## ABSTRACT

The design factor of surge tank installation is a practical issue in the management of pressurized pipeline systems. To determine the general criteria for surge tank design in pipeline systems, dimensionless governing equations for unsteady flow and their solutions were developed for two widely used pipeline systems equipped with surge tanks. One is the reservoir pipeline surge tank valve and the other is the pipeline system with a pumping station and check valve protected by the surge tank. Two distinct time-domain responses, point- and line-integrated pressure, can be used as objective functions to optimize the surge tank area. The developed formulations were integrated into a metaheuristic engine, particle swarm optimization, to explore a general solution for a wide range of dimensionless resistances that comprehensively address various flow features into one dimensionless parameter. Depending on the dimensionless location of the surge tank, the optimum dimensionless surge tank areas were delineated for a range of dimensionless resistances for the two pipeline systems with and without a pumping station protected by a surge tank.

## HIGHLIGHTS

Dimensionless solutions for pressure response were derived for pipeline systems with surge tanks.

The dimensionless resistance represents flow rate, friction, diameter, length and wave speed.

Both point- and line-integrated pressures were used as objective functions.

The frequency domain models were integrated into particle swarm optimization.

Dimensionless surge tank areas were delineated for dimensionless resistances.

## INTRODUCTION

Water hammers in pipeline systems can be generated by sudden valve closures, pump stoppages, or instant check valve actions. When a pressure wave is generated by an abrupt change in flow velocity, it introduces either overpressure or pressure along the pipeline. Although a high pressure can burst a weakened part of the pipeline, a low pressure can generate column separation and cavitation, which usually substantially damage the pipeline system. To protect the pipeline structure from hydraulic transient events, surge protection devices, such as surge tanks, have been used in front of the control valve and pump station.

To analyze surge events in pipeline systems, the characteristic method has been used (Wylie & Streeter 1993; Karney & Simpson 2007; Wan & Zhang 2018) and the size and location of surge protection devices have been determined to relax sudden pressure variations, considering the cost of hydraulic structures (Di Santo *et al.* 2002; Jung & Karney 2009; Duan *et al.* 2010; Martino & Fontana 2012; Skulovich *et al.* 2015).

Transient generation and its propagation and reflection along pipeline systems introduce pressure oscillations, which can be expressed by the surface water variation of the surge tank (Guo *et al.* 2017). The length of the main pipeline and the location, cross-sectional area, and connector length of the surge tank in the pipeline system are important variables for determining the resonance characteristics of the pressure response. The cross-sectional area of a surge tank appears to be an important variable for moderating the oscillations in pipeline systems (Liu *et al.* 2023). The application of the impulse–response method demonstrated the potential of the frequency domain approach in the context of resonance characterization for the design of hydraulic structures using transient analysis (Kim 2010). Assuming that the layouts of pipeline systems with a surge tank are simple and similar to each other (e.g., reservoir pipeline surge tank valve), the pressure wave propagation pattern of the pipeline system can be generalized through the dimensionless development of governing equations and its optimum solution in the time-domain response (Kim & Choi 2022). Analytical developments for pipeline systems with and without pumping stations have provided distinct phase differences between the two systems depending on the transient generation location (Kim 2023).

A dimensionless analysis of the pressure response in pipeline systems equipped with surge tanks needs to be further explored in the context of a unified dimensionless viewpoint that can provide a robust basis for a comprehensive understanding of the system behavior and better management for various transient scenarios. The implementation of the pressure response feature in the practical flow regime can be successfully addressed by considering a wide range of dimensionless resistances in the optimum design of the system and modeling.

Therefore, this study explores a solution for optimum surge tank installation in a dimensionless space that can provide a general criterion for surge protection in pipeline systems, both with and without a pumping system. The objective function for parameter optimization can be either a point or a line-integrated response of the pressure at any designated point or part along the pipeline. Dimensionless developments for pipeline systems were integrated into a particle swarm optimization (PSO) scheme, which provides a comprehensive solution in a dimensionless space for the design of surge tank installations for two distinct pipeline systems.

## METHODS

### Dimensionless governing equations

*t*) and distance (

*x*), and two dependent variables for mean velocity (

*V*) and pressure head (

*H*) (Wyile & Streeter 1993),where

*A*is the cross-sectional area,

*a*is the wave propagation speed,

*g*is the gravitational acceleration,

*f*is the Darcy–Weisbach friction factor and

*D*is the diameter.

The mean flow rate *Q* can be defined by multiplying *V* and *A* and the independent dimensionless variables can be defined as follows: and , where *L* is the length of the pipeline system. The dependent dimensionless variables can be defined as for the pressure head and for the flow rate:

Equations (3) and (4) indicate that the dimensionless variable represents the impact of friction, diameter, wave speed, and mean flow rate in response to the dimensionless flow rate and pressure head.

### Pipeline systems with surge tank with and without pumping stations

The upstream length between the upstream reservoir and surge tank, , can be converted into an upstream dimensionless length as , and the dimensionless downstream length can be defined as .

The dimensionless hydraulic impedance at the surge tank can be approximated by considering the flow rate fluctuation and pressure as follows: , where is the surge tank area. The disturbance of the pressure head () can be approximated as , where is the steady pressure head in the surge tank, and s is the frequency.

