## Abstract

The use of air vessels is an effective measure to control water hammers, and its volume selection has a certain blindness. This paper aims to reveal the surge wave characteristics and provide design guidelines for air vessels in long-distance water supply systems. First, the analytical formulas of water-level oscillations in the air vessel are derived based on the Krylov–Bogoliubov–Mitropolsky method. Then, an optimization model is constructed for selecting the optimal volume of the air vessels. Finally, the validation of the analytical formulas and the optimization of the model are conducted through two actual projects. The results show that the calculation error of the analytical formulas can be controlled within a very small range, and the process of selecting air vessel volume can be simplified with the provided model. In addition, increasing the air chamber height within its range can reduce the volume of air vessels with the same protection requirements. The optimization analysis results of the air vessel can provide guidance and reference for the design of actual projects.

## HIGHLIGHTS

The analytical formulas for water-level oscillations considering pipe friction are derived.

The volume optimization model can preliminarily estimate the minimum volume of the air vessel.

Under the same protection requirements, the volume of air vessels decreases with the increase of air chamber height.

### Graphical Abstract

## NOTATION

Change of water level in the air vessel (relative to the reference datum, positive direction is upward), m

Water level at the end of a transition process in the air vessel, m

Air chamber height at the end of a transition process in the air vessel, m

Water level of the outlet sump, m

Head loss of the pipeline, m

Absolute pressure of the gas in the air vessel, m

Absolute pressure of the gas at the end of a transition process, m

Local atmospheric pressure, m

Minimum allowable pressure of the pipeline, m

Discharge in the pipeline, m

^{3}/sDischarge of the pump, m

^{3}/sDischarge flow into or flow out of the air vessel (positive direction is flow out), m

^{3}/sCross-sectional area of the air vessel, m

^{2}Cross-sectional area of the pipeline, m

^{2}Total length of the pipeline, m

Distance between the outlet sump and pipe node

*i*, mFlow velocity in the pipeline, m/s

Initial flow velocity in the pipeline, m/s

Minimum of the water-level oscillations, m

Maximum of the water-level oscillations, m

Total height of the air vessel, m

Safe water depth in the air vessel, m

Wave velocity of water hammer, m/s

Acceleration of gravity, m/s

^{2}Polytropic index of the gas state process

Head loss coefficient of the connecting pipe

## INTRODUCTION

A long-distance water supply project is an effective method to reallocate and improve the utilization of water resources. Safety issues have become increasingly prominent with the increasing number of projects (Shi *et al.* 2019a). Suddenly stopping a pump or closing a valve can result in water-hammer events (Duan *et al.* 2020). With the propagation of the water-hammer wave, the local pressure along the pipeline may drop to the vaporization pressure, thereby column separation occurs. The pipeline system and pump units can be seriously damaged when the separated columns rejoin (Simpson & Bergant 1994; Kim *et al.* 2014). To date, surge tanks, one-way surge tanks, air valves and air vessels have been used to relieve these problems (Duan *et al.* 2010; Chaudhry 2014; Sun *et al.* 2016; Yazdi *et al.* 2019). However, the setting of surge tanks and one-way surge tanks is restricted by terrain conditions. An air valve is generally used as a safety reserve for negative pressure protection due to the difficulty of exhaust after air intake (Shi *et al.* 2019b). Air vessels have been widely used because they can be conveniently installed and managed. For a water supply project with air vessel protection, the pressure change along the pipeline is closely related to the surge wave (the water-level oscillation in the air vessel) in the air vessel. Therefore, many researchers have studied the basic parameters of air vessels in recent years. The main achievements are stated as follows: Stephenson (2002) points out that suitable connecting pipe diameters (including outlet and inlet diameters) were able to minimize the size of the air vessels. Besharat *et al.* (2017) investigated the size of an air pocket within an air vessel by an experimental method and given a specific range of air pockets for other real systems.

