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An air cushion chamber is a feasible and efficient hydraulic device to control water hammer for pressurised pipeline systems. The introduction and evolution and hydraulic transient depend on distinct features of pipeline systems such as length, diameter and the property of pipeline extension as physical properties; flowrates, pressure, boundary conditions as hydraulic conditions; and interaction with surge protection devices (air chamber) as hydraulic conditions. Considering all factors simultaneously can be incredibly difficult, even for a simple air chamber pipeline valve layout. This study introduces the dimensionless transfer function and expression for air chambers in dimensionless frequency domain in order to effectively address the water hammer generation and its counteracting processes. To comprehensively characterize hydraulic transients for pipeline systems equipped with air chambers, two representative dimensionless parameters were used. Along the pipeline system, it is possible to develop a frequency-independent expression for hydraulic impedance. A comparison between the developed method and other existing methods (e.g. characteristic method and impulse response method) revealed excellent agreement. Application of the dimensionless parameters to systems with different dimensions and hydraulic conditions shows that the proposed dimensionless parameters can address substantial ranges of real systems.

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  • A dimensionless expression for air chambers was developed for pressurized pipeline systems.

  • A dimensionless frequency domain expressions were developed for hydraulic impedance along the pipeline system.

  • Holistic behavior could be explained by two important dimensionless parameters.

dimensionless air chamber, general characterization of pipeline systems, dimensionless impedance method
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A, A′

cross-sectional areas for the pipeline and air chamber

a, a′

wave speeds in the pipeline and connector

C

capacitance

D, Dc

diameter of the pipeline and connector

f

Darcy Weisbach friction factor

g

gravitational acceleration

H,

real and dimensionless pressure heads

h′,

real and dimensionless pressure fluctuations

L, 11, l2, l3

pipeline lengths

m

polytrophic coefficient

Q,

real and dimensionless flowrates

q′,

real and dimensionless flowrate fluctuations

dimensionless resistance

dimensionless frequency for pipeline

t,

real and dimensionless time

⁠,

dimensionless volumes

ω,

real and dimensionless frequency for the air chamber

dimensionless hydraulic impedance

dimensionless characteristic impedance

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Variations in flow rate in pipelines can result in pressure variations from hydraulic structures such as valves and pumps. The pressure wave can be generated and propagated along pipeline extensions when sudden changes in dynamic boundary conditions occur. Depending on the pattern of pressure wave generation, a constant pressure head, such as a water tank or dead end, can bounce back a pressure wave either reversed or identical pressure, and even minor diversions in pipeline layouts (such as branches or loops) can dissipate or even increase pressure variations. Overpressure and underpressure in transient events can damage pipeline systems, causing bursts in relatively weak points in pipeline elements, or column separation due to cavitation. Various surge arresting devices (e.g. air chambers, surge tanks, and pressure relief valves) were used to protect the pipeline system from hydraulic transient events. The proper pipeline spacing between valves and the design of check valves has been studied to improve pipeline reliability (Kyriakopoulos et al. 2022; Soboleva et al. 2023). The air chamber is a closed chamber filled with compressed air and water that offers significant advantages in terms of installation, operation, and economics (Liu et al. 2023). In hydroelectric power plants, air chambers were the most commonly used water hammer protection device. Air chambers can be designed reliably by considering various transient scenarios and simulating hydraulic transients in a designated pipeline system.

