## Abstract

The final velocity was put forward to study the water flow characteristics inside the building drainage system; however, it is more suitable for low-rise and multi-storey buildings, not for high-rise buildings. This study revealed the drainage transient characteristics of a double stack drainage system in high-rise residential buildings. Based on the final velocity, the air-water interaction mechanism and two-phase flow conditions in high-rise residential drainage stacks were discussed. An influence model of drainage system flow rate on pressure fluctuation under the change of state parameters such as ventilation rate, pipe wall roughness and building height was established. The pressure limit and flow rate data were obtained through full-scale experiments. The pressure limit and flow rate model were simplified to . After the data were verified, the fitting coefficients A, B and C were linear to the floor height.

## HIGHLIGHTS

This study puts forward a model of drainage system flow rate on pressure fluctuation under the change of state parameters in double stack drainage system in high-rise residential buildings.

The pressure limit and flow rate data were obtained through full-scale experiments in high-rise experiment tower.

The fitting coefficient A, B and C of pressure fluctuation model were linear to the floor height.

### Graphical Abstract

## INTRODUCTION

As an indispensable feature of residential buildings, the building drainage system plays an important role in ensuring indoor environmental sanitation and safety (Andargie *et al.* 2019). The original purpose of designing the building drainage system was to collect domestic sewage and wastewater and discharge it into the municipal pipe network. Today, people have higher requirements for the comfort and safety of the living environment. In engineering projects, a large volume of feedback from construction units and property managements indicate that leakage, odour return and blockage in high-rise drainage systems have become the main complaints of residents. In some extreme conditions, returning odour may even cause more severe cases, such as the SARS virus outbreak in Amoy Gardens in Hong Kong's Kwun Tong in 2003; in this case, an unqualified floor drain spread the virus through the drainage stack. In 2020, a similar incident also occurred during the outbreak of COVID-19, which publicly exposed the prevailing security environment issues (Liu *et al.* 2020). Studies have found significant limitations of existing norms of building drainage systems because the height of high-rise residential is increasing, and the water seal is relatively more likely to be depleted, thereby increasing the probability of cross infection (Gormley *et al.* 2021). The high-rise building drainage system involves numerous building floors, resulting in a higher flow rate of the converged water flow in the lower floors of the high-rise building (Gormley & Kelly 2019). During the drainage process, the water flows with air in the stacks, which makes the entire flow process more complicated.

When a household starts to use water and the wastewater flows into the drainage pipe, the water flows downward under the action of three main forces: gravity, pipe wall friction and the resistance of the air in the pipe to the water flow. For a long time, scholars at home and abroad generally believed that after wastewater entered the drainage stack, it would form wall helical flow, water film flow or water plug flow according to the discharge flow rate. Early research results mainly include *water film flow* and *final velocity* for establishing mathematical and physical models on building drainage systems. Thereafter, some scholars proposed the concept of pipe filling rate.

Swaffield *et al.* (2004, 2005a, 2005b) and Campbell & MacLeod (2001) conducted a large number of theoretical studies and experiments in full-scale buildings and achieved significant results, as well as proposed a series of improvements on the applicability of the mathematical model. The researchers measured the transient pressure in the drainage stack. They proposed the existence of a fluctuating state of alternating positive and negative pressures in the drainage system. Then, Swaffield & Campbell (1995) studied the impact of domestic wastewater containing detergents on the pressure fluctuations in the drainage system during the discharge process, and obtained the mathematical models of air-pressure fluctuations under different detergent addition conditions. Swaffield & Wright (1998) tested the pressure fluctuations in the system under different drainage flows under various ventilation conditions, and found that the flow rate has a quadratic function relationship with the pressure limit value of the single riser drainage system through non-dimensional analysis. Swaffield & Jack (2004) also studied the factors that affect air flow in the drainage pipe through the density of the air-water mixture and water flow rate at different positions inside the drainage stack. The results suggested that the main factors affecting air flow include the shear stress caused by the velocity gradient between the water film and surface of the air column, and the drag effect of free water droplets. Based on the tests of the drainage system in low-rise buildings, Heriot-Watt University (Swaffield *et al.* 2004, 2005a, 2005b; Kelly & Gormley 2014) also conducted a series of studies on the relationship between the internal pressure fluctuations of the system and the water seal loss of water appliances. Continuity equations, momentum equations and quadratic partial differential equations were adopted to establish mathematical models that could be used to predict internal pressure fluctuations and water seal losses in the drainage system. By conducting a clean water test, Cheng *et al.* (2005) proposed a pressure peak prediction model for various drainage riser systems in which the analysis of pressure peaks should be divided into several sections according to the floor heights. Yu *et al.* (2019) used physical models to study the pressure changes in the horizontal pipe of a certain floor in the drainage system, and proposed four parameters that affect the pressure changes in the horizontal pipes, namely flow rate, inlet height, ventilation conditions and outlet conditions. Cheng *et al.* (2011) measured water-flow velocity in the drainage stack with a high-speed video camera. He suggested that the *final velocity* needs correction as the water quality should be considered during the drainage process.

