# Time-Variant Positive Air Pressure in Drainage Stacks as a Pathogen Transmission Pathway of COVID-19

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## Abstract

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^{−1}discharging at a height between 15 m to 33 m above the stack base. The maximum pressure and probabilistic positive air pressures in the discharging stack ventilation section with no water (Zone A of the discharging drainage stack) were determined. It was demonstrated that the positive air pressures were lower in frequency as compared with those in other stack zones and could propagate along the upper 1/3 portion of the ventilation pipe (H’ ≥ 0.63) towards the ventilation opening at the rooftop. As the probabilistic positive pressures at a stack height were found to be related to the water discharging height and flow rate, a risk model of positive air pressure is proposed. Taking the 119th, 124th, 140th and 11,547th COVID-19 cases of an epidemiological investigation in Hong Kong as a baseline of concern, excessive risk of system overuse was evaluated. The results showed that for a 20–80% increase in the frequency of discharge flow rate, the number of floors identified at risk increased from 1 to 9 and 1 to 6 in the 34- and 25-storey residential buildings, respectively. The outcome can apply to facilities planning for self-quarantine arrangements in high-rise buildings where pathogenic virus transmission associated with drainage system overuse is a concern.

## 1. Introduction

_{d}= 0.05–0.95). Figure 2 plots the occasional positive air pressures recorded within the discharging drainage stacks [12,13,14,15,16]. In Figure 2b, the probability of positive pressure is the fractional counts of all positive pressure values measured in a 24-h period for unsteady flow conditions [15,16]. For measurements made under steady flow conditions, the probability was estimated using the reported minimum pressures at the 99% confidence intervals in measurement time periods of 140 s–300 s [12,13,14].

## 2. Materials and Method

^{−1}to 4 Ls

^{−1}discharging at a height between 15 m to 33 m above the stack base was used for this study. It was reported by Pink that the steady flow conditions covered the maximum ventilation flow rate require for discharging stacks (34–104 m of height, 0.1–0.15 m of diameter) [20]. These flow rates were typical for drainage stacks in high-rise buildings [21,22].

_{d}, H’

_{p}, q) and its probability (occurrence) p~p(H’

_{d}, H’

_{p}, q) measured at a height H

_{p}(levels 2–12), along with the discharge points H

_{d}(at levels 6–12) at steady flow rates q = 1 Ls

^{−1}, 2 Ls

^{−1}, 3 Ls

^{−1}and 4 Ls

^{−1}, are expressed in Equation (1) as a normalized stack height H’ for generality [14].

^{−1}) at time period t

_{a}(s) is given by Equation (2), where P (m wg.) is the gauge air pressure, g (=9.81 ms

^{−2}) is gravity, t (s) is the period of a pressure variation cycle, and f (Hz) is the frequency given by Equation (3), for k = 0, 1, …, N − 1 [23,24].

_{max}and the probability of positive air pressure p in a contaminated drainage stack can be an indicator of higher pathogen transmission risk through the stack. Both the maximum positive air pressure P and the probability p of having positive air pressure are assumed correlated to the heights H’

_{p}, H’

_{d}and the flow rates q [25] as shown in Equation (4).

_{max}, p) and the predictors (H’

_{d}, H’

_{p}, q) can be measured by the sample correlation coefficient r and test to be significant for p ≤ 0.05 of t-test T given by Equation (5) [26].

_{max}≥ 0 in Equation (4) with the significant predictors, where H’

_{p}is the fractional stack height, q is the discharging flow rate and (H’

_{d}–H’

_{p}) is the fractional discharging height. The regression constants are determined by least-square fitting as shown in Equations (6) and (7) [26].

_{1}is the flow rate of the appliance, k is the frequency factor of probabilistic discharge of appliances being connected to the stack [17],

_{max}> P

_{1}and probability p > p

_{1}as shown in Equation (9), where P

_{1}and p

_{1}are respectively the reference positive air pressure and the reference probability of having positive air pressure of the transmission risk concerned.