In pipeline systems, the method of characteristics (MOC) has been widely used as a transient simulation method (Wylie & Streeter 1993). Under transient analysis, optimal designs of air vessels have been explored (Di Santo et al. 2002; Stephenson 2002; Martino & Fontana (2012) 2012; Kim et al. 2014; Moghaddas 2018; Zhou et al. 2020; Lin et al. 2021). Several discretization methods (such as the finite difference method, finite element method, and control volume method) can be used to improve the accuracy of computation with several adjustments for system approximation (Kendir & Ozdamar 2013; Wan et al. 2019; Liu et al. 2023). The simulation of gas–liquid flow across long pipeline systems was improved by incorporating three-dimensional computable fluid dynamics into MOC (Zhang et al. 2023). Grid-based approximation methods, however, shared limitations with regard to the accuracy of representation and computational cost of time–space domain representation. The impulse response method (IRM) for pipelines equipped with air chambers demonstrated the benefits of a frequency domain approach to designing hydraulic structures in several transient scenarios (Kim 2008). However, a unified criterion for system parameters (such as chamber volume, air volume, connector lengths and sizes, and polytrophic coefficient as compressed air) can be explored in the context of various transient conditions.

The hydraulic transients in a simple pipeline were studied with a dimensionless frequency domain model under a scale-free environment (Kim 2022). A dimensionless pressure and flowrate response pattern can also be generalized to pipeline systems equipped with pumps, check valves, and surge tanks (Kim 2023). Nevertheless, a dimensionless method needs to be developed for a pipeline system with an air chamber along with a representative dimensionless parameter for that system. Although a simple pipeline system consists of just five or six components (such as a reservoir, pipeline, air chamber, pipeline valve, and reservoir), the total number of parameters is much higher. Consequently, comprehensive analysis and characterization are virtually impossible.

The dimensionless modeling approach can be useful not only because it provides system response through a generalized platform in dimensionless number but also because this approach explains pressure response in the frequency domain with a fewer parameters, which represent distinct components of the system, such as the main pipeline and the air chamber.

The purpose of this study is to provide a holistic viewpoint of the transient behavior of a water supply system with an air chamber. A hydraulic transient governing equation can be described in the dimensionless frequency domain, and the corresponding time domain response can be obtained in the dimensionless time domain. A new approach to water hammer in pipeline systems equipped with air chambers under a scale-free environment will be developed, which systematically generalizes existing solutions.

The aim of this study is to characterize pipeline systems with air chambers based on the following two aspects.

The first step was to formulate a generalized expression for air chambers in the dimensionless frequency domain by combining differential equations of momentum and mass conservation. Using two existing methods, MOC and IRM, the developed method was validated. Several realistic systems were simulated to prove the representativeness of the developed dimensionless parameters.

Second, the impact of the Reynolds number on dimensionless hydraulic impedance response was investigated for both laminar and turbulent flow conditions. Under two distinct flow regimes, different combinations of two dimensionless parameters were explored to analyze the dimensionless pressure response patterns.

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Frequency domain hydraulic transient in a pipeline system with air chamber

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In pressurized pipeline systems, partial differential equations of continuity and momentum conservation can be expressed as follows (Wylie & Streeter 1993):
(1)
(2)
where Q is the mean flowrate, H is the piezometric head, a is the wave propagation speed, g is the gravitational acceleration, x is the distance, t is the time, f is the Darcy–Weisbach friction factor, and D is the diameter.
Application of perturbation theory or Laplace transform into Equations (1) and (2), the relationship between an upstream and downstream head-discharge point as a function of distance, x, can be expressed in the frequency domain as (Chaudhry 1987; Wylie & Streeter 1993)
(3)
(4)
where subscript U denotes an upstream section. The propagation constant is ⁠, the characteristic impedance is ⁠, and the complex frequency is ⁠, where the capacitance (C) is gA/a2, the inertance (L) is 1/gA, σ is a decay factor, ω is the frequency, and the resistance (R) is or depending on laminar or turbulent flow.

Air chamber expression in the frequency domain

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Assuming that the compressibility of water is negligible compared to the compressibility of the air in the air chamber (Figure 1), the impact of inertia and friction on the air chamber can be neglected. The reversible polytrophic relation between pressure and volume can be applied (Wylie & Streeter 1993).
(5)
where HA = Hs − elevation of Hs from datum + ⁠, = the barometric pressure head, = the gas volume, m = the polytrophic exponent, and = a constant.
Figure 1
Schematic of the air chamber. , air pressure; , volume of air; ,water surface head; , length of connector; , diameter of connector; , friction factor of connector.
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Schematic of the air chamber. ⁠, air pressure; ⁠, volume of air; ⁠,water surface head; ⁠, length of connector; ⁠, diameter of connector; ⁠, friction factor of connector.