The purpose of the above research on the basic parameters of air vessels is to obtain their optimal volume. Although an air vessel with a large volume can often achieve better water-hammer protection, a larger volume means more engineering investment (Moghaddas *et al.* 2017). At present, the method of characteristics (MOC) is commonly used to simulate the transient process of pipeline systems (Wylie *et al.* 1993). The volume of the air vessel usually needs to be determined according to the simulation results, whereas the calculation process requires repeated trial calculations or diagrams. For volume optimization of the air vessel, a theoretical and thorough understanding of the water-level oscillation processes in pump trip transients is the most important step. To investigate the characteristics of the water-level oscillations in the air vessel, the theoretical derivation of the analytical formula for surge waves is the most effective and helpful method (Chen *et al.* 2021). A water supply system with air vessels involves the nonlinearity of momentum equations, continuity equations and gas state equations; hence, it is very difficult to solve them directly and accurately (Moghaddas *et al.* 2017; Bostan *et al.* 2019).

The process of water-level oscillations in the air vessel is similar to that of surge tanks. Great achievements have been made in the study of surge tanks in hydropower stations (Zhang & Liao 2012; Guo *et al.* 2017). The representative literature is summarized as follows: Suo *et al.* (1998) established a mathematical model for the form optimization of the air cushion surge chamber, and the optimum solution was found through genetic algorithms. Using the nonlinear vibrational asymptotic method, Zhang *et al.* (2001) derived an explicit formula for calculating the surges in a surge tank following load rejection or load acceptance. Inspired by this, Miao *et al.* (2017) derived an analytical formula of the surge wave in an air vessel and proposed an approximate analysis method to optimize the size of the air vessel. Wang *et al.* (2019) derived the formula for selecting the optimal location of an air vessel based on the analytical formula of the surge wave.

It is worth mentioning that, for the above analytical formulas of the surge wave in air vessels, pipe friction is not considered. Therefore, the main objective of the present study is to improve the accuracy of the existing theoretical calculation formula of the surge wave in air vessels and reduce the workload in the process of air vessel volume optimization. To achieve this goal, the analytical formulas for water-level oscillations in air vessels with pipeline friction are derived based on the Krylov–Bogoliubov–Mitropolsky (KBM) method, and the mathematical model for the form optimization of air vessels is established with the minimum total volume as the objective function. Moreover, the rationality and accuracy of the analytical formulas are verified by comparison with the numerical results. Finally, the effects of the air chamber height on the volume of the air vessels are analyzed.

## METHODOLOGY

Figure 1 is a schematic diagram of a common water supply system with an air vessel. Three basic equations are used in the process of water-level oscillations, i.e., the momentum equation, continuity equation and gas state equation. The following assumptions are adopted for these equations: (1) the air vessel and pipe are rigid, and the water body is incompressible; (2) the gas in the air vessel is an ideal gas, and it changes according to the ideal gas law; (3) the friction of the water supply system is constant, and the friction coefficient does not change with time in the transient process; and (4) the initial water in the air vessel can be neglected compared with the water in the pipelines.

### Basic equations

*z*is the change of water level in the air vessel (relative to the reference datum, positive direction is upward); is the water level of the outlet sump;

*p*is the absolute pressure of the gas in the air vessel; is the local atmospheric pressure; is the minimum allowable pressure of the pipeline;

*Q*and are the discharge flows of the pipeline and pump, respectively; is the discharge flow into or flow out of the air vessel (positive direction is flow out);

*F*and are the cross-sectional area of the air vessel and pipeline, respectively;

*L*is the total length of the pipeline; ; ; is the head loss coefficient of the connecting pipe;

*d*is the diameter of the connecting pipe;

*R*is the hydraulic radius of the pipeline;

*n*is the roughness of the pipeline;

*g*is the acceleration of gravity;

*v*is the flow velocity in the pipeline; is the exponent in the gas state equation and

*C*is a constant. The value of can vary from 1.0 to 1.4 (‘isothermal’ to ‘adiabatic’ behavior). In this calculation, was taken because the transients are usually rapid at the beginning but slow near the end (Chaudhry 2014).

