Question 2 of 7: Grouped motor-pump sets — source addition, noise rating and permissible exposure

Question 2: Grouped motor-pump sets — source addition, noise rating and permissible exposure (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Four identical, incoherent pump sets clustered together produce a combined A-weighted level of 98 dB(A) at 2 m, with negligible background noise, and their spectrum is dominated by the 2 kHz octave band.

Find. The level with one pump running, the Noise Rating of the space, the permissible daily exposure time in each case, and two practical controls that would lengthen it.

Approach. Incoherent sources add on an energy basis, so removing three of four identical pumps subtracts $10\log_{10}4$; the Noise Rating is then read from the curve that passes through the dominant band, and each level is converted to a permissible time through the stated 5 dB exchange rule.

1. Add the four incoherent sources to establish the single-pump level. Four identical machines that are not phase-locked add their mean-square pressures, so the combined level exceeds a single unit's level by $10\log_{10} n$:$$L_{p,4}=L_{p,1}+10\log_{10} n \quad\Longrightarrow\quad L_{p,1}=L_{p,4}-10\log_{10} 4$$Substituting the measured value, $L_{p,1}=98-10\log_{10}4 = 98-6.02$, so $$\boxed{L_{p,1}=92.0\ \text{dB(A)}}$$Turning off three of four pumps buys only about 6 dB — a useful reminder that noise reduction by removing duplicate machines is slow going.
2. Read the Noise Rating from the dominant octave band. The NR rating of a space is the number of the lowest NR curve that lies at or above every octave-band level. Because all other bands are stated to be significantly lower, the 2 kHz band alone sets the rating. Along the standard NR family the band level varies with the rating as $L_p = a_f + b_f N$; at 2 kHz, $a_{2k}=-3.5$ and $b_{2k}=1.015$, so$$N=\frac{L_p-a_{2k}}{b_{2k}}=\frac{98-(-3.5)}{1.015}=100.0$$Read graphically in Figure 2.1, the 98 dB point at 2 kHz sits midway between the printed NR 95 curve (92.9 dB at 2 kHz) and the NR 105 curve (103.1 dB), which places it on$$\boxed{\text{NR }100}$$
3. Convert the four-pump level to a permissible exposure time. With a base level $L_0$ permitted for $T_0=8$ h and an exchange rate $q$, the allowable time halves for every $q$ decibels of excess:$$T=\frac{T_0}{2^{(L-L_0)/q}}=\frac{8}{2^{(98-85)/5}}=\frac{8}{2^{2.6}}=\frac{8}{6.063}$$so that$$\boxed{T_{4\ \text{pumps}}=1.32\ \text{h}\approx 1\ \text{h}\ 19\ \text{min}}$$A worker may therefore spend only about one-sixth of a shift at this position without hearing protection.
4. Repeat for the single-pump condition. Using the level from Step 1,$$T=\frac{8}{2^{(92.0-85)/5}}=\frac{8}{2^{1.396}}=\frac{8}{2.632}$$which gives$$\boxed{T_{1\ \text{pump}}=3.04\ \text{h}\approx 3\ \text{h}\ 2\ \text{min}}$$The 6.0 dB reduction has multiplied the permissible time by $2^{6.02/5}=2.30$ — the exchange rate, not the number of machines, sets the payoff.
5. Propose two engineering changes. The two highest-value measures at this installation are (a) enclose or isolate the pump sets — a sealed, absorptively lined acoustic enclosure over each motor-pump set, mounted on vibration isolators and with the pipework flexibly connected, readily returns 15–25 dB in the 2 kHz band because the dominant path is airborne casing radiation; and (b) treat the reverberant field and increase separation — adding absorptive lining to the ceiling and upper walls of the large space lowers the reverberant contribution, while relocating the operator station (or the pumps) so the working position is well outside the near field further reduces the direct contribution. A 13 dB reduction, which either measure can approach, restores a full 8-hour shift.

Administrative controls — job rotation, or hearing protection with an adequate attenuation rating — are legitimate supplements but rank below engineering controls in the hierarchy of controls used in Canadian OH&S practice, and would be applied only while the engineering work is being carried out.