Figure 1
Schematic of the air chamber. , air pressure; , volume of air; ,water surface head; , length of connector; , diameter of connector; , friction factor of connector.
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Schematic of the air chamber. ⁠, air pressure; ⁠, volume of air; ⁠,water surface head; ⁠, length of connector; ⁠, diameter of connector; ⁠, friction factor of connector.

Close modal
The air pressure and volume can be divided into two components the average and the oscillatory part such as
(6)
Combining Equations (6) and (5) and applying the binomial formula provides the following relationship by neglecting high-order terms.
(7)
Introducing ⁠, ⁠, and dividing complex discharge to Equation (7) provides the impedance at the air chamber as
(8)
where i represents the imaginary part.

Dimensionless governing equation

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Assuming a reservoir pipeline air chamber pipeline valve (RPAPV) system shown in Figure 2, Equations (1) and (2) can be converted into the dimensionless equation as the dependent variable equation is changed into dimensionless dependet variable one ⁠, where g is the gravitational acceleration, A is the cross-sectional area, a is the wave speed, and is the steady flowrate for the dimensionless pressure head, and for the dimensionless flow rate, and two independent variables and can be expressed as follows:
(9)
(10)
where is the dimensionless resistance, which can be estimated as under laminar flow and for turbulent flow conditions, where f is the Darcy–Weisbach friction coefficient under the assumption of steady friction.
Figure 2
Schematic of the RPAPV system.
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Schematic of the RPAPV system.

Figure 2
Schematic of the RPAPV system.
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Schematic of the RPAPV system.

Close modal
Applying the perturbation theory to the dimensionless pressure head and flow rate expressed in Equations (9) and (10) yields the trigonometric relationship between the upstream and downstream dimensionless frequency as follows:
(11)
(12)
where the dimensionless propagation constant, ⁠, can be expressed as
(13)
where is the dimensionless frequency, that can be defined as ⁠.

Dimensionless hydraulic impedance for the air chamber

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Dimensionless terms need to be obtained by properly normalizing the important variables in the air chamber. Given the importance of cross-sectional area and its function to mean flowrate and wave speed, one can express the dimensionless expression for pressure head variation as follows: ⁠; the dimensionless mean pressure of air can be expressed as ⁠, where is the mean pressure of air chamber and is the cross-sectional area of the air chamber. The dimensionless volume variation can be defined as ⁠. Introducing the relationship between pressure head variation and volume variation in an air chamber (Wylie & Streeter 1993), the dimensionless pressure head variation can be expressed as follows.
(14)
The relationship between dimensionless variables and can be expressed in dimensionless frequency and time domain as follows.
(15)
where ⁠, ⁠, and and are wave speed and length of the connector, respectively.
Combing Equations (10) and (11) provides the following dimensionless relationship between the dimensionless pressure head and dimensionless volume of the air chamber:
(16)
Assuming the dimensionless flowrate change as and the application of dimensionless frequency and time into Equation (7) provides the following expression:
(17)
Therefore, the dimensionless hydraulic impedance at the outlet of the air chamber connector can be expressed as follows:
(18)
where

Dimensionless hydraulic impedances for the RPAPV system

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As shown in Figure 2, the schematic for the RPAPV system can be converted into a dimensionless system through the introduction of dimensionless variables as follows: and ⁠.