### Analytical formulas for water-level oscillations in air vessel

#### Analytical formulas without pipe friction

*et al.*2017; Wang

*et al.*2019). In addition, the Taylor expansion is used for Equation (3), and the first-order approximation is taken. Thus, the square term of Equation (6) is neglected. Thus, Equation (7) is simplified to the harmonic oscillation in Equation (8):

#### Analytical formulas considering pipe friction

*et al.*2020). Therefore, the KBM method can be used for an approximate solution (Bogoliubov & Mitropolsky 1961). This method combines the advantages of the average method and the perturbation method, which can reduce the amount of calculation and improve the calculation accuracy (Nayfeh 1973; Ou

*et al.*2020). The idea of the KBM method is to find a series solution with asymptotic properties according to the quasiperiodic property of the oscillations in a nonlinear system. The solution of the equation and the derivatives of the amplitude and phase angle are expressed as a power series with small parameters. Hence, Equation (7) is similar to the following differential equation:where is a small parameter.

### Optimization model for air vessel volume

The numerical simulation method is commonly used to optimize the size of air vessels in a new water supply project. However, the numerical results require repeated trial calculations and cannot guarantee that the optimized air vessel is the optimal volume. In this section, the formula of the water-level oscillation with pipe friction derived from Equation (20) is used to propose a new method for calculating the volume of an air vessel. To ensure the safety of the water supply system, the following conditions should be met when pumps trip: (1) the air vessel must be large enough to supply necessary water to prevent the air from entering the pipeline; (2) the pressure along the pipeline meets the minimum pressure requirements of the water supply system; and (3) the size of the air vessel meets the requirements of manufacture and installation.

The pipelines are arranged according to the terrain in the water supply system. Although it is difficult to express the relationship between the pipeline elevation and distance by a continuous function, the trend of the pipeline can be reflected by discrete elevation nodes. The set of lengths between the elevation control nodes and the outlet sump is denoted as . Likewise, the sets of pipe center elevations and the minimum piezometric heads in the transition process are denoted as and , respectively. In Figure 1, when power failure occurs, water is drawn from the air vessel into the pipeline, and the volume of gas in the vessel expands, causing the gas pressure in the air vessel to drop. Subsequently, the low-pressure wave generated by the drop of gas pressure propagates to the downstream outlet sump, which is reflected by the outlet sump and returns as a positive pressure wave. For node (ranging from 1 to ) in the pipeline, the pressure drops until the negative pressure wave reaches node . Therefore, the pipeline pressure at node is approximately equal to the minimum value of pressure at this point when the high-pressure wave reaches node .

*i*satisfies Equation (26). The values of , and are determined by Equations (27), (28) and (29), respectively. The formula for calculating the minimum piezometric head (MPH) at node

*i*is finally written as Equation (30):

The expressions for are the same as above, and their values vary with the change of *l*_{0} and *F*. In Equation (27), is the correction value of the water lever deviation from the reference datum in the vessel when the pumps trip.

Taking Equations (26)–(30) as additional conditions, the optimal solution of Equation (25) satisfying the constraint condition of Equation (26) can be obtained. Given a value of *l*_{0}, there is always a minimum value of *F* that corresponds to it. To obtain the relationship between the volume of the air vessel and the height of the air chamber, the interval steps and of *l*_{0} and *F* are adopted, respectively. The following procedure is used to calculate the optimal volume of the air vessel, which is schematically shown in Figure 2. When a water supply project is given, we first need to input the basic parameters such as pipe diameter, flow and length of the water supply project and then set an interval step of *l*_{0} and *F* as needed. Finally, through the optimization calculation, the optimal volume of the corresponding air vessel under different air chamber heights can be obtained.

### Model validation

To verify the analytical formula and optimization model, two actual water supply projects are conducted. These two water supply projects are denoted as WSP-1 and WSP-2, respectively. The rated parameters of the pump units for WSP-1 and WSP-2 are shown in Table 1. The other calculation data of WSP-1 are *L* = 7.5 km, *Q* = 1.2 m^{3}/s, *H _{d}* = 278.0 m,

*f*= 0.875 m

^{2},

*n*= 0.012 and

*c*= 800 m/s. The other calculation data of WSP-2 are

*L*= 12.6 km,

*Q*= 2.0 m

^{3}/s,

*H*= 708.4 m,

_{d}*f*= 1.540 m

^{2},

*n*= 0.012 and c = 1,000 m/s. Additionally, the minimum allowable pressure of the pipeline in both projects is 2.0 m, namely,

*p*= 2.0 m.