In fact, the *final velocity* was put forward by European countries from the experiments in low-rise and multi-storey buildings in the past. The probability of simultaneous discharge in low-rise and multi-storey buildings is smaller than high-rise buildings. For example, when the discharge water from horizontal pipe of 4th floor section enters the drainage stack, it will not be disturbed and fall directly, and it can easily reach the stable state after gravity balanced friction. However, in high-rise buildings, for example, a 30-floor building, the water flow in the drainage stack will be disturbed by the lower floor discharge from time to time, and the upward air flow in the drainage stack is blocked. Stewart *et al.* (2021) established a steady-state pressure model in high-rise wastewater drainage networks to reveal the air ﬂowrates and discharge water ﬂowrates effects. As they described, it is difficult to analyse the margins in two-phase annular flow models. It is difficult for the discharge water to reach the force balance, and the water flow is constantly in the process of alternating movement of acceleration and deceleration in high-rise residential buildings.

This study aims to reveal the drainage transient characteristics of a double stack drainage system in high-rise residential buildings. Based on the final velocity, we discuss the air-water interaction mechanism and two-phase flow conditions in high-rise residential drainage stacks. An influence model of drainage system flow rate on pressure fluctuation under the change of state parameters such as ventilation rate, pipe wall roughness and building height was established. The pressure limit and flow rate data were obtained through full-scale experiments, and the model was verified.

### Mathematical model of pressure fluctuation with ventilation inside a dual-stack drainage system

According to the *water film flow* and *final velocity*, the flow pattern of the water inside the drainage stack could be classified into three categories, namely walled spiral flow, water film flow and water plug flow. However, the *final velocity* discussed the water flows under the premise, ignoring the influence of the cross branch tee and the air in the vent pipe. In this study, we establish a dynamic pressure model of the water-air flow in the vented building drainage system. To simplify the model, we suppose that the pressure value at any position of the drainage stack is the equation of the air flow pressure fluctuation and the central air column of the drainage stack; this value is equal to the sum of all the pressures that cause the head loss at that location.

The cross point position of the yoke pipe connected to the drainage stack was set as a base point for establishing the mathematic model, as shown in Figure 1. In the experiment system in this study, the yoke pipe was connected to the drainage every floor. The cross point of yoke pipe and drainage stack was 20 cm below the cross point of horizontal branch and drainage stack. The analysis points of the top layer were named , followed by the lower floors , , …, , . The air supplement amounts from vent pipe to lateral branch pipes were named ,, …, , , respectively.

The total pressure of has been analysed as follows.

*k*is the rough height of the inner wall of the pipeline, m; is the kinematic viscosity coefficient of air, ; and is the ventilation volume at the top of the drainage stack vent cap, .