The hydraulic impedance of the main pipeline upstream of the connecting element (Figure 2) can be approximated as follows:
(19)
where and ⁠.
Considering the hydraulic impedance at the air chamber, the hydraulic impedance downstream of the connecting element (Figure 2) can be expressed as follows:
(20)
where and ⁠,
The hydraulic impedance of the main pipeline at the downstream of the connecting element (Figure 2) can be evaluated as follows
(21)
Therefore, the hydraulic impedance at the downstream valve can be expressed as follows:
(22)
Equations (11) and (12) can be expressed in terms of and ⁠. Dividing by provides the pressure head response between the downstream valve and air chamber connector for flow discharge variation, which can be expressed as
(23)
where the dimensionless length ⁠.
By implementing Equation (23) as a dimensionless hydraulic impedance for air chambers and extending it to the upstream direction, the pressure head response between upstream reservoirs and air chamber connectors for flow discharge variation can be expressed as
(24)
where ⁠, ⁠, and dimensionless length ⁠, ⁠; is the distance between the air chamber and downstream valve.

The transformation of dimensionless impedance into time domain response can be done through inverse fast Fourier . Convolution of the dimensionless impedance with the downstream dimensionless flowrate provides a temporal response of the pressure head to any point (Kim 2022).

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The proposed method was validated using a simple pipeline system with an air chamber. With a length of 150 m, the pipeline system consists of a typical reservoir pipeline air chamber pipeline downstream valve system. Pipelines and connectors had diameters of 0.02 m, and the air chamber was 50 m from downstream valves. The Darcy–Weisbach friction factor was 0.03. The wave propagation speed in the pipeline was assumed as 1210.5 m/s, which was measured from an experimental pipeline system (Kim, 2022).

The steady flowrates (water) can be manipulated as the pressure heads of upstream and downstream reservoir were modulated, which varied between 1.57 × 10−6 and 1.57 × 10−3 m3/s. The diameter of the air chamber was assumed to be 2 m, and the initial air volume was 4 m3 with the polytrophic coefficient as 1.4 with the length of the connector as 0.5 m. Transient hydraulic transients can be introduced by abruptly closing downstream valves, which can be modeled by the convolution procedure in Kim (2022). The dimensions of the system can be altered depending on the field conditions and the preferences of the system designer (e.g. pipeline length, diameter, location of the air chamber, air volume, polytrophic coefficient, and connector length). The impact of boundary conditions from various hydraulic structures such as reservoir, valve, and pump with check valve can be easily implemented as the complex pressure head or flowrate equal to zero, simplifying the development of dimensionless impedance expressions along pipeline systems (Kim 2022).

Comparison of the developed method with existing approaches

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The developed method can be validated by comparing its performance with that of existing methods. For pipeline systems equipped with an air chamber, there have traditionally been two distinct methods of transient simulation. One is the MOC that solves the hyperbolic partial differential equations of momentum and mass conservation through discretization of pipeline layout under the condition of a Courant number equal to one (Wylie & Streeter, 1993). Another way to extend IRMs is to use the frequency domain implementation of an air chamber to translate it into a time domain response (Kim 2008). The locations of the air chambers were one-third away from the downstream boundary conditions. Assuming the steady flowrate is 2.32 × 10−5 m3/s, a transient was generated through the slow valve closure in 0.3 s (Kim 2022), which can address the wave reflection and interaction during the transient generation period. The pressure resonance of the system can, therefore, be used to check the flow regime during transient events. We compared pressure variations at a point 100 m upstream from the downstream valve. This is because the analytical development from the upstream to the downstream valve and its backward formulation derivation including the air chamber can completely validate the proposed method at the point between the upstream reservoir and the air chamber. It can be seen in Figure 3 that the normalized pressure variations for MOC, IRM, and dimensionless impedance method (DIM) show excellent agreement with each other. To compare MOC and IRM, the dimensionless time for DIM was converted into the real-time scale (s). DIM and IRM differed less than an order of 10−6 in terms of normalized pressure response. Therefore, DIM's development was correct and can be used for the general characterization of pipeline systems with an air chamber.
Figure 3
Normalized pressure responses at 100 m upstream from downstream point simulated by MOC, IRM, and DIM.
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Normalized pressure responses at 100 m upstream from downstream point simulated by MOC, IRM, and DIM.

