is the speed of sound in air (m/s)

`a` is the larger cross-sectional dimension of rectangular duct (m)

`d` is the diameter of a round duct (m)

Table 7 ASHRAE gives sound absorption coefficients for common plenum materials. These are incorporated into the calculation and the appropriate material can be chosen.

At the frequencies above f_{co}, waves that propagate in the duct are called cross or spinning modes. The TL in this higher frequency range is predicted using:

`TL = b [(S_(out) Q)/(4 pi r^2) + (S_(out) (1 - alpha_a))/(S alpha_a)]^n`

`TL = b (K_f)^n`

where,

`TL` is the transmission loss (dB)

`b = 3.505`

`n = -0.359`

`K_f` is the attenuation coefficient

`S_(out)` is the area of plenum outlet (m^{2})

`S` is the total inside surface area of plenum minus inlet and outlet areas (m^{2})

`r` is the distance between centers of inlet and outlet of plenum (m)

`Q` is the directivity factor; taken as 2 for opening near centre of wall, 4 for opening near corner of plenum

`alpha_a` is the average absorption coefficient of plenum lining given by

`alpha_a = (S_1 alpha_1 + S_2 alpha_2)/(S_1 + S_2)`

where,

`alpha_1` is the sound absorption coefficient of any bare or unlined inside surfaces of plenum

`S_1` is the surface area of any bare or unlined inside surfaces of plenum (m^{2})

`alpha_2` is the sound absorption coefficient of acoustically lined inside surfaces of plenum

`S_2` is the surface area of acoustically lined inside surfaces of plenum (m^{2})

For lower frequencies below f_{co} (with a limit of 50 Hz), TL is calculated as:

`TL = 10.76 * A_f S + W_e`

where,

`10.76` is a conversion factor from square metres to sqaure feet

`A_f` is the surface area coefficient (dB/ft^{2}), referenced according to plenum size. NOTE: the coefficients in Table 13 in the SI version of the Handbook are incorrect - the coefficients from the IP Handbook are used instead, and the 10.76 correction factor is applied.

`W_e` is the wall effect (dB), selected by the user as the type of plenum wall lining

`cos(theta) = l/r = l/sqrt(l^2 + r_w^2 + r_h^2)`

where,

`theta` is the offset angle representring `r` to long axis `l` of duct

`l` is the length of plenum (m)

`r_w` is the vertical offset between axes of plenum inlet and outlet (m)

`r_h` is the horizontal offset between axes of plenum inlet and outlet (m)

Note that `r_w` and `r_h` are interchangeable.

For a given offset angle, Table 9 of the ASHRAE guide includes a TL correction for angles up to 45°.

References:

- Arup Sound and Vibration in Mechanical Services Design, p46
- ASHRAE 2007 Handbook, 47.12 - 47.15
- ASHRAE 2011 Handbook, 48.18 - 48.21

Comments or suggestions to strutt@arup.com