Equation (10) is the pressure equation of the drainage stack under ventilation. According to this equation, the pressure value at the node is related to the roughness of the inner wall of the pipeline, the resistance coefficient for the air that passes through the annular membrane flow when entering the drainage stack from the cross-branch, and the ventilation volume. For a normally ventilated double-stack drainage system, under the action of gravity, the air flow rate at the bottom of the pipe increases under the entrainment of water flow; thus, . Therefore, the value obtained by Equation (10) is negative. Based on the assumption that the air flow velocity is equal to the water flow velocity at this location, as the drainage flow rate increased, the pipeline pressure increased in a negative direction. According to the traditional *final velocity*, when the drainage flow rate was a higher value, the water film attached to the inner wall of the pipeline maintained a constant thickness after reaching the final flow velocity. In the absence of external air supply, the pipeline pressure remains constant. In the double-stack drainage system, due to the existence of supplementary air, the item existed and continued to increase, so the negative pressure of the drainage stack would keep increasing. When the water film in the drainage stack was excessively thick, the air inside the vent pipe could not be supplied to the drainage stack, and the pipeline pressure line would present a curve where the negative pressure on the high floors was larger and that on the low floors was smaller. For the non-ventilated drainage system, the local resistance coefficient at the air cap , high floor areas would face a huge negative pressure due to lack of air supplementation. The conclusion derived from Equation (10) was consistent with the pressure trend from experimental observations.

*P*can be divided into three major terms. The first term is the sum of the square terms of the air flow, assuming that the air supplements on each floor were equal, and at the same time, ignoring the local head loss difference of the three-tee connection installation. Then, the coefficient of this term can be simplified to . Although the thickness of the water film on the inner wall of the pipeline is uneven, the difference between each floor of the

_{n}*X*term in is not too small to be ignored. The second term is , where the main variables of the term are the initial ventilation velocity and air flow terms, and the other coefficients were constant. The third term was the velocity-related term, which was related to the location of the floor. According to the flow-pressure limit curve in the work of Guan

*et al.*(2020b), Equation (10) can be simplified as

In the simplified instantaneous pressure value model, the main influencing parameters of coefficient A are the local head losses at each node of the pipeline and the water film thickness. For coefficient B, the main influencing parameter is the local head loss of the nodes. The main influencing parameter of coefficient C is the cumulative effect of air velocity at the nodes. In the two experiment systems in this study, the value range for coefficient A is [−80.104,121.019], the value range for coefficient B is [−142.187,101.000], and the value range for coefficient C is [−108.060,209.810].

## METHODS AND MATERIALS

The experimental work of this study was carried out in Suns's experimental tower in Shanxi Province, China, which was 60 m high with 20 3-meter high floors (19 floors aboveground and one floor underground). The test system, pipe and joint materials and connection modes can be referred to Guan's studies (Guan *et al.* 2020a, 2020b). The installation methods of horizontal branch pipes on the test floor were shown in Figure 2. According to the *Standard for testing the drainage capacity of risers in building drainage system*, CJJ/T 245-2016, the drainage system height is decided by the discharge height, and this experimental tower could act as a high-rise building to discuss the discharge characteristic.

The installations of drainage stack, horizontal branch pipe, vent pipe, vent cap and discharge pipes were in accordance with the relevant provisions of *Standard for design of building water supply and drainage,* GB50015-2019.

In this experiment, comparative analysis was made on different drainage system connection modes (Figure 2) and vent pipe diameters. The specific comparison of each working condition was shown in Table 1. All tests were conducted at fixed discharge rates.

Test . | Bottom connection position . | Drainage stack diameter (mm) . | Vent pipe diameter (mm) . | Yoke pipe lifting height (m) . | Maximum discharge rate (L/s) . |
---|---|---|---|---|---|

JH 1 | Horizontal main | 100 | 50 | 1.5 | 6.0 |

JH 2 | Horizontal main | 100 | 50 | 1.0 | 5.5 |

JH 3 | Horizontal main | 100 | 50 | 0.5 | 5.5 |

JH 4 | Horizontal main | 100 | 75 | 1.5 | 8.0 |

HX 1 | Horizontal main | 100 | 100 | 1.5 | 17.0 |

HX 2 | Drainage stack | 100 | 100 | 1.5 | 8.5 |

Test . | Bottom connection position . | Drainage stack diameter (mm) . | Vent pipe diameter (mm) . | Yoke pipe lifting height (m) . | Maximum discharge rate (L/s) . |
---|---|---|---|---|---|

JH 1 | Horizontal main | 100 | 50 | 1.5 | 6.0 |

JH 2 | Horizontal main | 100 | 50 | 1.0 | 5.5 |

JH 3 | Horizontal main | 100 | 50 | 0.5 | 5.5 |

JH 4 | Horizontal main | 100 | 75 | 1.5 | 8.0 |

HX 1 | Horizontal main | 100 | 100 | 1.5 | 17.0 |

HX 2 | Drainage stack | 100 | 100 | 1.5 | 8.5 |

The test evaluation criteria was refered to the *Standard for testing the drainage capacity of risers in building drainage system*, CJJ/T 245-2016. The detail could also be found in Guan's studies (Guan *et al.* 2020a, 2020b).

