The size parameters of an air vessel, mainly including its initial gas volume and total volume, account for its performance in hydraulic transients. Traditionally, the size of an air vessel in a given water supply system is estimated by trial and error, usually aided by computers. To simplify the preliminary estimation of the air vessel's volume, in this paper, an approximate analytical method is proposed and derived on the basis of incompressible flow theory and simple harmonic vibration theory. With this method, a theoretically optimal size of an air vessel, which is able to meet the protective requirements in hydraulic transients, could easily be obtained in an analytical way. Besides, a real engineering case is introduced in this paper to illustrate that the proposed method could give guidance and reference in the preliminary work of designing an air vessel in practice.

## INTRODUCTION

Water supply systems are usually protected by protective devices such as surge tanks, air valves and air vessels that are capable of keeping the operating pressure within the limited value after any valve or pump operations. Among these devices, the air vessel generally alleviates negative pressures more effectively than others, and it can maintain a positive pressure in the line at all stages following a pump trip. Generally, the performance of an air vessel mainly depends on its size parameters, including its initial gas volume, total volume and inlet/outlet pipe diameter etc. (Stephenson 2002), and a better protective effect will be obtained with a larger volume of air vessel (Liang et al. 2005; Kim et al. 2015). However, an overlarge volume may lead to more cost. Therefore, an air vessel with optimal size is necessary, because it cannot only show its reliable protective effect on water hammer protection, but also avoid unnecessary cost by a large extent. Currently, more and more concerns have been emphasizing the sizing of an air vessel rationally and economically. A number of methods to size an air vessel have been proposed in recent years by numerous researchers (Thorley & Lastowiecki 1985; Stephenson 2002; Jung & Karney 2006; Zaki & Elansary 2011). However, most of these methods are either based on numerical computations or on various practical engineering experiences.

In this paper, an approximate analytical method to size an air vessel is introduced and derived. It allows designers to obtain a reasonable size of air vessel in an analytical and quick way at the preliminary stage of designing an air vessel. The verification of this method is also conducted by studying the practical case introduced in this paper.

### Mathematical model

For a given water supply system equipped with an air vessel, shown in Figure 1, on the basis of the incompressibility theory of water, the mathematical model of the system could be obtained and is shown in the Appendix (available with the online version of this paper).
When a pump trip occurs, the valve behind the pump will be closed rapidly to avoid reverse flow, and the discharge near the pump outlet can be neglected. Then the continuity equation can be written as:Taking Equations (1)–(3) into the motion equation of the system in the Appendix, we have:where , .

Equation (4) describes the variation of water level z in the air vessel when the pump trip happens, and there is no explicit solution to it. At the preliminary stage of designing an air vessel in a given water system, we tend to initially estimate a proper size of the air vessel which could be regarded as guidance for the following further optimization of the vessel size. Then, in order to make the initial estimation more reasonable, an approximate analytical method is proposed in the following section of this paper.

### Approximate analytical method of sizing an air vessel

#### Conditions for safe operation of a water supply system

In a water supply system equipped with an air vessel as a protective device, shown in Figure 2, in order to guarantee the safe operation of the system, the following three conditions must be met: (1) the surge in the vessel must not go below the minimum limit so that the vessel would not be empty of water and the gas in the vessel would not run into the pipe; (2) the pressure in the system must not exceed the limit that brings damage to the pipeline; (3) the pressure in the system must remain positive and there should be some safety margin for it, otherwise negative pressure may cause water column separation resulting in overpressures. The above three conditions are closely associated with the air vessel's size parameters. Thus, it is critical to determine the size parameters properly to guarantee the safe operation of the system.

#### Approximate analytical method to size an air vessel

It should be noticed that the third term in Equation (4) stands for the impact of pipe friction and throttling. In order to find an analytical solution, we suggest neglecting pipe friction and throttling in the preliminary stage of designing an air vessel. In this way, we have:Equation (5) is obviously the simple harmonic vibration equation, through which we can respectively obtain the surge period △T, the wave amplitude in the vessel △Z* and the bottom pressure of the vessel △H in the Appendix.

Equation (12) describes the theoretical minimum air volume which is required for the safe operation of the pipeline system. For a pipeline system that rises uniformly in profile, dominates the value of , and it can make sure that the positive water hammer pressure due to pump failure is within the maximum limit. Whereas, when it comes to a pipeline with convex points in profile, then determines the value of , and it can guarantee that no negative pressures appear along the pipeline.

The approximate analytical size of the air vessel can eventually be described as:where k is the correction coefficient to make up for the approximation in the derivation; its value is 1.1. VT and Va are respectively the total volume and the air volume of the air vessel. As long as the air height l0 is determined, the cross-sectional area of the vessel F can be obtained by F = Va/l0.