### Pipeline systems having multiple branches with surge tanks

### Integration with metaheuristic engine

*et al.*2023). Considering the substantial impact of local topographical features such as the elevation distribution along pipeline extension, the potential locations of surge tank installation can be varied between 0.1 and 0.9 in terms of dimensionless distance from upstream to downstream. The developed formulations were used to determine the optimum surge tank area for the available range of dimensionless resistance. To ensure fast computation and optimization, an algorithm was designed considering the iterative evaluation of the dimensionless resistance (Figure 4) and its incorporation into the PSO (Kennedy & Eberhart 1995).

*n*is the number of dimensionless time steps and is the dimensionless pressure response.

## RESULTS AND DISCUSSION

### Results

The dimensionless approach provides an identical solution for different flows, frictions, and lengths and diameters of the pipeline, as long as the dimensionless parameters between the two different systems are identical. As presented in Equations (3) and (4), a dimensionless parameter, dimensionless resistance, can holistically characterize system behavior. This implies that we can effectively explore the general response features of the system behavior by changing the dimensionless resistance. Considering that the location of the surge tank is frequently determined by field conditions, such as feasibility and elevation distribution, the size of the surge tank is the most important parameter for the design of surge tank protection from hydraulic transients.

These complex optimization results may be associated with hydraulic structures that are more complicated than those shown in Figure 1. Both the downstream and upstream pumping stations with check valves can be potential exciters of system response, and the optimum surge tank area can be affected by the interaction between multiple surge generation devices and pipeline features in terms of dimensionless resistance.

## DISCUSSION

Kim (2023) validated the dimensionless approach by comparing the existing method with a developed scheme in the time-domain response. This study explored the optimization of the surge tank design using a dimensionless formulation. The optimization results shown in Figures 5–8 are based on the pressure response minimization in the time-domain. A surge event through instant velocity changes and abrupt pressure variations is a widely used transient event for the safe design of surge tanks. However, the field conditions may be more flexible than the hypothetical consideration of water hammer events. Modulated valve maneuvers can be used in many pipeline systems, even though sudden pump stoppage due to power failure can generate a shape surge event. Therefore, an alternative approach can be considered for the optimization of pipeline systems in the context of a wider scope of unsteady events. The kernel function for the convolution of the time-domain response can be obtained through an inverse fast Fourier transformation of the response functions, and its amplitude of kernel function depends on the frequency response functions (for example, Equations (16)–(23)). In other words, the optimization of the surge tank may not necessarily be limited to the time-domain response. The developed frequency response functions can be used to optimize pipeline systems. Depending on the system characteristics in terms of frequency response, a designated frequency range can be used for the objective function. This implies that the system can be designed based on the response feature of dimensionless response in the frequency domain. Further exploration of system optimization in the frequency domain has not yet been attempted, but the potential of frequency domain optimization may provide a wider comprehensive solution for all possible frequency ranges of the impulse input. Considering the substantial factors, assumptions, and uncertainties in the frequency domain model, further developments in modeling should be made for the optimum design of the system using an enhanced frequency objective function. Therefore, the optimal design of pipeline systems with a dimensionless frequency response is a topic for future research.

In this study, the parameters of the system were optimized based on the assumption that the water hammer was generated by an instantaneous valve closure. The optimization should be extended if the transient introduced is different from the transient caused by abrupt flow rate changes. Friction is another factor to consider. A steady friction model can provide satisfactory results when the water hammer is caused by an instant change of boundary condition, as the surge wave is mainly caused by the initial pressure wave. The attenuation of subsequent pressure waves can be important if the transient introduction is slow. Further development incorporating an unsteady friction model can be a future research topic, especially for transients caused by a slow valve action.

Experimental validation of dimensionless transient models for pipeline systems with surge tanks is both critical and challenging. There are numerous real-life systems that can exist for a dimensionless solution designed for a specific dimensionless system, which is an important underlying assumption for dimensionless approaches. Real systems with different physical dimensions and properties will need to be tested under identical dimensionless time scales, which will be different time scales between the various systems. Therefore, systematic studies of dimensionless transient models can be important research topics for the future.

## CONCLUSIONS

In this study, a dimensionless solution for a transient event was used for the optimum design of pipeline systems with surge tanks. The developed dimensionless equations indicate that a one dimensionless parameter, dimensionless resistance, can holistically address the effects of friction, diameter, length, wave speed, and mean flow rate. The PSO scheme was incorporated into the time-domain response of the pressure from the developed dimensionless expressions. Both point- and line-integrated pressures can be used for the objective function to minimize the pressure response. Considering the importance of the surge tank area in the design, the distribution of the optimum dimensionless surge tank area can be obtained for the possible dimensionless locations of the surge tank and the dimensionless resistance. Depending on the system features and objective function, the optimum dimensionless surge tank area exhibited distinct responses for the dimensionless resistance range. The optimization results provide a general solution for the surge tank area if the dimensionless resistance and location are identical to those of multiple real-life systems. The further development of a dimensionless model for a more general design of the system response could be a future research topic for various transient impulses in the frequency domain.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.