In this study, a dual-stack drainage system with yoke pipe for ventilation and dual-stack drainage system with auxiliary vent pipe were adopted to conduct the experiments in the drainage experimental tower.

The dual-stack drainage system with yoke pipe for ventilation had a vent pipe and a drainage stack, and the yoke pipes were adopted to connect the vent pipe and drainage stack for ventilation. The diameter of drainage stack was DN100 (DN is the abbreviation of ‘nominal diameter’, DN100 refers to the nominal diameter 100 mm, the same below) and the horizontal branch pipe diameter was DN100. The diameters of the vent pipe were DN50 and DN75 for comparison. The yoke pipe had the same diameter as the vent pipe. The lifting heights of the yoke pipe were 1.5, 1.0 and 0.5 m for comparison.

The vent pipe and drainage stack of the dual-stack drainage system with auxiliary vent pipe were connected by horizontal branches, with the two continuous pipe branches forming a circuit. The diameter of the drainage stack was DN100 and that of the horizontal branch pipe was DN100. The auxiliary vent pipe diameter was DN100. In Test JHs and HX1, the bottom of the vent pipe was directly connected to the horizontal main (Liu *et al.* 2012), whereas in Test HX2, the bottom of the vent pipe was connected to the drainage stack, as shown in Figure 2. The vent pipe terminal and drainage stack were connected to vent caps.

In the experiments, the floors from 14th to 18th were set as the discharge floors. During the discharge process, the pressure fluctuation was directly caused by the discharge flow, thus the pressure fluctuations in discharge floors were not discussed in this paper.

## RESULTS

In this section, fitting analysis is conducted on the relationship between the pressure value and flow rate of each floor under different ventilation conditions.

Table 1 presents the maximum discharge rates sustained in various test conditions.

### Analysis of experimental results of dual-stack drainage system with yoke pipe for ventilation

The tested pressure limit value and discharge rate data of the dual-stack drainage system with combined ventilation are fitted in Equation (11). Test JHl was set as an example to discuss the fitting results. The fitting curves of the 2nd to the 13th floors are presented in Figure 3.

To analyse the fitting results more intuitively, the parameters A, B and C values (Equation (11)) and the curve fitting coefficient *R ^{2}* of the fitting curves obtained in test JHl are listed in Appendix, Table A1.

Table A1 shows that in the fitting results of test JH1, the coefficient R^{2} of the negative pressure value is basically greater than 0.9, indicating that the overall fitting result is reasonable while the positive pressure value did not fit well. We can infer that Equation (11) is unsuitable for the positive pressure results. We can discuss the correlation between pressure limit value and flowrate from drainage process. When the horizontal branch pipe drains, the water flows downward, forming a local cross-section at the tee. So the pipe below the discharge position is temporarily separated from the atmosphere. The water in the drainage stack flows downward forming a negative pressure area below the water film on the cross section. Therefore, the air in drainage stack will draw air from the horizontal branch pipe and the vent pipe to compensate for the loss of negative pressure. When the negative pressure is large enough, it will cause water seal loss of sanitary appliances. When the water flows down to another horizontal branch cross point discharging, the air channels are blocked and the down-flow water produces positive pressure. However, positive pressure causes the opposite flow direction of sanitary appliances’ water seal to negative pressure, which flows towards indoor. What's more, only when positive pressure is much higher than the safety limit value, which means the water seal splash to the indoor, the water seal would be damaged. From Guan *et al.*’s study (Guan *et al.* 2020b), the negative pressure is linearly correlated with water seal loss. Therefore, the subsequent analysis only performed a fitting analysis on the relationship between the negative pressure limit value and the flow rate. Also, Figure 3 indicates the negative pressure was declining with the increase of flow rate from 2nd to 13th floor. The greater the drainage flow, the greater the flow velocity, thus the greater the kinetic energy of the falling sewage at the connection between the drainage stack and the horizontal branch. The stronger the impact flow generated at the node position causes more air drawn from the vent pipe, so a greater negative pressure is generated.