### Case study

The engineering parameters of a pump-station water supply system located in China are shown in Table 1, and the schematic diagram of the pipeline system is shown in Figure 3. The pipeline rises uniformly in profile. According to some particular engineering principles in China, the limited rate of pressure rise in the pipeline system φ2 = 30%. The polytropic exponent m = 1.2. Since this water supply project is still under construction currently, no measured data about the pipeline system could be presented for now. Therefore, the simulation numerical computation, which is on the basis of the method of characteristics (Streeter & Wylie 1963; Wylie et al. 1993; Chaudhry 2014), was given instead of measured data to make a comparison with the theoretical result.
Table 1

Parameters of pipeline system

 Upstream water level (m) 110 Wave speed (m/s) 1,000 Downstream water level (m) 170 Roughness coefficient 0.012 Installation elevation of air vessel (m) 100 Pump speed (r/min) 1,450 Pipe cross-sectional area (m2) 1 Motor power (kW) 710 Pipe length (m) 3,000 Pump head (m) 60 Rated flow discharge (m3/s) 1 Moment of inertia (kg m2) 374
 Upstream water level (m) 110 Wave speed (m/s) 1,000 Downstream water level (m) 170 Roughness coefficient 0.012 Installation elevation of air vessel (m) 100 Pump speed (r/min) 1,450 Pipe cross-sectional area (m2) 1 Motor power (kW) 710 Pipe length (m) 3,000 Pump head (m) 60 Rated flow discharge (m3/s) 1 Moment of inertia (kg m2) 374

## RESULTS AND DISCUSSION

Since the pipeline in this case rises uniformly in profile, was adopted to calculate the analytical size of the air vessel, and Table 2 shows the corresponding size parameters of the vessel. In addition, in Figures 4 and 5, the theoretical result for the amplitudes of the bottom pressure of the vessel and amplitudes of the water depth of the vessel are given. In the numerical computation, the analytical size of the air vessel is adopted, and the area of the impedance orifice is 0.5 m, as large as 20% of the pipeline's (Gong et al. 2013), and the orifice-metering coefficient is 0.6. Pipe friction and throttling were considered during the numerical computation, and the numerical result is shown in Figures 4 and 5.
Table 2

Size parameters of the air vessel

Total volume (m3)Water volume (m3)Gas volume (m3)Cross-sectional area (m2)Water height (m)Gas height (m)Water–gas ratio
89 16 73 14.6 1.1 5.0 0.22
Total volume (m3)Water volume (m3)Gas volume (m3)Cross-sectional area (m2)Water height (m)Gas height (m)Water–gas ratio
89 16 73 14.6 1.1 5.0 0.22
Table 3

Statistics of results

ResultsInitial pressure (m)Maximum pressure (m)Minimum pressure (m)Initial water depth (m)Maximum water depth (m)Minimum water depth (m)
Theoretical 70.0 90.0 50.0 1.1 2.1 0.1
Numerical 70.0 83.6 54.2 1.1 1.7 0.2
ResultsInitial pressure (m)Maximum pressure (m)Minimum pressure (m)Initial water depth (m)Maximum water depth (m)Minimum water depth (m)
Theoretical 70.0 90.0 50.0 1.1 2.1 0.1
Numerical 70.0 83.6 54.2 1.1 1.7 0.2

Through the comparison between the theoretical result and the numerical result, we can see that the surge periods of these two results can match well with each other, whereas the amplitudes of the bottom pressure are different in varying degrees in these two results, as well as the amplitudes of the water depth. In addition, the differences between the two results grow increasingly as time goes on.

The reason that could explain the differences between the above two results is that pipe friction and throttling are neglected in the theoretical result, while they are considered in the numerical result. In the numerical result, water hammer due to pump trip will be increasingly relieved by pipe friction and throttling as time goes on, whereas water hammer in the theoretical result will not. As a result, the analytical size of air vessel is relatively larger than expected in practical cases, and it can leave some safety margin to the pipeline system. Table 3 shows that the analytical size of the air vessel left a margin of about 7 m for the system pressure and a margin of 0.15 m for the water depth in the vessel. Moreover, it could be implied that, as the length of the pipeline increases, the impact of pipe friction on relieving water hammer becomes larger, and that the analytical size of the air vessel obtained by this analytical method will be increasingly larger than the one actually required in the practical case.

Despite the fact that this analytical method may make the air vessel volume relatively larger than actually needed in practical cases, it is able to provide a reference to size an air vessel in a safe and conservative way. At the preliminary stage of designing an air vessel, we can use this analytical method to approximately estimate the size of the air vessel, giving guidance for the following optimization of the air vessel. In addition, it must be pointed out that, in this paper, little safety margin for the water level in the vessel has been taken into consideration during the derivation of this method, because we wanted to seek a theoretical minimum volume of the air vessel by this method, not a practical one. Therefore, in practical engineering cases, the analytical size of the air vessel should be properly increased by some extent for enough safety margin.

## CONCLUSIONS

The following conclusions can be drawn in this paper:

1. An analytical method to size an air vessel, which allows designers to size an air vessel in an analytical way, is proposed and derived on the basis of incompressible flow theory and simple vibration theory.

2. This analytical method is verified by studying the comparison between the theoretical and numerical results of a practical case introduced in this paper. The results showed that the analytical method is capable of providing a reasonable size of air vessel which can be regarded as guidance for the subsequent optimization of the air vessel. Additionally, since no safety margin was considered in the derivation of the analytical method, the analytical volume of air vessel obtained by this method, especially the initial volume that contains water, should be increased by some extent for enough safety margin in practical cases.

3. Currently, the practical project introduced in this paper is still under construction. Therefore, no measured data are available to make comparison with the theoretical and numerical results. Further research will be carried out to improve this analytical method after the measured data are next obtained in the future.

## ACKNOWLEDGEMENTS

The present project is financially supported by the National Natural Science Foundation of China (No. 51379064), the Fundamental Research Funds for the Central Universities (Grant No. 2015B00414 and No. 2016B10814) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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