_{s}Parameters . | Rated head (m) . | Rated flow (m^{3}/s)
. | Rated speed (r/min) . | Motor power (kW) . | Moment of inertia (kg·m^{2})
. |
---|---|---|---|---|---|

WSP-1 | 138.2 | 1,480 | 0.4 | 900 | 179.7 |

WSP-2 | 240.0 | 990 | 0.5 | 1,800 | 674.6 |

Parameters . | Rated head (m) . | Rated flow (m^{3}/s)
. | Rated speed (r/min) . | Motor power (kW) . | Moment of inertia (kg·m^{2})
. |
---|---|---|---|---|---|

WSP-1 | 138.2 | 1,480 | 0.4 | 900 | 179.7 |

WSP-2 | 240.0 | 990 | 0.5 | 1,800 | 674.6 |

Since the installation elevation and connecting pipe are not the main factors affecting the volume of the air vessel, the central elevation of the pipeline is selected as the installation elevation of the air vessel; likewise, half of the pipeline cross-sectional area is selected as the cross-sectional area of the connecting pipe. Hence, the installation elevations of the air vessels in WSP-1 and WSP-2 are 152.0 and 482.0 m, respectively, and the cross-sectional areas of the connecting pipes in WSP-1 and WSP-2 are 0.7 and 1.0 m, respectively. The values of , and may be different in different water supply projects. For WSP-1 and WSP-2, these values are m, m and m. To meet the calculation accuracy, the interval steps are m and .

## RESULTS AND DISCUSSION

In this section, WSP-1 and WSP-2 are used for validation and analysis. The optimal shape of the air vessel is obtained by the optimization model first. Then, the precision of Equations (20)–(22) is verified by comparing the analytical and numerical results. Finally, the rationality of Equation (30) and the optimization model are verified in a similar way.

Then, the optimization program in Figure 2 is transformed into the FORTRAN language, and the optimal shape parameters of the air vessels are finally obtained, as shown in Table 2.

Parameters . | L_{0} (m)
. | H (m)
. _{v} | F (m^{2})
. | V_{air} (m^{3})
. |
---|---|---|---|---|

WSP-1 | 5.27 | 7.50 | 12.32 | 92.40 |

WSP-2 | 5.38 | 7.50 | 23.22 | 174.15 |

Parameters . | L_{0} (m)
. | H (m)
. _{v} | F (m^{2})
. | V_{air} (m^{3})
. |
---|---|---|---|---|

WSP-1 | 5.27 | 7.50 | 12.32 | 92.40 |

WSP-2 | 5.38 | 7.50 | 23.22 | 174.15 |

### Validation of analytical formulas

The water-level oscillation processes in the air vessels obtained by Equations (10) and (20) and the numerical simulation are shown in Figure 3. The comparisons of the extremum values of water-level oscillations and their calculation errors between the analytical results and numerical results are shown in Table 3. Note that the numerical results are obtained by the MOC through FORTRAN programming. The boundary conditions of the suction and outlet sump, pump node, valve node, bifurcation node and air vessel are included in the program. Likewise, the elasticity of the pipe wall and water as well as pipe friction are considered.

Parameters . | NR . | Equation (9) . | Equations (21) and (22) . | |||||
---|---|---|---|---|---|---|---|---|

Extremum (m) . | Extremum (m) . | Absolute error (m) . | Relative error (%) . | Extremum (m) . | Absolute error (m) . | Relative error (%) . | ||

WSP-1 | A_{1} | −1.672 | −1.885 | 0.213 | 12.74 | −1.724 | 0.052 | 3.11 |

A_{2} | 1.060 | 1.885 | 0.825 | 77.83 | 1.102 | 0.042 | 3.96 | |

WSP-2 | A_{1} | −1.583 | −1.633 | 0.050 | 3.16 | −1.618 | 0.035 | 2.21 |

A_{2} | 1.080 | 1.633 | 0.583 | 53.98 | 1.148 | 0.068 | 6.80 |

Parameters . | NR . | Equation (9) . | Equations (21) and (22) . | |||||
---|---|---|---|---|---|---|---|---|