The fitting results of test JH2, JH3 and JH4 are shown in Figures 4–6 and Appendix Tables A2, A3 and A4.

Figure 7 showed more intuitive curve fitting results. We can conclude from Figure 7 that with the increase of floor height, coefficient B is rising while coefficients A and C show a downward trend. In tests JHl and JH2, coefficient B is positive while coefficients A and C are negative, but for tests JH3 and JH4, coefficient B is negative, and coefficients A and C are positive. According to Equation (11), the main parameters that affect coefficient B are initial ventilation velocity and initial ventilation . Furthermore, coefficient A is affected by the partial resistance coefficient, and coefficient C is mainly affected by the air/water flow velocity , and the trend of coefficients A and C is consistent. The flow patterns of tests JH1 and JH2 are different from those of tests JH3 and JH4.

Tests JHl, JH2 and JH3 retained the same features with different lifting heights. Differences exist between the lifting height of 0.5, 1.0 and 1.5 m from the fitting curve, indicating that the lifting height has a certain influence on the fitting coefficient value, but the difference of the fitting coefficients between tests JH1 and JH2 is not apparent, indicating that the effect of lifting height is not significant enough. The variation trend of fitting A and coefficient B of tests JH1 and JH4 are the same, and the coefficient value of the different floors is similar, indicating that the vent pipe diameter has a limited effect on the fitting coefficient value, thereby showing a linear relationship.

### Analysis of experimental results of dual-stack drainage system with auxiliary vent pipe

The tested limit pressure value and total system discharge rate data of the dual-stack drainage system with auxiliary vent pipe are fitted in Equation (11). The fitting curves of the 2nd-13th floors in test HX1 and HX2 are shown in Figure 8–10 and Appendix, Tables A5 and A6. The variation tendency of dual-stack drainage system with auxiliary vent pipe was consistent with that of dual-stack drainage system with yoke pipe for ventilation. The greater the drainage flow, the greater the negative pressure. However, the maximum discharge rate of HX1 is 2.83 times of JH1, which could related to the ventilation. In the dual-stack drainage system with auxiliary vent pipe, the vent pipe diameter is 100 mm, which is the first factor that cause the maximum discharge rate difference. The larger the vent pipe, the smaller the local resistance, and more air could be sucked into the drainage stack. What's more, from the point of system structure, the horizontal branch serves as the connecting pipe of the drainage stack and the vent pipe, and the sanitary ware in the horizontal branch can directly provide air through the vent pipe. Therefore, the drainage process is slightly affected by negative pressure. The bottom of vent pipe connected to the horizontal main in HX1 could form an air loop pathway that is much larger than that of a traditional double stack drainage system. In that case, air circulation is better, so the improvement of the system's discharge ability is more pronounced.

Different from test HX2, the value of coefficient A in test HX1 is relatively stable, while the value of A in test HX2 shows a downward trend overall. The bottom connection mode of the test HX1 is different from that of test HX2, which indicates that different bottom installation modes directly affect the change of coefficient A.

In addition, with the increase of floors, the fitting coefficient B of HX1 increases with the increase of floor height, but the increase is slower than that of HX2.

The variation trend of coefficient C is similar to that of the dual-stack drainage system with yoke pipe for ventilation, both of them decreasing with the floor height increasing.

The fluctuation of C value in HX1 is greater than that in HX2.

In the dual-stack drainage system with auxiliary vent pipe, the coefficient C of HX2 is basically positive, while that of most floors under the HX1 condition is less than 0.

## DISCUSSION

Table 2 shows the fitting results of coefficients A, B and C values bound to floor height in dual-stack drainage system with yoke pipe for ventilation and in dual-stack drainage system with auxiliary vent pipe.