Figure 3
Normalized pressure responses at 100 m upstream from downstream point simulated by MOC, IRM, and DIM.
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Normalized pressure responses at 100 m upstream from downstream point simulated by MOC, IRM, and DIM.

Close modal

General characterization of system response through dimensionless parameters

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An efficient characterization of system response can be achieved through the comprehensive consideration of distinct parts of the system with a fewer dimensionless parameters. A dimensionless impedance evaluation in the frequency domain can be used to characterize comprehensive responses for pipeline systems. This study evaluated the dimensionless impedance responses at downstream valves and compared different systems having identical dimensionless parameters. Dimensionless resistance, ⁠, can be defined as a function of pipeline length, diameter, friction factor, and connector length to match laminar and turbulent steady flow conditions. Through hydraulic impedance at downstream points, the pressure response can be addressed under the assumption that air chamber parameters, such as polytrophic constant, air pressure, and air volume, are identical.

As shown in Figure 4(a), dimensionless impedance amplitudes at downstream valves are plotted for three distinct systems with pipe lengths of 150, 100, and 3,750 m; diameters of 0.02, 0.016, and 0.1 m; and connector lengths of 0.5, 0.33, and 12.5 m. Air chamber locations were 3% of the total pipeline length away from downstream boundary conditions. Therefore, three distinct systems have identical dimensionless resistance with a steady flow rate of 2.32 × 10−5 m3/s. As shown in Figure 4(a), hydraulic impedances in the dimensionless frequency domain exhibit well-matched amplitude distributions.
Figure 4
Dimensionless hydraulic impedance at the downstream point for identical dimensionless resistance and layout for three distinct pipeline systems with air chamber under laminar steady flow condition (a) and turbulent steady flow condition (b).
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Dimensionless hydraulic impedance at the downstream point for identical dimensionless resistance and layout for three distinct pipeline systems with air chamber under laminar steady flow condition (a) and turbulent steady flow condition (b).

Figure 4
Dimensionless hydraulic impedance at the downstream point for identical dimensionless resistance and layout for three distinct pipeline systems with air chamber under laminar steady flow condition (a) and turbulent steady flow condition (b).
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Dimensionless hydraulic impedance at the downstream point for identical dimensionless resistance and layout for three distinct pipeline systems with air chamber under laminar steady flow condition (a) and turbulent steady flow condition (b).

Close modal

The amplitudes of dimensionless impedance are shown in Figure 4(b) for three distinct systems under turbulent conditions (steady flowrate = 9.28 × 10−5 m3/s) having pipeline lengths of 150, 225, and 433 m; diameters as 0.02, 0.015, and 0.03 m; connector lengths as 0.5, 0.75, and 1.45 m; and Darcy–Weisbach friction factors as 0.03, 0.02, and 0.025, respectively. In this case, the air chamber was also positioned 3% of the length from the downstream valve to the upstream valve. Dimensionless hydraulic impedance distributions for three distinct systems are also completely matched (Figure 4(b)). Consequently, the dimensionless resistance, ⁠, can be used to characterize the pressure response in terms of system dimensions (e.g. length, diameter, and connector) and layout (system composition and its relative placement).

The dimensionless impedance of an air chamber, expressed as Equation (18), is another important dimensionless parameter. Using a steady flow rate of 9.28 × 10−5 m3/s, the identical dimensionless hydraulic impedance for the air chamber can be fabricated by combining polytrophic coefficient, air volume, connecting wave speed, dimensionless air pressure, and connector length. For the evaluation of dimensionless hydraulic impedance at downstream valves assuming air pressure and wave speed in the connector are identical, three distinct polytrophic coefficients were used: 1.4, 1.0, and 1.2; connector lengths: 0.6, 0.7, and 0.5 m; and air volume: 4, 4, and 6.86 m3.