Extremum (m) . | Extremum (m) . | Absolute error (m) . | Relative error (%) . | Extremum (m) . | Absolute error (m) . | Relative error (%) . | ||

WSP-1 | A_{1} | −1.672 | −1.885 | 0.213 | 12.74 | −1.724 | 0.052 | 3.11 |

A_{2} | 1.060 | 1.885 | 0.825 | 77.83 | 1.102 | 0.042 | 3.96 | |

WSP-2 | A_{1} | −1.583 | −1.633 | 0.050 | 3.16 | −1.618 | 0.035 | 2.21 |

A_{2} | 1.080 | 1.633 | 0.583 | 53.98 | 1.148 | 0.068 | 6.80 |

Figure 3(a) shows that the oscillation period of Equation (10) is very close to that of Equation (20), and both of them are slightly smaller than that of the numerical results. The reason for the difference is that the compressibility of water and the elasticity of the pipe wall are neglected in Equations (10) and (20), while they are considered in the numerical simulation. In addition, the oscillation curve of Equation (20) is in good agreement with that of the numerical results, especially in the first decline period. However, there is a large difference between the results of Equation (10) and the numerical calculations, and the difference grows increasingly over time. For the minimum of water-level oscillations in Table 3, the numerical result is −1.672 m. The absolute error of Equation (9) is 0.213 m, and that of Equation (21) is 0.052 m. The relative error of the analytical formula is reduced from 12.74 to 3.11% when the friction of the water supply system is considered. Similarly, the numerical result of the maximum is 1.060 m. The absolute error of Equation (9) is 0.825 m, and that of Equation (21) is 0.042 m. The relative error of the analytical formula is reduced from 77.83 to 3.96% when the friction of the water supply system is considered. For WSP-2, the results are similar to those for WSP-1.

The above analysis results indicate that when pipe friction is considered, the analytical formula for water-level oscillation can better fit the numerical results. The calculation error of the analytical formula for minimum and maximum values can be controlled within 7%, which is greatly improved compared with the previous analytical formulas (Miao *et al.* 2017). Hence, Equations (20)–(22) are reasonable and have good precision, which can provide some reference for the initial design of the air vessel.

### Validation of optimization model

The MPH along the pipeline can be obtained by Equation (30). The comparison of the MPH between the analytical results and numerical results is shown in Figure 4, where is the piezometric head under steady flow, is the numerical result of the MPH, is the analytical result of the MPH, is the MPH without protective measures and is the center elevation of the pipeline.

For WSP-1, assuming no water-hammer protective measures were taken in the water supply system, most of the MPHs are below the center elevation of the pipeline when pumps trip, as shown in Figure 4 (note that: the difference between the MPH and the center elevation only represents the severity of the negative pressure in the pipeline. Water has already vaporized when the negative pressure of the pipeline is below – 10 m). When the optimized air vessels in Table 2 are used as the protective measure, the analytical results and numerical results of the MPHs are above the center elevation of the pipeline. Moreover, the analytical results and numerical results of the minimum internal pressure are 0.76 and 2.0 m, respectively, and the positions of the minimum pressure are located 6 and 930 m behind the pump. In addition, the errors between the analytical results and numerical results increase with increasing . Hence, the error is the lowest at the outlet sump and the largest at the check valve, which are 0 and 6.45 m, respectively. Similar results are obtained for WSP-2; the analytical results and numerical results of the minimum internal pressure are 0.91 and 2.0 m, respectively, and the positions of the minimum pressure are both located 11 and 890 m behind the pump. Moreover, the maximum and minimum errors are 0 and 4.12 m, respectively.

The above analysis results indicate that air vessels can effectively protect pipelines from water hammers, which is consistent with previous research conclusions (Verhoeven *et al.* 1998). Although the minimum internal pressure obtained by the analytical formula is less than the numerical result, the optimized air vessel in Table 2 can meet the requirements of the water supply system when *p _{s}* has a small safety margin. Additionally, the pressure control point along the pipeline is generally located at the end of the water supply system, where the analytical results have good accuracy. Therefore, the optimization model is reasonable and feasible, and Equation (30) can provide a certain guidance value for the laying of a water supply pipeline. From Equation (30), we can conclude that avoiding local uplift or reducing the elevation of the pipeline tail is beneficial to the design of water-hammer protection measures in a long-distance water supply system.