. | Test . | Intercept . | Intercept standard deviation . | Slope . | Slope standard deviation . | R^{2}
. |
---|---|---|---|---|---|---|

A | JH1 | −9.78 | 9.54 | −4.59 | 1.16 | 0.57 |

JH2 | 66.41 | 23.36 | −13.45 | 2.83 | 0.66 | |

JH3 | 125.60 | 10.52 | −13.12 | 1.27 | 0.91 | |

JH4 | 32.42 | 4.17 | −3.61 | 0.51 | 0.82 | |

HX1 | 8.80 | 1.24 | −0.89 | 0.15 | 0.76 | |

HX2 | 62.57 | 6.53 | −3.68 | 0.79 | 0.65 | |

B | JH1 | 8.78 | 10.41 | 5.66 | 1.26 | 0.63 |

JH2 | −112.88 | 33.42 | 19.64 | 4.05 | 0.67 | |

JH3 | −195.88 | 15.21 | 18.89 | 1.84 | 0.90 | |

JH4 | −56.69 | 6.33 | 5.49 | 0.77 | 0.82 | |

HX1 | −18.05 | 2.22 | 1.64 | 0.27 | 0.77 | |

HX2 | −105.93 | 10.26 | 6.03 | 1.24 | 0.67 | |

C | JH1 | 4.22 | 8.22 | −3.16 | 1.00 | 0.45 |

JH2 | 158.40 | 31.21 | −23.28 | 3.78 | 0.77 | |

JH3 | 209.94 | 18.15 | −17.98 | 2.20 | 0.86 | |

JH4 | 66.76 | 14.67 | −6.34 | 1.78 | 0.52 | |

HX1 | 85.79 | 16.12 | −10.29 | 1.95 | 0.71 | |

HX2 | 192.96 | 23.64 | −12.14 | 2.86 | 0.61 |

. | Test . | Intercept . | Intercept standard deviation . | Slope . | Slope standard deviation . | R^{2}
. |
---|---|---|---|---|---|---|

A | JH1 | −9.78 | 9.54 | −4.59 | 1.16 | 0.57 |

JH2 | 66.41 | 23.36 | −13.45 | 2.83 | 0.66 | |

JH3 | 125.60 | 10.52 | −13.12 | 1.27 | 0.91 | |

JH4 | 32.42 | 4.17 | −3.61 | 0.51 | 0.82 | |

HX1 | 8.80 | 1.24 | −0.89 | 0.15 | 0.76 | |

HX2 | 62.57 | 6.53 | −3.68 | 0.79 | 0.65 | |

B | JH1 | 8.78 | 10.41 | 5.66 | 1.26 | 0.63 |

JH2 | −112.88 | 33.42 | 19.64 | 4.05 | 0.67 | |

JH3 | −195.88 | 15.21 | 18.89 | 1.84 | 0.90 | |

JH4 | −56.69 | 6.33 | 5.49 | 0.77 | 0.82 | |

HX1 | −18.05 | 2.22 | 1.64 | 0.27 | 0.77 | |

HX2 | −105.93 | 10.26 | 6.03 | 1.24 | 0.67 | |

C | JH1 | 4.22 | 8.22 | −3.16 | 1.00 | 0.45 |

JH2 | 158.40 | 31.21 | −23.28 | 3.78 | 0.77 | |

JH3 | 209.94 | 18.15 | −17.98 | 2.20 | 0.86 | |

JH4 | 66.76 | 14.67 | −6.34 | 1.78 | 0.52 | |

HX1 | 85.79 | 16.12 | −10.29 | 1.95 | 0.71 | |

HX2 | 192.96 | 23.64 | −12.14 | 2.86 | 0.61 |

From Table 2 the coefficients A, B and C are all linearly correlated with the floor height. To be more precise, the coefficient A and C decrease with the floor height, while the coefficient B does the opposite. For the dual-stack drainage system with yoke pipe for ventilation and the drainage system with an auxiliary vent pipe, the change trends of coefficient A, B and C are consistent.

The change trend of coefficient A is relatively flat in the drainage system with an auxiliary vent pipe than that in the dual-stack drainage system with yoke pipe. As mentioned above, coefficient A represents the local head losses at each node of the pipeline and the water film thickness. The tendency of coefficient A to decrease with increasing floor height means that less local head losses, the smaller the pressure limit value. It is worth noting that the pressure limit values discuss here are the most negative. The other influence parameters for calculating coefficient A are all positive. The correlation coefficient of fitting curves of tests JH3 and JH4 are higher than JH1 and JH2. Also, the change trend of coefficient A of tests HX1 and HX2 can be explained that test HX1 has better performance than HX2 at the bottom floor because the local head loss was decreased by connecting the vent pipe bottom to the horizontal main.