In Figure 5, identical distributions of hydraulic impedance amplitudes are shown in a dimensionless frequency domain. Thus, the dimensionless hydraulic impedance for the air chamber is a general parameter that can be used to represent air chamber impact. By manipulating feasible parameters (e.g. air pressure and air volume), the air chamber can function correctly for designated transient events. Using unified criteria, it is possible to reduce both the cost of air chambers (e.g. air chamber volume) and the mechanical fitting of the air chambers (connector length and property).
Figure 5
Dimensionless hydraulic impedances at the downstream point for three identical dimensionless air chamber hydraulic impedances with different combinations of polytrophic coefficients, connector lengths, and air volumes.
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Dimensionless hydraulic impedances at the downstream point for three identical dimensionless air chamber hydraulic impedances with different combinations of polytrophic coefficients, connector lengths, and air volumes.

Figure 5
Dimensionless hydraulic impedances at the downstream point for three identical dimensionless air chamber hydraulic impedances with different combinations of polytrophic coefficients, connector lengths, and air volumes.
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Dimensionless hydraulic impedances at the downstream point for three identical dimensionless air chamber hydraulic impedances with different combinations of polytrophic coefficients, connector lengths, and air volumes.

Close modal

Dimensionless hydraulic impedance in various flow regimes

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Reynolds number is important to describe flow regime in a closed conduit flow. Steady flowrate plays an important role in dimensionless resistance, dimensionless hydraulic impedance, and dimensionless pressure. Using the same pipeline system as shown in Figure 5, different steady flowrates were used to calculate the dimensionless hydraulic impedance at the downstream valve. Figure 6(a) and 6(b) shows the dimensionless hydraulic impedance at the downstream point for four distinct Reynolds numbers (Re) in laminar and turbulent flows. Under laminar flow conditions, the distributions of impedance amplitudes (Re = 100, 300, 700, 1,500) were identical, while those under turbulent flow conditions (Re = 4,000, 10,000, 50,000, 100,000) showed different responses. As a result, while resistance under laminar conditions depends on diameter and viscosity, resistance under turbulent conditions substantially depends on mean flowrate and fiction, which is also influenced by Re. As shown in Figure 6(c), the higher the Re, the greater the damping in peak impedance responses. Because friction has a greater impact in higher Re, the pressure response is mitigated in all dimensionless frequency ranges, even if the points of discontinuity in the dimensionless hydraulic impedance are the same in all conditions. A pipeline's dimensions and layout determine the systematic resonance behavior, and flow conditions impact pressure response mitigation. Actually, the impact of Re can be addressed through the dimensionless resistance, ⁠, defined as under laminar flow and for turbulent flow conditions. Under laminar flow, dimensionless resistance remains constant regardless of flowrate and friction factor. As a result, dimensionless resistance can be used to address the impact of Re on the response of dimensionless hydraulic impedance under specific conditions.
Figure 6
Amplitude of dimensionless hydraulic impedances for four different Reynolds numbers (Re) under laminar flow (a) and for different Re under turbulent flow conditions (b) and enlargement of (b) for a designated dimensionless frequency range between 37 and 47 (c).
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Amplitude of dimensionless hydraulic impedances for four different Reynolds numbers (Re) under laminar flow (a) and for different Re under turbulent flow conditions (b) and enlargement of (b) for a designated dimensionless frequency range between 37 and 47 (c).

Figure 6
Amplitude of dimensionless hydraulic impedances for four different Reynolds numbers (Re) under laminar flow (a) and for different Re under turbulent flow conditions (b) and enlargement of (b) for a designated dimensionless frequency range between 37 and 47 (c).
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Amplitude of dimensionless hydraulic impedances for four different Reynolds numbers (Re) under laminar flow (a) and for different Re under turbulent flow conditions (b) and enlargement of (b) for a designated dimensionless frequency range between 37 and 47 (c).