## 3. Results and Discussion

_{max}(m, water gauge pressure of denoted as ‘m wg’) along a normalized stack height H’

_{p}(=0.08, 0.16, 0.24, 0.32, 0.39, 0.47, 0.55, 0.63, 0.71, 0.79, 0.87) for a steady flow rate q (=1, 2, 3, 4 Ls

^{−1}) discharged at 7 normalized height H

^{’}

_{q}(=0.39, 0.47, 0.55, 0.63, 0.71, 0.79 & 0.87) is shown in Figure 4a, where H (=38 m) is the stack height, H

_{q}(m) is the discharge height and H

_{p}(m) is the height of pressure measurement. Results of 28 cases (4 discharging flow rates at 7 discharging locations) were presented.

_{min}| > P

_{max}. Second, the time average air pressures over the measurement period were negative in all cases.

_{p}≤ 0.55 at a water discharge flow rate of ≤2 Ls

^{−1}.

_{p}. The fractional number of 28 test cases having positive air pressure recorded is shown in Figure 5b. The average maximum local positive air pressure did not vary significantly in Zone A, except for the test condition discharging the highest flow rate of 4 Ls

^{−1}. The average maximum pressure is from 0.002(±0.000) to 0.045(±0.005) m wg for a discharge flow rate of 1 to 4 Ls

^{−1}at 0.55 ≤ H’

_{p}≤ 0.87.

_{p}in Zone A. Probability of positive air pressure at Zone A was calculated by the fractional case in Figure 4b divided by the maximum fractional cases (numeric values are shown in the figure) were 0.82 (±0.01) at 0.55 ≤ H’

_{p}≤ 0.87 and 0.5 (±0.00) at 0.39 ≤ H’

_{p}≤ 0.47.

_{p}≥ 0.55 and at a discharging flow rate of ≥ 2 Ls

^{−1}. Although positive air pressure was recorded at 0.39 ≤ H’

_{p}≤ 0.47, the occurrence was not noticeable at a test flow rate < 4 Ls

^{−1}. The probability did not vary significantly in Zone A and was found linear increasing with the discharging water flow rate (p < 0.01, t-test) 1 Ls

^{−1}≤ q ≤ 4 Ls

^{−1}.

_{p}≥ 0.55 is 0.019 m wg., with a slight downtrend of 0.001 m wg. for an increasing height of 0.1 H’ (p < 0.01, t-test). The average probability of positive pressure at 0.63 ≤ H’

_{p}≤ 0.87 is 0.14 with a slight uptrend of 0.01 as for an increased height of 0.1H’ (p < 0.1, t-test). The results indicated that the upper portion (H’ ≥ 0.63) of a discharging stack would influence the positive air pressure propagation towards the ventilation pipe open at the roof (H’ = 1).

^{−1}at H’

_{d}= 0.63 and q = 3 Ls

^{−1}at H’

_{d}= 0.55 respectively. As the measurement period was 5 min at a sampling frequency of 0.01 s, the period t in the figure determined by the fast Fourier transformation was 0.02 s to 20 s. The figures showed that energy concentrate at t = 0.06−1 s in Zone A, as a result of discharging water flows, compressed air and air turbulent energy dissipated the stack. In Zones B and C, the figures showed a clear decreasing trend of energy from time 1 to 0.02 s. However, in Zone D, as a contrast to Zones B and C, the energy kept at a level over the range of t = 0.02−0.06 s. It was noted that positive air pressure was recorded in both Zones A and D. In Zone D, however, turbulent eddies were small, quickly dissipated and the positive air pressure influence was confined near the stack bottom part (H’

_{d}= 0.08−0.16). In Zone A, larger turbulent eddies would take a longer period of turbulent energy absorption by breaking down into smaller and smaller eddies. Thus the positive air pressure influence sustained for a longer, upper portion of a stack (H’

_{d}= 0.63−0.87 at H’

_{q}= 0.63 and H’

_{d}= 0.55−0.87 at H’

_{q}= 0.55), as shown in Figure 7a,b.