### Effects of air chamber height on the volume of air vessels

To investigate the relationship between the air chamber height and the minimum volume of the air vessel, WSP-1 and WSP-2 are selected as engineering cases in this section. The relationship between *F*, , and can be obtained through the idea in Figure 2, as shown in Figure 5, where *F*, and are the minimum values satisfying the constraint Equation (25) under the corresponding .

In Figure 5, the curves of *F* with respect to *l*_{0} in WSP-1 are similar to hyperbolas. The values of *F* decrease with increasing *l*_{0}, and the decreasing rate decreases gradually. By contrast, the curves of *H _{v}* with respect to

*l*

_{0}are similar to straight lines, and the values of

*H*increase with increasing

_{v}*l*

_{0}. In addition, the values of

*V*

_{air}decrease with the increase of

*l*

_{0}. A slight change of

*l*

_{0}causes a drastic change of

*V*

_{air}when

*l*

_{0}< 2 m, while the decreasing rate of

*V*

_{air}gradually decreases with the increase of

*l*

_{0}when

*l*

_{0}≥ 2 m, and finally, the curve tends to be horizontal. For WSP-2, there are the same conclusions as for WSP-1. The reason for the above results is as follows: When the air chamber height is very small, the cross-sectional area of the air vessel is large to ensure the safety of the water supply system. Due to the safe water depth left in the air vessel, most of the volume of the vessel is used as a safety reserve, which has no effect on water-hammer protection. In addition, the influence of the gas volume variation on the absolute gas pressure decreases with increasing air chamber height.

The above calculations imply that the air chamber height has a significant effect on the water-hammer protection characteristics of the air vessel. Moreover, increasing the air chamber height can reduce the total volume of air vessels with the same protection requirements, which is similar to the result of Besharat's experiment (Besharat *et al.* 2017). Therefore, a slender air vessel has a better water-hammer protection effect than a squat vessel, which can provide guidance for the design of air vessels.

## CONCLUSIONS

The use of air vessels is a common measure to prevent water hammers in long-distance water supply systems. The selection of the air vessel has a certain blindness, which is often accompanied by much trial work. In this paper, the analytical formulas of the water-level oscillations in the air vessel were derived based on the KBM method considering pipe friction. Based on nonlinear programming theory, an optimization model of the air vessel volume was established, and the analytical formulas of the MPH along the pipeline were derived. The rationality and accuracy of the analytical formulas and the volume optimization model were verified by comparison with the numerical simulation results. The effects of air chamber height on the volume of air vessels were analyzed. Several conclusions can be drawn from this study:

- (1)
The calculated results of Equations (20)–(22) are in good agreement with the numerical simulation results, which indicates that the analytical formulas of water-level oscillations and their extremum values in air vessels considering pipe friction are reasonable.

- (2)
The optimization model of the air vessel volume can preliminarily estimate the minimum volume of the vessel required by the water supply system. Compared with the traditional MOC numerical simulation method, the optimization model in this paper can simplify the selection process of the air vessel volume.

- (3)
For a fixed volume, the water-hammer protection effect of a slender air vessel is better than that of a squat vessel. In other words, increasing the air chamber height of the vessel and reducing its cross-sectional area can reduce the volume of the vessel with the same protective effect. In addition, avoiding local uplift or reducing the elevation of the pipeline tail is also beneficial to water-hammer protection in long-distance water supply systems.

- (4)
The main factors that determine the volume of the air vessel are the cross-sectional area, air chamber height and water-gas ratio, which are considered in the optimization model. The influence of the connecting pipe diameter and installation elevation will be considered in future research to obtain a more reliable volume of air vessels.

## ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos 51879087 and 51839008) and the Culturing Funds for the fifth ‘333 Project’ of Jiangsu Province (Grant No. BRA2018061).

## DATA AVAILABILITY STATEMENT

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