Close modal

Pressure responses for different dimensionless parameters under laminar and turbulent conditions

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It can be explored whether relative magnitudes between and to the distribution of dimensionless hydraulic impedance impact pressure response patterns. Using combinations of two different and under laminar flow conditions, Figure 7(a) illustrates a part of a dimensionless hydraulic impedance at downstream valves (steady flow = 2.13 × 10−5 m3/s). As presented in Figure 7(a), legends BR and SR stand for = 0.066 and 0.0066, those for BZ and SZ correspond to = 4.3 × 10−9 and 1.45 × 10−9i/⁠, respectively. It is evident from Figure 7(a) that the impedance frequency response patterns of BR and SR are substantially different. However, there seems to be a minor phase shift and occasional amplitude increase between BZ and SZ, which is explained by the resonance between the main pipeline and the connector. Figure 7(a) shows similar amplitude responses for all four combinations between 10 and 1,000.
Figure 7
Amplitude of dimensionless hydraulic impedances for four different combinations of = 0.066 and 0.0066 as BR and SR, respectively; = 4.3 × 10−9 and 1.45 × 10−9i/ as BZ and SZ, respectively, under laminar flow (a), and those for four different combinations of = 0.464 and 0.046 as BR and SR, respectively; = 5.18 × 10−8 and 1.744 × 10−8i/ as BZ and SZ, respectively, under turbulent flow (b).
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Amplitude of dimensionless hydraulic impedances for four different combinations of = 0.066 and 0.0066 as BR and SR, respectively; = 4.3 × 10−9 and 1.45 × 10−9i/ as BZ and SZ, respectively, under laminar flow (a), and those for four different combinations of = 0.464 and 0.046 as BR and SR, respectively; = 5.18 × 10−8 and 1.744 × 10−8i/ as BZ and SZ, respectively, under turbulent flow (b).

Figure 7
Amplitude of dimensionless hydraulic impedances for four different combinations of = 0.066 and 0.0066 as BR and SR, respectively; = 4.3 × 10−9 and 1.45 × 10−9i/ as BZ and SZ, respectively, under laminar flow (a), and those for four different combinations of = 0.464 and 0.046 as BR and SR, respectively; = 5.18 × 10−8 and 1.744 × 10−8i/ as BZ and SZ, respectively, under turbulent flow (b).
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Amplitude of dimensionless hydraulic impedances for four different combinations of = 0.066 and 0.0066 as BR and SR, respectively; = 4.3 × 10−9 and 1.45 × 10−9i/ as BZ and SZ, respectively, under laminar flow (a), and those for four different combinations of = 0.464 and 0.046 as BR and SR, respectively; = 5.18 × 10−8 and 1.744 × 10−8i/ as BZ and SZ, respectively, under turbulent flow (b).

Close modal

In Figure 7(b), distributions of dimensionless hydraulic impedance are shown for four combinations under turbulent conditions (steady flow = 2.36 × 10−4 m3/s). The legends BR and SR indicate = 0.464 and 0.046, those for BZ and ZS are = 5.18 × 10−8 and 1.744 × 10−8i⁠, respectively, for Figure 7(b). The impact of dimensionless resistance on amplitude appears substantial. Dimensionless impedance amplitudes of BR are 102 and 103, while those of SR are 101 and 10−1. A large phase shift and amplitude for two different s appear in big (BR), whereas those for SR appear negligible even with a much smaller amplitude distribution.

There is a fundamental difference between laminar and turbulent conditions in the relative impact of and ⁠, as illustrated in Figure 7(a) and 7(b). In laminar flow, dimensionless resistance affects the frequency response pattern. In turbulent flow, dimensionless resistance affects both amplitude range and frequency response pattern. There is also a difference between laminar and turbulent flow conditions in the impact of the dimensionless hydraulic impedance of air chambers. The dimensionless parameter is sensitive to primary amplitude ranges of dimensionless hydraulic impedance, as explained by interaction, which tends to be stronger in greater conditions, as shown in Figure 7(b).