_{p}, discharging flow rate q and fractional discharging height H’

_{d}–H’

_{p}(p < 0.05, t-test) by regression analysis [26] and Equations (6) and (7) were tested to be significant (p < 0.01, t-test).

## 4. Application Examples

_{p}= 0.85–0.9) where unsteady local positive air pressure was recorded in a discharging stack experimentally at H’

_{p}≥ 0.55. Second, 34/F unit’s WC was connected to the stack Zone A when 32/F unit’s WC discharging. Later confirmed case and tested positive samples were reported above 32/F (Zone A) but not floors below it (Zone B). As the experiments recorded local positive air pressure only in Zone A but not in Zone B.

_{p}= 0.80–0.88). Discharges at a lower level (i.e., 20/F and below) into a stack contaminated by pathogens generated local positive air pressure (Zone A) and posed excessive transmission risk to the upper levels (i.e., 21/F & 22/F).

_{1}= 1.8 L/s, the probably maximum simultaneous demands in the stack calculated by Equation (8) were used as the discharging flow rate in Equations (6) and (7) for the floor levels of H’

_{p}& H’

_{d}, with k = 0.5 the frequency factor for intermittent use (typical dwelling, guesthouse and office) [27,28]. The design flow rate q for intermittent use allows each installed WC to discharge once in every 20 min in the daily rush hour [27,28]. For frequent use like hotel, restaurant, school and hospital, k = 0.7, taking account of the discharge once every 10 min, is adopted.

_{max}≤ 0.03 and 0.2 ≤ p ≤ 0.3. Higher pathogen transmission risk was identified for higher floors where the higher occurrence of positive air pressure was predicted as shown in Figure 9ii. Taking a reference risk level of pressure P (=0.025 m wg.) for a stack height H = 103 m at the nominal q (=100%), the occurrences of p (=0.25) on 33/F and 32/F are 0.51 and 0.42 respectively (Figure 9aii). Approximately, the unit occurrence of p covers 23/F to 34/F (i.e., 11 floors; 0.09 per floor) for H = 103 m or 16/F to 25/F (i.e., 9 floors; 0.11 per floor) for H = 79 m. Higher transmission risk was identified for a taller stack and thus the top floors. Occurrences of p (=0.25) at P (=0.025 m wg.) predicted for the topmost floor were 0.6 and 0.23 for H = 103 m and 79 m respectively.

## 5. Conclusions

_{p}≥ 0.63) towards the ventilation opening at the rooftop. The average maximum positive pressure and the probability of positive pressures were found to be related to the water discharging height and flow rate, and mathematical expressions were proposed to indicate excessive risk due to the positive air pressures. Based on the epidemiological investigation reported for the 119th, 124th, 140th and 11,547th COVID-19 cases in Hong Kong, the baseline risk was taken as the positive air pressure inside the involved units. An excessive risk of drainage system overuse was evaluated. It was demonstrated that with an additional 20–80% in the frequency of discharge flow rate, the number of floors identified at risk increased from 1 to 9 and 1 to 6 in the 34- and 25-storey residential buildings respectively. The results would be useful for understanding the effects of positive air pressure on trap seals connected to a drainage stack in a high-rise building and for measuring the probabilistic positive air pressure in high-rise drainage systems. It would also be useful for facilities planning of self-quarantine arrangements in high-rise buildings where pathogenic virus transmission associated with drainage system overuse is a concern. Nevertheless, these estimates were made based on the results measured in a 100-mm diameter, 38-m tall drainage stack with a ventilation stack of 50 mm in diameter. Despite full-scale experiments are resources demanding, further measurements in an extended range of stack heights are required to reduce the uncertainty of predictions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

E | unit potential energy (J kg^{−1}) |

F() | function of |

f | frequency (Hz) |

f_{k} | Fourier transformation at frequencies n = 0, 1, 2,…, n − 1 and k = 0, 1, 2,…, n − 1 |

g | gravity (=9.81 ms^{−2}) |

H, H’ | stack full height (m), normalized stack full height H’ = 1 |

H_{d}, H’_{d} | stack discharging height (m), normalized stack discharging height (-) |