Coefficient B refers to , which relates to the initial ventilation volume. As all the parameters for calculating coefficient B are all constant. This item should be constant. But the fitting curves show changes. One possible reason is that during the experiments, the atmosphere temperature was changing and the length from vent cap to the first discharge point was changing. Another possible reason is that the local head loss at vent cap point was changing, which need further research to verify.

The main influencing parameter of coefficient C is the cumulative effect of air velocity at the nodes. The decreasing tendency of coefficient C means that the velocities at each node are not constant. It also illustrates that the downward flow velocity is continuing accelerating during the discharge process. This conclusion shows that the *final velocity* is not appropriate to explain the discharge characteristic in the drainage system of high-rise residential buildings, as under experiment discharge rate, there were seldom film flow conditions in drainage stacks. The *final velocity* does not consider the impact of the vent pipe on the discharge process. It is not difficult to draw from the previous studies that a drainage system with a vent pipe has higher discharge capacity than the system without a vent pipe.

In fact, *final velocity* was proposed in the experiments of buildings with lower floors in European countries in the past. Since the floors are low, the probability of simultaneous discharge of the multi-storey floors is small. That means when the horizontal pipes of the upstairs discharge into the drainage stack, the water flow will not be disturbed and directly fall down, and it reaches the *final velocity* state after falling for a certain period of time. However, in high-rise buildings, the large water flow in the drainage stack will be disturbed by the drainage of the lower horizontal branch from time to time, and the upward air flow in the drainage stack is blocked, so it is difficult to form a stable water film flow. The drainage cannot reach the force balance, and the water flow is constantly in the process of alternating acceleration and deceleration, so it is difficult to reach the *final velocity*. In addition, there are more floors connected to the drainage stack in the high-rise residential buildings, the water flow has to meet with a tee-joint in every floor. The location of the tee-joint and horizontal pipe cavity itself will affect the water flow, cause pressure fluctuations and lead to velocity change. This is also not in conformity with original *final velocity* assumes.

## CONCLUSIONS

Based on the traditional *final velocity*, taking the air flow in the drainage system as the main research object, this study establishes the airflow-pressure relationship model starting from the pipeline hydrodynamics and energy equation. This model considers the influence of parameters such as the inner wall roughness of the pipeline, local resistance coefficient and pipe diameter at any node of the building drainage system. According to this model, the *final velocity* of the dual-stack drainage system has been deepened and improved. The model indicates that the roughness of the inner wall and the local resistance coefficient mainly influence pressure fluctuation in the system. The pressure equation of the drainage stack under ventilation is simplified according to the previous study. The main influencing parameters of coefficient A are the local head losses at each node of the pipeline and the water film thickness. For coefficient B, the main influencing parameter is the local head loss of the nodes. The main influencing parameter of coefficient C is the cumulative effect of air velocity at the nodes.

The data obtained through the full-scale experiment are adopted to verify the simplified mathematical model (Equation (10)), and certain differences exist in the trends of fitting results obtained by various drainage systems. For the dual-stack drainage system with yoke pipe for ventilation, the lifting height of the vent pipe has a minimal effect on the fitting coefficient value, and the diameter of the vent pipe is linearly related to the values of the fitting coefficients A and B. For the drainage system with an auxiliary vent pipe, the pressure limit value in the drainage system is related to the installation structure. The local head loss caused by the bottom connection plays a crucial role in affecting the pressure of the drainage system. Overall, the coefficient A, B, and C are linearly correlated with the floor height. The decreasing tendency of coefficient C indicates that the *final velocity* is not applicable under maximum discharge flow rate in the drainage system of high-rise residential buildings

## ACKNOWLEDGEMENTS

This research has been supported by the National Natural Science Foundation of China (grant no. 51978536); the Fundamental Research Funds for the Central Universities (grant no. 2042021kf0059); and the China Postdoctoral Science Foundation (grant no. 2021M702529). The National Building Drainage Piping System Technology Center in Gaoping, Shanxi Province, China provided us massive supports, including the experimental equipment and technical guidance. The SUNS Industrial Group provided the test materials needed.

## CONFLICT OF INTEREST STATEMENT

The authors are not affiliated with or involved with any organisation or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this paper.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.