Dimensionless solutions for a pipeline system with an air chamber provide feasible hydraulic response configurations for two distinct regimes. One is the dimensionless resistance, which compiles information for the main pipeline, and the other is the contribution of the air chamber section to the dimensionless hydraulic impedance for the air chamber. As a result, the transient characteristics of a pipeline with an air chamber can be modeled under a scale-free environment. Independent matching of two dimensionless parameters can satisfy the similarity of hydraulic transient response. A dimensionless resistance can be calculated by manipulating the diameter and length of the pipeline, the wave speed, the friction factor, and the mean flowrate. By adjusting the polytrophic constant, the length and wave speed of the connector, the air pressure, and the volume of the air chamber, the effect of the air chamber can be addressed. This makes pipeline systems with air chambers significantly more feasible to install and operate.

Hydraulic similarity and air chamber design

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Using two representative frequency domain parameters, and ⁠, respectively, we can holistically address the influence of pipelines and air chambers on hydraulic transients. Figure 4(a) and 4(b) illustrate that alternative combinations of real system parameters with identical dimensionless numbers produce identical hydraulic impedance responses. Under two identical dimensionless parameters, the pipeline system and air chamber design can be flexible, which can lead to significant system feasibility. A polytrophic coefficient, air volume, and connector length can, for instance, be adjusted according to the field conditions while considering efficiency in the system, the length of the main pipeline, and the diameter and friction factor to satisfy two conditions related to system resonance. According to extended numerical tests under wide ranges of Reynolds numbers, the impact of higher Reynolds numbers can be explained by the mitigation of peak pressure responses with no changes in phase response. Under turbulent flow conditions, dimensionless resistance and air chamber hydraulic impedance seem to have a notable influence, which should be carefully considered when designing a system.

The hydraulic similarity in transient response is achieved when two systems have identical two dimensionless numbers and identical system layouts. This means that a physical model with different dimensions can replicate the hydraulic behavior of a prototype. It is possible to fabricate a scaled-down experimental air chamber as a safety device. Moreover, multiple choices of parameters seem to be available as long as dimensionless numbers are matched. Operators can use a model, mostly scaled down, based on this approach to test new hydraulic scenarios for prototypes, considering identical pressure resonance between two systems.

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Hydraulic transients in pipeline systems with air chambers were characterized by dimensionless solutions. By comparing pressure time series by MOC, IRM, and DIM, the proposed method was validated. To demonstrate universal representativeness for pipelines and air chambers, two important parameters, dimensionless resistance, and dimensionless air chamber hydraulic impedance, were identified. We explored the response pattern of dimensionless hydraulic impedance under both laminar and turbulent flow conditions. The impact of the Reynolds number was examined under laminar and turbulent flow conditions, indicating the robustness of dimensionless resistance.

This study provides a generalized solution to the water hammer problem in pipeline systems equipped with air chambers under scale-free environments. The similarity in hydraulic behavior can be obtained regardless of the scale up or down from the original system as long as two dimensionless parameters are identical between the model and prototype. A potential research topic is the incorporation of an unsteady friction model such as acceleration-based model into a dimensionless solution that improves prediction accuracy under slow transient condition even though the highest impulse (first response) can be adequately modeled using a steady fiction model. The hydraulic transient in complicated pipeline structures, such as loops and branches, can also be a challenging issue not only because resonances between many distinct pipeline elements can be complicated but also because the dimensionless expression for different wave propagation sectors can be difficult to be derived for general solution. The dimensionless approach will enable better analysis and simplification of such a challenging problem.

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This study was partially supported by the Basic Research Laboratory from the Korea Research Foundation (2022R1A4A5028840).

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The author designed the research, developed the theory, performed the analysis, and wrote the manuscript.

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

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The authors declare there is no conflict.

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