H_{p}, H’_{p} | stack height of air pressure measurement (m), normalized stack height of air pressure measurement (m wg.) |

k | frequency factor |

max | maximum value |

n | number of samples |

p | probability (-) |

P | air pressure (m wg.) |

$\overline{\mathit{P}}$ | average air pressure (m wg.) |

q | stack water discharging flow rate (Ls^{−1}) |

R | risk index defined in Equation (9) (-) |

r | correlation coefficient (-) |

T | T-statistics for t-test (-) |

t | time period of a fluctuation cycle (s) |

Subscript | |

1 | of reference condition of transmission risk concerned |

m | of measured value |

max | of maximum value |

p | of predicted value |

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**Figure 2.**Positive air pressures within the discharging drainage stacks (H = 38–115 m) of an experimental tower and some in-use buildings: (

**a**) maximum air pressure; (

**b**) probability of positive air pressure.

**Figure 3.**Development of risk index R for pathogen transmission risk in a contaminated drainage stack.

**Figure 4.**Positive air pressure within a discharging drainage stack (H = 38 m) at a normalized stack height H’

_{p}with a normalized stack discharging height H’

_{d}. (

**a**) Maximum air pressure; (

**b**) probability of positive air pressure.

**Figure 5.**Positive air pressure occurrence profiles along with a drainage stack (H = 38 m). (

**a**) Maximum positive pressure (m wg.); (

**b**) fractional cases having positive pressure; (

**c**) fractional discharging time having positive pressure in a case.

**Figure 6.**Measured quantity at a fractional height H’ of a discharging stack. (

**a**) Average maximum air pressure ${\overline{P}}_{max}$; (

**b**) probability p.

**Figure 7.**Energy spectrum density examples of air pressure along with a stack (measurement time = 20.48 s). (

**a**) Discharge at H’ = 0.63 at a flow rate of 4 Ls

^{−1}; (

**b**) discharge at H’ = 0.55 at a flow rate of 3 Ls

^{−1}.

**Figure 8.**Predicted quantity at Zone A of a discharging drainage stack at H’ ≥ 0.47. (

**a**) Maximum air pressure; (

**b**) the probability of positive pressure.

**Figure 9.**Zone A positive air pressures and occurrences inside the stacks of two high-rise residential buildings with confirmed samples of COVID-19 virus. (

**a**) Height H = 103 m; (

**b**) height H = 79 m; (

**i**) maximum positive air pressure; (

**ii**) probability of positive pressure.

**Figure 10.**Sensitivity risk increment due to 20% more discharge flow rate in two example buildings. (

**a**) Height H = 103 m; (

**b**) height H = 79 m.

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## Share and Cite

**MDPI and ACS Style**

Wong, L.-T.; Mui, K.-W.; Cheng, C.-L.; Leung, P.H.-M.
Time-Variant Positive Air Pressure in Drainage Stacks as a Pathogen Transmission Pathway of COVID-19. *Int. J. Environ. Res. Public Health* **2021**, *18*, 6068.
https://doi.org/10.3390/ijerph18116068

**AMA Style**

Wong L-T, Mui K-W, Cheng C-L, Leung PH-M.
Time-Variant Positive Air Pressure in Drainage Stacks as a Pathogen Transmission Pathway of COVID-19. *International Journal of Environmental Research and Public Health*. 2021; 18(11):6068.
https://doi.org/10.3390/ijerph18116068

**Chicago/Turabian Style**

Wong, Ling-Tim, Kwok-Wai Mui, Cheng-Li Cheng, and Polly Hang-Mei Leung.
2021. "Time-Variant Positive Air Pressure in Drainage Stacks as a Pathogen Transmission Pathway of COVID-19" *International Journal of Environmental Research and Public Health* 18, no. 11: 6068.
https://doi.org/10.3390/ijerph18116068